\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 61, pp. 1--51.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/61\hfil Self-similar decay]
{Self-similar decay to the  marginally stable  ground state
in a model for film flow over inclined wavy  bottoms}

\author[T. H\"acker, G. Schneider, H. Uecker \hfil EJDE-2012/61\hfilneg]
{Tobias H\"acker, Guido Schneider, Hannes Uecker}  % in alphabetical order

\address{Tobias H\"acker \newline
Institut f\"ur Analysis, Dynamik und Modellierung,
Universit\"at Stuttgart, Pfaffenwaldring 57,
D-70569 Stuttgart, Germany}
\email{tobias.haecker@mathematik.uni-stuttgart.de}

\address{Guido Schneider \newline
Institut f\"ur Analysis, Dynamik und Modellierung,
Universit\"at Stuttgart, Pfaffenwaldring 57,
D-70569 Stuttgart, Germany}
\email{guido.schneider@mathematik.uni-stuttgart.de}

\address{Hannes Uecker \newline
Institut f\"ur Mathematik, Carl von Ossietzky Universit\"at Oldenburg,
D-26111 Oldenburg, Germany}
\email{hannes.uecker@uni-oldenburg.de}

\thanks{Submitted October 27, 2010. Published April 12, 2012.}
\subjclass[2000]{35Q35, 37E20, 35B35}
\keywords{Diffusive stability; renormalization; IBL system; periodic media}

\begin{abstract}
 The integral boundary layer system (IBL) with spatially periodic
 coefficients arises as a long wave approximation for the flow of a
 viscous incompressible fluid down a wavy inclined plane.
 The  Nusselt-like stationary solution of the IBL is linearly at best
 marginally stable; i.e., it has essential spectrum at least up to the
 imaginary axis. Nevertheless, in this stable  case we show that
 localized perturbations of the ground state decay  in a self-similar way.
 The proof uses the renormalization group method in Bloch variables
 and the fact that in the stable case the Burgers equation is the
 amplitude equation for long waves of small amplitude  in the IBL.
 It is the first time that such a proof is given for a quasilinear
 PDE with spatially periodic coefficients.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\allowdisplaybreaks

%\tableofcontents

\section{Introduction}

The gravity driven free surface flow of a viscous incompressible fluid
down an inclined plate plays an important role in heat exchanging devices.
Numerous applications are found in coating processes ranging from
the production of compact discs to photographic industries.
For a flat bottom, the inclined film problem has been extensively studied
experimentally, numerically, and analytically; see \cite{cd02} for a review.
In particular, it is well known that for a given film height the underlying
Navier-Stokes equations possess a stationary solution with a parabolic
velocity profile and a flat surface. Denoting the inclination angle by
$\alpha$, this  so-called Nusselt solution is spectrally stable
if the Reynolds number ${\rm R}$ is below the
critical value ${\rm R}_{\rm crit} = 5/6 \cot \alpha$, and unstable to long waves
for ${\rm R}>{\rm R}_{\rm crit}$, cf.~\cite{Benjamin_57, Yih_63}.
Nonlinear diffusive stability
in the sense of the present paper
in the spectrally stable case
was shown in \cite{Uecker_07}, while for ${\rm R}>{\rm R}_{\rm crit}$ surface waves
are generated, which pass through a number of secondary instabilities
until turbulence occurs at high Reynolds numbers; see \cite{Cheng_95},
for instance.

In many applications the bottom is not perfectly flat but rather has
a wavy profile. This may be due to natural irregularities or by design,
for example in cooling processes. Thus, it is of interest to study the
impact of an undulated bottom on the film flow. However, to study the
stability of stationary solutions, the Navier-Stokes equations in
combination with the free surface are hard to handle and thus there
has been much effort to derive simpler model  equations.
Starting from the 2D Navier-Stokes equations in curvilinear
coordinates,
in \cite{HU09} we derived a 2-dimensional  system
with periodic coefficients for the film thickness $F=F(t,x)\in\mathbb{R}$ and the
local flow rate $Q=Q(t,x):= \int_0^{F(t,x)} U(t,x,z)\,{\rm d} z$, where
$U$ is the velocity in direction parallel to the bottom. In \cite{HU09}
this system is called
weighted residual integral boundary layer system, here IBL in short,
and may be written as
\begin{gather}
  \partial_t F =  -\frac{1}{1+\kappa F}\partial_x Q, \label{FT3}\\
\begin{aligned}
  \partial_t Q &=
  \frac{5}{2{\rm R}}\left(\frac{\sin(\alpha{-}\theta)}{\sin\alpha}F-\frac{Q}{F^2}
  -\frac{\cos(\alpha{-}\theta)}{\sin\alpha}\partial_x F\, F
  -\frac{3}{8}\frac{\sin(\alpha{-}\theta)}{\sin\alpha}\partial_x\theta \, F^2\right) \\
  &\quad +\frac{5}{6}{\rm W}(\partial_x^3F- \partial_x\kappa)F -\frac{17}{7} \frac{Q}{F} \partial_x Q
  +\frac{9}{7} \frac{Q^2}{F^2}\partial_x F{-}\frac{1}{210}{\rm R}(\partial_x Q)^2 Q \\
  &\quad +\frac{1}{{\rm R}}\left(\frac{9}{2}\partial_x^2Q
+\frac{45}{16} \kappa \frac{Q}{F}{+}4 \frac{Q}{F^2}(\partial_x
  F)^2{-}6\frac{Q}{F}\partial_x^2F {-} \frac{9}{2}\frac{1}{F}\partial_x Q \,\partial_x F\right).
\end{aligned}   \label{QT3}
\end{gather}
Here $t\ge 0$ denotes time, $x\in\mathbb{R}$ corresponds to  arclength along
the bottom, and we
simplified notation of the IBL used in
\cite[(31),(32)]{HU09} by  redefining
the spatial variable $ X $, the temporal variable $ T $, and the
curvature $ K $ used in   \cite[(31),(32)]{HU09} via
\begin{equation}
  x:=\frac{1}{\delta}X, \quad t := \frac{1}{\delta}T,
\quad \kappa := \delta\zeta K, \label{51_50}
\end{equation}
where $\delta>0$ is a dimensionless wave number, $\zeta\ge 0$ describes
the bottom waviness, and $\kappa=\kappa(x)$ is the curvature of the bottom
which is periodic  with period $\gamma>0$.
For the surface tension effects here we replaced the inverse Bond
number ${\rm B}_{\rm i}$ from \cite{HU09} by the Weber number ${\rm W}$, defined by
${\rm W}:=3 \delta^{-2}{\rm B}_{\rm i} {\rm R}^{-1}$. Finally, $\alpha>0$ is the mean
inclination angle such that $\alpha-\theta$, with $ \theta = \theta(x) $
 is the $\gamma$-periodic
local inclination angle, and ${\rm R}$ is the
Reynolds number which measures the ratio between inertia and viscous
forces.


\begin{remark} \label{51_r1} \rm
(a) From the non-dimensionalization and derivation in
\cite{HU09} we have that $F\approx 1$ and $1+\kappa F\approx 1$ and thus
the denominators in \eqref{FT3}, \eqref{QT3} are bounded from
below by, e.g.,  $1/2$.

(b) In \cite{HU09} we also considered a regularized version (rIBL) of
\eqref{FT3}, \eqref{QT3}, mainly to correct some unphysical behaviour
of \eqref{FT3}, \eqref{QT3} for ${\rm R}\gg {\rm R}_{\rm crit}$. Here we are interested
in ${\rm R}\le {\rm R}_{\rm crit}$ where the difference between \eqref{FT3}, \eqref{QT3} and
the rIBL is very small.
In particular, the two versions only differ by terms which for ${\rm R}<{\rm R}_{\rm crit}$ are
asymptotically irrelevant. Therefore we stick to
the slightly simpler version \eqref{FT3}, \eqref{QT3}, but nevertheless
\eqref{FT3}, \eqref{QT3} is a quasilinear parabolic system with spatially
periodic coefficients.
\end{remark}

Numerical simulations for \eqref{FT3}, \eqref{QT3} showed very
good agreement with  data available from experiment and full Navier-Stokes
simulations. In particular, \eqref{FT3}, \eqref{QT3} can be used to
approximate stationary solutions of the original Navier-Stokes systems,
even with eddies, see \cite{HU09}.
Moreover, from linear stability analysis one can again find a critical
Reynolds number ${\rm R}_{\rm crit}$ beyond which
the free surface of stationary solutions undergoes a long wave
instability \cite{Acta_05}, and again the numerical
stability results from \cite{HU09} for the IBL agree very well with
\cite{Acta_05}.

Thus, here we use the IBL as a model problem to study
nonlinear stability of Nusselt-like stationary $\gamma$-periodic
solutions $(f_s,q_s)$ in
the spectrally stable case.  For stationary solutions $q_s$ is
constant, and it turns out
that we always have families of stationary solutions which can be
parametrized by $q_s$.
Therefore, the stability of any spectrally stable $(f_s,q_s)$ is
nontrivial since linearizations around such $(f_s,q_s)$ always have essential
spectrum up to the imaginary axis. Thus, we
cannot conclude stability from the linearization alone but have
to take into account the nonlinearity.

If we restrict to spatially localized perturbations,
dissipative systems often show dynamics which are similar to those of
linear diffusion equations. To be more precise,
denoting the solution by $v(t,x)$, the rescaled solution
$\sqrt{t}v(t,\sqrt{t}x)$ converges towards a Gaussian limit.
In this case, the nonlinearity is called
asymptotically irrelevant.  However,
if the nonlinearity has an advection term $\partial_x(v^2)$, then it becomes
relevant and the resulting non-Gaussian limit of the rescaled solution is determined by the
Burgers equation, see \cite{bkl94}, for instance.

Here we show a similar result for the IBL,
namely that localized
perturbations of spectrally stable stationary solutions $(f_s,q_s)^\top$
decay in a universal manner,
which is determined by the Burgers equation. The proof relies on
renormalization group (RG) methods \cite{bkl94} for nonlinear parabolic
PDEs, which have been used for systems like the Ginzburg-Landau
equation, see \cite{bk92,cee92,bk94,gm98}, or pattern forming systems,
see \cite{gs96a,gs98,ue99,gsu04,su03poi}.
Also for film flow over flat
inclines RG methods were used to show nonlinear stability of spectrally stable
stationary solutions, namely in \cite{ue04ibl} for an IBL  and
in \cite{Uecker_07} for the full Navier-Stokes system.

Mathematically, \eqref{FT3}, \eqref{QT3} can be classified as a
quasilinear second order parabolic system.
Besides the quasilinearity, which makes the local existence theory
difficult, we have the following issues. First, in contrast to the Nusselt
solution over flat bottoms, over wavy bottoms the stationary
solutions are not known in closed form. Second, Fourier analysis,
which is an essential tool in the stability proofs for flat inclines,
has to be replaced by Bloch wave analysis. This was used
in \cite{uw07} to prove nonlinear stability
for a semilinear model problem, namely a
spatially periodic Kuramoto-Shivashinsky equation.

\subsection*{Notation} For $m,r\in\mathbb{R}$ the weighted Sobolev
spaces $H^r(m)$ are defined as
\begin{equation}
  H^r(m) := \{ v:\mathbb{R} \to \mathbb{C} \mid \|v\|_{H^r(m)}
  = \|\varrho^m v\|_{H^r} < \infty \}  \text{ with } \varrho(x)
  = (1+x^2)^{1/2}.  \label{521_5}
\end{equation}
Fourier transform $ \mathcal{F} $ is defined by
\begin{equation}
\mathcal{F} v(k) = \frac{1}{2 \pi}\int_\mathbb{R} v(x) {\rm e}^{-{\rm i} kx} {\rm d} x, \quad
 v(x) = \mathcal{F}^{-1} \hat{v} (x) = \int_\mathbb{R}\hat{v}(k) {\rm e}^{{\rm i} kx} {\rm d} k,
\label{521_6}
\end{equation}
and is an isomorphism between $H^r(m)$ and $H^m(r)$.

Our main result now reads as follows, where for notational
convenience we take initial conditions for \eqref{FT3}, \eqref{QT3} at
$t=1$, and where the spectral stability  assumptions will be discussed
below
in  Assumption \ref{ass_spec}.

\begin{theorem}\label{main_th2}
 Let $p\in(0,1/2)$, $3<r<4$, and let $(f_s,q_s)^\top$ be a
spectrally stable stationary solution of the IBL
\eqref{FT3}, \eqref{QT3}, cf.~Assumption \ref{ass_spec} below.
Then there exist constants $C_1, C_2>0$ such that the
following holds. If $\|f_0\|_{H^r(2)}+\|q_0\|_{H^{r-1}(2)}\leq C_1$,
then there exists a unique global solution
$(F,Q)^\top = (f_s,q_s)^\top+(f,q)^\top$ of the
IBL  \eqref{FT3}, \eqref{QT3} with
$(f,q)^\top\big|_{t=1} = (f_0,q_0)^\top$ and
  \begin{equation}\label{mre}
    \sup_{x\in\mathbb{R}}\left|(f,q)^\top - t^{-1/2}f_{z_0}(t^{-1/2}(x+c_1 t))
\Phi^1(0,x)\right| \leq C_2 t^{-1+p/2}
  \end{equation}
  for $ t \in [1,\infty)$. Here,
$\Phi^1(0,\cdot) = ({\rm d} f_s / {\rm d} q_s,1)^\top$ is the critical eigenfunction
of the linearization of \eqref{FT3}, \eqref{QT3} around  $(f_s,q_s)^\top$,
and
\begin{equation}
   f_{z_0}(y) = \frac{\sqrt{c_2}}{d}\frac{z_0 \operatorname{erf}'(y/(2\sqrt{c_2}))}
{4+2z_0\left(1+\operatorname{erf}(y/(2\sqrt{c_2}))\right)}, \label{522_1a}
\end{equation}
denotes the non-Gaussian profile determined by the Burgers equation, where
$c_1<0$, $c_2>0$ and $d<0$
are likewise determined by the linearization around $(f_s,q_s)^\top$, while
$z_0>-1$ can be given explicitly in terms of the excess mass
$\int_\mathbb{R} f_0 \,{\rm d} x$, see \eqref{63_20}.
\end{theorem}

The behaviour of the function $v_z(t,x):= t^{-1/2} f_z(t^{-1/2} x)$ is
shown in Fig.~\ref{522_fig1}. 


 \begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} %{burg.pdf}
\end{center}
\caption{Sketch of self-similar decay of the amplitude in a
comoving frame in \eqref{mre}}
 \label{522_fig1}
 \end{figure}

\begin{figure}[ht]
\begin{center}
(a) Decay of the film thickness $F$ \hspace{15mm}
(b) Decay of the flow rate \\
\includegraphics[width=0.45\textwidth]{fig2a}  %fdec1.eps 
 \quad
\includegraphics[width=0.45\textwidth]{fig2b} %qdec1.eps
\\[4mm]
 \begin{tabular}{l} (c)
Bottom profile and free surface in\\ 
Cartesian coordinates (mm), $t=1$
\end{tabular}
\hspace{10mm}  \begin{tabular}{l}
(d) Evolution to a long pulse\\ 
in the unstable case, $Q(t,x)$ \end{tabular} \\
\includegraphics[width=0.45\textwidth]{fig2c} % cart.eps
\quad
\includegraphics[width=0.45\textwidth]{fig2d} % unstab1.eps
\end{center}

\caption{Numerical simulations of \eqref{FT3}, \eqref{QT3}}
\label{522_fig2}
 \end{figure}

Figure \ref{522_fig2} 
shows numerical simulations of \eqref{FT3}, \eqref{QT3} in the stable
case (a)-(c) and the unstable case (d), with periodic boundary conditions.
This is also intended to relate
 \eqref{FT3}, \eqref{QT3} to the underlying physics.
In (a)-(c) we used a sinusoidal bottom with amplitude $a=0.4$mm
and wavelength $\lambda{=}10$mm (bottom profile
$\hat{b}(\hat x)=a\cos(\frac{2\pi}{\lambda}\hat{x})$).
The mean film thickness is $h{\approx}0.06$mm,
inclination angle $\alpha=60^\circ$,
and the fluid parameters correspond to water, which yields
$\delta \approx 0.037$, $\zeta = 0.25$,
${\rm B}_{\rm i}\approx 3.25$ and ${\rm R}=0.6$. The initial condition
is $F=f_s+2/\cosh((x-50)/5)$, $Q=q_s\equiv 1$. Although ${\rm R}$
is larger than the critical
Reynolds number over flat bottom, which is ${\rm R}_{\rm crit}\approx 0.48$,
the stationary solution is stable and the perturbation decays in the
self-similar way predicted by \eqref{mre}. (a) shows $F$ at times as
indicated, while (b) shows the evolution of $Q$. In the latter we directly
see the envelope $t^{-1/2}f_{z_0}(t^{-1/2}(x+c_1 t))$ since
$\Phi^1_2(0,\cdot)\equiv 1$, while $\Phi^1_1(0,x)=\frac{{\rm d} f_s}{{\rm d} q_s}(x)$
is $\gamma$-periodic. To illustrate the physical situation,
panel (c) shows the bottom contour and the free surface
at initial time $t=1$ in dimensional (mm) cartesian coordinates, between the
$4^{\text{th}}$ and $6^{\text{th}}$ bottom wave. Finally, panel (d)
shows $Q$ (for large time) after
we increased $\alpha$ to $90^\circ$. Here $(f_s,q_s)^\top$ has become
unstable: the perturbation does not decay to $0$, but instead evolves into
a long pulse. We expect that this situation can be described by a
generalized KS equation, see, e.g., \cite{cd02, psu07} for
the situation over flat bottoms, and \cite{uw07} for a model problem
for wavy bottoms.


The plan of this article is as follows. First we make precise the assumptions
on spectral stability of $(f_s,q_s)^\top$, review basics
of the RG method and of Bloch transform,
and formally derive the Burgers equation from \eqref{FT3}, \eqref{QT3}.
Then, using maximal regularity results we first prove local
existence for \eqref{FT3}, \eqref{QT3} and then use the RG method
to prove Theorem \ref{main_th2}.
The RG method is worked out here for the first time
for a realistic quasilinear
fluid dynamical system with spatially periodic coefficients
in which the renormalized solution converges to a non-Gaussian limit.
We expect that the analysis is useful for a number of similar
problems, for instance the full Navier-Stokes film flow problem
over wavy bottom, and other parabolic systems with spatially
periodic coefficients
and a nonlinearity with lowest order terms of convective type.


\section{Background and result}\label{presec}
\subsection{Stationary solutions}\label{ssec}

From \eqref{FT3}, for $\gamma$-periodic stationary solutions\\
 $(F,Q)^\top=(f_s,q_s)^\top$, we immediately obtain that $\partial_x q_s \equiv 0$.
Plugging $\partial_t q_s = \partial_x q_s = 0$ into \eqref{QT3} and multiplying
it by $f_s^2$, we obtain
\begin{equation} \begin{aligned}
0 &=
  \frac{5}{2{\rm R}}\left(\frac{\sin(\alpha{-}\theta)}{\sin\alpha}f_s^3-q_s
    -\frac{\cos(\alpha{-}\theta)}{\sin\alpha}\partial_x f_s\, f_s^3
    -\frac{3}{8}\frac{\sin(\alpha{-}\theta)}{\sin\alpha}\partial_x\theta \, f_s^4\right) \\
  &\quad+\frac{5}{6}{\rm W}(\partial_x^3f_s- \partial_x\kappa)f_s^3 +\frac{9}{7}q_s^2\,\partial_x
  f_s+ \frac{1}{{\rm R}}\left( \frac{45}{16} \kappa q_s f_s {+}4 q_s(\partial_x
    f_s)^2{-}6 q_s \, \partial_x^2f_s \, f_s \right).
\end{aligned}\label{ss1}
\end{equation}
If the bottom waviness $\zeta$ is zero, the coefficients
$\kappa$ and $\theta$ vanish and we have the well known Nusselt
solution $f_s = f_N$ with constant film thickness $f_N =q_s^{1/3}$.
Thus, one possibility to obtain solutions of \eqref{ss1} is to continue
$(f_N,q_s)$ for $\zeta>0$ using the implicit function theorem.
Since $x$ is
measured in curvilinear coordinates, the periodicity $\gamma$ of
$x$ depends on the bottom waviness $\zeta$, and in order to
apply the implicit function theorem in a function space with
fixed periodicity we temporarily replace $x$ by $k_0 x$, where we set
$k_0=2\pi/\gamma$\label{page_k0}. This yields
\begin{equation} \begin{aligned}
0 &=
  \frac{5}{2{\rm R}}\left(\frac{\sin(\alpha{-}\theta)}{\sin\alpha}f_s^3-q_s
    -\frac{\cos(\alpha{-}\theta)}{\sin\alpha}k_0\partial_x f_s\, f_s^3
    -\frac{3}{8}\frac{\sin(\alpha{-}\theta)}{\sin\alpha}k_0\partial_x\theta \, f_s^4\right)
\\
  &\quad +\frac{5}{6}{\rm W}(k_0^3\partial_x^3f_s- k_0\partial_x\kappa)f_s^3
 +\frac{9}{7}k_0q_s^2\,\partial_x f_s  \\
  &\quad +\frac{1}{{\rm R}}\left( \frac{45}{16} \kappa q_s f_s {+}4k_0^2
    q_s(\partial_x f_s)^2{-}6k_0^2 q_s \, \partial_x^2f_s \, f_s
  \right).
  \end{aligned}\label{51_52}
 \end{equation}
To solve this equation  we fix the
parameters $\alpha, \delta, {\rm R}, {\rm W}$ and the flow rate $q_s$.
For $\zeta \ge 0$, we write \eqref{51_52} as $S(f_s,
\zeta)=0$. Assuming that the bottom contour is in
$H_\text{per}^{s}(0,2\pi)$ with $s\geq 3$, we obtain $\partial_x\kappa \in
H_\text{per}^{s-3}(0,2\pi)$, and thus,
\[
S \in C^1\left(H_\text{per}^s(0,2\pi) \times U,
  H_\text{per}^{s-3}(0,2\pi)\right)
\]
 with $U \subset \mathbb{R}_0^+$.
For $A_0:= \partial_f S(f_N,0), \ H_\text{per}^{s}(0,2\pi) \to
H_\text{per}^{s-3}(0,2\pi)$ we have
\[
A_0 = \frac{15}{2{\rm R}}q_s^{2/3} + \left(
    \frac{9}{7}q_s^2-\frac{5}{2{\rm R}}\cot(\alpha) \, q_s
  \right)k_0\partial_x - \frac{6}{{\rm R}}k_0^2q_s^{4/3}\partial_x^2 +
  \frac{5}{6}k_0^3{\rm W} q_s \partial_x^3,
\]
and the eigenfunctions of this constant coefficient linear
differential operator are ${\rm e}^{{\rm i} kx}$, $k \in\mathbb{Z}$.
The real part of the eigenvalue $\omega_k$ is given by
\[
  \operatorname{Re} \omega_k = \frac{15}{2{\rm R}}q_s^{2/3} + \frac{6}{{\rm R}}k_0^2
  q_s^{4/3}k^2;
\]
i.e., the spectrum is bounded away from
zero. Therefore, $A_0$ is an isomorphism between
$H_\text{per}^{s}(0,2\pi)$ and $H_\text{per}^{s-3}(0,2\pi)$, and the
implicit function theorem yields that for each $\zeta$ small enough
the equation $S(f_s,\zeta) = 0$ has a unique solution $f_s(\zeta) \in
H_\text{per}^{s}(0,2\pi)$ which depends continuously on
$\zeta$. Altogether, for each constant flow rate $q_s>0$ and for small
bottom waviness $\zeta$ there exists a unique stationary solution of
the IBL  \eqref{FT3}, \eqref{QT3}.

\begin{remark} \rm
The implicit function theorem yields the existence
of $f_s$ for small values of $\zeta$. This can be extended
  until a bifurcation occurs, but it is not clear for which parameters
  the stationary solution $f_s$ for fixed $q_s$ is unique. However,
  numerically this was the case in our simulations in \cite{HU09}
  up to moderate ${\rm R}$ much larger than the critical
  Reynolds number, beyond which the branch of Nusselt-like solutions
  becomes unstable. Thus, it is mainly this branch that we have in
  mind here. However, we shall
  prove a general nonlinear stability result for all spectrally stable
$(f_s,q_s)$. Thus, instead
of discussing the existence and spectral properties of stationary
solutions in more detail, we simply postulate the pertinent properties in
Assumptions \ref{ass_stat} and \ref{ass_spec}.
\end{remark}

\begin{assumption} \label{ass_stat} \rm
For fixed $\alpha,{\rm R},{\rm W}>0$ and $\kappa\in H^{s-3}_{per}(0,\gamma), s\ge 3$,
 the IBL
\eqref{FT3}, \eqref{QT3} has a family of
  $\gamma$-periodic stationary solutions $(f_s,q_s)^\top$ with
\begin{equation}
    f_s \in H_\text{per}^{s}(0,\gamma), \quad q_s = \text{const.},
\end{equation}
  which can be parametrized by the flow rate $q_s \in (q_{s,\min},
q_{s,\max})$, where $q_{s,\min,\max}$ may depend on the branch considered.
\end{assumption}

\subsection{Perturbation of stationary solutions}\label{sec_540}
Let $(f_s,q_s)^\top$ be a fixed stationary solution of the
IBL  \eqref{FT3}, \eqref{QT3}. Then the perturbation
 $(f,q)^\top := (F-f_s,Q-q_s)^\top$ satisfies
\begin{equation}
  \partial_t f =  -\frac{1}{1+\kappa(f_s+f)}\partial_x q \label{FT4}
\end{equation}
and
\begin{equation} \begin{aligned}
  \partial_t q &=
  \frac{5}{2{\rm R}}\Big(\frac{\sin(\alpha{-}\theta)}{\sin\alpha}f
   +\frac{-f_s^2q{+}2f_sq_sf{+}q_sf^2}{f_s^2(f_s{+}f)^2}
  -\frac{\cos(\alpha{-}\theta)}{\sin\alpha}(\partial_x f_s\, f{+}f_s\,\partial_x f{+} \partial_x
  f\, f) \\
    & \quad\ -\frac{3}{8}\frac{\sin(\alpha{-}\theta)}{\sin\alpha}\partial_x\theta
    (2f_s f+f^2)\Big) +\frac{5}{6}{\rm W} f_s\,\partial_x^3f
   + \frac{5}{6}{\rm W}(\partial_x^3 f_s + \partial_x^3 f - \partial_x\kappa)f  \\
 &\quad -\frac{17}{7} \frac{q_s+q}{f_s+f}\partial_x q +\frac{9}{7}
  \frac{(q_s+q)^2}{(f_s+f)^2}\partial_x f + \frac{9}{7}\frac{2f_s^2q_s q - 2f_sq_s^2f
  + f_s^2q^2 - q_s^2f^2}{f_s^2(f_s+f)^2}\partial_x f_s  \\
&\quad +\frac{1}{{\rm R}}\Big(\frac{9}{2}\partial_x^2 q+\frac{45}{16} \kappa
\frac{f_sq-q_sf}{(f_s+f)f_s} + 4 \frac{q_s+q}{(f_s+f)^2}(2\partial_x f_s \,\partial_x f
+ (\partial_x f)^2)   \\
&\quad + 4\frac{f_s^2q - 2f_sq_sf-q_sf^2}{f_s^2(f_s+f)^2}(\partial_x f_s)^2 -6\frac{q_s+q}{f_s+f}\partial_x^2f - 6\frac{f_sq-q_sf}{(f_s+f)f_s}\partial_x^2 f_s  \\
    & \quad - \frac{9}{2}\frac{\partial_x q(\partial_x f_s + \partial_x f)}{f_s + f}\Big)
    -\frac{1}{210}{\rm R}(\partial_x q)^2(q_s+q).
\end{aligned}\label{QT4}
\end{equation}
The denominators in \eqref{QT4} are bounded from below since $F$ is of order
$1$, cf.~Remark \ref{51_r1}.
The linearization of  \eqref{FT4}, \eqref{QT4} around
$(f,q)^\top=0$ reads
\begin{equation}
  \partial_t\begin{pmatrix}f\\q\end{pmatrix} = \begin{pmatrix}
  0 & -\frac{1}{1+\kappa f_s}\partial_x \\
  \tilde a_{10}+\tilde a_{11}\partial_x+\tilde a_{12}\partial_x^2+\tilde a_{13}\partial_x^3
  & a_{20}+a_{21}\partial_x+a_{22}\partial_x^2\end{pmatrix}
  \begin{pmatrix}f\\q\end{pmatrix}, \label{51_6}
\end{equation}
with the $\gamma$-periodic coefficients
\begin{gather}
 \begin{aligned}
  \tilde a_{10} &=  \frac{5}{2{\rm R}}\left( \frac{\sin(\alpha-\theta)}{\sin\alpha}
  + 2\frac{q_s}{f_s^3} - \frac{\cos(\alpha-\theta)}{\sin\alpha}\partial_x f_s
  -\frac{3}{4} \frac{\sin(\alpha-\theta)}{\sin\alpha}\partial_x\theta\, f_s \right) \\
 &\quad + \frac{5}{6}{\rm W}(\partial_x^3f_s - \partial_x\kappa)
 - \frac{18}{7}\frac{\partial_x f_s\, q_s^2}{f_s^3}
 - \frac{45}{16{\rm R}}\kappa\frac{q_s}{f_s^2}
 - 8\frac{1}{{\rm R}}\frac{(\partial_x f_s)^2q_s}{f_s^3}
 + 6\frac{1}{{\rm R}}\frac{\partial_x^2 f_s\, q_s}{f_s^2},
 \end{aligned} \label{51_10}
\\
 \tilde a_{11} =  -\frac{5}{2}\frac{1}{{\rm R}} \frac{\cos(\alpha-\theta)}{\sin\alpha}f_s
 + \frac{9}{7}\frac{q_s^2}{f_s^2} + 8 \frac{1}{{\rm R}}\frac{\partial_x f_s\, q_s}{f_s^2},
 \quad \tilde a_{12} = -6\frac{1}{{\rm R}}\frac{q_s}{f_s}, \quad
  \tilde a_{13} =  \frac{5}{6} {\rm W} f_s, \label{51_13}
\\
  a_{20} =  -\frac{5}{2{\rm R}}\frac{1}{f_s^2}
  + \frac{18}{7}\frac{\partial_x f_s\, q_s}{f_s^2} + \frac{45}{16{\rm R}}\kappa \frac{1}{f_s}
  + 4 \frac{1}{{\rm R}}\frac{(\partial_x f_s)^2}{f_s^2}
  - 6\frac{1}{{\rm R}}\frac{\partial_x^2 f_s}{f_s},\label{51_20}
\\
  a_{21} =  -\frac{17}{7} \frac{q_s}{f_s} - \frac{9}{2{\rm R}}\frac{\partial_x f_s}{f_s}, \quad
  a_{22} = \frac{9}{2{\rm R}}. \label{51_22}
\end{gather}
By the transformation
\begin{equation}
  H := F + \frac{1}{2}\kappa F^2 \label{51_1}
\end{equation}
the nonlinear equation \eqref{FT3} becomes linear, namely $\partial_t H = - \partial_x Q$.
The transformation \eqref{51_1} is one-to-one if the film
thickness $F$ is of order $1$, and we can express $F$ by $H$ as
\begin{equation}
  F  = \frac{-1+\sqrt{1+2\kappa H}}{\kappa}
  = H - \frac{1}{2}\kappa H^2 + \mathcal{O}(H^3).  \label{51_3}
\end{equation}
The family of stationary solutions $(f_s, q_s)^\top$ from Assumption
\ref{ass_stat} is transformed into $(h_s, q_s)^\top$\label{page_hs},
where $h_s = f_s + \frac{1}{2}\kappa f_s^2$. Setting $h := H - h_s$ we obtain
\begin{equation} \begin{aligned}
  f & = F-f_s = \frac{-1+\sqrt{1+2\kappa H}}{\kappa}
   - \frac{-1+\sqrt{1+2\kappa h_s}}{\kappa}\\
&= \frac{1}{\kappa}\left( \sqrt{1+2\kappa (h_s+h)} - \sqrt{1+2\kappa h_s} \right) \\
& = \frac{1}{(1+2\kappa h_s)^{1/2}}h - \frac{\kappa}{2(1+2\kappa h_s)^{3/2}}h^2
+ \mathcal{O}(h^3)\\
&= \frac{1}{1+\kappa f_s}h - \frac{\kappa}{2(1+\kappa f_s)^3}h^2 + \mathcal{O}(h^3),
\end{aligned}\label{51_2}
\end{equation}
while the inverse transformation is given by
\begin{equation}
  h = (1+\kappa f_s)f + \frac{1}{2}\kappa f^2. \label{51_8}
\end{equation}
For the time derivative of  the perturbation's total mass
\begin{equation}
  M = \int_\mathbb{R} \int_{f_s}^{f_s+f}(1+\kappa z) \,{\rm d} z\, {\rm d} x
  = \int_\mathbb{R} f(1+\kappa(f_s+f/2))\,{\rm d} x \label{51_14}
\end{equation}
we obtain
\begin{equation}
  \partial_t M = \int_\mathbb{R} \partial_t f(1+\kappa(f_s+f))\,{\rm d} x
  = -\int_\mathbb{R} \partial_x q \,{\rm d} x = 0. \label{51_9}
\end{equation}
Thus, the total mass of perturbations is conserved.
This simply reads $\frac{{\rm d}}{{\rm d} t} \int_\mathbb{R} h \, {\rm d} x=0$,
and the IBL  \eqref{FT4}, \eqref{QT4} is equivalent to solving
$\partial_t h = - \partial_x q$ together with \eqref{QT4}, where $f$ must be
replaced everywhere according to \eqref{51_2}. For the linear terms
we write in short
\begin{equation}
  A(\partial_x)\begin{pmatrix}h\\q\end{pmatrix}
  := \begin{pmatrix}0 & -\partial_x \\ a_{10}+a_{11}\partial_x+a_{12}\partial_x^2+a_{13}\partial_x^3
  & a_{20}+a_{21}\partial_x+a_{22}\partial_x^2\end{pmatrix}\begin{pmatrix}h\\q\end{pmatrix}, \label{51_5}
\end{equation}
where
$ a_{10} =  \tilde a_{10}\beta + \tilde a_{11} \partial_x \beta
  + \tilde a_{12} \partial_x^2 \beta + \tilde a_{13}\partial_x^3 \beta$,
$a_{11}=  \tilde a_{11}\beta + 2\tilde a_{12} \partial_x \beta + 3\tilde a_{13}
 \partial_x^2 \beta$,
$a_{12}=  \tilde a_{12}\beta + 3\tilde a_{13} \partial_x \beta$,
$a_{13}=  \tilde a_{13}\beta$,
with
$  \beta(x) := \frac{1}{1+\kappa(x) f_s(x)}$.
Since all fractions in \eqref{QT4} are finite for
small perturbations with $\|f\|_{L^\infty}<\|f_s\|_{L^\infty}/2$,
they can be expanded in powers of $f$, and thus, in powers of $h$.
Hence we can write the transformed IBL  as
\begin{equation}
  \partial_t \begin{pmatrix}h\\q\end{pmatrix} = A(\partial_x)\begin{pmatrix}h\\q\end{pmatrix} + N(h,q), \label{540_1}
\end{equation}
where $N$ contains the nonlinear terms. The first component of $N$ vanishes,
since the equation for $\partial_t h$ is linear.
We look for a solution $(h,q)^\top$ of \eqref{540_1} with
$(h(t),q(t))^\top \in H^r(2)\times H^{r-1}(2)$ for fixed $t$ and
$r\geq3$ in order to avoid Sobolev spaces with negative orders.
Due to the weight we will achieve $C^1$-regularity with respect
to the wave number $\ell$ in Bloch space, which is necessary to
expand the critical mode in terms of $\ell$ in Section \ref{sec_63}.

\subsection{Bloch transform}\label{sec_541}

Considering a bottom with fixed wavelength $\gamma$ and setting $k_0:= 2\pi/\gamma$,
 we define for $v \in H^r(m)$ the Bloch transform $\mathcal{J} v$ as
\begin{equation}
  \mathcal{J} v(\ell,x) = \tilde{v}(\ell,x) := \sum_{j \in \mathbb{Z}} {\rm e}^{{\rm i} j k_0 x} \hat{v}(k_0 j + \ell). \label{541_1}
\end{equation}
From \eqref{541_1} we have that $\mathcal{J} v(\ell,x+\gamma) = \mathcal{J} v(\ell,x)$, and that
Bloch transform is an isomorphism between the weighted Sobolev space $H^r(m)$ and
the Bloch space $B(m,r)$ defined by
\begin{equation}
\begin{gathered}
  B(m,r) = H^m((-k_0/2,k_0/2), H_\text{per}^r(0,\gamma)),\\
 \| \tilde v \|_{B(m,r)} :=
 \Big( \sum_{j\leq m} \int_{I_{k_0}}\| \partial_{\ell}^j \tilde v(\ell, \cdot)
 \|_{H^r(I_\gamma)}^2 \,{\rm d} \ell \Big)^{1/2},
\end{gathered} \label{541_13}
\end{equation}
where $I_\delta := (-\delta/2, \delta/2)$\label{page_I_del}.
The inverse Bloch transform is given by
\begin{equation}
  v(x) = \int_{I_{k_0}}{\rm e}^{{\rm i}\ell x}\mathcal{J} v(\ell,x)\,{\rm d}\ell. \label{541_3}
\end{equation}
We collect some useful properties of Bloch transform.
For a real-valued function $v$, we have
\begin{equation}
  \mathcal{J} v(-\ell,x) = \overline{\mathcal{J} v(\ell,x)}. \label{541_4}
\end{equation}
If $a:\mathbb{R} \to \mathbb{R}$ is $\gamma$-periodic, then
\begin{equation}
  \mathcal{J}(av)(\ell,x) = a(x) \mathcal{J} v(\ell,x). \label{541_5}
\end{equation}
Thus, Bloch transform is invariant under multiplication with
$\gamma$-periodic coefficients. So far, functions in Bloch space
are only defined for $\ell \in (-k_0/2,k_0/2]$. In order to transform
products $uv$ with $u,v \in H^r(m)$ we extend the domain of $\tilde{v} \in B(r,m)$
corresponding to \eqref{541_1}; i.e.,
\[
  \tilde{v}(\ell+k_0,x) = \sum_{j\in\mathbb{Z}} {\rm e}^{{\rm i} j k_0 x} \hat{v}(k_0 j + \ell+k_0)
  = {\rm e}^{-{\rm i} k_0 x}\sum_{j\in\mathbb{Z}} {\rm e}^{{\rm i} j k_0 x} \hat{v}(k_0 j + \ell)
  = {\rm e}^{-{\rm i} k_0 x} \tilde{v}(\ell,x).
\]
Then, multiplication in $x$-space corresponds to convolution in Bloch space;
 i.e.,
\begin{equation}
  \mathcal{J}(uv)(\ell,x) = \int_{-k_0/2}^{k_0/2}\mathcal{J} u(l-k,x) \mathcal{J} v(k,x)\,{\rm d} k
  =: (\mathcal{J} u *_1 \mathcal{J} v)(\ell,x). \label{541_6}
\end{equation}
Therefore we adapt the definition of $B(m,r)$ in \eqref{541_13} to
\begin{equation}
  B(m,r) := \Big\{\tilde{v} \mid \tilde{v}\big|_{\ell \in I_{k_0}} \in H^m(I_{k_0},
  H_\text{per}^r(0,\gamma)) \text{ and } \tilde{v}(\ell+k_0,x)
  = {\rm e}^{-{\rm i} k_0 x} \tilde{v}(\ell,x)\Big\}. \label{541_2}
\end{equation}
The notation $*_1$ in \eqref{541_6} becomes clear in \eqref{62_2},
where we define a more general convolution operator.
If there is no ambiguity we omit the subscript in the following
and write $\mathcal{J} u * \mathcal{J} v$. Due to the extension in \eqref{541_2} convolution
becomes commutative. From \eqref{541_3} we obtain
\begin{equation}
  \partial_x v(x) = \int_{-k_0/2}^{k_0/2} {\rm e}^{{\rm i}\ell x}(\partial_x + {\rm i}\ell) \mathcal{J} v(\ell,x) \,{\rm d}\ell;
\end{equation}
i.e., $\partial_x$ in $x$-space corresponds to the operator
 $(\partial_x + {\rm i}\ell)$ in Bloch space. Thus, setting $\tilde{h} := \mathcal{J} h$ and
$\tilde{q} := \mathcal{J} q$ the IBL  \eqref{540_1} is equivalent to
\begin{equation}
	\partial_t \begin{pmatrix}\tilde{h} \\ \tilde{q}\end{pmatrix}
 = A(\partial_x+{\rm i}\ell)\begin{pmatrix}\tilde{h} \\ \tilde{q}\end{pmatrix}
	+ \tilde{N}(\tilde{h},\tilde q) \label{541_10}
\end{equation}
in Bloch space, where
\begin{equation}
	\tilde N(\tilde{h},\tilde q)  := \mathcal{J} N(\mathcal{J}^{-1}\tilde{h}, \mathcal{J}^{-1}\tilde{q}). \label{541_11}
\end{equation}
Since Bloch transform is an isomorphism between $H^r(2)$ and $B(2,r)$,
we look for a solution $(\tilde{h},\tilde{q})^\top$ of \eqref{541_10} with
$(\tilde{h}(t),\tilde{q}(t))^\top \in B(2,r)\times B(2,r-1)$ for fixed $t$
and $r\geq 3$.

\subsection{Spectral situation and mode filters}\label{sec_542}
\textbf{Spectral situation.}

By \eqref{51_1} the family of stationary solutions $(f_s,q_s)^\top$
from Assumption \ref{ass_stat} is transformed into a family of stationary
solutions $(h_s,q_s)^\top$ of the IBL  for
$(H,Q)^\top$, which we write in short as
\begin{equation}
  \partial_t\begin{pmatrix}H\\ Q \end{pmatrix}=
 \mathcal{G}(H,Q) = \begin{pmatrix}\mathcal{G}_1(H,Q)\\ \mathcal{G}_2(H,Q)\end{pmatrix}. \label{542_11}
\end{equation}
Since the $\gamma$-periodic stationary solutions are parametrized by the $x$-independent flow rates $q_s$, we have
\begin{equation}
  \mathcal{G}(h_s(q_s),q_s) = 0 \quad \text{ for all } q_s
  \in (q_{s,\text{min}},q_{s,\text{max}}),
\end{equation}
and differentiating with respect to $q_s$ gives
\begin{equation}
  0  = \frac{{\rm d}}{{\rm d} q_s}\mathcal{G}(h_s(q_s),q_s)
= \begin{pmatrix}
\frac{\partial \mathcal{G}_1}{\partial H}(h_s(q_s),q_s)
& \frac{\partial \mathcal{G}_1}{\partial Q}(h_s(q_s),q_s)\\
\frac{\partial \mathcal{G}_2}{\partial H}(h_s(q_s),q_s)
& \frac{\partial \mathcal{G}_2}{\partial Q}(h_s(q_s),q_s)
\end{pmatrix}
\begin{pmatrix} \frac{{\rm d} h_s}{{\rm d} q_s}(q_s)\\1 \end{pmatrix}. \label{542_9}
\end{equation}
The linear differential operator on the right-hand side of \eqref{542_9}
also occurs in the linearization of the IBL  \eqref{542_11} around a
stationary solution: Choosing in the following $q_s$ fixed, the
perturbation $(h,q)^\top=(H-h_s,Q-q_s)^\top$ satisfies
$\partial_t(h,q)^\top = \mathcal{G}(h_s+h,q_s+q)$. Thus, the linearization around
$(h,q)^\top=0$ reads
\begin{equation}
  \partial_t\begin{pmatrix}h\\q\end{pmatrix}
= \frac{\partial\mathcal{G}}{\partial(H,Q)}(h_s,q_s)\begin{pmatrix}h\\q\end{pmatrix}, \label{542_10}
\end{equation}
which we have already expressed in \eqref{51_5} with the help of the
differential operator $A(\partial_x)$. Therefore, combining \eqref{542_9}
and \eqref{542_10} gives
\begin{equation}
  A(\partial_x)\begin{pmatrix}\frac{{\rm d} h_s}{{\rm d} q_s}(q_s)\\1\end{pmatrix} = 0 \quad \text{for all }
  q_s \in (0,q_{s,\text{max}}). \label{542_12}
\end{equation}
Transferring the IBL  to Bloch space,
we know from \eqref{541_10} that the linear operator in the evolution
equation for $(\tilde{h},\tilde{q})^\top$ is given by $A(\partial_x+{\rm i}\ell)$.
Corresponding to \eqref{542_12},
$({\rm d} h_s/{\rm d} q_s,1)^\top \in H_\text{per}^{s}(0,\gamma)\times
H_\text{per}^{s-1}(0,\gamma)$ is an eigenfunction of $A(\partial_x+{\rm i}\ell)$
to the eigenvalue $\lambda_1(0) = 0$ for $\ell=0$. Thus, in Bloch space
the linearization of the IBL  around a stationary solution has always
a zero eigenvalue. This property corresponds to the free surface in the
underlying physical problem. Furthermore,
for fixed $\ell \in (-k_0/2, k_0/2)$ the differential operator
$A(\partial_x+{\rm i}\ell):H_\text{per}^{s}(0,\gamma)\times H_\text{per}^{s-1}(0,\gamma)
\to H_\text{per}^{s-2}(0,\gamma)\times H_\text{per}^{s-3}(0,\gamma)$
is elliptic, and thus we obtain countable many curves
of eigenvalues $\lambda_n$ with $\operatorname{Re}\lambda_n(\ell) \to -\infty$ for $n\to\infty$.
Like for the stationary solutions,
instead of calculating the spectrum of $A(\partial_x + {\rm i}\ell)$,
we state an assumption based on the properties derived above.
A typical spectrum is then sketched in Figure \ref{542_fig1}.

\begin{assumption}[Spectral stability]\label{ass_spec} \rm
  Let $s$ be the bottom regularity from Assumption \ref{ass_stat}.
We assume that
$A(\partial_x + {\rm i}\cdot)$ with  $A(\partial_x+{\rm i}\ell):H_\text{per}^{s}(0,\gamma)
\times H_\text{per}^{s-1}(0,\gamma) \to H_\text{per}^{s-2}(0,\gamma)
\times H_\text{per}^{s-3}(0,\gamma)$ has countable many curves of eigenvalues
$\lambda_n:(-k_0/2,k_0/2)\\ \to\mathbb{C}$, $n \in \mathbb{N}$, with eigenfunctions
$\ell\mapsto\phi^n(\ell,\cdot)\in H_\text{per}^{s}(0,\gamma)\times
H_\text{per}^{s-1}(0,\gamma)$ and
  \begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}
    \item $\lambda_1(\ell) = c_1 {\rm i}\ell - c_2 \ell^2 + \mathcal{O}(\ell^3)$ with $c_1 \in \mathbb{R}$,
$\operatorname{Re} c_2 > 0$,
    \item $\operatorname{Re}\lambda_1(\ell) < -\tilde{c}_2 \ell^2$ for $|\ell|\leq 4 r_\chi$ and a
 $\tilde{c}_2<c_2$,
    \item $\operatorname{Re}\lambda_1(\ell)<-\sigma_0<0$ for $|\ell|> 4 r_\chi$ and $\operatorname{Re}\lambda_1(\ell)>-\sigma_0$
for $|\ell|< 4 r_\chi$,
    \item $\operatorname{Re}\lambda_n(\ell) < -\sigma_0$ for all $n\geq2$,  $\ell \in (-k_0/2,k_0/2)$.
  \end{enumerate}
\end{assumption}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig3}  % spec.eps,
\end{center}
\caption{Sketch of the spectral situation and the cut-off function $\chi$}
 \label{542_fig1}
\end{figure}

\subsection*{Eigenfunctions}
The relation between the eigenvalues and eigenvectors of the
two versions of the IBL, namely system \eqref{FT4}, \eqref{QT4}
for $(f,q)^\top$ and system \eqref{540_1}
for $(h,q)^\top$, is as follows.  Let us denote the
linearized $(f,q)$-system \eqref{51_6} by
$\partial_t (f,q)^\top = \hat A(\partial_x)(f,q)^\top$. Since
$h = \frac{1}{\beta}f + \mathcal{O}(f^2)$, see \eqref{51_5}, we have
\[
  A(\partial_x)\begin{pmatrix}h\\q\end{pmatrix}
= \begin{pmatrix}1/\beta & 0\\0 & 1\end{pmatrix}\hat A(\partial_x)\begin{pmatrix}\beta h\\q\end{pmatrix},
\]
where $\beta(x) = 1/(1+\kappa(x) f_s(x))$, see \eqref{51_5}.
Thus, for each eigenvalue $\lambda_n$ of $A(\partial_x + {\rm i}\ell)$ we obtain
\[
  \lambda_n \phi^n  = A(\partial_x + {\rm i}\ell)\phi^n = \begin{pmatrix}1/\beta & 0\\0 & 1\end{pmatrix}
  \hat A(\partial_x + {\rm i}\ell) \begin{pmatrix}\beta \phi^n_1\\ \phi^n_2\end{pmatrix};
\]
i.e.,
\[
  \lambda_n\begin{pmatrix}\beta \phi^n_1\\ \phi^n_2\end{pmatrix} = \hat A(\partial_x + {\rm i}\ell)
  \begin{pmatrix}\beta \phi^n_1\\ \phi^n_2\end{pmatrix}.
\]
Therefore, in Bloch space the two systems for $(f,q)^\top$ and
 $(h,q)^\top$ have exactly the same eigenvalues, where the eigenvectors
 of the $(f,q)$-system are given by
\begin{equation}
  \Phi^n := \begin{pmatrix}\beta & 0\\0 & 1\end{pmatrix}\phi^n. \label{542_8}
\end{equation}
In particular, the critical eigenfunctions read
\begin{equation}
  \phi^1(\ell,\cdot) = \begin{pmatrix}\frac{{\rm d} f_s}{{\rm d} q_s} \\ 1 \end{pmatrix} + \mathcal{O}(\ell), \quad
  \Phi^1(\ell,\cdot) = \begin{pmatrix}\frac{1}{1+\kappa f_s}\frac{{\rm d} h_s}{{\rm d} q_s}\\1\end{pmatrix}
  = \begin{pmatrix}\frac{{\rm d} f_s}{{\rm d} q_s}\\1\end{pmatrix}  + \mathcal{O}(\ell). \label{542_7}
\end{equation}
This property is used in the proof of Theorem \ref{main_th2},
where the universal decay behavior for the $(h,q)$-system is
transferred back to the original $(f,q)$-system.

Since the IBL  \eqref{541_10} in Bloch space has a zero eigenvalue,
we have to split $(\tilde{h}, \tilde{q})^\top$ into its stable part and into
a multiple of the critical eigenvector $\phi^1$. On the linear level,
the critical curve $\lambda_1(\ell) = c_1 {\rm i}\ell - c_2 \ell^2 + \mathcal{O}(\ell^3)$
for the mode $\phi^1(\ell,\cdot)$ corresponds to
$\partial_t v = (c_1 \partial_x +c_2 \partial_x^2) v$, which is the linear diffusion equation
in the comoving frame $y = x + c_1 t$. However, going into this comoving
frame in \eqref{540_1} leads to a time dependent differential operator,
which would make the subsequent analysis more complicated. Therefore,
we introduce the rotated variable $\tilde{w}$ by
\begin{equation}
  \tilde{w}(t,\ell,x) = \begin{pmatrix}\tilde{w}_1\\ \tilde{w}_2\end{pmatrix}(t,\ell,x)
  := {\rm e}^{-c_1{\rm i}\ell t} \begin{pmatrix}\tilde{h}\\ \tilde{q}\end{pmatrix}(t,\ell,x), \label{542_17}
\end{equation}
which satisfies
\begin{equation}
  \partial_t \tilde{w}(t,\ell,x) = \tilde{A}(\ell) \tilde{w}(t,\ell,x) + \tilde{N}(\tilde{w})(t,\ell,x) \label{542_25}
\end{equation}
with
\begin{equation} \begin{aligned}
   \tilde{A}(\ell) &:=  -c_1{\rm i}\ell + A(\partial_x + {\rm i}\ell)  \\
  & =
{\scriptsize
\begin{pmatrix}-c_1{\rm i}\ell & -(\partial_x{+}{\rm i}\ell) \\ a_{10}{+}a_{11}
  (\partial_x{+}{\rm i}\ell){+}a_{12}(\partial_x{+}{\rm i}\ell)^2{+}a_{13}(\partial_x{+}{\rm i}\ell)^3
  & (a_{20}{-}c_1{\rm i}\ell){+}a_{21}(\partial_x{+}{\rm i}\ell){+}a_{22}(\partial_x{+}{\rm i}\ell)^2\end{pmatrix}.
}
\end{aligned}\label{542_13}
\end{equation}
The nonlinearity $\tilde{N}$ is exactly the same as for the
$(\tilde{h},\tilde{q})$-system in \eqref{541_10} since
$(\tilde{v}_i * \tilde{v}_j)(\ell) = \int_{-k_0/2}^{k_0/2}\tilde{w}_i(\ell-k)
{\rm e}^{c_1 {\rm i}(\ell-k)t}\tilde{w}_j(k){\rm e}^{c_1{\rm i} k t}\,{\rm d} k
= {\rm e}^{c_1{\rm i}\ell t} (\tilde{w}_i*\tilde{w}_j)(\ell)  $
  for $\tilde{v} := {\rm e}^{c_1{\rm i}\ell t} \tilde{w}$ and $i,j \in \{1,2\}$.
Clearly, $\tilde{A}$ has the same eigenfunctions $\phi^n$ as
$A(\partial_x + {\rm i}\ell)$ with eigenvalues $\mu_n(\ell) = \lambda_n(\ell) - c_1{\rm i}\ell$
\label{page_mu}. In particular, for the critical eigenvalue we obtain
\begin{equation}
  \mu_1(\ell) = - c_2 \ell^2 + \mathcal{O}(\ell^3). \label{542_27}
\end{equation}

\subsection*{Mode filters}
We introduce mode filters to extract the critical mode $\phi^1$.
Let $\chi:\mathbb{R} \to [0,1]$ be a smooth cut-off function\label{page_chi}
with $\chi(\ell)=1$ for $|\ell| \leq r_\chi$ and $\chi(\ell)=0$ for
$|\ell| \geq 2r_\chi$, see Figure \ref{542_fig1}.
Due to Assumption \ref{ass_spec} the curve of critical eigenvalues
$\mu_1$ is isolated from the rest of the spectrum for $|\ell| < 4r_\chi$.
Thus, denoting the scalar product in $L^2(0,\gamma)$ by
$\langle\cdot,\cdot\rangle$; i.e.,
\[
  \langle u,v\rangle := \int_0^\gamma u\cdot \bar v\,{\rm d} x,
\]
where the ``$\cdot$'' stands for the standard scalar product in
$\mathbb{R}^2$, we can define the critical mode filter $\tilde{E}_c$ by
\begin{equation}
  (\tilde E_c \tilde{w})(\ell,x) := \chi(\ell)
\bigl\langle \tilde{w}(\ell,\cdot),\psi^1(\ell,\cdot)\bigr\rangle\phi^1(\ell,x).\label{542_15}
\end{equation}
Here $\psi^1(\ell,\cdot)$ is an eigenfunction of the $L^2(0,\gamma)$-adjoint
operator $\tilde{A}^*(\ell)$ to the eigenvalue $\bar\mu_1(\ell)$.
The $L^2(0,\gamma)$-adjoint operator of a differential operator
$L = a(x)(\partial_x + {\rm i}\ell)$ with a $\gamma$-periodic coefficient $a$ is
given by $L^*v = -(\partial_x+{\rm i}\ell)(\bar a v)$. Thus, for the critical
eigenfunction we obtain $\psi^1(0,x) = (c_0,0)^\top$; i.e.,
\begin{equation}
  \psi^1(\ell,x) = c_0\begin{pmatrix}1\\0\end{pmatrix} + \mathcal{O}(\ell), \label{542_24}
\end{equation}
and we choose $\psi^1$ such that
$\langle \phi^1(\ell,\cdot), \psi^1(\ell,\cdot)\rangle = 1$ for all
$\ell \in (-4r_\chi,4r_\chi)$. Additionally to $\tilde{E}_c$,
we define the scalar mode filter $\tilde{E}_c^*$ and the stable mode
filter $\tilde{E}_s$ by
\begin{equation}
    (\tilde E_c^* \tilde{w})(\ell)  := \chi(\ell)\langle \tilde{w}(\ell,\cdot),
    \psi^1(\ell,\cdot) \rangle,\quad
    \tilde E_s := \operatorname{Id} - \tilde E_c.\label{542_1}
\end{equation}
Moreover, we define auxiliary mode filters
\begin{equation}
  (\tilde E_c^h \tilde{w})(\ell,x)  := \chi(\ell/2)\langle \tilde{w}(\ell,\cdot),\psi^1(\ell,\cdot)
  \rangle\phi^1(\ell,x), \quad
   \tilde E_s^h  := \operatorname{Id} - \tilde E_c^h \label{542_14}
\end{equation}
such that $\tilde{E}_c^h\tilde{E}_c = \tilde{E}_c$ and $\tilde{E}_s^h\tilde{E}_s = \tilde{E}_s$,
which is used to substitute for missing projection properties of
$\tilde{E}_c$ and $\tilde{E}_s$. Setting
$\tilde{\alpha}(t,\ell) := (\tilde{E}_c^* \tilde{w}(t))(\ell), \ \tilde{w}_s(t,\ell,x)
:= (\tilde{E}_s\tilde{w}(t))(\ell,x)$, we obtain the splitting
\begin{equation}
  \tilde{w}(t,\ell,x) = \tilde{\alpha}(t,\ell)\phi^1(\ell,x) + \tilde{w}_s(t,\ell,x) \label{542_16}
\end{equation}
into the critical mode $\tilde{\alpha}\phi^1$ and the stable component $\tilde{w}_s$.

\begin{remark} \label{542_r1} \rm
  The idea of this splitting is that due to the spectral properties of
   $\tilde{E}_s^h\tilde{A}(\ell)$, $w_s$ is linearly exponentially damped. Thus,
we expect the dynamics of \eqref{542_25} to be governed by the dynamics
of the critical mode $\tilde{\alpha}\phi^1$.
\end{remark}

Altogether, after applying mode filters, the IBL  in Bloch space reads
\begin{gather}
  \partial_t \tilde{\alpha}(t,\ell)  = \mu_1(\ell)\tilde{\alpha}(t,\ell) + \tilde B_c(\tilde{\alpha}(t))(\ell)
  + \tilde H_c(\tilde{\alpha}(t),\tilde{w}_s(t))(\ell), \label{542_al}
\\
  \partial_t \tilde{w}_s(t,\ell,x)  = \tilde{A}_s(\ell) \tilde w_s(t,\ell,x)
   + \tilde H_s(\tilde{\alpha}(t),\tilde{w}_s(t))(\ell,x), \label{542_ws}
\end{gather}
where
\begin{gather}
  \tilde B_c(\tilde{\alpha})(\ell) := {\rm i} d\ell\chi(\ell)(\tilde{\alpha}^{*2})(\ell), \\
  \tilde H_c(\tilde{\alpha},\tilde{w}_s)(\ell) := \tilde{E}_c^*
  \left(\tilde{N}(\tilde{\alpha}\phi^1 + \tilde{w}_s) \right)(\ell) - {\rm i} d
  \ell\chi(\ell)\tilde{\alpha}^{*2}(\ell), \label{542_22}
\\
\tilde{A}_s(\ell)  := \tilde{E}_s^h\tilde{A}(\ell), \quad
  \tilde H_s(\tilde{\alpha},\tilde{w}_s)(\ell,x) := \tilde E_s
  \left(\tilde{N}(\tilde{\alpha} \phi^1 + \tilde w_s) \right) (\ell,x), \label{542_23}
\end{gather}
with $d$ specified subsequently in \eqref{543_9}.
Below we will see that cubic terms as well as those
involving $\tilde{w}_s$ are asymptotically irrelevant. Thus, the only
dangerous terms are the quadratic ones in $\tilde{N}(\tilde{\alpha}\phi^1)$,
 which are not damped by the decay of $\tilde{w}_s$. In the formal derivation
in \S\ref{sec_543} we will see that these terms have the ``derivative-like''
structure ${\rm i} d\ell\chi(\ell)\tilde{\alpha}^{*2}$ with $d \in \mathbb{R}$, which leads to a
Burgers-like decay. There also occur terms of the order of
$\mathcal{O}(\ell^2)\tilde{\alpha}^{*2}$, but as they turn out to be irrelevant due to
the additional factor $\ell$, we put them into $\tilde{H}_c$ and denote by
$\tilde{B}_c$ the term ${\rm i} d\ell\chi(\ell)\tilde{\alpha}^{*2}$, which is the only relevant one.

\subsection*{Function spaces}

It remains to choose appropriate function spaces for $\tilde{\alpha}$ and $\tilde{w}_s$.
For fixed $t$ we have $(h,q)^\top \in H^r(2) \times H^{r-1}(2)$ if and only
if $\tilde{w} \in B(2,r) \times B(2,r-1)$; i.e., both $\tilde{\alpha}\phi^1$ and
$\tilde{w}_s \in B(2,r) \times B(2,r-1)$.

Thus, in a first step we assume that $\tilde{\alpha}\phi^1\in B(2,r)\times B(2,r-1)$.
In the following let the bottom profile be at least in $H^r_\text{per}(0,\gamma)$,
such that due to Assumption \ref{ass_spec} we have
$\phi^1(\ell) \in H^r_\text{per}(0,\gamma)\times H^{r-1}_\text{per}(0,\gamma)$
for fixed $\ell$. Since the critical eigenvalue $\mu_1(\ell)$ is isolated from the
 rest of the spectrum for $|\ell| < 4r_\chi$, the eigenfunction $\phi^1$ is
smooth with respect to $\ell$ in this interval. In particular, we have
$\phi^1\in H^2((-2r_\chi,2r_\chi), H_\text{per}^r(0,\gamma)\times
H^{r-1}_\text{per}(0,\gamma))$. Since the same is true for the adjoint eigenfunction
$\psi^1$, the definition of the critical mode filter in \eqref{542_1} leads to
\begin{equation}
  \tilde{\alpha}\in H^2(\mathbb{R}), \quad \operatorname{supp}\tilde{\alpha} \in [-2r_\chi,2r_\chi].
\end{equation}
Next, we conversely assume that $\tilde{\alpha}\in H^2(\mathbb{R})$ with
 $\operatorname{supp}\tilde{\alpha} \in [-2r_\chi,2r_\chi]$, and $\tilde{w}_s \in B(2,r) \times B(2,r-1)$.
 It immediately follows that $\tilde{\alpha}\phi^1$ is in
$H^2(I_{k_0}, H^r_\text{per}(0,\gamma) \times  H^{r-1}_\text{per}(0,\gamma))$,
 but not in $B(2,r) \times B(2,r-1)$ since the extension property from \eqref{541_2}
is missing, which is required to calculate convolutions. However, since $\tilde{\alpha}$
has compact support, this is not needed. On the one hand, in convolutions like
\[
  \int_{-k_0/2}^{k_0/2} \tilde{\alpha}(\ell-k)\phi^1(\ell-k) \tilde{v}(k) \,{\rm d} k
= \int_{-k_0/2}^{k_0/2} \tilde{\alpha}(k)\phi^1(k) \tilde{v}(\ell-k) \,{\rm d} k,
\]
with $\tilde{v} \in B(2,r) \times B(2,r-1)$, we can use the extension property of
$\tilde{v}$ such that $\tilde{\alpha}\phi^1$ must only be evaluated for $\ell \in I_{k_0}$.
On the other hand, for convolutions
\[
  \int_{-k_0/2}^{k_0/2} \tilde{\alpha}(\ell-k)\phi^1(\ell-k) \tilde{\alpha}(k)\phi^1(k) \,{\rm d} k
 = \int_{-2r_\chi}^{2r_\chi} \tilde{\alpha}(\ell-k)\phi^1(\ell-k) \tilde{\alpha}(k)\phi^1(k) \,{\rm d} k
\]
we have to extend $\tilde{\alpha}\phi^1$ to $|\ell| \leq k_0/2 + 2r_\chi$ by
$(\tilde{\alpha}\phi^1)(\ell+k_0) = {\rm e}^{-{\rm i} k_0x}(\tilde{\alpha}\phi^1)(\ell)$.
Thus, $\tilde{\alpha}\phi^1$ is extended with values of
$(\tilde{\alpha}\phi^1)(\ell)$, $\ell \in (-1/2,-1/2+2r_\chi]\cup(1/2-2r_\chi,1/2]$,
where $\tilde{\alpha}$ and hence $\tilde{\alpha}\phi^1$ is zero. Thus, there is no difference
if we extend $\tilde{\alpha}\phi^1$ according to the extension rule in \eqref{541_2}
or if we use $\tilde{\alpha} \in H^2(\mathbb{R})$ with compact support. If necessary,
we must replace $r_\chi$ in Assumption \ref{ass_spec} by a smaller value
 depending on the final degree of the nonlinearity since each convolution
enlarges the support of $(\tilde{\alpha}\phi^1)^{*j}$. Altogether, we obtain the equivalence
\begin{equation} \begin{aligned}
 \tilde{w} \in B(2,r){\times}B(2,r-1)
   &\Leftrightarrow \ \tilde{\alpha} \in H^2(\mathbb{R}),
 \ \operatorname{supp}\tilde{\alpha} \in [-2r_\chi,2r_\chi], \\
& \text{ and }  \tilde{w}_s \in B(2,r){\times}B(2,r-1). \label{542_28}
\end{aligned}
\end{equation}
Moreover, since $\tilde{\alpha}$ is independent of $x$ we obtain
\begin{align*}
  \|\tilde{\alpha}\|_{B(2,r)}^2
 &= \sum_{j\leq2}\int_{-k_0/2}^{k_0/2}\|\partial_{\ell}^j \tilde{\alpha}(\ell)\|_{H^r(I_\gamma)}^2\,{\rm d}\ell\\
 &= \sum_{j\leq2}\int_{-k_0/2}^{k_0/2}\gamma^2|\partial_{\ell}^j \tilde{\alpha}(\ell)|^2\,{\rm d}\ell
 = \gamma^2\|\tilde{\alpha}\|_{H^2(I_{k_0})}^2
\end{align*}
for all $r\geq 0$. Therefore, and since it does
 not matter how $\tilde{\alpha}$ is extended to $|\ell|>k_0/2$,
 $\tilde{\alpha} \in H^2(\mathbb{R})$ in \eqref{542_28}
can be substituted by $\tilde{\alpha} \in B(2,r)$.
Thus, we look for a solution $(\tilde{\alpha},\tilde{w}_s)$ of \eqref{542_al}, \eqref{542_ws}
with $\tilde{\alpha}(t) \in B(2,r)$ and $\tilde{w}(t) \in B(2,r) \times B(2,r-1)$
for fixed $t$ and $r \geq3$.

\subsection{Self-similar decay in the viscous Burgers equation}
\label{sec_522}

The idea behind the splitting of $\tilde{w}$ into $\tilde{\alpha}$ and $\tilde{w}_s$
is that $\tilde{\alpha}$ will fulfill a perturbed Burgers equation while
$\tilde{w}$ is linearly exponentially damped. Here we collect some basic
facts about the  dynamics of the Burgers equation, mainly from
\cite{bkl94}, see also \cite{ue04ibl, Uecker_07} for more details.

 By the Cole-Hopf transformation
$ \eta(t,\xi) = \exp\left( \frac{d}{c_2}\int_{-\infty}^{\sqrt{c_2}\xi}v(t,x) {\rm d} x \right)$,
 the viscous Burgers equation
\begin{equation}\label{522_burg}
  \partial_t v = c_2 \partial_x^2 v + d \partial_x(v^2), \ x \in \mathbb{R}, \, t \geq 0
\end{equation}
is transformed into  the linear diffusion equation
 $\partial_t\eta = \partial^2_\xi \eta$.
The inverse transformation is given by
\[
  v(t,x) = \frac{\sqrt{c_2}}{d}
\frac{\partial_\xi\eta(t,x/\sqrt{c_2})}{\eta(t,x/\sqrt{c_2})}.
\]
By construction, we have $\lim_{\xi\to-\infty}\eta(t,\xi)=1$ for all $t\geq0$.
Setting $\lim_{\xi\to\infty}\eta(0,\xi)=1+z$ for the initial condition,
it is well known that
\[
  \eta(t,\xi) = 1+\frac{z}{2}\Big( 1+\operatorname{erf}\big(\frac{\xi}{2\sqrt t}\big) \Big)
 \text{ with } \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x {\rm e}^{-y^2} {\rm d} y
\]
is an exact solution of the linear diffusion equation. Thus, for every $z>-1$ there
exists a self-similar solution of the Burgers equation \eqref{522_burg} given by
\begin{equation}
   v_z(t,x) := t^{-1/2}f_z(t^{-1/2}x) \text{ with } f_z(y) = \frac{\sqrt{c_2}}{d}\frac{z \operatorname{erf}'(y/(2\sqrt{c_2}))}{4+2z\left(1+\operatorname{erf}(y/(2\sqrt{c_2}))\right)}, \label{522_1}
\end{equation}
where $\ln(z+1) = \frac{d}{c_2}\int_\mathbb{R}  v_z(t,x) {\rm d} x$.

Moreover, if we consider an arbitrary initial condition
$\eta\big|_{t=0} = \eta_0 \in L^\infty$ with the boundary conditions
$\lim_{\xi \to -\infty}\eta_0(\xi) = 1$ and
$\lim_{\xi \to \infty}\eta_0(\xi) = 1+z$, the solution can be written as
\[
  \eta(t,\xi) = \frac{1}{\sqrt{4\pi t}}\int_\mathbb{R} {\rm e}^{-(\xi-y)^2/(4t)}\eta_0(y) {\rm d} y.
\]
If we assume that $\eta_0$ decays sufficiently fast to $ 1 $ for
$\xi \to \pm \infty$, we have $\varphi_0 := \partial_\xi \eta_0 \in L^1$,
and $\varphi := \partial_\xi \eta$ satisfies the linear diffusion equation
with the localized initial condition $\varphi(0,\xi) = \varphi_0(\xi)$. Then
$  \sup_{\xi\in\mathbb{R}}|\varphi(t,\xi) - \sqrt{\pi/t}\,
\hat\varphi_0(0){\rm e}^{-\xi^2/(4t)}| \le Ct^{-1}$,
which, by integration with respect to $\xi$, yields
\[
  \sup_{\xi\in\mathbb{R}}\big|\eta(t,\xi) - 1-\frac{z}{2}
\Big( 1+\operatorname{erf}\big(\frac{\xi}{2\sqrt t}\big) \Big)\big| \le Ct^{-1/2}.
\]
Therefore, the renormalized solution of the Burgers equation \eqref{522_burg}
with initial condition $v\big|_{t=0} = v_0 \in L^1$ satisfies
\begin{equation}
  \sup_{x\in\mathbb{R}}|t^{1/2}v(t,t^{1/2}x) - f_z(x)| \le Ct^{-1/2}, \label{522_2}
\end{equation}
where $\ln(z+1) = \frac{d}{c_2}\int_\mathbb{R} v_0(x) {\rm d} x$. Thus, solutions of the
Burgers equation to localized initial conditions converge to a non-Gaussian
profile, see Fig.~\ref{522_fig1}.
This behaviour is stable under suitable perturbations of the Burgers equation,
cf., e.g., \cite[Theorem 1.5]{Uecker_07}.

\begin{lemma}
  Let $p\in(0,1/2)$ and
$h(v,\partial_x v, \partial_x^2 v) = v^{q_1}(\partial_x v)^{q_2}(\partial_x^2 v)^{q_3}$ with
$d_h=q_1+2q_2+3q_3>3$, $q_j \in \mathbb{N}_0$, and $q_3\leq 1$.
Then there exist $C_1, C_2>0$ such that the following holds.
If $\|v_0\|_{H^2(2)}\leq C_1$, then the perturbed Burgers equation
  \[
    \partial_t v = c_2 \partial_x^2 v + d \partial_x(v^2) + h(v,\partial_x v, \partial_x^2 v)
\]
  with $c_2>0, \, d\neq0$ has a unique solution $v$ with $v\big|_{t=1} = v_0$.
For a $z > -1$ it satisfies
  \begin{equation}\label{bdec1}
    \|\sqrt{t}v(t,\sqrt{t}x) - f_z(x)\|_{H^2(2)} \leq C_2t^{-1/2+p}
  \end{equation}
  for all $t\geq 1$, where $f_z$ is the non--Gaussian profile from \eqref{522_1}.
\end{lemma}

In particular, nonlinearities $h$ with degree $d_h>3$, or more general
nonlinearities (not necessarily monomials) such that  \eqref{bdec1} holds,
are called asymptotically irrelevant.

\subsection{Derivation of the Burgers equation}\label{sec_543}
\textbf{Splitting of the nonlinearity.}
To distinguish relevant from asymptotically irrelevant terms we split the
nonlinearity $N$ from \eqref{540_1} into $N=B+G$, where the second component of
$B(h,q)$ contains all quadratic terms without a factor $\partial_x q$.
The terms in $B$ turn out to have a ``derivative-like'' structure and hence
 lead to a Burgers-like decay, see Remark \ref{543_r1} below.
For all other terms, which we collect in $G(h,q)$, we later show that they are
irrelevant. By construction,
\begin{equation} \begin{aligned}
  B(h,q)&= \begin{pmatrix}0\\ \mathcal{B}_2(h,q)\end{pmatrix} \\
    & =
{\scriptsize
\begin{pmatrix}0\\ b_{00}h^2{+}b_{01}h\,\partial_x h{+}b_{02} h\,\partial_x^2 h{+}b_{03}h\,
 \partial_x^3 h{+}b_{11}(\partial_x h)^2{+}b_1 h q{+}b_2 \partial_x h\, q{+}b_3 \partial_x^2 h\, q{+}b_4 q^2\end{pmatrix},
}
\end{aligned} \label{543_5}
\end{equation}
where again all coefficients are $\gamma$-periodic in $x$ and depend on the
stationary solution $(h_s,q_s)$.

Since the equation for $\partial_t h$ is linear, also in $G(h,q)$ the first component vanishes.
The terms in the second component of $G$ can be characterized as follows:
\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}
  \item Terms in $B_2(h,q)$, multiplied by $h^j, \,j \geq1$.
  \item $(\partial_x q)^2, \, h^j\, \partial_x q \, q, \, h^{j+1} \,\partial_x q, \, h^j\, \partial_x h \,\partial_x q$ with $j\geq 0$.
  \item $(\partial_x q)^2q, \, h^j\, \partial_x h \, q^2, \, h^j (\partial_x h)^2 q$ with $j\geq 0$.
\end{enumerate}
The terms in (i) are due to the expansions $1/(f_s+f) = \sum_{j\geq0}c_j f^j$ and
$f = \sum_{j\geq1}\tilde c_jh^j$. They are at least cubic and contain the quasilinear
terms $h^j \,\partial_x^3 h, \, j \geq 2$. The terms in (ii) are the quadratic ones
in \eqref{QT4} which contain a factor $\partial_x q$. Except of the first one,
they also occur multiplied by powers of $h$ due to the denominator $1/(f_s+f)$.
 Finally, the terms in (iii) originate from the terms in \eqref{QT4} having a
 cubic numerator.
Altogether, we can write the IBL  \eqref{540_1} for $(h,q)^\top$ as
\begin{equation}
  \partial_t \begin{pmatrix}h\\q\end{pmatrix} = A(\partial_x)\begin{pmatrix}h\\q\end{pmatrix}
 + B(h,q) + G(h,q). \label{543_4}
\end{equation}
Setting $\tilde{B}(\tilde{h},\tilde{q}) = \mathcal{J} B(\mathcal{J}^{-1}\tilde{h}, \mathcal{J}^{-1}\tilde{q})$ and
$\tilde{G}(\tilde{h},\tilde{q}) = \mathcal{J} G(\mathcal{J}^{-1}\tilde{h}, \mathcal{J}^{-1}\tilde{q})$,
this corresponds to
\begin{equation}
  \partial_t \tilde{w}(t,\ell,x) = \tilde{A}(\ell) \tilde{w}(t,\ell,x) + \tilde{B}(\tilde{w})(t,\ell,x)
+ \tilde{G}(\tilde{w})(t,\ell,x)\label{543_12}
\end{equation}
in Bloch space, cf.~\eqref{542_25}.


\begin{remark} \label{543_r1}  \rm
Heuristically, the reason for splitting the
  nonlinearity into $B$ and $G$ is the following.  To project the
  nonlinearity $\tilde{N}$  onto the critical eigenfunction
we take  the scalar product of $\tilde{N}(\ell,\cdot)$ with
the eigenvector of the adjoint linear operator
  $\tilde{A}^*(\ell,\cdot)$, which, by \eqref{542_24}, reads
$\psi^1(\ell) = (c_0,0)^\top +
  \mathcal{O}(\ell)$. Thus, since the equation for $\partial_t h$ is linear, the
  critical component of the nonlinearity obtains an additional factor
  $\ell$ in Bloch space, which increases its degree by
  $1$. This is the reason why terms like $h^2$ turn out
  to have the same degree as the nonlinearity
  $\partial_x(v^2)$ in the Burgers equation. As the IBL  has
  non-constant coefficients, a $\partial_x$ in $x$-space, which corresponds
  to $(\partial_x + {\rm i}\ell)$ in Bloch space, does not automatically increase
  the degree. Therefore, also terms like $h\,\partial_x^3h$,
  which at first view appear to be irrelevant, make an
  contribution to the relevant terms. On the other hand, since the
  $q$-component of the critical eigenvector $\phi^1$ is independent of
  $x$ at wave number $\ell=0$, a factor $\partial_x q$ leads to a further
  factor $\ell$ after projecting it onto the critical eigenvector, and
  thus to an asymptotically irrelevant term. That is why quadratic
  terms with a factor $\partial_x q$ are assigned to $G$. These
  considerations are made rigorous in \S\ref{sec_61}.
\end{remark}

\subsection*{Formal derivation of the Burgers equation}
Following Remarks \ref{542_r1} and \ref{543_r1} we formally
derive the Burgers equation for $\tilde{\alpha}$ by ignoring $\tilde{w}_s$ as well as the
 nonlinearity $\tilde{G}$. Thus, setting
 $\tilde{w} = (\tilde{w}_1,\tilde{w}_2)^\top = \tilde{\alpha} \phi^1$, \eqref{542_al} becomes
\begin{equation}
  \partial_t \tilde{\alpha}(t, \ell) = \mu_1(\ell)\tilde{\alpha}(t,\ell) + \tilde{E}_c^*(\tilde{B}(\tilde{\alpha}(t)\phi^1))(\ell). \label{543_11}
\end{equation}
Since the equation for $\partial_t h$ is linear, the nonlinearity reads
\[
  \tilde{E}_c^*(\tilde{B}(\tilde{\alpha}\phi^1))(\ell)
= \chi(\ell)\int_0^\gamma\tilde{B}_2(\tilde{\alpha}\phi^1)(\ell,x)\,\bar\psi^1_2(\ell,x)\,{\rm d} x,
\]
where \eqref{543_5} yields
\begin{equation} \begin{aligned}
  \tilde{B}_2(\tilde{w})
&=  b_{00}\tilde{w}_1^{*2} + b_{01}\tilde{w}_1*[(\partial_x+{\rm i}\ell)\tilde{w}_1]
+ b_{02}\tilde{w}_1*[(\partial_x+{\rm i}\ell)^2\tilde{w}_1] \\
&\quad + b_{03}\tilde{w}_1*[(\partial_x+{\rm i}\ell)^3\tilde{w}_1] + b_{11}[(\partial_x+{\rm i}\ell)\tilde{w}_1]^{*2}
 + b_1\tilde{w}_1*\tilde{w}_2 \\
&\quad + b_2[(\partial_x+{\rm i}\ell)\tilde{w}_1]*\tilde{w}_2 + b_3[(\partial_x+{\rm i}\ell)^2\tilde{w}_1]*\tilde{w}_2
+ b_4\tilde{w}_2^{*2}.
\end{aligned}
\end{equation}
We study in detail only the first term of $\tilde{B}_2(\tilde{w})$ and show afterwards that
 all other terms can be treated the same way. We have
\begin{equation} \begin{aligned}
  & \int_0^\gamma b_{00}(x)\tilde{w}_1^{*2}(\ell,x)\bar\psi^1_2(\ell,x)\,{\rm d} x\\
  &= \int_0^\gamma b_{00}(x)\int_{-k_0/2}^{k_0/2}\tilde{\alpha}(\ell-k)\phi^1_1(\ell-k,x)\tilde{\alpha}(k)
  \phi^1_1(k,x)\,{\rm d} k \,\bar\psi^1_2(\ell,x)\,{\rm d} x \\
  & = \int_{-k_0/2}^{k_0/2}\tilde{\alpha}(\ell-k)\tilde{\alpha}(k)\int_0^\gamma b_{00}(x)
  \phi^1_1(\ell-k,x)\phi^1_1(k,x)\bar\psi^1_2(\ell,x)\,{\rm d} x\,{\rm d} k \\
  & =: \int_{-k_0/2}^{k_0/2}\tilde{\alpha}(\ell-k)\tilde{\alpha}(k) K(\ell,\ell-k,k) \,{\rm d} k.
\end{aligned} \label{543_6}
\end{equation}
Before we expand the kernel $K(\ell,\ell-k,k)$ in terms of $\ell$, we state the
following useful properties.

\begin{lemma}
The adjoint eigenfunction $\psi^1$ satisfies
$\psi^1_2(0,x) = 0$ and $\partial_{\ell}\psi^1_2(0,x)\in {\rm i}\mathbb{R}$.
\end{lemma}

\begin{proof}
  The first property immediately follows from \eqref{542_24}.
Differentiating the eigenvalue equation
$\tilde{A}^*(\ell) \psi^1(\ell,x) = \bar\mu_1(\ell)\psi^1(\ell,x)$ with respect to $\ell$ gives
  \[
    \partial_{\ell} \tilde{A}^*(\ell)\psi^1(\ell,x) + \tilde{A}^*(\ell)\partial_{\ell}\psi^1(\ell,x)
= \partial_{\ell} \bar\mu_1(\ell)\psi^1(\ell,x) + \bar\mu_1(\ell)\partial_{\ell} \psi^1(\ell,x);
\]
  i.e., from the locally parabolic shape of
 $\bar\mu_1(\ell) = -c_2\ell^2 + \mathcal{O}(\ell^3)$ it follows that
  \[
    \partial_{\ell}\tilde{A}^*(0) \psi^1(0,x) + \tilde{A}^*(0)\partial_{\ell}\psi^1(0,x) = 0.
\]
  Since $\partial_{\ell}\tilde{A}^*(0)\psi^1(0,x) = ({\rm i} c_0c_1, {\rm i} c_0)^\top$, cf.~\eqref{542_13},
we obtain $\tilde{A}^*(0)\partial_{\ell}\psi^1(0,x) \in {\rm i}\mathbb{R}^2$.
As all coefficients of $\tilde{A}^*(0)$ are real, we obtain
$\partial_{\ell}\psi^1(0,x) \in {\rm i}\mathbb{R}^2 + \ker\tilde{A}^*(0) = {\rm i}\mathbb{R}^2 + \mathbb{C}\psi^1(0,x)$ and
  thus $\partial_{\ell}\psi^1_2(0,x) \in{\rm i}\mathbb{R}$.
\end{proof}

Returning to \eqref{543_6} we can write the integral kernel as
\[
  K(\ell,\ell-k,k) = \partial_1K(\mathbf{0})\ell + \partial_2K(\mathbf{0})(\ell-k)
 + \partial_3K(\mathbf{0})k + \mathcal{O}(|\ell|^2 + |\ell-k|^2 + |k|^2),
\]
since $K(\mathbf{0})=0$ due to $\psi^1_2(0,x) = 0$. For the same reason we
obtain $\partial_2K(\mathbf{0}) = \partial_3K(\mathbf{0}) = 0$,
while the first term reads
\begin{equation}
  \partial_1K(\mathbf{0})\ell
= \int_0^\gamma b_{00}(x) \phi^1_1(0,x)\phi^1_1(0,x)
 \partial_{\ell} \bar\psi^1_2(0,x)\,{\rm d} x \,\ell =: {\rm i} K_1\ell \in {\rm i}\mathbb{R}, \label{543_8}
\end{equation}
since $\phi^1(0,x) \in \mathbb{R}^2$, see \eqref{542_7}. Altogether, we have
\begin{equation}
  K(\ell,\ell-k,k) = {\rm i} K_1\ell + \mathcal{O}(|\ell|^2 + |\ell-k|^2 + |k|^2). \label{543_7}
\end{equation}
Therefore,
\begin{align*}
&\int_0^\gamma b_{00}(x)\tilde{w}_1^{*2}(\ell,x)\bar\psi^1_2(\ell,x)\,{\rm d} x \\
  & = \int_{-k_0/2}^{k_0/2}\tilde{\alpha}(\ell-k)\tilde{\alpha}(k)\left({\rm i} K_1\ell
+ \mathcal{O}(|\ell|^2 + |\ell-k|^2 + |k|^2)\right)\,{\rm d} k \\
  & = {\rm i} K_1\ell \tilde{\alpha}^{*2}(\ell) + \mathcal{O}(\ell^2) \tilde{\alpha}^{*2}(\ell).
\end{align*}
For the remaining terms in $\tilde{B}_2(\tilde{w})$ the same considerations lead to
\[
  \int_0^\gamma \tilde{B}_2(\tilde{\alpha}\phi^1)(\ell,x)\bar\psi^1_2(\ell,x)\,{\rm d} x
 = \int_{-k_0/2}^{k_0/2}\tilde{\alpha}(\ell-k)\tilde{\alpha}(k) K(\ell,\ell-k,k) \,{\rm d} k,
\]
where the integral kernel $K$ is given as sum of terms of the type
\[
  \int_0^\gamma b_{ij}(x) \phi^1_{j_1}(\ell-k,x)({\rm i} k)^{n_1}\partial_x^{n_2}
\phi^1_{j_2}(k,x)\bar\psi^1_2(\ell,x)\,{\rm d} x.
\]
Thus, there exists a $d \in \mathbb{R}$ such that we can write
\begin{equation}
  K(\ell,\ell-k,k) = {\rm i} d\ell + \mathcal{O}(|\ell|^2 + |\ell-k|^2 + |k|^2), \label{543_9}
\end{equation}
which yields
\begin{equation}
  \tilde{E}_c^*(\tilde{B}(\tilde{\alpha}\phi^1))(\ell)
= {\rm i} d \ell \chi(\ell) \tilde{\alpha}^{*2}(\ell) + \mathcal{O}(\ell^2)\tilde{\alpha}^{*2}(\ell). \label{543_res}
\end{equation}
Altogether \eqref{543_11} leads to
\begin{equation}
  \partial_t \tilde{\alpha}(t,\ell) = -c_2\ell^2 \tilde{\alpha}(t,\ell) + {\rm i} d \ell\chi(\ell)\tilde{\alpha}^{*2}(t,\ell)
 + \mathcal{O}(\ell^3)\tilde{\alpha}(t,\ell) + \mathcal{O}(\ell^2)\tilde{\alpha}^{*2}(t,\ell). \label{543_10}
\end{equation}
Since $\partial_x$ in $x$-space corresponds to ${\rm i}\ell$ in Fourier space and since a
derivative increases the degree of irrelevance by one, this reminds us
strongly of an asymptotically irrelevant perturbation of the Fourier transformed
Burgers equation $\partial_t \hat{v} = -c_2 k^2\hat{v} + {\rm i} d k\hat{v}^{*2}$.
This formally explains why in the main Theorem \ref{main_th2} the
comoving non-Gaussian profile $t^{-1/2}f_{z_0}(t^{-1/2}(x+c_1 t)) \Phi^1(0,x)$
 governs the asymptotics of the IBL  at lowest order.

\subsection{The result}\label{sec_55}
\S\ref{sec_522} about the Burgers equation and the formal calculations in
\S\ref{sec_543} motivate the formulation of the following theorem about
nonlinear stability of stationary solutions of the IBL.

\begin{theorem} \label{main_th1}
	Let $p\in(0,1/2)$, $3<r<4$, and let $(f_s,q_s)^\top$ be a spectrally stable
stationary solution of the IBL  \eqref{FT3}, \eqref{QT3};
i.e., Assumption \ref{ass_spec} is fulfilled.
Then there exist constants $C_1, C_2>0$ such that the following holds.
If $\|h_0\|_{H^r(2)}+\|q_0\|_{H^{r-1}(2)}\leq C_1$, then there exists
a unique global solution $(h,q)$ of the transformed IBL  \eqref{540_1}
with $(h,q)\big|_{t=1} = (h_0,q_0)$ and
	\begin{equation}
\sup_{x\in\mathbb{R}}\left|(h,q)^\top - t^{-1/2}f_{z_0}(t^{-1/2}(x+c_1 t)) \phi^1(0,x)\right|
 \leq C_2 t^{-1+p/2} \label{55_3}
	\end{equation}
  for $ t \in [1,\infty)$. Here, $z_0 >-1$, $f_{z_0}$ denotes the non-Gaussian
profile from \eqref{522_1}, and $\phi^1(0,\cdot) = ({\rm d} h_s / {\rm d} q_s,1)^\top$
is an eigenfunction to the critical eigenvalue
$\lambda_1(\ell) = c_1{\rm i}\ell - c_2\ell^2 + \mathcal{O}(\ell^3)$ from Assumption \ref{ass_spec}.
\end{theorem}

Theorem \ref{main_th1} follows from
the subsequent Theorem \ref{main_th3} about nonlinear stability in rescaled Bloch spaces.
To transfer Theorem \ref{main_th1} to the original $(F,Q)$-system
\eqref{FT3}, \eqref{QT3} we note that \eqref{55_3} yields
$\sup_{x\in\mathbb{R}}h^2 \leq Ct^{-1}$. Since due to \eqref{51_2} the
transformation for the film thickness reads $f=\beta h + \mathcal{O}(h^2)$, we can write
\begin{align*}
	& |(f,q)^\top - t^{-1/2}f_{z_0}(t^{-1/2}(x+c_1 t)) \Phi^1(0,x)| \\
	& = \left| \begin{pmatrix}\beta&0\\0&1\end{pmatrix}\left( (h,q)^\top
- t^{-1/2}f_{z_0}(t^{-1/2}(x+c_1 t)) \phi^1(0,x) \right) \right| + \mathcal{O}(h^2),
\end{align*}
where $\Phi^1$ is the eigenvector corresponding to the critical eigenvalue
$\lambda_1$ in the $(f,q)$-system, see \eqref{542_8}.
This yields Theorem \ref{main_th2}.


\section{Existence of a local solution}
For the proof of Theorem \ref{main_th1} we use the  RG method \cite{bkl94}
for \eqref{542_al}, \eqref{542_ws}. The main steps consist in
a proof of local existence using maximal regularity methods, and
in a careful estimate of the nonlinear terms. The local existence and uniqueness
of solutions
is carried out via resolvent estimates in $x$-space, while the RG method is
set up in Bloch space.

\subsection{Function spaces depending on time and space}\label{sec_71}
In the following, we always assume that $X$ is a Hilbert space and
$t_0, t_1 \in \mathbb{R} \cup \{-\infty,\infty \}$. If not stated otherwise, $H^r$ stands for
$H^r(\mathbb{R})$.

\begin{definition}\label{71_d2} \rm
  $L^2((t_0,t_1),X)$ denotes the space of (strongly) measurable
functions $u$ with values in $X$ such that the norm
$\|u\|_{L^2((t_0,t_1),X)} :=
\Big( \int_{t_0}^{t_1}\|u(t)\|_X^2 \,{\rm d} t \Big)^{1/2}$  is finite.
For $m \in \mathbb{N}$ we write
  \begin{gather*}
    H^m((t_0,t_1),X) := \{u \mid \partial_t^j u \in L^2((t_0,t_1),X)
 \text{ for } 0\leq j\leq m\}, \\
    \|u\|_{H^m((t_0,t_1),X)} :=
\Big( \sum_{j=0}^m \| \partial_t^j u \|_{L^2((t_0,t_1),X)}^2 \Big)^{1/2}.
  \end{gather*}
\end{definition}

In the special case $(t_0,t_1)=\mathbb{R}$ and $X=H^r(\mathbb{R}), r \in \mathbb{R}^+$ we find the
equivalent norm
\begin{equation}
  \|u\|_{H^m(\mathbb{R},H^r)} \sim
\Big( \int_\mathbb{R} \int_\mathbb{R} (1+\tau^2)^m(1+k^2)^r|\mathcal{F}_{tx}u(\tau,k)|^2 \,{\rm d}\tau\,{\rm d} k \Big)^{1/2},
  \label{71_1}
\end{equation}
where $\mathcal{F}_{tx}u$ denotes the Fourier transform of $u$ with respect to time and space.
 Obviously, this  definition can be extended to all $m \in \mathbb{R}^+$.

\begin{lemma}\label{71_l1}
  Let $s\geq0$. Then, we have
  \begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}
    \item $u \in H^s(\mathbb{R},X)  \Leftrightarrow  (1+\tau^2)^\frac{s}{2} \mathcal{F}_t u \in L^2(\mathbb{R},X)$.
    \item $H^s((t_0,t_1),X)$ coincides with the space of restrictions to $(t_0,t_1)$
of the elements in $H^s(\mathbb{R},X)$. Extension and restriction are both continuous operators.
  \end{enumerate}
\end{lemma}

For a proof see \cite[p. 58 and Theorem 9.1]{LM1}. By (i) we see that \eqref{71_1}
is an equivalent norm in $H^m(\mathbb{R},H^r)$ also for non-integer values of $m$.
The next lemma shows that the regularity of space and time derivatives
is the same as in the scalar valued case. Here we denote by $[s]$
the integer part of $s$.

\begin{lemma}\label{71_l2}
  Let $u \in H^s((t_0,t_1),H^r)$, $j \leq [s]$, $l \leq [r]$. Then
  $$
 \partial_x^l\partial_t^j u \in H^{s-j}((t_0,t_1),H^{r-l}).
$$
\end{lemma}

\begin{proof}
  We assume $(t_0,t_1) = \mathbb{R}$. Then by \eqref{71_1} we have
  \begin{align*}
    \|\partial_x^l\partial_t^j u\|_{H^{s-j}(\mathbb{R},H^{r-l})}^2
& \leq C \int_\mathbb{R} \int_\mathbb{R} (1+\tau^2)^{s-j}(1+k^2)^{r-l}|\mathcal{F}_{tx}[\partial_x^l\partial_t^ju](\tau,k)|^2 \,{\rm d}\tau \,{\rm d} k \\
      & = C \int_\mathbb{R} \int_\mathbb{R} (1+\tau^2)^{s-j}(1+k^2)^{r-l}|k|^{2l}|\tau|^{2j}
 |\mathcal{F}_{tx}u(\tau,k)|^2 \,{\rm d}\tau \,{\rm d} k \\
      & \leq C \|u\|_{H^s(\mathbb{R},H^r)}.
  \end{align*}
\end{proof}

As we will see in Lemma \ref{71_l6} below, for $u \in H^s((t_0,\infty),L^2)$,
$s > 1/2$, there exist traces $\partial_t^ju(t_0, \cdot) \in L^2(\mathbb{R})$ for all
 $j \in \mathbb{N}_0$ with $j < s - 1/2$. Thus, we can define the following subspace of
 $H^s((t_0,\infty),H^r)$.

\begin{definition} \rm
\[
  H_0^s((t_0,\infty),H^r) := \{u \in H^s((t_0,\infty),H^r)
\mid \|\partial_t^j u(t_0,\cdot)\|_{L^2}{=}0 \text{ for } {j} < s{-}\frac{1}{2},
j {\in}\mathbb{N}_0 \}.
\]
\end{definition}

By the following lemma these functions can be extended by zero for $t\leq0$.

\begin{lemma}\label{71_l3}
  Let $s\geq0$ be not a half integer, $t_0 \in \mathbb{R}$, $u \in H_0^s((t_0,\infty),H^r)$, and
  \begin{equation*}
    u_0(t,\cdot) := \begin{cases}
u(t,\cdot) & \text{for } t>t_0,\\
0 &\text{for } t \leq t_0. \end{cases}
  \end{equation*}
Then $u \mapsto u_0$ is a continuous mapping from $H_0^s((t_0,\infty),H^r)$
into $H^s(\mathbb{R},H^r)$; i.e., there exist $C_1, C_2 >0$ such that
 $$
 C_1\|u\|_{H^s((t_0,\infty),H^r)} \leq \|u_0\|_{H^s(\mathbb{R},H^r)}
 \leq C_2\|u\|_{H^s((t_0,\infty),H^r)}.
$$
\end{lemma}

For a proof see \cite{LM1}, in particular Theorem 11.4. Next we characterize
functions $u$ in $H_0^s((0,\infty),H^r)$. Since they can be extended by
zero for $t \leq 0$ we can apply Fourier transform in time.
The problem is that without making further demands on the regularity
of $\mathcal{F}_t u$ we can not guarantee that the inverse Fourier transform is
again in $H_0^s$. The following two lemmas show conditions based
upon the Paley-Wiener Theorem which ensure that the inverse Fourier
transform maps back to functions vanishing on the negative time axis.
As $u$ is only defined for $t>0$ and $\mathcal{F}_t u$ must be treated as function
on $\tau \in \mathbb{C}$ it is common to replace Fourier transform in time by
Laplace transform:
\begin{equation}
  \mathcal{L} u(\tau) := \frac{1}{2\pi} \int_0^\infty u(t) {\rm e}^{-t\tau}\,{\rm d} t. \label{71_12}
\end{equation}
The relation to Fourier transform is
\[ %\label{71_14}
  \mathcal{L} u (\tau_1 + {\rm i} \tau_2,x)
= \frac{1}{2\pi}\int_\mathbb{R} u_0(t,x){\rm e}^{-t\tau_1}{\rm e}^{-{\rm i} t\tau_2} \,{\rm d} t
 = \mathcal{F}_t[{\rm e}^{-\cdot\tau_1}u_0(\cdot,x)](\tau_2).
\]

\begin{lemma}\label{71_l4}
  Let $s\geq 0$ be not a half integer, $r\geq 0$. If $u \in H_0^s((0,\infty),H^r)$,
then the Laplace transform $\mathcal{L} u$ satisfies
  \begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}
    \item $\tau \mapsto \mathcal{L} u(\tau,x)$ is holomorphic in the half-plane $\operatorname{Re} \tau > 0$ for almost every $x \in \mathbb{R}$.
    \item $\displaystyle \sup_{\tau_1>0} \int_\mathbb{R} |\mathcal{L} u(\tau_1 + {\rm i} \tau_2, x)|^2 \,{\rm d} \tau_2 < \infty$ for almost every $x \in \mathbb{R}$.
    \item $\displaystyle \|u\|_{H^s((0,\infty),H^r)} \sim \left( \int_\mathbb{R} (1+\tau_2^2)^s\|\mathcal{L} u({\rm i} \tau_2, \cdot)\|_{H^r}^2 \,{\rm d}\tau_2\right)^{1/2}$.
  \end{enumerate}
\end{lemma}

\begin{proof}
  Since $u \in H_0^s((0,\infty),H^r)$, we have $u \in L^2((0,\infty),L^2)$. Thus
  \begin{equation*}
    \int_\mathbb{R}\int_0^\infty |u(t,x)|^2\,{\rm d} t\,{\rm d} x
= \int_0^\infty\int_\mathbb{R} |u(t,x)|^2\,{\rm d} x\,{\rm d} t < \infty
  \end{equation*}
  which yields $\int_0^\infty |u(t,x)|^2\,{\rm d} t < \infty$ for almost every
$x\in \mathbb{R}$. Applying the Paley-Wiener Theorem, see \cite{Yosida} for instance,
 gives the first property. Now let $\tau_1>0$. Parseval's identity implies
  \begin{align*}
   & \int_\mathbb{R}|\mathcal{L} u(\tau_1+{\rm i}\tau_2,x)|^2\,{\rm d}\tau_2 \\
&= \int_\mathbb{R}|\mathcal{F}_t[{\rm e}^{-\cdot\tau_1}u_0(\cdot,x)](\tau_2)|^2\,{\rm d}\tau_2 \\
      & \leq C \int_\mathbb{R}|{\rm e}^{-t\tau_1}u_0(t,x)|^2\,{\rm d} t
   \leq C \int_0^\infty|u(t,x)|^2\,{\rm d} t < \infty \text{ for almost every } x \in \mathbb{R}
  \end{align*}
independently of $\tau_1$. This shows the second property.
The third property follows with the help of Lemma \ref{71_l3} and
Lemma \ref{71_l1}; i.e.,
  \begin{align*}
    \int_\mathbb{R} (1+\tau_2^2)^s\|\mathcal{L} u({\rm i} \tau_2, \cdot)\|_{H^r}^2 \,{\rm d}\tau_2
& = \int_\mathbb{R}(1+\tau_2^2)^s\|\mathcal{F}_tu_0(\tau_2,\cdot)\|_{H^r}^2\,{\rm d}\tau_2 \\
& \sim \|u_0\|_{H^s(\mathbb{R},H^r)}^2 \sim \|u\|_{H^s((0,\infty),H^r)}^2.
  \end{align*}
\end{proof}

The following lemma shows that regularity in $x$ is preserved under Laplace transform.

\begin{lemma}\label{71_l12}
  Let $r \geq 0$. If $u \in L^2((0,\infty),H^r)$, then
  \[
    \sup_{\tau_1\geq0}\int_\mathbb{R} \|\mathcal{L} u(\tau_1 + {\rm i}\tau_2, \cdot) \|_{H^r}^2 \,{\rm d} \tau_2
\leq C \|u\|_{L^2((0,\infty),H^r)}^2.
\]
  In particular, $\mathcal{L} u(\tau,\cdot) \in H^r$ for almost every $\tau$ with $\operatorname{Re}\tau \geq 0$.
\end{lemma}

\begin{proof}
  Let $\tau_1 \geq0$. Then
\begin{align*}
    \int_\mathbb{R} \|\mathcal{L} u(\tau_1 + {\rm i}\tau_2, \cdot) \|_{H^r}^2 \,{\rm d} \tau_2
& = \int_\mathbb{R} \|\mathcal{F}_t[{\rm e}^{-t\tau_1} u_0(t,\cdot)](\tau_2)\|_{H^r}^2 \,{\rm d}\tau_2 \\
&\leq C \int_\mathbb{R} \|{\rm e}^{-t\tau_1} u_0(t,\cdot) \|_{H^r}^2 \,{\rm d} t\\
&= C \int_0^\infty e^{-2t\tau_1} \|u(t,\cdot)\|_{H^r}^2 \,{\rm d} t
       \leq C \|u\|_{L^2((0,\infty),H^r)}^2
\end{align*}
  independently of $\tau_1$.
\end{proof}

Lemma \ref{71_l4} has the following inverse.

\begin{lemma}\label{71_l5}
  Let $s\geq 0$ be not a half integer, $r\geq 0$, and  assume
$f:\mathbb{C}\times\mathbb{R} \to \mathbb{C}$ fulfills the following conditions.
  \begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}
    \item $f(\tau,x)$ is holomorphic in the half-plane $\operatorname{Re} \tau > 0$
for almost every $x \in \mathbb{R}$.
    \item $\displaystyle \sup_{\tau_1>0} \int_\mathbb{R} |f(\tau_1
+ {\rm i} \tau_2, x)|^2 \,{\rm d} \tau_2 < \infty$ for almost every $x \in \mathbb{R}$.
    \item $\displaystyle \int_\mathbb{R} (1+\tau_2^2)^s\|f({\rm i} \tau_2, \cdot)\|_{H^r}^2
\,{\rm d} \tau_2 < \infty$.
  \end{enumerate}
  Then the inverse Fourier transform
$g(t,x) = \int_\mathbb{R} f(i\tau_2,x){\rm e}^{{\rm i} t\tau_2}\,{\rm d} \tau_2$ satisfies
  \begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}\addtocounter{enumi}{3}
    \item $g\big|_{\mathbb{R}^+\times\mathbb{R}} \in H_0^s((0, \infty), H^r)$\quad and \quad
     $\mathcal{L} g = f$.
  \end{enumerate}
\end{lemma}

\begin{proof}
  Due to (i) and (ii), for almost every $x \in \mathbb{R}$ we can apply the Paley-Wiener
Theorem and obtain $g(t,x)=0 \text{ for } t<0, \mathcal{L} g(\tau,x) = f(\tau,x)$,
  see \cite{Yosida}, for instance.
  It remains to prove that $g\big|_{\mathbb{R}^+\times\mathbb{R}} \in H_0^s((0, \infty), H^r)$.
 Due to Lemma \ref{71_l1} we have
  \begin{align*}
    \|g\|_{H^s(\mathbb{R},H^r)}^2 & \leq C \int_\mathbb{R} \|(1+\tau_2^2)^{\frac{s}{2}}\mathcal{F}_t g(\tau_2, \cdot)\|_{H^r}^2\,{\rm d}\tau_2
      = C \int_\mathbb{R}(1+\tau_2^2)^s\|\mathcal{L} g({\rm i}\tau_2,\cdot)\|_{H^r}^2 \,{\rm d} \tau_2 \\
      & = C \int_\mathbb{R}(1+\tau_2^2)^s\|f({\rm i} \tau_2,\cdot)\|_{H^r}^2 \,{\rm d} \tau_2  < \infty,
  \end{align*}
  thus $g \in H^s(\mathbb{R},H^r)$. Let $j \in \mathbb{N}$ such that $s-j > 1/2$.
Lemma \ref{71_l2} gives $\partial_t^j g \in H^{s-j}(\mathbb{R},H^r)$, and due to
standard Sobolev embedding we have $\partial_t^j g \in C(\mathbb{R},H^r)$. Since $g(t,x) = 0$
for $t<0$ and almost every $x \in \mathbb{R}$ we achieve $\|\partial_t^j g(0,\cdot)\|_{L^2} = 0$.
This shows the first property.
\end{proof}

To prove local existence for the IBL \eqref{540_1}
we use the following Sobolev spaces.

\begin{definition} \rm
 Let $ H^{r,s}((t_0,t_1)) := L^2((t_0,t_1),H^r(\mathbb{R})) \cap H^s((t_0,t_1),L^2(\mathbb{R}))$,
 with norm
$$
 \|u\|_{H^{r,s}((t_0,t_1))} :=
\Big( \int_{t_0}^{t_1} \|u(t,\cdot)\|_{H^r}^2\,{\rm d} t + \|u\|_{H^s((t_0,t_1),L^2)}^2
\Big)^{1/2}.
$$
\end{definition}

Applying \eqref{71_1} we can state an equivalent norm for the case $(t_0,t_1) = \mathbb{R}$,
namely
\begin{equation}
  \|u\|_{H^{r,s}(\mathbb{R})} \sim
\Big( \int_\mathbb{R}\int_\mathbb{R}\big( (1+\tau^2)^{\frac{s}{2}} + (1+k^2)^{\frac{r}{2}} \big)^2
|\mathcal{F}_{tx}u(\tau,k)|^2 \,{\rm d} \tau \,{\rm d} k \Big)^{1/2}.
\end{equation}
Functions in $H^{r,s}(\mathbb{R})$ also belong to
``intermediate spaces'' with intermediate regularities in time and space.

\begin{lemma}\label{71_l7}
  Let $r,s \geq 0$, $\vartheta \in (0,1)$. Then
$H^{r,s}(\mathbb{R})$ is continuously embedded into $H^{\vartheta s}(\mathbb{R},H^{(1-\vartheta)r})$.
\end{lemma}

\begin{proof}
  For $u \in H^{r,s}(\mathbb{R})$, Lemma \ref{71_l1} yields
  \begin{equation*}
    \|u\|_{H^{\vartheta s}(\mathbb{R},H^{(1-\vartheta)r})}^2
\leq C \int_\mathbb{R} \int_\mathbb{R} (1+\tau^2)^{\vartheta s}(1+k^2)
^{(1-\vartheta)r}|\mathcal{F}_{tx}u(\tau,k)|^2 \,{\rm d}\tau\,{\rm d} k.
  \end{equation*}
  By Young's inequality,
  $(1+\tau^2)^{\vartheta s}(1+k^2)^{(1-\vartheta)r}
\leq \vartheta (1 + \tau^2)^s + (1-\vartheta)(1+k^2)^r$
  we obtain
$
    \|u\|_{H^{\vartheta s}(\mathbb{R},H^{(1-\vartheta)r})}^2 \leq C \|u\|_{H^{r,s}(\mathbb{R})}^2.
 $
\end{proof}


Later we need estimates of the $H^{\frac{r}{s}(s-1/2)}$-norm for fixed times.
Lemma \ref{71_l7} particularly yields that $H^{r,s}(\mathbb{R})$ is continuously
embedded into $H^{1/2}(\mathbb{R},H^{\frac{r}{s}(s-1/2)})$, but by standard Sobolev
embedding theory this does not allow any conclusion for fixed $t$. However,
by interpolation theory the following trace theorem can be shown.
\begin{lemma}\label{71_l6}
  Let $u \in H^{r,s}((t_0,t_1))$, $r \geq 0$, $s>1/2$. Then for all integers
 $j < s - \frac{1}{2}$ there exists the trace
  \begin{equation*}
    \partial_t^j u(t_0,\cdot) \in H^{p_j}(\mathbb{R}), \ p_j = \frac{r}{s}\big(s-j-\frac{1}{2}\big).
  \end{equation*}
The mappings $H^{r,s}((t_0,t_1)) \to H^{p_j}(\mathbb{R}):u \mapsto \partial_t^j u$ are continuous.
Furthermore, the mapping $u \mapsto (\partial_t^j u(t_0,\cdot))_{0\leq j< s-\frac{1}{2}}$
from $H^{r,s}((t_0,t_1))$ into $\prod_{0\leq j< s-\frac{1}{2}} H^{p_j}$ is surjective.
\end{lemma}

A proof can be found in \cite[Theorem 4.2.1]{LM2}.
For the surjectivity of the trace operator see \cite[Theorem 4.4.2]{LM1}
 with $X=H^r, Y=L^2$. Since the trace operator is continuous, we have the following
corollary, see \cite[Theorem 1.3.1]{LM1} and the proof of \cite[Theorem 1.4.2]{LM1}.

\begin{corollary}\label{71_c1}
  Let $u \in H^{r,s}((t_0,t_1))$, $r \geq 0$, $s>1/2$. Then there exists a $C>0$ such that
  \[
    \sup_{t\in [t_0,t_1]}\| u(t,\cdot) \|_{H^{r-1}} < C \| u \|_{H^{r,s}((t_0,t_1))}.
 \]
\end{corollary}

It turns out, that the IBL is a
second-order parabolic evolution system, and therefore the spaces $H^{r,s}$
always occur with $s=r/2$ and usually consist of functions defined only for $t\geq0$.
Hence we set

\begin{definition}\label{71_d4} \rm
$ K^r((t_0,t_1)) := H^{r,\frac{r}{2}}((t_0,t_1))$, and
$$
K_0^r((t_0,t_1)) := \{u \in K^r((t_0,t_1))
\mid \| \partial_t^j u(t_0,\cdot)\|_{L^2} = 0 \text{ for } j \in \mathbb{N}_0 \text{ with } 2j < r-1\}.
$$
\end{definition}

Thus we have $K_0^r((0,\infty)) = H_0^{\frac{r}{2}}((0,\infty),L^2)
\cap L^2((0,\infty),H^r)$ and with the help of Lemma \ref{71_l4} and
Lemma \ref{71_l5} we can characterize $K_0^r((0,\infty))$ in Fourier space.

\begin{theorem}\label{71_t1}
  Let $r\geq0$, $(r+1)/2 \notin \mathbb{N}$. Then $u \in K_0^r((0,\infty))$
if and only if the Laplace transform $\mathcal{L} u$ fulfills
  \begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}
    \item $\mathcal{L} u(\tau,x)$ is holomorphic in the half-plane $\operatorname{Re} \tau > 0$ for almost every $x \in \mathbb{R}$.
    \item $\displaystyle \sup_{\tau_1>0} \int_\mathbb{R} |\mathcal{L} u(\tau_1 + {\rm i} \tau_2, x) |^2 \,{\rm d} \tau_2 < \infty$ for almost every $x \in \mathbb{R}$.
    \item $\displaystyle \left(\int_\mathbb{R} (\|\mathcal{L} u({\rm i} \tau,\cdot)\|_{H^r}^2 + |\tau|^r \|\mathcal{L} u({\rm i} \tau,\cdot)\|_{L^2}^2 )\,{\rm d}\tau\right)^{1/2} < \infty$. \label{71_10}
  \end{enumerate}
  The left-hand side in \emph{(iii)} defines a norm equivalent to
$\|\cdot\|_{K^r((0,\infty))}$.
\end{theorem}

\begin{proof}
  Due to Lemmas \ref{71_l4} and  \ref{71_l5} it remains to show the equivalence of norms.
Since $r/2$ is not a half integer we have
  \[
    \|u\|_{K^r((0,\infty))}^2 \sim \int_\mathbb{R}\|\mathcal{L} u({\rm i}\tau,\cdot)\|_{H^r}^2\,{\rm d} \tau + \int_\mathbb{R}(1+\tau^2)^{\frac{r}{2}}\|\mathcal{L} u({\rm i} \tau,\cdot)\|_{L^2}^2\,{\rm d} \tau.
\]
  Now using $(1+\tau^2)^{\frac{r}{2}} \sim 1 + |\tau|^r$ yields the result.
\end{proof}

Next, we collect some useful properties of the $K^r$-spaces, concerning
derivatives and nonlinear interaction.

\begin{lemma}\label{71_l9}
  Let $r{>}0$ and $l,j{\in}\mathbb{N}$ with $l{+}2j\leq r$. If $u{\in}K^r((t_0,t_1))$
then
  $  \partial_x^l\partial_t^j u \in K^{r-l-2j}((t_0,t_1))$.
\end{lemma}

\begin{proof}
  Applying Lemma \ref{71_l7} with $\vartheta=1-\frac{l}{r}$ and
$\vartheta=\frac{2j}{r}$, respectively,  we obtain
 $ u \in H^{\frac{r-l}{2}}((t_0,t_1),H^l) \cap H^j((t_0,t_1),H^{r-2j})$.
  By Lemma \ref{71_l2} it follows that
$
    \partial_x^l\partial_t^j u \in H^{\frac{r-l}{2}-j}((t_0,t_1),L^2) \cap L^2((t_0,t_1),H^{r-l-2j}).
 $
\end{proof}

\begin{lemma}\label{71_l10}
  Let $r>3/2$, $r\geq s\geq 0$. If $u \in K^r((t_0,t_1))$ and $v \in K^s((t_0,t_1))$, then $uv \in K^s((t_0,t_1))$ and there exists a $C>0$ such that
  \begin{equation}\label{71_11}
    \|uv\|_{K^s((t_0,t_1))} \leq C \|u\|_{K^r((t_0,t_1))}\|v\|_{K^s((t_0,t_1))}.
  \end{equation}
  If $u\in K_0^r$ or $v \in K_0^s$, then $uv \in K_0^s$.
\end{lemma}

A proof of \eqref{71_11} can be found in \cite[Lemma 5.1]{Beale84}
 while the second statement is obvious.
We need function spaces with weights in the spatial variable,
namely $H^s((t_0,t_1),H^r(n))$ where $H^r(n)$ is the
weighted Sobolev space introduced in \eqref{521_5}; i.e.,
$\|u\|_{H^r(n)} = \|\varrho^n u\|_{H^r}$ with
$\varrho(x):= (1+x^2)^{1/2}$.
A natural description equivalent to Definition \ref{71_d2}
is given by the following obvious lemma.

\begin{lemma}\label{71_l11}
  Let $s,r \geq 0$, $n \in \mathbb{N}$. Then
$ u \in H^s((t_0,t_1),H^r(n)) \Leftrightarrow \varrho^n u \in H^s((t_0,t_1),H^r)$.
\end{lemma}

\begin{definition} \rm
For $r,s,n\ge 0$ let
\[
 H^{r,s}((t_0,t_1),n) :=  L^2((t_0,t_1),H^r(n))
\cap H^s((t_0,t_1),L^2(n)),
\]
$K^r((t_0,t_1),n) := H^{r,\frac{r}{2}}((t_0,t_1),n)$.
\end{definition}

\begin{remark}\label{71_r1} \rm
  Due to Lemma \ref{71_l11}, Theorem \ref{71_t1} also holds
for $u \in K_0^r((0,\infty),n)$ if we replace $H^r$ by $H^r(n)$ and $L^2$
by $L^2(n)$ in property (iii). The same is true for
Lemmas \ref{71_l3}, \ref{71_l12}, \ref{71_l6}, \ref{71_l9}, \ref{71_l10}
 if we replace the respective Sobolev spaces by weighted ones.
\end{remark}

\subsection{Existence of a local solution}\label{sec_60}
Taking into account that the space regularity of $h$
should be taken higher
than that of $q$, we introduce the vector-valued function spaces
\begin{gather}
  \mathcal{H}^r(m)  := H^r(m) \times H^{r-1}(m), \label{600_3}\\
  \mathcal{K}^{r+1}((t_0,t_1),m)  := K^{r+1}((t_0,t_1),m) \times K^{r}((t_0,t_1),m). \label{600_4}
\end{gather}
To prove that spectrally stable stationary solutions of the IBL  are nonlinearly
stable we first need local existence in a given time interval $(t_0,t_1)$.

\begin{theorem}[Local existence]\label{600_th1}
  Let $3<r<4$ and fix some $t_0<t_1$. Then there exist $C_1,C_2>0$ such that
the following holds. If $(h_0,q_0)^\top \in \mathcal{H}^r(2) = H^r(2)\times H^{r-1}(2)$ satisfies
  \[
    \rho := \|(h_0,q_0)^\top\|_{\mathcal{H}^r(2)} \leq C_1,
 \]
  then there exists a unique local solution
\[
(h,q)^\top \in \mathcal{K}^{r+1}((t_0,t_1),2) = K^{r+1}((t_0,t_1),2)\times K^{r}((t_0,t_1),2)
\]
of  IBL  \eqref{540_1} with
  \begin{equation}
    \|(h,q)^\top\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} \leq C_2 \rho \label{600_1}
 \end{equation}
and $ (h,q)^\top|_{t= t_0} = (h_0,q_0)^\top $.
  Moreover, for $t_0 < \tilde t_0 < t_1$and any $m \in \mathbb{N}$ we have
 $(h,q)^\top \in \mathcal{K}^{r+m}((\tilde t_0,t_1),2)$, and there exists $C_3=C_3(\tilde t_0,m)$ such that
  \begin{equation}
    \|(h,q)^\top\|_{\mathcal{K}^{r+m}((\tilde t_0,t_1),2)} \leq C_3 \rho. \label{600_2}
  \end{equation}
\end{theorem}

To prove this theorem we need to apply maximal regularity results based on Laplace  transform.
First we solve the linearized problem with inhomogeneous right-hand side and zero initial condition.
This requires resolvent estimates for the linear operator $A$. Due to the periodic coefficients,
these cannot be shown by applying Fourier transform in space. Instead, we have to test in $x$-space
with appropriate test functions. This is carried out in detail in \S\ref{sec_601}.
The higher regularity in the time interval $[\tilde{t}_0, t_1]$ then follows from a bootstrapping
argument in \S\ref{sec_602}; i.e., since \eqref{600_1}
yields $(h(\tilde{t}),q(\tilde{t}))^\top \in \mathcal{H}^{r+1}$ for almost every $\tilde{t} \in (t_0,t_1)$,
we can start again at $t=\tilde{t}$. This gives $(h,q)^\top \in \mathcal{K}^{r+2}$,
and iterating this argument shows \eqref{600_2}.

\subsection{Resolvent estimates}\label{sec_601}

The resolvent equation is obtained by Laplace transform with respect to time
 of the linear inhomogeneous equation $(\partial_t - A(\partial_x))u = g$ and
reads $(\lambda -A(\partial_x))u= g$ with $g\in H^{r-1}\times H^{r-2}$
and the linear operator $A$ from \eqref{51_5}; i.e.,
\begin{equation}\label{601_2}
	\lambda \begin{pmatrix}h\\q\end{pmatrix}
 - \begin{pmatrix}0 & -\partial_x \\ a_{10}+a_{11}\partial_x+a_{12}\partial_x^2+a_{13}\partial_x^3
& a_{20}+a_{21}\partial_x+a_{22}\partial_x^2 \end{pmatrix}
\begin{pmatrix}h\\q\end{pmatrix} = \begin{pmatrix}g_1\\g_2\end{pmatrix}.
\end{equation}

\begin{theorem}\label{601_thres1}
	Let $r\geq2$. Then there exist $C,a>0$ such that for all
 $g = (g_1,g_2)^\top \in H^{r-1} \times H^{r-2}$ and all $\lambda$ with
 $\operatorname{Re} \lambda \geq a$ the resolvent equation $(\lambda - A(\partial_x))(h,q)^\top = g$
has a unique solution, which satisfies
	\begin{equation} \begin{aligned}
		& \|h\|_{H^{r+1}} + |\lambda|^{(r+1)/2}\|h\|_{L^2} + \|q\|_{H^r}
+ |\lambda|^{r/2}\|q\|_{L^2} \\
		& \leq C\left( \|g_1\|_{H^{r-1}} + |\lambda|^{(r-1)/2}\|g_1\|_{L^2}
+ \|g_2\|_{H^{r-2}} + |\lambda|^{(r-2)/2}\|g_2\|_{L^2} \right).
\end{aligned} \label{601_1}
\end{equation}
\end{theorem}

For the proof we give separate estimates for $q$ and $h$, and moreover first
restrict to $r=2$.

\subsection*{Estimates for $q$}

From the first equation in \eqref{601_2} we obtain
\begin{equation}
	h = \frac{-\partial_x q + g_1}{\lambda}, \label{601_23}
\end{equation}
and plugging this into the second equation yields
\begin{equation}
	(\lambda + a_0)q + a_1\partial_x q + a_2\partial_x^2 q + a_3 \partial_x^3 q + a_4\partial_x^4 q
= g_2 + \frac{1}{\lambda}(a_{10} + a_{11}\partial_x + a_{12}\partial_x^2 + a_{13}\partial_x^3)g_1. \label{601_3}
\end{equation}
The $\gamma$-periodic coefficients are given by
\begin{equation}
	a_0 = -a_{20}, \quad a_1 = -a_{21} + \frac{a_{10}}{\lambda}, \quad
a_2  = -a_{22} + \frac{a_{11}}{\lambda}, \quad a_3 = \frac{a_{12}}{\lambda}, \quad
 a_4 = \frac{a_{13}}{\lambda}. \label{601_8}
\end{equation}
To solve \eqref{601_3} we define on $H^2\times H^2$ the bilinear form
\begin{align*}
	b(q,\varphi)
&:= \ \int_\mathbb{R}\Big( (\lambda+a_0)q \bar\varphi + a_1\partial_x q\, \bar\varphi + a_2\partial_x^2 q\,\bar\varphi
- \partial_x a_3\, \partial_x^2 q\,\bar\varphi - a_3\partial_x^2 q \,\partial_x \bar\varphi   \\
& \quad + \partial_x^2 a_4\, \partial_x^2 q\,\bar\varphi + 2\partial_x a_4\, \partial_x^2 q\, \partial_x \bar\varphi
+ a_4 \partial_x^2 q\, \partial_x^2 \bar\varphi \Big){\rm d} x.
\end{align*}
Using integration by parts,
$q \in H^2$ is a weak solution of \eqref{601_3} if and only if
\begin{equation}
	b(q,\varphi) = \int_\mathbb{R} \Big( g_2 + \frac{1}{\lambda}(a_{10} + a_{11}\partial_x
+ a_{12}\partial_x^2 + a_{13}\partial_x^3)g_1 \Big)\bar\varphi \,{\rm d} x \label{601_9}
\end{equation}
for all $\varphi \in H^2$. To prove the existence of a unique weak
solution we apply the Lemma of Lax-Milgram.
Therefore, we have to show that the bilinear form $b$ is continuous and elliptic.
Since all coefficients of $b$ are in $L^\infty$, the continuity is obvious.
To verify the ellipticity of $b$ we have to estimate $b(q,q)$, which reads
\begin{equation}
\begin{aligned}
	b(q,q) & = \int_\mathbb{R}(\lambda+a_0)|q|^2 \,{\rm d} x + \int_\mathbb{R} a_1 \partial_x q\, \bar{q} \,{\rm d} x
 - \int_\mathbb{R} a_2|\partial_x q|^2\,{\rm d} x - \int_\mathbb{R} \partial_x a_2 \,\partial_x q\, \bar{q}\,{\rm d} x  \\
		& \quad - \int_\mathbb{R}\partial_x a_3\, \partial_x^2 q \,\bar{q}\,{\rm d} x - \int_\mathbb{R} a_3 \partial_x^2 q\,\partial_x \bar{q} \,{\rm d} x
 + \int_\mathbb{R}\partial_x^2 a_4\, \partial_x^2 q \,\bar{q}\,{\rm d} x  \\
		& \quad + 2\int_\mathbb{R} \partial_x a_4\,\partial_x^2 q \,\partial_x \bar{q} \,{\rm d} x + \int_\mathbb{R} a_4|\partial_x^2 q|^2 \,{\rm d} x  \\
	& = \int_\mathbb{R}(\lambda+a_0)|q|^2 \,{\rm d} x - \int_\mathbb{R} a_2|\partial_x q|^2\,{\rm d} x
+ \int_\mathbb{R} a_4|\partial_x^2 q|^2 \,{\rm d} x \\
		& \quad + \int_\mathbb{R} \left(a_1{-}\partial_x a_2\right) \partial_x q\, \bar{q} \,{\rm d} x
+ \int_\mathbb{R} \left(\partial_x^2 a_4{-}\partial_x a_3 \right) \partial_x^2 q \,\bar{q} \,{\rm d} x \\
&\quad + \int_\mathbb{R}\left( 2\partial_x a_4{-}a_3 \right) \partial_x^2 q\, \partial_x \bar{q} \,{\rm d} x.
\end{aligned} \label{601_4}
\end{equation}
We begin with estimating the real part of the first three integrals on the right-hand
side of \eqref{601_4}, related to the $H^2$-norm of $q$, and which
for $\operatorname{Re} \lambda$ large enough also absorb the mixed terms.
For $\operatorname{Re}\lambda \geq \|a_{20}\|_{L^\infty}$ we have
\begin{equation}\label{601_5}
	\operatorname{Re} \int_\mathbb{R}(\lambda+a_0)|q|^2\,{\rm d} x \geq (\operatorname{Re}\lambda - \|a_{20}\|_{L^\infty})\|q\|_{L^2}^2.
\end{equation}
Since $\operatorname{Re}(1/\lambda) = \operatorname{Re}\lambda/|\lambda|^2$ we  obtain from \eqref{51_13}
$-\operatorname{Re} a_2 \geq \frac{9}{2{\rm R}} - \frac{\operatorname{Re}\lambda}{|\lambda|^2}\|a_{11}\|_{L^\infty}$.
For $\operatorname{Re}\lambda \geq (4{\rm R}/9) \|a_{11}\|_{L^\infty}$
the second integral in \eqref{601_4} can be estimated by
\begin{equation}\label{601_6}
	-\operatorname{Re} \int_\mathbb{R} a_2|\partial_x q|^2 \, {\rm d} x \geq \frac{9}{4{\rm R}}\|\partial_x q\|_{L^2}^2.
\end{equation}
By \eqref{51_13} we obtain
$a_4 = \frac{a_{13}}{\lambda} = \frac{5{\rm W}}{6\lambda}\frac{f_s}{1+\kappa f_s}$, and
since $\kappa$ is small we have $C_1:=\min_{x\in[0,\gamma]}a_{13}(x)>0$,
 and therefore $\operatorname{Re} a_4 \geq C_1 \operatorname{Re}\lambda/|\lambda|^2$. Thus,
\begin{equation}\label{601_7}
	\operatorname{Re} \int_\mathbb{R} a_4|\partial_x^2 q|^2 \,{\rm d} x \geq C_1 \frac{\operatorname{Re}\lambda}{|\lambda|^2}\|\partial_x^2 q\|_{L^2}^2.
\end{equation}
Next, we estimate the mixed terms in \eqref{601_4} by applying Young's inequality
$ ab \leq \frac{\epsilon}{2}a^2 + \frac{1}{2\epsilon}b^2$
for $\epsilon > 0$. Looking at \eqref{601_5}-\eqref{601_7} we find that in case
$|\lambda| \to \infty$ the inequalities for $\|q\|_{L^2}^2$, $\|\partial_x q\|_{L^2}^2$,
and $\|\partial_x^2 q\|_{L^2}^2$ get worse the more derivatives we have.
Thus, we have to choose $\epsilon$ with care such that the mixed terms can be absorbed
by \eqref{601_5}-\eqref{601_7} without losing the positive coefficients.
Therefore, we start with the integral containing the highest derivatives; i.e.,
\[
  \left| \int_\mathbb{R}\left( 2\partial_x a_4 - a_3 \right) \partial_x^2 q\, \partial_x \bar{q} \,{\rm d} x \right|
\leq \frac{1}{2\epsilon} \int_\mathbb{R}|2\partial_x a_4 - a_3|^2|\partial_x^2 q|^2 \,{\rm d} x
+ \frac{\epsilon}{2}\int_\mathbb{R}|\partial_x q|^2 \,{\rm d} x.
\]
Choosing $\epsilon = 9/(8{\rm R})$, we obtain
\[
  \frac{1}{2\epsilon}|2\partial_x a_4 - a_3|^2
\leq \frac{1}{|\lambda|^2} \frac{4{\rm R}}{9}\|2\partial_x a_{13}-a_{12}\|_{L^\infty}^2 =: C_2 \frac{1}{|\lambda|^2},
\]
and thus,
\begin{align*}
  \left| \int_\mathbb{R}\left( 2\partial_x a_4 - a_3 \right) \partial_x^2 q \,\partial_x \bar{q} \,{\rm d} x \right|
& \leq C_2 \frac{1}{|\lambda|^2} \|\partial_x^2 q\|_{L^2}^2 + \frac{9}{16{\rm R}}\|\partial_x q\|_{L^2}^2\\
&\leq C_1 \frac{\operatorname{Re}\lambda}{4|\lambda|^2} \|\partial_x^2 q\|_{L^2}^2 + \frac{9}{16{\rm R}}\|\partial_x q\|_{L^2}^2
\end{align*}
for $\operatorname{Re}\lambda \geq 4C_2/C_1$. Analogously, for
$\operatorname{Re}\lambda \geq \|\partial_x^2 a_{13} - \partial_x a_{12}\|_{L^\infty}^2/(C_1\|a_{20}\|_{L^\infty}) =: C_3/C_1$ we obtain
\begin{equation}
  \left|\int_\mathbb{R} \left(\partial_x^2 a_4 - \partial_x a_3 \right) \partial_x^2 q\, \bar{q} \,{\rm d} x\right|
\leq C_1 \frac{\operatorname{Re}\lambda}{4|\lambda|^2} \|\partial_x^2 q\|_{L^2}^2 + \|a_{20}\|_{L^\infty}\|q\|_{L^2}^2.
\end{equation}
Finally, for the fourth integral in \eqref{601_4} we use the inequality
\begin{align*}
  \|a_1-\partial_x a_2\|_{L^\infty}
& \leq \|-a_{21} + \partial_x a_{22}\|_{L^\infty} + \frac{1}{|\lambda|}\|a_{10} - \partial_x a_{11}\|_{L^\infty} \\
& \leq \|-a_{21} + \partial_x a_{22}\|_{L^\infty} + \frac{1}{\|a_{20}\|_{L^\infty}}\|a_{10}
- \partial_x a_{11}\|_{L^\infty} =: \sqrt{\frac{9}{4{\rm R}}C_4}
\end{align*}
for $\operatorname{Re}\lambda \geq \|a_{20}\|_{L^\infty}$. Using again $\epsilon = 9/(8{\rm R})$ in Young's inequality,
we obtain
\begin{align*}
  \big| \int_\mathbb{R} \left(a_1{-}\partial_x a_2\right) \partial_x q\, \bar{q} \,{\rm d} x \big|
& \leq \frac{1}{2\epsilon} \|a_1{-}\partial_x a_2\|_{L^\infty}^2 \|q\|_{L^2}^2
 + \frac{\epsilon}{2} \|\partial_x q\|_{L^2}^2 \\
& \leq C_4 \|q\|_{L^2}^2 + \frac{9}{16{\rm R}}\|\partial_x q\|_{L^2}^2.
\end{align*}
Altogether,  we have
\begin{equation}
  \operatorname{Re} b(q,q) \geq (\operatorname{Re} \lambda - 2\|a_{20}\|_{L^\infty}-C_4)\|q\|_{L^2}^2
 + \frac{9}{8{\rm R}}\|\partial_x q\|_{L^2}^2
 + \frac{C_1\operatorname{Re}\lambda}{2|\lambda|^2} \|\partial_x^2 q\|_{L^2}^2 \label{601_20}
\end{equation}
for all $\lambda$ with
\[
  \operatorname{Re}\lambda \geq \max\big\{ 2\|a_{20}\|_{L^\infty} + C_4, \frac{4{\rm R}}{9}\|a_{11}\|_{L^\infty},
\frac{4C_2}{C_1}, \frac{C_3}{C_1} \big\}.
\]
This shows the ellipticity of $b$; i.e.,
 there exist $a,C>0$ such that for all $\lambda$ with $\operatorname{Re}\lambda\geq a$ we have
  \[
    C\operatorname{Re} b(q,q) \geq (\operatorname{Re} \lambda - a/2)\|q\|_{L^2}^2 + \|\partial_x q\|_{L^2}^2
+ \frac{\operatorname{Re}\lambda}{|\lambda|^2} \|\partial_x^2 q\|_{L^2}^2.
 \]
Thus, by the Lax-Milgram Lemma, there exists a unique weak solution $q \in H^2$ of \eqref{601_3} if
 $\operatorname{Re}\lambda \geq a$. Furthermore, from the weak formulation \eqref{601_9} we obtain the estimate
\[
  (\operatorname{Re} \lambda - a/2)\|q\|_{L^2}^2 + \|\partial_x q\|_{L^2}^2
+ \frac{\operatorname{Re}\lambda}{|\lambda|^2} \|\partial_x^2 q\|_{L^2}^2 \leq C\|g_2\|_{L^2} \|q\|_{L^2}
+ C \frac{\operatorname{Re}\lambda}{|\lambda|^2}\|g_1\|_{H^1} \|q\|_{H^2}.
\]
To estimate the $H^2$-norm of $q$, we can use $\operatorname{Re}\lambda - a/2 \geq a/2$.
Thus, the coefficient in front of $\|q\|_{L^2}^2$ can be estimated from below
independently of $\lambda$. However, the coefficient of $\|\partial_x^2 q\|_{L^2}^2$
converges to zero for $|\lambda|\to \infty$. Therefore, it is necessary to test
the resolvent equation \eqref{601_3} not only with $q$ itself, but also with $\partial_x^2 q$.
 However, since $g_1$ is only in $H^1$, on the right-hand side of the weak formulation \eqref{601_9}
there occurs the integral $\int_\mathbb{R} \partial_x g_1\, \partial_x^4 \bar{q} \,{\rm d} x$, for instance.
This can only be estimated with the help of $\|q\|_{H^4}$, which is not helpful
for estimating $\|q\|_{H^2}$. Therefore, we split $q$ into $q = q_0 + \tilde{q}$,
 where the two components are supposed to fulfill
\begin{gather}
  (\lambda{+}a_0) q_0    + (a_1\partial_x{+}a_2\partial_x^2 {+} a_3\partial_x^3 {+} a_4\partial_x^4)
 q_0
 = \frac{1}{\lambda}(a_{10} {+} a_{11}\partial_x {+} a_{12}\partial_x^2 {+} a_{13}\partial_x^3)g_1, \label{601_10}\\
  (\lambda{+}a_0) \tilde{q}  + (a_1\partial_x {+} a_2\partial_x^2 {+} a_3\partial_x^3 {+} a_4\partial_x^4)
 \tilde{q}   = g_2. \label{601_11}
\end{gather}
Since the right-hand side of \eqref{601_10} has a leading factor $1/\lambda$, it is sufficient
to test with $q_0$. In \eqref{601_11}, the right-hand side is in $L^2$, thus it can be
tested with $\partial_x^2 \tilde{q}$, which leads to an estimate of $\|\tilde{q}\|_{H^2}$
independent of $\lambda$.

We begin with estimating $q_0$. By the considerations above we find a unique weak
solution $q_0$ of \eqref{601_10} with
\[
  (\operatorname{Re} \lambda - a/2)\|q_0\|_{L^2}^2 + \|\partial_x q_0\|_{L^2}^2
+ \frac{\operatorname{Re}\lambda}{|\lambda|^2} \|\partial_x^2 q_0\|_{L^2}^2
\leq C \frac{\operatorname{Re}\lambda}{|\lambda|^2}\|g_1\|_{H^1} \|q_0\|_{H^2}
\]
for $\operatorname{Re}\lambda \geq a$. Since $(\operatorname{Re}\lambda-a/2)/\operatorname{Re} \lambda \geq 1/2$ and since
$|\lambda|^2/\operatorname{Re}\lambda \geq |\lambda|$, we obtain
\begin{equation}
  |\lambda|^2\|q_0\|_{L^2}^2 + |\lambda|\|\partial_x q_0\|_{L^2}^2 + \|\partial_x^2 q_0\|_{L^2}^2
\leq C \|g_1\|_{H^1} \|q_0\|_{H^2}. \label{601_12}
\end{equation}
As $|\lambda|\geq a$ is bounded from below, it follows that
\begin{equation}
\|q_0\|_{H^2} \leq C \|g_1\|_{H^1}. \label{601_13}
\end{equation}
In particular, together with \eqref{601_12} this leads to
$ |\lambda|^2 \|q_0\|_{L^2}^2 \leq C \|g_1\|_{H^1}\|q_0\|_{H^2} \leq C \|g_1\|_{H^1}^2$; i.e.,
\begin{equation}\label{601_17}
  |\lambda|\|q_0\|_{L^2} \leq C \|g_1\|_{H^1}.
\end{equation}
Next, we look for the corresponding estimates for $\tilde{q}$. Exactly as for $q_0$,
by testing \eqref{601_11} with $\tilde{q}$ and taking the real part we obtain
\begin{equation}
  (\operatorname{Re}\lambda - a/2) \|\tilde{q}\|_{L^2}^2 + \|\partial_x \tilde{q}\|_{L^2}^2
+ \frac{\operatorname{Re}\lambda}{|\lambda|^2} \|\partial_x^2 \tilde{q}\|_{L^2}^2 \leq C \|g_2\|_{L^2} \|\tilde{q}\|_{L^2} \label{601_15}
\end{equation}
for $\operatorname{Re}\lambda \geq a$. Testing \eqref{601_11} with $\partial_x^2 \tilde{q}$; i.e.,
$ b(\tilde{q},\partial_x^2 \tilde{q}) = \int_\mathbb{R} g_2\, \partial_x^2 \bar{\tilde{q}} \,{\rm d} x$,
and integration by parts leads to
\begin{equation} \begin{aligned}
&  -b(\tilde{q},\partial_x^2\tilde{q}) \\
& = \int_\mathbb{R}(\lambda+a_0)|\partial_x\tilde{q}|^2\,{\rm d} x + \int_\mathbb{R}(-a_2 {+} \partial_x a_3 {-} \partial_x^2 a_4)|\partial_x^2\tilde{q}|^2 \,{\rm d} x
+ \int_\mathbb{R} a_4 |\partial_x^3\tilde{q}|^2 \,{\rm d} x  \\
& \quad + \int_\mathbb{R} \partial_x a_0 \,\tilde{q}\, \partial_x \bar{\tilde{q}} \,{\rm d} x - \int_\mathbb{R} a_1 \partial_x \tilde{q} \,\partial_x^2 \bar{\tilde{q}} \,{\rm d} x
+ \int_\mathbb{R}(a_3 - \partial_x a_4) \partial_x^2 \tilde{q}\, \partial_x^3 \bar{\tilde{q}} \,{\rm d} x.
\end{aligned} \label{601_14}
\end{equation}
Since both $a_3$ and $a_4$ have a leading $1/\lambda$ and since $-a_2>0$ is bounded
from below, \eqref{601_14} can be estimated similarly to \eqref{601_4} by applying Young's inequality.
An exception is the integral $\int_\mathbb{R} \partial_x a_0\, \tilde{q}\, \partial_x \bar{\tilde{q}} \,{\rm d} x$,
which cannot be absorbed  by the first three integrals,  and therefore
\begin{equation} \begin{aligned}
  & (\operatorname{Re}\lambda - a/2)\|\partial_x \tilde{q}\|_{L^2}^2 + \|\partial_x^2 \tilde{q}\|_{L^2}^2
+ \frac{\operatorname{Re}\lambda}{|\lambda|^2}\|\partial_x^3 \tilde{q}\|_{L^2}^2  \\
  & \leq C \|g_2\|_{L^2} \|\partial_x^2 \tilde{q}\|_{L^2} + C\int_\mathbb{R} (|\tilde{q}|^2 + |\partial_x \tilde{q}|^2) \,{\rm d} x
  \leq C \|g_2\|_{L^2} \|\partial_x^2 \tilde{q}\|_{L^2} + C \|g_2\|_{L^2} \| \tilde{q}\|_{L^2}.
\end{aligned}\label{601_16}
\end{equation}
Here, we used \eqref{601_15} in the second estimate. Combining \eqref{601_15}
and \eqref{601_16} yields the resolvent estimate
\begin{equation}
  \|\tilde{q}\|_{H^2} \leq C \|g_2\|_{L^2} \label{601_19}
\end{equation}
for $\operatorname{Re}\lambda \geq a$.

\begin{remark} \rm
  To test \eqref{601_11} with $\partial_x^2 \tilde{q}$ we actually have to test with smooth functions
which are dense in $H^2$ and then extend the resulting resolvent estimates continuously
to the respective Sobolev spaces.
\end{remark}
Finally,  to estimate $|\lambda|\|\tilde{q}\|_{L^2}$, we also have to estimate the imaginary
 part of $b(\tilde{q},\tilde{q})$. Using $\operatorname{Im}(1/\lambda)=-\operatorname{Im}\lambda/|\lambda|^2$ we obtain
from \eqref{601_4} and \eqref{601_11},
\begin{equation} \begin{aligned}
  &\left|\int_\mathbb{R} g_2 \bar{\tilde{q}}\,{\rm d} x \right| \geq \operatorname{Im} b(\tilde{q},\tilde{q})  \\
    & \geq (\operatorname{Im}\lambda - \|a_{20}\|_{L^\infty}) \|\tilde{q}\|_{L^2}^2
+\frac{\operatorname{Im}\lambda}{|\lambda|^2} \int_\mathbb{R} a_{11} |\partial_x \tilde{q}|^2 \,{\rm d} x
-\frac{\operatorname{Im}\lambda}{|\lambda|^2} \int_\mathbb{R} a_{13} |\partial_x^2 \tilde{q}|^2 \,{\rm d} x  \\
    & \quad + \operatorname{Im}\int_\mathbb{R} \left(a_1-\partial_x a_2\right) \partial_x \tilde{q}\, \bar{\tilde{q}} \,{\rm d} x
+ \operatorname{Im}\int_\mathbb{R} \frac{1}{\lambda}\left(\partial_x^2 a_{13} - \partial_x a_{12} \right) \partial_x^2 \tilde{q}\, \bar{\tilde{q}} \,{\rm d} x  \\
    & \quad + \operatorname{Im}\int_\mathbb{R} \frac{1}{\lambda}\left( 2\partial_x a_{13}
- a_{12} \right) \partial_x^2 \tilde{q}\, \partial_x \bar{\tilde{q}} \,{\rm d} x.
\end{aligned} \label{601_18}
\end{equation}
This estimate is less powerful than the one for the real part, since $a_{11}$ and $a_{13}$
have an undefined or even the wrong sign. However, since the coefficients $a_{11}$, $a_{12}$,
and $a_{13}$ have a leading $\operatorname{Im}(1/\lambda)$, it allows for $\operatorname{Im}\lambda \geq a$ the inequality
\begin{align*}
  &(\operatorname{Im}\lambda - a/2) \|\tilde{q}\|_{L^2}^2 \\
  & \leq C\frac{1}{|\lambda|}(\|\partial_x\tilde{q}\|_{L^2}^2 + \|\partial_x^2\tilde{q}\|_{L^2}^2)
+ \|a_1-\partial_x a_2\|_{L^\infty}\int_\mathbb{R}  |\partial_x \tilde{q}| |\tilde{q}| \,{\rm d} x
+ \left|\int_\mathbb{R} g_2 \bar{\tilde{q}}\,{\rm d} x \right|\\
  & \leq C\frac{1}{|\lambda|}(\|\partial_x\tilde{q}\|_{L^2}^2 + \|\partial_x^2\tilde{q}\|_{L^2}^2)
+ \frac{\operatorname{Im}\lambda}{8}\int_\mathbb{R} |\tilde{q}|^2 \,{\rm d} x
+ \frac{2}{\operatorname{Im}\lambda}\|a_1-\partial_x a_2\|_{L^\infty}^2\int_\mathbb{R}  |\partial_x \tilde{q}|^2 \,{\rm d} x \\
  & \quad + \frac{\operatorname{Im}\lambda}{8}\int_\mathbb{R} |\tilde{q}|^2 \,{\rm d} x
+ \frac{2}{\operatorname{Im}\lambda}\int_\mathbb{R}  |g_2|^2 \,{\rm d} x \\
  & \leq C\frac{1}{\operatorname{Im}\lambda}(\|\partial_x\tilde{q}\|_{L^2}^2 + \|\partial_x^2\tilde{q}\|_{L^2}^2)
+ \frac{\operatorname{Im}\lambda}{4} \|\tilde{q}\|_{L^2}^2 + \frac{2}{\operatorname{Im}\lambda}\|g_2\|_{L^2}^2,
\end{align*}
where we used Young's inequality twice. Thus,
\[
  \big(\frac{3}{4}\operatorname{Im}\lambda - a/2\big) \|\tilde{q}\|_{L^2}^2
\leq C \frac{1}{\operatorname{Im}\lambda}(\|\partial_x\tilde{q}\|_{L^2}^2 + \|\partial_x^2\tilde{q}\|_{L^2}^2)
+ \frac{2}{\operatorname{Im}\lambda}\|g_2\|_{L^2}^2.
\]
Since $(3/4)\operatorname{Im}\lambda = ((3/4)\operatorname{Im} \lambda - a/2) + a/2 \leq ((3/4)\operatorname{Im} \lambda - a/2) + \operatorname{Im}\lambda/2$,
we have $\operatorname{Im}\lambda \leq 4((3/4)\operatorname{Im}\lambda -a/2)$, and as we have already estimated
$\|\tilde{q}\|_{H^2}$ in \eqref{601_19}, we obtain
\[
  \operatorname{Im}\lambda \, \|\tilde{q}\|_{L^2}^2 \leq C\frac{1}{\operatorname{Im}\lambda}(\|\partial_x\tilde{q}\|_{L^2}^2
+ \|\partial_x^2\tilde{q}\|_{L^2}^2 + \|g_2\|_{L^2}^2) \leq C\frac{1}{\operatorname{Im}\lambda} \|g_2\|_{L^2}^2.
\]
Considering ${-}\operatorname{Im} b(\tilde{q},\tilde{q})$ in \eqref{601_18} gives the same estimate
for ${-}\operatorname{Im}\lambda$ instead of $\operatorname{Im}\lambda$. Thus, for $|\operatorname{Im}\lambda|>a$ we have
\[
  |\operatorname{Im}\lambda| \, \|\tilde{q}\|_{L^2} \leq C \|g_2\|_{L^2}.
\]
By \eqref{601_15}, the same estimate is true if we replace $|\operatorname{Im}\lambda|$ by $\operatorname{Re}\lambda$,
since $\operatorname{Re}\lambda \leq (\operatorname{Re}\lambda - a/2) + \operatorname{Re}\lambda/2$, which implies $\operatorname{Re}\lambda \leq 2 (\operatorname{Re}\lambda -a/2)$.
Altogether, for $\operatorname{Re}\lambda > a$ we obtain
\begin{equation}
  |\lambda|\|\tilde{q}\|_{L^2} \leq C \|g_2\|_{L^2}. \label{601_21}
\end{equation}
Combining \eqref{601_13}, \eqref{601_17}, \eqref{601_19}, and \eqref{601_21} yields
for $q = q_0 + \tilde{q}$ the resolvent estimate
\begin{equation}
  |\lambda|\|q\|_{L^2} + \|q\|_{H^2} \leq C\left( \|g_1\|_{H^1} + \|g_2\|_{L^2}\right). \label{601_22}
\end{equation}

\subsection*{Estimates for $h$}
It remains to estimate the $L^2$- and the $H^3$-norm of $h$. Identity \eqref{601_23} leads to
\begin{equation}\label{601_24}
  \|h\|_{L^2} \leq \frac{1}{|\lambda|}\left( \|\partial_x q\|_{L^2} + \|g_1\|_{L^2}\right),
\quad \|\partial_x h\|_{L^2} \leq \frac{1}{|\lambda|}\left( \|\partial_x^2 q\|_{L^2} + \|\partial_x g_1\|_{L^2}\right),
\end{equation}
thus, by applying \eqref{601_22} we obtain
\begin{equation}\label{601_29}
  \|h\|_{L^2} + \|\partial_x h\|_{L^2} \leq C\left( \|g_1\|_{H^1} + \|g_2\|_{L^2}\right).
\end{equation}
From the second equation in the resolvent equation \eqref{601_2} we obtain
\begin{equation} \begin{aligned}
 & a_{12}\partial_x^2 h + a_{13} \partial_x^3 h  \\
 & = -g_2 + (\lambda{-}a_{20})q - a_{21}\partial_x q - a_{22} \partial_x^2 q - a_{10}h - a_{11} \partial_x h  \\
 & = -g_2 + (\lambda{-}a_{20})q - a_{21}\partial_x q - a_{22} \partial_x^2 q
 + \frac{1}{\lambda} a_{10}(\partial_x q - g_1) + \frac{1}{\lambda}a_{11}(\partial_x^2 q - \partial_x g_1).
\end{aligned} \label{601_25}
\end{equation}
Since all coefficients $a_{ij}$ are real, testing \eqref{601_25} with $(-\partial_x^2 h + \partial_x^3 h)$
 yields for the left-hand side
\begin{equation} \begin{aligned}
  & \operatorname{Re} \int_\mathbb{R} (a_{12}\partial_x^2 h + a_{13} \partial_x^3 h)(-\partial_x^2 \bar{h} + \partial_x^3 \bar{h})\,{\rm d} x \\
    &= -\int_\mathbb{R} a_{12}|\partial_x^2 h|^2 \,{\rm d} x + \int_\mathbb{R} a_{13} |\partial_x^3 h|^2 \,{\rm d} x
+ \operatorname{Re} \int_\mathbb{R} a_{12} \partial_x^2 h \,\partial_x^3 \bar{h} \,{\rm d} x \\
&\quad - \operatorname{Re} \int_\mathbb{R} a_{13} \partial_x^3 h \,\partial_x^2 \bar{h} \,{\rm d} x \\
    &= -\int_\mathbb{R} a_{12}|\partial_x^2 h|^2 \,{\rm d} x + \int_\mathbb{R} a_{13} |\partial_x^3 h|^2 \,{\rm d} x
+ \operatorname{Re} \int_\mathbb{R} (a_{12}-a_{13}) \partial_x^2 h\, \partial_x^3 \bar{h} \,{\rm d} x.
\end{aligned} \label{601_26}
\end{equation}
Integration by parts leads to
\[
  \int_\mathbb{R}(a_{12} - a_{13}) \partial_x^2 h\, \partial_x^3 \bar{h} \,{\rm d} x = - \int_\mathbb{R}(\partial_x a_{12}
- \partial_x a_{13})|\partial_x^2 h|^2 \,{\rm d} x - \int_\mathbb{R} (a_{12} - a_{13})\partial_x^3 h\, \partial_x^2 \bar{h} \,{\rm d} x,
\]
and thus,
\[
  \operatorname{Re} \int_\mathbb{R}(a_{12} - a_{13}) \partial_x^2 h\, \partial_x^3 \bar{h} \,{\rm d} x
= -\frac{1}{2} \int_\mathbb{R}(\partial_x a_{12} - \partial_x a_{13})|\partial_x^2 h|^2 \,{\rm d} x.
\]
Due to the definitions of $a_{12}, a_{13}$ in \eqref{51_5}, there exist positive
constants $C_1, \tilde C_1$ with $\min_{x\in [0,\gamma]}a_{13} = C_1$,
 $\min_{x\in [0,\gamma]}(-a_{12}) = 2\tilde C_1$. Hence, \eqref{601_26} can be estimated by
\begin{equation} \begin{aligned}
  & \operatorname{Re} \int_\mathbb{R} (a_{12}\partial_x^2 h + a_{13} \partial_x^3 h)(-\partial_x^2 \bar{h} + \partial_x^3 \bar{h})\,{\rm d} x \\
  & \geq 2 \tilde C_1 \|\partial_x^2 h\|_{L^2}^2 + C_1\|\partial_x^3 h\|_{L^2}^2 - \frac{1}{2}
\left(\|\partial_x a_{12}\|_{L^\infty} + \|\partial_x a_{13}\|_{L^\infty}  \right)
 \|\partial_x^2h\|_{L^2}^2.
\end{aligned}\label{601_27}
\end{equation}
On closer inspection of \eqref{51_5} we find that due to the additional $x$-derivative,
the coefficients $\partial_x a_{12}, \partial_x a_{13}$ are of the order of $\mathcal{O}(\varepsilon)$,
where $\varepsilon$ is proportional to the bottom waviness. Hence, without loss of
generality we may assume that
$\frac{1}{2}\left(\|\partial_x a_{12}\|_{L^\infty} + \|\partial_x a_{13}\|_{L^\infty}\right) < \tilde C_1$.
Then \eqref{601_27} reads
\begin{equation}\label{601_28}
  \operatorname{Re} \int_\mathbb{R} (a_{12}\partial_x^2 h + a_{13} \partial_x^3 h)(-\partial_x^2 \bar{h} + \partial_x^3 \bar{h})\,{\rm d} x
\geq \tilde C_1 \|\partial_x^2 h\|_{L^2}^2 + C_1\|\partial_x^3 h\|_{L^2}^2.
\end{equation}
Thus, testing \eqref{601_25} with $(-\partial_x^2 h + \partial_x^3 h)$ yields
\begin{align*}
  & \tilde C_1 \|\partial_x^2 h\|_{L^2}^2 + C_1\|\partial_x^3 h\|_{L^2}^2 \\
  &\leq\Big|\int_\mathbb{R}\Big( -g_2{-}\frac{1}{\lambda}a_{10}g_1{-}\frac{1}{\lambda}a_{11}\partial_x g_1
{+}(\lambda{-}a_{20})q {+}\big(\frac{1}{\lambda}a_{10}{-}a_{21} \big)\partial_x q\\
&\quad  {+} \big(\frac{1}{\lambda}a_{11}{-}a_{22} \big)\partial_x^2 q \Big)
  (-\partial_x^2 \bar{h} + \partial_x^3 \bar{h})\,{\rm d} x \Big|\\
  & \leq C \left( \|g_2\|_{L^2} + \|g_1\|_{H^1} + |\lambda| \|q\|_{L^2} + \|q\|_{H^2}
\right) \left( \|\partial_x^2h\|_{L^2} + \|\partial_x^3 h\|_{L^2} \right),
\end{align*}
together with \eqref{601_22} we obtain
$\|\partial_x^2h\|_{L^2} + \|\partial_x^3 h\|_{L^2} \leq C\left( \|g_1\|_{H^1} + \|g_2\|_{L^2} \right)$.
Combination with \eqref{601_29} yields
\begin{equation}
  \|h\|_{H^3} \leq C\left( \|g_1\|_{H^1} + \|g_2\|_{L^2}\right), \label{601_30}
\end{equation}
and \eqref{601_24} finally implies
\begin{equation}
  |\lambda|^{3/2}\|h\|_{L^2} \leq C|\lambda|^{1/2}(\|\partial_x q\|_{L^2} + \|g_1\|_{L^2}). \label{601_31}
\end{equation}
Since integration by parts gives
\[
|\lambda|\|\partial_x q\|_{L^2}^2  = -|\lambda| \int_\mathbb{R} q \,\partial_x^2 \bar{q} \,{\rm d} x
\leq \frac{1}{2}|\lambda|^2 \|q\|_{L^2}^2 + \frac{1}{2} \|\partial_x^2 q\|_{L^2}^2,
\]
we have $|\lambda|^{1/2} \|\partial_x q\|_{L^2} \leq C(|\lambda| \|q\|_{L^2} + \|\partial_x^2 q\|_{L^2})$,
and \eqref{601_31} reads
\begin{equation}
  |\lambda|^{3/2}\|h\|_{L^2} \leq C\left(|\lambda|^{1/2} \|g_1\|_{L^2} + \|g_1\|_{H^1}
 + \|g_2\|_{L^2}\right). \label{601_32}
\end{equation}
This proves Theorem \ref{601_thres1} for $r=2$.
For $r\geq 3$, $r \in \mathbb{N}$, the proof works the same way as above by
testing with the respective derivatives of $h,q$. For non-integer values of $r$,
 the resolvent estimate follows by interpolation theory, see \cite{LM1}, for instance.


\subsection*{Analytic semigroup}

With a few additional expenses the proof of Theorem \ref{601_thres1} allows to show
that the linear operator $A$ is sectorial; i.e., there exists a $\vartheta \in (0,\pi/2)$
such that a slightly modified resolvent estimate can be extended to the sector
$S_{a,\vartheta} := \{ \lambda \mid 0 \leq |\arg(a-\lambda)| \leq \vartheta+\pi/2 \}$
covering the half-plane $\operatorname{Re}\lambda \geq a$.
Setting $X := \mathcal{H}^1=H^1\times L^2$, the domain of $A$ is $\mathcal{D}(A) = \mathcal{H}^3 = H^3\times H^2\subset X$.
 Moreover, let $g \in \mathcal{D}(A)$ and $\operatorname{Re}\lambda' \geq a$. By Theorem \ref{601_thres1} there exists
a unique solution of the resolvent equation $(\lambda' - A)u = g$, and from
\eqref{601_22}, \eqref{601_24} it follows the estimate
\begin{equation}
  \|(\lambda'-A)^{-1}g\|_X \leq \frac{M}{|\lambda'|}\|g\|_X
\end{equation}
for a $M>0$ independent of $\lambda'$ and $g$. It remains to extend this estimate to the
sector $S_{a,\vartheta}$ by a perturbation argument, which we recall for completeness
in the following. Let $\lambda \in S_{a,\vartheta}$ with $\operatorname{Re}\lambda < a$, where $\vartheta$ is
specified later. Setting $\lambda' := a+ {\rm i} (\operatorname{Im} \lambda)$ yields
\begin{equation}
  \lambda-A = \lambda' - A+\lambda-\lambda' = (\lambda' - A)\left(\operatorname{Id} + (\lambda' -A)^{-1}(\lambda-\lambda')\right). \label{601_40}
\end{equation}
Choosing $\vartheta \in (0,\pi/2)$ small enough we can always ensure that
\[
  \frac{|\lambda-\lambda'|}{|\lambda'|} \leq \frac{|\lambda-\lambda'|}{|\operatorname{Im} \lambda|}
\leq \tan\vartheta < \frac{1}{M}.
\]
Hence,
\[
  \|(\lambda' - A)^{-1}(\lambda-\lambda')\|_{\mathscr{L}(X,X)} \leq \frac{M}{|\lambda'|}|\lambda-\lambda'| < 1,
\]
and the Neumann series
\[
\left(\operatorname{Id}+(\lambda'-A)^{-1}(\lambda-\lambda')\right)^{-1}
= \sum_{j=0}^\infty\left( -(\lambda'-A)^{-1}(\lambda-\lambda') \right)^j
\]
converges in $\mathscr{L}(X,X)$. By \eqref{601_40}, there exists the inverse
\begin{equation}
  (\lambda-A)^{-1} = \left(\operatorname{Id} + (\lambda'-A)^{-1}(\lambda-\lambda')\right)^{-1} (\lambda'-A)^{-1}
\end{equation}
with
\begin{equation}
  \|(\lambda-A)^{-1}g\|_X \leq C\|(\lambda' - A)^{-1}g\|_X \leq C\frac{M}{|\lambda'|}\|g\|_X
\leq C\frac{M}{|\lambda|}\|g\|_X.
\end{equation}

\begin{lemma}\label{cinf-lem}
  Let $X = H^1 \times L^2$. Then the operator $A: \mathcal{D}(A) \to X$ from \eqref{51_5}
is sectorial; i.e., there exist $M,a>0$ and $\vartheta \in (0,\pi/2)$ such that
the sector $S_{a,\vartheta} = \{ \lambda \mid 0 \leq |\arg(a-\lambda)| \leq \vartheta+\pi/2 \}$
is part of the resolvent set and
  \[
    \|(\lambda-A)^{-1}g\|_X \leq \frac{M}{|\lambda|}\|g\|_X
  \]
  for all $\lambda \in S_{a,\vartheta}$ and $g \in X$.
\end{lemma}

Thus, the linear operator $A$ generates an analytic semigroup, see \cite{Henry}, for instance.

\subsection*{Weighted Sobolev spaces}

Since we will need some decay rate in $x$, which corresponds to some regularity with
respect to the wave number in Bloch space, we transfer the result of
Theorem \ref{601_thres1} to the case of weighted Sobolev spaces.

\begin{theorem}[Resolvent estimate in weighted Sobolev spaces]\label{601_thres2}
	Theorem \ref{601_thres1} \\
also holds for the weighted spaces $H^r(2)$;
i.e., \eqref{601_1} becomes
\begin{align*}
		& \|h\|_{H^{r+1}(2)} + |\lambda|^{(r+1)/2}\|h\|_{L^2(2)} + \|q\|_{H^r(2)}
+ |\lambda|^{r/2}\|q\|_{L^2(2)} \\
		& \leq C \left( \|g_1\|_{H^{r-1}(2)} + |\lambda|^{(r-1)/2}\|g_1\|_{L^2(2)}
+ \|g_2\|_{H^{r-2}(2)} + |\lambda|^{(r-2)/2}\|g_2\|_{L^2(2)} \right).
\end{align*}
\end{theorem}

\begin{proof}
  In contrast to the proof of \eqref{601_1}, we have to multiply the test functions
by $\varrho(x) = (1+x^2)^{1/2}$ before testing. Differentiating the weight leads to
 additional terms in the estimates of the weak formulation. However, since derivatives
of $\varrho$ are of lower order, the additional mixed terms can be controlled by the
terms in which $\varrho$ occurs without a derivative if we choose $a$ and $C$ larger
than in Theorem \ref{601_thres1}. Details for a related problem can be found
in \cite[Appendix A.2]{Uecker_07}, for instance.
\end{proof}

\subsection{Maximal regularity}\label{sec_602}

With the resolvent estimate from Theorem \ref{601_thres2} we are now able to prove
Theorem \ref{600_th1} concerning local existence. For this purpose we fix some
 times $t_0<t_1$ and denote again by $A$ the linear operator from \eqref{51_5}.
Furthermore, let $r> 2$ be not an integer such that both $(r+1)/2$ and $r/2$
are not half integers in order to apply Theorem \ref{71_t1} and use Laplace transform in time.

\subsection*{The linear inhomogeneous problem}
We begin with the linear inhomogeneous equation
\begin{equation}
  Mu := (\partial_t - A)u = g, \quad u\big|_{t=t_0} = 0, \label{602_1}
\end{equation}
where $g \in \mathcal{K}_0^{r-1}((t_0,t_1),2) := K_0^{r-1}((t_0,t_1),2) \times K_0^{r-2}((t_0,t_1),2)$.
Due to Lemma \ref{71_l1} we can identify $g$ with its extension to $[t_0,\infty)$.
Thus, without loss of generality, we can write $g \in \mathcal{K}_0^{r-1}((t_0,\infty),2)$.
For a $\sigma_1 > 0$ chosen below we set
\begin{equation}
  U(t,x) := {\rm e}^{-\sigma_1 t}u(t+t_0,x), \quad G(t,x) := {\rm e}^{-\sigma_1 t}g(t+t_0,x).
\end{equation}
Then $G\in \mathcal{K}_0^{r-1}((0,\infty),2)$, and \eqref{602_1} is equivalent to solving
\begin{equation}
  (\partial_t + \sigma_1 - A)U = G, \quad U\big|_{t=0} = 0. \label{602_2}
\end{equation}
Since $U\big|_{t=0} = 0$, the Laplace transform of $U$ satisfies
\[
  \mathcal{L} (\partial_t v)(\tau) = \frac{1}{2\pi}\int_0^\infty \partial_t v(t){\rm e}^{-t\tau}\,{\rm d} t = \tau \mathcal{L} v(\tau),
\]
and \eqref{602_2} becomes
\begin{equation}
  (\tau + \sigma_1 - A)\mathcal{L} U(\tau,x) = \mathcal{L} G(\tau,x). \label{602_3}
\end{equation}
From Lemma \ref{71_l12} and Remark \ref{71_r1} it follows that
$\mathcal{L} G \in H^{r-1}(2)\times H^{r-2}(2)$ for almost every $\tau$ with $\operatorname{Re}\tau \geq 0$.
Thus, according to Theorem \ref{601_thres2}, for almost every $\tau$ with
 $\operatorname{Re} \tau + \sigma_1 \geq a$ there exists a unique solution of the resolvent
equation \eqref{602_3}. Choosing $\sigma_1 \geq a$ and setting $U=(U_1, U_2)^\top$,
 $G=(G_1,G_2)^\top$, we achieve the estimate
\begin{equation}
\begin{aligned}
  & \|\mathcal{L} U_1\|_{H^{r+1}(2)} + |\tau|^{(r+1)/2}\|\mathcal{L} U_1\|_{L^2(2)} + \|\mathcal{L} U_2\|_{H^r(2)}
+ |\tau|^{r/2}\|\mathcal{L} U_2\|_{L^2(2)} \\
  & \leq C\Big( \|\mathcal{L} G_1\|_{H^{r-1}(2)} + |\tau|^{(r-1)/2}\|\mathcal{L} G_1\|_{L^2(2)}\\
&\quad + \|\mathcal{L} G_2\|_{H^{r-2}(2)} + |\tau|^{(r-2)/2}\|\mathcal{L} G_2\|_{L^2(2)} \Big)
\end{aligned}
\end{equation}
for almost every $\tau$ with $\operatorname{Re}\tau \geq 0$. To apply Theorem \ref{71_t1},
which yields $U \in \mathcal{K}_0^{r+1}((0,\infty),2)$, we additionally have to show
for $j \in \{1,2\}$ that
\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}
    \item $\tau \mapsto \mathcal{L} U_j(\tau,x)$ is holomorphic in the half-plane $\operatorname{Re} \tau > 0$
for almost every $x \in \mathbb{R}$,
    \item $ \sup_{\tau_1>0} \int_\mathbb{R} |\mathcal{L} U_j(\tau_1 + {\rm i} \tau_2, x) |^2 \,{\rm d} \tau_2 < \infty$
for almost every $x \in \mathbb{R}$.
  \end{enumerate}
Property (ii) immediately follows from the corresponding estimate for $\mathcal{L} G$; i.e.,
\begin{align*}
  & \sup_{\tau_1>0} \int_\mathbb{R}\left( |\mathcal{L} U_1(\tau_1+{\rm i}\tau_2,x)|^2
+ |\mathcal{L} U_2(\tau_1+{\rm i}\tau_2,x)|^2 \right){\rm d}\tau_2  \\
  & \leq C\sup_{\tau_1>0} \int_\mathbb{R}\left( \|\mathcal{L} U_1(\tau_1+{\rm i}\tau_2,\cdot)\|_{H^2(2)}^2
+ \|\mathcal{L} U_2(\tau_1+{\rm i}\tau_2,\cdot)\|_{H^2(2)}^2 \right){\rm d}\tau_2  \\
  & \leq C\sup_{\tau_1>0} \int_\mathbb{R}\left( \|\mathcal{L} G_1(\tau_1+{\rm i}\tau_2,\cdot)\|_{L^2(2)}^2
+ \|\mathcal{L} G_2(\tau_1+{\rm i}\tau_2,\cdot)\|_{L^2(2)}^2 \right){\rm d}\tau_2 < \infty
\end{align*}
due to Lemma \ref{71_l12}. In order to show that $\mathcal{L} U$ is holomorphic,
we set $\tau = \tau_r + {\rm i}\tau_i$ and $\mathcal{L} U=\mathcal{U}_r+{\rm i} \mathcal{U}_i$,
thus the resolvent equation \eqref{602_3} reads
\[
  (\tau_r + \sigma_1 + {\rm i}\tau_i -A) (\mathcal{U}_r+{\rm i} \mathcal{U}_i) = \mathcal{L} G.
\]
Differentiating with respect to $\tau_r$ and $\tau_i$ and using on the right-hand
side that $\mathcal{L} G$ is holomorphic, we obtain
\begin{equation}
  (\tau_r + \sigma_1 + {\rm i}\tau_i -A)\left[ \partial_{\tau_r}\mathcal{U}_r - \partial_{\tau_i}\mathcal{U}_i
 + {\rm i}\left( \partial_{\tau_r}\mathcal{U}_i + \partial_{\tau_i}\mathcal{U}_r \right) \right] = 0 \label{602_4}
\end{equation}
for $\tau_r >0$. Due to Theorem \ref{601_thres2}, there exists a unique solution of \eqref{602_4},
given by
\begin{equation}
  \partial_{\tau_r}\mathcal{U}_r - \partial_{\tau_i}\mathcal{U}_i + {\rm i}\left( \partial_{\tau_r}\mathcal{U}_i
+ \partial_{\tau_i}\mathcal{U}_r \right) = 0.
\end{equation}
Thus, $\mathcal{L} U$ fulfills the Cauchy-Riemann differential equations for $\operatorname{Re}\tau >0$.
Transferring the results back to $u,g$ proves the following lemma.

\begin{lemma}\label{602_l1}
  Let $r>2$ be not an integer, and fix some $t_1>t_0$. Then there exists a $C>0$ such
that for $g \in \mathcal{K}_0^{r-1}((t_0,t_1),2)$ there exists a unique solution
$u \in \mathcal{K}_0^{r+1}((t_0,t_1),2)$ of
  \[
    Mu = (\partial_t - A)u = g, \quad u\big|_{t=t_0} = 0
  \]
  with $\|u\|_{\mathcal{K}_0^{r+1}((t_0,t_1),2)} \leq C \|g\|_{\mathcal{K}_0^{r-1}((t_0,t_1),2)}$.
\end{lemma}

\subsection*{The nonlinear problem}
To prove Theorem \ref{600_th1} we look for a solution $u=(h,q)^\top\in \mathcal{K}^{r+1}((t_0,t_1),2)$
of the nonlinear problem \eqref{540_1}; i.e.,
\begin{equation}
  \partial_t \begin{pmatrix}h\\q\end{pmatrix} = A\begin{pmatrix}h\\q\end{pmatrix} + N(h,q). \label{602_5}
\end{equation}
As initial condition we take
\begin{equation}
  u\big|_{t=t_0} = u_0 = (h_0,q_0)^\top \in \mathcal{H}^r(2) = H^r(2)\times H^{r-1}(2). \label{602_6}
\end{equation}
Since the nonlinearity $N$ contains a third derivative of $h$, we restrict our calculations
to the case $r\geq 3$. According to Lemma \ref{71_l6} there exists a
function $v \in \mathcal{K}^{r+1}((t_0,t_1),2)$ with $v\big|_{t=t_0}=u_0$.
Setting $u = v+w$ the initial value problem \eqref{602_5}, \eqref{602_6}
is equivalent to $\partial_t v + \partial_t w = Av + Aw +N(v+w)$ satisfied by
$w \in \mathcal{K}^{r+1}((t_0,t_1),2)$ with homogeneous initial condition; i.e.,
\begin{equation}
  Mw = N(v+w) - Mv, \quad w\big|_{t=t_0} = 0. \label{602_7}
\end{equation}
In the next step we assume that $w \in \mathcal{K}_0^{r+1}((t_0,t_1),2)$.
To invert the operator $M$ on the left-hand side of \eqref{602_7}, we have to
show that $N(v+w) - Mv \in \mathcal{K}_0^{r-1}((t_0,t_1),2)$.
As Lemma \ref{602_l1} does not work for integers, we take $r>3$ in the following.
 The highest derivatives occurring in the nonlinearity $N(v+w)$ are $\partial_x^3 h$ and $\partial_x q$.
Since $v,w \in \mathcal{K}^{r+1}((t_0,t_1),2) = K^{r+1}((t_0,t_1),2) \times K^{r}((t_0,t_1),2)$,
 Lemma \ref{71_l9} yields $\partial_x^3 h \in K^{r-2}((t_0,t_1),2)$ and $\partial_x q \in K^{r-1}((t_0,t_1),2)$.
Thus, we can apply Lemma \ref{71_l10} which gives $N(v+w) \in \mathcal{K}^{r-1}((t_0,t_1),2)$.
Due to Lemma \ref{71_l9} the same is true for $Mv$, hence
\begin{equation}
  N(v+w)-Mv \in \mathcal{K}^{r-1}((t_0,t_1),2).
\end{equation}
According to Definition \ref{71_d4} it remains to show that
\begin{equation}
  \partial_t^j\left(N(v+w) - Mv)\right)\big|_{t=t_0} = 0 \quad \text{for all } j < (r-2)/2. \label{602_8}
\end{equation}
Restricting the regularity to $3<r<4$, we have to check \eqref{602_8} only for $j=0$.
Since $v \in K^{r+1}((t_0,t_1),2) \times K^{r}((t_0,t_1),2)$,
Lemma \ref{71_l6} additionally allows to choose $\partial_t^j v\big|_{t=t_0}$ for $j < (r-1)/2$ arbitrarily.
Hence we set $\partial_t v\big|_{t=t_0} = Au_0 + N(u_0)$, which yields
$
\left(N(v+w) - Mv\right)\big|_{t=t_0} = N(u_0) - \partial_t v\big|_{t=t_0} + Au_0=0$,
thus, $N(v+w)-Mv \in \mathcal{K}_0^{r-1}((t_0,t_1),2)$ if $w \in \mathcal{K}_0^{r+1}((t_0,t_1),2)$.
Therefore, we can apply Lemma \ref{602_l1} and write \eqref{602_7} as fixed point equation, namely
\begin{equation}
  w = M_0^{-1}\left(N(v+w) - Mv\right), \label{602_9}
\end{equation}
where we denote the solution operator of Lemma \ref{602_l1} by $M_0^{-1}$.
 The choice of $v$ is not unique, but by applying a cut-off function in time we can always
 ensure that $\|v\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} < C \|u_0\|_{\mathcal{H}^r(2)}$ for a fixed $C>0$.
Setting $\|u_0\|_{\mathcal{H}^r(2)} = \epsilon^2 < 1$ and assuming
$\|w\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} \leq \epsilon$ we obtain
\begin{equation} \begin{aligned}
&\|M_0^{-1}\left(N(v+w) - Mv\right)\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} \\
&\leq C \|N(v+w) - Mv\|_{\mathcal{K}^{r-1}((t_0,t_1),2)}  \\
& \leq C\left( \|w\|_{\mathcal{K}^{r+1}((t_0,t_1),2)}^2 + \|v\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} \right) \\
&\leq C \epsilon^2 = C \|u_0\|_{\mathcal{H}^r(2)} < \epsilon
\end{aligned}\label{602_10}
\end{equation}
for $\epsilon>0$ small enough. Therefore, the right-hand side of \eqref{602_9} maps a small
ball in $\mathcal{K}_0^{r+1}((t_0,t_1),2)$ into itself if the initial condition $u_0$ is small enough.
For $w_1$ and $w_2$ in this ball, additionally we have
\begin{equation} \begin{aligned}
  &\|M_0^{-1}\left( N(v+w_1) - Mv \right) - M_0^{-1}\left( N(v+w_2)
 - Mv \right)\|_{\mathcal{K}^{r+1}((t_0,t_1),2)}  \\
  & \leq C \|N(v+w_1) - N(v+w_2)\|_{\mathcal{K}^{r-1}((t_0,t_1),2)}
   \leq \frac{1}{2} \|w_1-w_2\|_{\mathcal{K}^{r+1}((t_0,t_1),2)},
\end{aligned}\label{602_14}
\end{equation}
since $N$ is at least quadratic and pure $v$-terms drop out.

Thus, for a sufficiently small initial condition the right-hand side of \eqref{602_9}
defines a contraction in $\mathcal{K}_0^{r+1}((t_0,t_1),2)$, and the contraction mapping theorem
yields the existence of a $w \in \mathcal{K}_0^{r+1}((t_0,t_1),2)$ satisfying \eqref{602_7}.
Since from \eqref{602_10} we obtain $\|w\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} \leq C \|u_0\|_{\mathcal{H}^r(2)}$,
 there exists a solution $u=v+w$ of \eqref{602_5}, \eqref{602_6} with
\[
  \|u\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} \leq C \|u_0\|_{\mathcal{H}^r(2)}.
\]

\subsection*{Uniqueness}
To show uniqueness of $u$ suppose there are two solutions $u_1, u_2$.
Then the difference $u_1 - u_2$ fulfills
\begin{equation}
  M(u_1 - u_2) = N(u_1) - N(u_2), \quad (u_1 - u_2)\big|_{t=t_0} = 0. \label{602_11}
\end{equation}
Since $N(u_1)\big|_{t=t_0} = N(u_2)\big|_{t=t_0}$, we have
 $N(u_1) - N(u_2) \in \mathcal{K}_0^{r-1}((t_0,t_1),2)$. Thus, we can write
$u_1-u_2 = M_0^{-1}\left(N(u_1)-N(u_2)\right)$, and similarly to \eqref{602_14} we obtain
\[
    \|u_1-u_2\|_{\mathcal{K}^{r+1}((t_0,t_1),2)}
 \leq C \|N(u_1) - N(u_2)\|_{\mathcal{K}^{r-1}((t_0,t_1),2)}
\leq \frac{1}{2} \|u_1-u_2\|_{\mathcal{K}^{r+1}((t_0,t_1),2)},
\]
if the initial condition $u_0$ is small enough. Thus, $u_1=u_2$.

\subsection*{Higher regularity}
The higher regularity in the time interval $t \in [\tilde t_0, t_1]$ for
$t_0<\tilde{t}_0<t_1$ follows from a bootstrapping argument, which we sketch next.
As $u \in L^2((t_0,t_1),\mathcal{H}^{r+1}(2))$, there exists a
 $\tilde{t} \in \left[ \frac{t_0 + \tilde{t}_0}{2},\tilde{t}_0 \right]$ with
\[
  \|u(\tilde{t}, \cdot)\|_{\mathcal{H}^{r+1}(2)} \leq \frac{2}{\tilde{t}_0-t_0}\|u\|_{L^2((t_0,\tilde{t}_0),\mathcal{H}^{r+1}(2))},
\]
since otherwise we had
\[
  \int_{\frac{t_0+\tilde{t}_0}{2}}^{\tilde{t}_0}\|u(\tilde{t},\cdot)\|_{\mathcal{H}^{r+1}(2)}^2\,{\rm d}\tilde{t}
 > \|u\|_{L^2((t_0,\tilde{t}_0),\mathcal{H}^{r+1}(2))}^2.
\]
Starting again at $t=\tilde{t}$ yields $u \in \mathcal{K}^{r+2}((\tilde{t},t_1),2)$ with
\[
  \|u\|_{\mathcal{K}^{r+2}((\tilde{t},t_1),2)} \leq C \|u(\tilde{t},\cdot)\|_{\mathcal{H}^{r+1}(2)}
 \leq C \|u\|_{\mathcal{K}^{r+1}((t_0,t_1),2)} \leq C \|u_0\|_{\mathcal{H}^r(2)}.
\]
In particular, we have
\begin{equation}
  \|u\|_{\mathcal{K}^{r+2}((\tilde{t}_0,t_1),2)} \leq C \|u_0\|_{\mathcal{H}^r(2)}.
\end{equation}
Iterating this procedure for $m \in \mathbb{N}$ yields
 $\|u\|_{\mathcal{K}^{r+m}((\tilde{t}_0,t_1),2)} \leq C \|u_0\|_{\mathcal{H}^r(2)}$,
and hence the second assertion in Theorem \ref{600_th1}.

\begin{remark} \rm
  In \eqref{602_8} we had to choose $r<4$ in order to achieve
$(N(v+w)-Mv) \in \mathcal{K}_0^{r-1}((t_0,t_1),2)$, hence it is not obvious why the bootstrapping
 argument can be applied to initial conditions with higher regularity.
However, considering the two components of $N(v+w) - Mv$ separately,
condition \eqref{602_8} can be substituted by
\begin{gather}
    \partial_t^{j_1}\left(\partial_t v_1 + \partial_x v_2\right)\big|_{t=t_0}
  = 0 \quad \text{for all } j_1 < (r-2)/2, \label{602_12}\\
    \partial_t^{j_2}\left(N_2(v+w) - \partial_t v_2 +A_2v\right)\big|_{t=t_0}
  = 0 \quad \text{for all } j_2 < (r-3)/2, \label{602_13}
 \end{gather}
where $A_2v$ denotes the second component of $Av$.
 According to Lemma \ref{71_l6}, we can choose the time derivatives
\[
  \partial_t^{j_1}v_1\big|_{t=t_0}, \quad \partial_t^{j_2}v_2\big|_{t=t_0}
\]
arbitrarily for all $j_1 < r/2$ and $j_2 < (r-1)/2$.
The essential property is that the regularities of $v \in \mathcal{K}^{r+1}$ and
$N(v+w) - Mv \in \mathcal{K}^{r-1}$ differ by two, such that there is always one degree
of freedom left in the choice of $v\big|_{t=t_0}$ to fulfill \eqref{602_12}, \eqref{602_13}.
 For instance, if $5<r<6$, we additionally have to fulfill \eqref{602_12} for $j_1=1$
and \eqref{602_13} for $j_2=1$, which is no problem since we can arbitrarily
choose $\partial_t^2 v_1$ and $\partial_t^2 v_2$ at $t=t_0$. Thus, the restriction to $r<4$ above is only for
notational convenience.
\end{remark}

\section{Renormalization}\label{sec_61}

To make the formal calculations in \S\ref{sec_543} rigorous and hence
prove Theorem \ref{main_th1} we establish a renormalization process as
in \cite{bkl94, uw07}. Additional to iterating the application of the
local existence and uniqueness theorem, the
key issue is to extract the leading order behavior formally described by
the Burgers equation \eqref{543_10}.
Therefore we now consider the IBL in Bloch space which is
split in  \eqref{542_al} for
the linearly diffusive mode $\tilde\alpha\phi^1$ and in \eqref{542_ws} for the linearly
exponentially damped remainder. Here
rescaled Bloch spaces with different weights in $\ell$ turn out to be useful.

\subsection{Basic setup}

For $m \in \mathbb{N}_0$, $r,b \geq 0$, and $L>0$ we set
\begin{equation} \begin{aligned}
  B_L(m,r,b)
& := H^m(I_{Lk_0}, H_\text{per}^r(I_\gamma)), \
  \| \tilde v \|_{B_L(m,r,b)}  \\
&:= \Big( \sum_{j\leq m} \int_{I_{Lk_0}}(1{+}\ell^2)^b\| \partial_{\ell}^j \tilde v(\ell, \cdot)
 \|_{H^r(I_\gamma)}^2 \,{\rm d} \ell \Big)^{1/2},
\end{aligned} \label{61_13}
\end{equation}
where again $I_\delta = (-\delta/2, \delta/2)$.
Note that the spaces adhere to the fixed choice of periodicity $\gamma = 2\pi/k_0$.
Let $B(m,r,b):= B_1(m,r,b)$. Regarding the original Bloch spaces
from \S\ref{sec_541} we have $B(m,r) = B(m,r,0)$.
At first view, the introduction of weights in the $\ell$-variable seems
dispensable since all norms $\|\cdot\|_{B_L(m,r,b_1)}$ and $\|\cdot\|_{B_L(m,r,b_2)}$
are equivalent due to the compact support in $\ell$. But as constants depend on $L$ this
step is crucial in \S \ref{sec_62} to control nonlinear interaction without losing
powers of $L^{-1}$. For $L>0$ we define the renormalization operator $\mathcal{R}_{1/L}$ by
\begin{equation}
	\mathcal{R}_{1/L}: B(m,r,b) \to B_L(m,r,b), \quad \mathcal{R}_{1/L}\tilde v(\ell,x)
:= \tilde v(\ell/L,x). \label{61_10}
\end{equation}
Note that only $\ell$ is rescaled, and thus there is no matching
rescaling in $x$-space. For $L\geq1$ we have
\begin{equation}
	L^{\frac{1-2m}{2}}\|\tilde{v}\|_{B(m,r,b)}
\leq \|\mathcal{R}_{1/L}\tilde{v}\|_{B_{L}(m,r,b)}
\leq L^{\frac{1+2b}{2}}\|\tilde{v}\|_{B(m,r,b)}
\leq  C L^{\frac{1+2b}{2}}\|\tilde{v}\|_{B(m,r,0)}. \label{61_7}
\end{equation}
We will mainly need the second inequality for $b=2$, which yields an additional
factor $L^{5/2}$ in the estimates.

For a fixed  $p \in (0,1/2)$ we introduce the renormalized variables
\begin{equation}
	\alpha_n(t,\ell)  := \mathcal{R}_{L^{-n}}\tilde \alpha(L^{2n}t,\ell), \quad
	w_n(t,\ell,x) := L^{(1-p)n} \mathcal{R}_{L^{-n}}\tilde{w}_s(L^{2n}t,\ell,x). \label{61_19}
\end{equation}
Since we suppose the stable component to decay like $t^{-1}$ and since time
is scaled by $L^{2n}$, we blow up $w_n$ by multiplying it with $L^{(1-p)n}$.
The factor $L^{pn}$ is needed later to control
some constants. From the IBL  \eqref{542_al}, \eqref{542_ws} in Bloch space we obtain
\begin{gather}
 \begin{aligned}
	\partial_t\alpha_n(t,\ell)
&=  L^{2n}\mu_1(L^{-n}\ell)\alpha_n(t,\ell)  + L^{2n}B_n^c(\alpha_n(t))(\ell)\\
&\quad  + L^{2n}H_n^c(\alpha_n(t),L^{-(1-p)n}w_n(t))(\ell),
\end{aligned}  \label{61_1}
\\
	\partial_t w_n(t,\ell,x) =   L^{2n}\tilde{A}_s(L^{-n}\ell)w_n(t,\ell,x)
+ L^{(3-p)n}H_n^s(\alpha_n(t),L^{-(1-p)n}w_n(t))(\ell,x), \label{61_2}
\end{gather}
where
\begin{equation}   \label{61_nl}
\begin{gathered}
	B_n^c(\alpha_n)  := \mathcal{R}_{L^{-n}}\tilde{B}_c\left( \mathcal{R}_{L^n}\alpha_n \right), \quad
	H_n^c(\alpha_n,w_n)  := \mathcal{R}_{L^{-n}}\tilde{H}_c\left( \mathcal{R}_{L^n}\alpha_n,\mathcal{R}_{L^n}w_n \right),
\\
	H_n^s(\alpha_n,w_n)  := \mathcal{R}_{L^{-n}}\tilde{H}_s\left( \mathcal{R}_{L^n}\alpha_n,\mathcal{R}_{L^n}w_n \right).
\end{gathered}
\end{equation}
Thus, solving \eqref{542_al}, \eqref{542_ws} with the initial condition
 $(\tilde{\alpha},\tilde{w}_s)\big|_{t=1} = (\alpha_0,w_0)$ is equivalent to iterating the
 following renormalization process:
For $n \in \mathbb{N}$  solve \eqref{61_1}, \eqref{61_2} for $t \in [1/L^2,1]$
 with the initial condition
\begin{equation}
\alpha_n(1/L^2,\ell) = \alpha_{n-1}(1,\ell/L),
	\quad w_n(1/L^2,\ell,x) = L^{1-p}w_{n-1}(1,\ell/L,x). \label{61_6}
\end{equation}
We take $(\alpha_n, w_n) \in X_{L^n}(2,r,b)$, where
\begin{equation} \begin{gathered}
  X_{L^n}(m,r,b)  := B_{L^n}(m,r,b){\times}\mathcal{B}_{L^n}(m,r,b), \\
\mathcal{B}_{L^n}(m,r,b) := B_{L^n}(m,r,b){\times}B_{L^n}(m,r{-}1,b),
\end{gathered} \label{61_14}
\end{equation}
with $r \geq 3$ and $b>0$ to be chosen later. Note that the value of $r$
does not play any role in the critical component $\tilde{\alpha}$ since $\tilde{\alpha}$
is independent of $x$. We introduce
\begin{equation}
	\tilde K_L^r((t_0,t_1),m,b)
 := H^{r/2}((t_0,t_1), B_L(m,0,b)) \cap L^2((t_0,t_1),B_L(m,r,b)). \label{61_16}
\end{equation}
Finally, let
\begin{gather}
  \tilde{\mathcal{K}}_L^{r+1}((t_0,t_1),m,b)
:= \tilde{K}_{L}^{r+1}((t_0,t_1),m,b) \times \tilde{K}_{L}^{r}((t_0,t_1),m,b), 
\label{61_17} \\
  \mathcal{X}_{L}^{r+1}((1/L^2,1),2,2)
:= \tilde{K}_{L}^{r+1}((1/L^2,1),2,2) \times\tilde {\mathcal{K}}_{L}^{r+1}((1/L^2,1),2,2). 
\label{61_18}
\end{gather}
Then, due to Theorem \ref{600_th1} concerning local existence in the original system,
 we expect local solutions of \eqref{61_1}, \eqref{61_2} in the space
 $\mathcal{X}_{L^n}^{r+1}((1/L^2,1),2,2)$.

\begin{theorem}[Local existence in the renormalized system]\label{61_th1}
	Let $3<r<4$. \\
There exist $L_0>1$ and $C_1, C_2>0$ such that for all $L>L_0$
the following holds.
Let
	\[
		\rho_{n-1} := \|(\alpha_{n-1}(1), w_{n-1}(1)) \|_{X_{L^{n-1}}(2,r,2)} \leq C_1L^{-5/2}.
	\]
	Then there exists a unique local solution $(\alpha_n, w_n) \in \mathcal{X}_{L^n}^{r+1}((1/L^2,1),2,2)$
of \eqref{61_1}, \eqref{61_2} with
\begin{equation}
		\|(\alpha_n,w_n)\|_{\mathcal{X}_{L^n}^{r+1}((1/L^2,1),2,2)} \leq C_2 L^{5/2} \rho_{n-1}. \label{61_21}
\end{equation}
	Moreover, for any $m \in \mathbb{N}$ we have $(\alpha_n, w_n) \in \mathcal{X}_{L^n}^{r+m}((1/2,1),2,2)$ and there exists a $C_3 > 0$ such that
\[
		\|(\alpha_n,w_n)\|_{\mathcal{X}_{L^n}^{r+m}((1/2,1),2,2)} \leq C_3 L^{5/2} \rho_{n-1}.
\]
\end{theorem}

\begin{proof}
  The proof can be adapted from Theorem \ref{600_th1}. The crucial point is to have $C_2,C_3$
independent of $n$, which depends on suitable resolvent estimates of the linear
 parts $L^{2n}\mu_1(L^{-n}\ell)$ and $L^{2n}\tilde{A}_s(L^{-n}\ell)$, and on estimates for
the nonlinearities. The latter is worked out in detail in \S \ref{sec_62} in a
slightly different form suitable to obtain more detailed asymptotics. Thus, here we only
  sketch the main ideas. First, we consider the linear inhomogeneous system
  \begin{gather}
    \partial_t\alpha_n-L^{2n}\mu_1(L^{-n}\ell)\alpha_n  =g_{n,1},  \label{61_11}\\
    \partial_t w_n-L^{2n}\tilde{A}_s(L^{-n}\ell)w_n  =g_{n,2}.  \label{61_12}
\end{gather}
  Since \eqref{61_11} is independent of $x$ no smoothing properties are needed,
and hence, as well as \eqref{61_1}, it can be solved by the variation-of-constants formula.
 For \eqref{61_12} we find resolvent estimates for $\lambda-L^{2n}\tilde{A}_s(L^{-n}\ell)$ which
correspond to Theorem \ref{601_thres2} transferred to Bloch space. Since we are in the
stable part, we can choose $a=0$ independently of $n$, which yields
  \begin{equation}
    \|(\alpha_n,w_n)\|_{\mathcal{X}_{L^n}^{r+1}((1/L^2,1),2,2)}
\leq C_2 \|(g_{n,1}, g_{n,2})\|_{\mathcal{X}_{L^n}^{r-1}((1/L^2,1),2,2)} \label{61_20}
\end{equation}
  with $C_2$ independent of $n$. Note that \eqref{61_20} could be improved by
choosing $a=-L^{2n}\sigma_0/2$, but to show the local existence result \eqref{61_21},
$a=0$ is sufficient here. The estimates for the nonlinearities, see \S \ref{sec_62},
 together with Banach's fixed point theorem and \eqref{61_7} then yield the first result,
 while the higher regularity follows as in Theorem \ref{600_th1} by a bootstrapping argument.
\end{proof}

The local existence Theorem \ref{61_th1} turns out to be a fundamental step in the proof
of the following nonlinear stability result.

\begin{theorem} \label{main_th3}
  Let $p\in(0,1/2)$ and $3<r<4$. In the spectrally stable case,
cf.~Assumption \ref{ass_spec}, there exist $C_1, C_2>0$ such that the following holds.
If $\|\alpha_0\|_{B(2,r,2)}+\|w_0\|_{\mathcal{B}(2,r,2)}\leq C_1$, then there exists a unique global
solution $\tilde{w}=\tilde{\alpha} \phi^1+\tilde{w}_s$ of the IBL  \eqref{542_al}, \eqref{542_ws} in Bloch
space with $(\tilde{\alpha},\tilde{w}_s)\big|_{t=1} = (\alpha_0,w_0)$. Moreover, we have
  \begin{equation}\label{55_1}
    \big\| (\ell,x) \mapsto \left( \tilde{w}(t,t^{-1/2}\ell,x) - \chi(t^{-1/2}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x)
\right) \big\|_{\mathcal{B}_{\sqrt{t}}(2,r,1)} \leq C_2 t^{-(1-p)/2},
\end{equation}
  where $\hat{f}_{z_0}$ is the Fourier transformed
profile from \eqref{522_1}, $\phi^1$ is the eigenvector to the critical eigenvalue $\lambda_1$,
see \eqref{542_7}, and $\ \ln(z_0+1)= 2\pi\frac{d}{c_2}\alpha_0(0)$ with $d$ from \eqref{543_9}.
\end{theorem}

Theorem \ref{main_th3} is proved in \S\ref{sec_62} and \S\ref{sec_63} by
an iteration scheme for the renormalized system. Here, we translate \eqref{55_1}
back to $x$-space in order to show Theorem \ref{main_th1}.

\subsection*{Proof of Theorem \ref{main_th1}}
We have
\begin{align*}
  (h,q)^\top(t,x)
& = \int_{-1/2k_0}^{1/2k_0} {\rm e}^{{\rm i}\ell x} (\tilde{h},\tilde{q})^\top(t,\ell,x) \,{\rm d}\ell
    = \int_{-1/2k_0}^{1/2k_0} {\rm e}^{{\rm i}\ell (x+c_1t)}\tilde{w}(t,\ell,x) \,{\rm d}\ell \\
& = t^{-1/2} \int_{-1/2k_0\sqrt{t}}^{1/2k_0\sqrt{t}} {\rm e}^{{\rm i}\ell t^{-1/2}(x+c_1t)}
 \tilde{w}(t,t^{-1/2}\ell,x) \,{\rm d}\ell.
\end{align*}
With the inverse Fourier transform
$f_{z_0}(t^{-1/2}(x+c_1t)) = \int_\mathbb{R} {\rm e}^{{\rm i}\ell t^{-1/2}(x+c_1t)} \hat{f}_{z_0}(\ell) \,{\rm d}\ell $
we obtain
\begin{equation} \begin{aligned}
  & (h,q)^\top(t,x) - t^{-1/2} f_{z_0}(t^{-1/2}(x+c_1t))\phi^1(0,x) \\
  & = t^{-1/2} \int_{-1/2k_0\sqrt{t}}^{1/2k_0\sqrt{t}} {\rm e}^{{\rm i}\ell t^{-1/2}(x+c_1t)}
\left( \tilde{w}(t,t^{-1/2}\ell,x) - \chi(t^{-1/2}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x) \right) {\rm d}\ell \\
  & \quad - t^{-1/2} \int_{\mathbb{R}} {\rm e}^{{\rm i}\ell t^{-1/2}(x+c_1t)}\hat{f}_{z_0}(\ell)
\left(1-\chi(t^{-1/2}\ell)\right){\rm d}\ell\, \phi^1(0,x).
\end{aligned} \label{55_2}
\end{equation}
The first integral on the right-hand side of \eqref{55_2} can be estimated by
\begin{align*}
  & t^{-1/2}\sup_{x\in\mathbb{R}}
\Big| \int_{-1/2k_0\sqrt{t}}^{1/2k_0\sqrt{t}} {\rm e}^{{\rm i}\ell t^{-1/2}(x+c_1t)}
\left( \tilde{w}(t,t^{-1/2}\ell,x) - \chi(t^{-1/2}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x) \right) {\rm d}\ell \Big| \\
  & \leq t^{-1/2} \int_{-1/2k_0\sqrt{t}}^{1/2k_0\sqrt{t}} \sup_{x\in[0,\gamma]}|
\tilde{w}(t,t^{-1/2}\ell,x) - \chi(t^{-1/2}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x)| {\rm d}\ell \\
  & \leq Ct^{-1+p/2}.
\end{align*}
Since $f_{z_0}$ is analytic, the Fourier transform $\hat{f}_{z_0}$ is exponentially decaying.
Thus, the second integral in \eqref{55_2} can be estimated as
\[
  t^{-1/2}\sup_{x\in\mathbb{R}}\Big|\int_{\mathbb{R}} {\rm e}^{{\rm i}\ell t^{-1/2}(x+c_1t)}\hat{f}_{z_0}(\ell)
\left(1-\chi(t^{-1/2}\ell)\right){\rm d}\ell\, \phi^1(0,x)\Big| \leq C t^{-1}.
\]
Altogether, this proves Theorem \ref{main_th1}. Thus, it remains
to prove Theorem \ref{main_th3}.



\subsection{Estimates}\label{sec_62}

We write the solution of the renormalized IBL  \eqref{61_1}, \eqref{61_2} in Bloch space
 with the help of the variation-of-constants formula; i.e.,
\begin{gather}
\begin{aligned}
&\alpha_n(t,\ell) \\
& = {\rm e}^{(t-L^{-2})L^{2n}\mu_1(L^{-n}\ell)}\alpha_{n-1}(1,L^{-1}\ell)  \\
&\quad + L^{2n}\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\mu_1(L^{-n}\ell)}
\left( B_n^c(\alpha_n(s))(\ell) + H_n^c(\alpha_n(s),L^{-(1-p)n}w_n(s))(\ell)  \right)
{\rm d} s,
\end{aligned}\label{62_vdk1}
\\
\begin{aligned}
& w_n(t,\ell,x)\\
 & = L^{1-p}{\rm e}^{(t-L^{-2})L^{2n}\tilde{A}_s(L^{-n}\ell)}w_{n-1}(1,L^{-1}\ell,x)  \\
&\quad + L^{(3-p)n}\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\tilde{A}_s(L^{-n}\ell)} H_n^s(\alpha_n(s),
L^{-(1-p)n}w_n(s))(\ell,x)\, {\rm d} s,
\end{aligned} \label{62_vdk2}
\end{gather}
where ${\rm e}^{tL^{2n}\tilde{A}_s(L^{-n}\ell)}$ stands for the analytic semigroup
generated by $L^{2n}\tilde{A}_s(L^{-n}\ell)$, cf.~Lemma \ref{cinf-lem}, which
clearly transfers from $ \tilde{A}_s(\ell) $ to $L^{2n}\tilde{A}_s(L^{-n}\ell)$.
To prove existence of a solution for $t\in[1/L^2,1]$
we need estimates of the linear semigroups and the nonlinearities.
These are shown in Lemma \ref{62_l1}, Lemma \ref{62_l6}, and Lemma \ref{62_l4}, respectively.

\begin{lemma}\label{62_l1}
  For $0\leq b_1 \leq b_2, \, 0 \leq j \leq 2$ there exists a $C>0$ such that for
$\alpha \in B_{L^n}(2,r,b_1)$ independent of $x$  we have
  \begin{equation}
    \|{\rm e}^{tL^{2n}\mu_1(L^{-n}\cdot)}\alpha\|_{B_{L^n}(2,r,b_2)}
\leq C \max\{1,t^{-(b_2 - b_1)/2}\}\| \alpha\|_{B_{L^n}(2,r,b_1)} \label{62_1}
\end{equation}
  in the critical part.
  The stable part is linearly exponentially damped; i.e.,
there exists a $\sigma_1 >0$ such that for $u \in B_{L^n}(2,r-j,b)\times B_{L^n}(2,r-1-j,b)$ we have
  \begin{equation} \begin{aligned}
    &\|{\rm e}^{tL^{2n}\tilde{A}_s(L^{-n}\cdot)}u\|_{B_{L^n}(2,r,b)\times B_{L^n}(2,r-1,b)}  \\
    & \ \leq C {\rm e}^{-\sigma_1 tL^{2n}}\max\{1,(L^{2n}t)^{-j/2}\}\|u\|_{B_{L^n}(2,r-j,b)
\times B_{L^n}(2,r-1-j,b)}.
\end{aligned} \label{62_8}
\end{equation}
\end{lemma}

\begin{proof}
  Estimate \eqref{62_1} follows from the locally parabolic shape of
 $L^{2n}\mu_1(L^{-n}\ell) = -c_2\ell^2 + \mathcal{O} (L^{-n}\ell^3)$ around $\ell=0$,
see Assumption \ref{ass_spec}. Since $\operatorname{Re} \mu_n(\ell) = \operatorname{Re} \lambda_n(\ell) < -\sigma_0$
for all $n\geq 2$ and $\ell \in (-k_0/2,k_0/2)$ and since $\operatorname{Re} \mu_1(\ell) < -\sigma_0$
for all $|\ell|>4r_\chi$, the real part of the spectrum of $\tilde{A}_s$ is bounded from
above by $-\sigma_0$. Thus, we have inequality \eqref{62_8} for a $\sigma_1 < \sigma_0$,
which avoids the treatment of Jordan blocks.
\end{proof}

To control the integral in \eqref{62_vdk2}  we use the following lemma.

\begin{lemma}\label{62_l5}
  There exists a $C>0$ such that for $t_0 \in [1/L^2,1]$ and $0\leq j \leq 1$ we have
$
    \int_{t_0}^1{\rm e}^{-\sigma_1(1-s)L^{2n}}(1{-}s)^{-j/2}{\rm d} s{\leq}C L^{-(2-j)n}.
$
  For $0 \leq j \leq 2$ we obtain
$
    \int_{1/L^2}^{1/2}{\rm e}^{-\sigma_1(1{-}s)L^{2n}}(1-s)^{-j/2}{\rm d} s{\leq}C L^{-2n}.
$
\end{lemma}

The next lemma exploits the role of leading $\ell$'s in the critical part of the
nonlinearity, cf.~\cite[Lemma 14]{gs96a}. Remember again that $k_0 := 2\pi/\gamma$,
where $\gamma$ denotes the bottom periodicity.

\begin{lemma}\label{62_l2}
  Let $\beta \in C^2([-k_0/2,k_0/2),C^2((0,\gamma),\mathbb{C}))$ with
$\|\beta(\ell,\cdot)\|_{C^2((0,\gamma),\mathbb{C})} \leq C \ell^{b_2-b_1}$ for a $b_1 \in [0,b_2]$.
 Then there exists a $C > 0$ such that for all $L>1$ we have
  \[
    \|(\mathcal{R}_{1/L} \beta)u\|_{B_L(2,r,b_1)}
\leq C L^{-(b_2-b_1)} \|\beta\|_{C^2([-k_0/2,k_0/2),C^2((0,\gamma),\mathbb{C}))} \|u\|_{B_L(2,r,b_2)}.
\]
\end{lemma}

The idea of the lemma is as follows. If the nonlinearity in \eqref{61_1} exhibits, e.g.,
a leading $(\ell/L^n)^j$, $j>0$,  we can extract the factor
$\ell^{b_2 - b_1}/L^{(b_2 - b_1)n}, \, b_2 - b_1\leq j$. While in
\eqref{62_vdk1} the factor $\ell^{b_2-b_1}$ can be balanced by the linear semigroup
in Lemma \ref{62_l1}, the factor $L^{-(b_2 - b_1)n}$ increases the degree of irrelevance.
 However, according to the term $t^{-(b_2 - b_1)/2}$ in \eqref{62_1} this is only
possible as long as $b_2-b_1$ is bounded from above by, e.g., $2-p$.

As products in $x$-space correspond to convolutions in Bloch space,
the nonlinearities in \eqref{61_nl} produce terms of the type
$\mathcal{R}_{L^{-1}}(\mathcal{R}_{L}u * \mathcal{R}_{L}v)$. Thus, we define an adapted convolution
operator $*_L$ for $u,v \in B_L(m,r,b)$ as follows:
\begin{equation}
  u*_Lv  := \int_{-Lk_0/2}^{Lk_0/2} u(\ell-m)v(m){\rm d} m
     = L \mathcal{R}_{L^{-1}}\left( (\mathcal{R}_{L}u) * (\mathcal{R}_{L}v) \right). \label{62_2}
\end{equation}
To estimate convolutions we use the following lemma,
based on standard Sobolev embeddings.

\begin{lemma}\label{62_l3}
  Let $b_2>1/2, \, b_2\geq b_1 \geq 0$. There exists a $C>0$ such that,  for all $L\geq1$,
  \[
    \|u*_Lv\|_{B_L(2,r,b_1)} \leq C\|u\|_{B_L(2,r,b_1)} \|v\|_{B_L(2,r,b_2)}.
 \]
\end{lemma}

\begin{remark} \rm
  Before estimating the nonlinearities in detail we want to summarize the different
effects in a descriptive way. By combining \eqref{62_2} and Lemma \ref{62_l3},
each convolution produces a factor $L^{-n}$. Due to the rescaling in \eqref{61_19},
each factor $w_n$ gives a further $L^{-(1-p)n}$. In the critical component,
a factor $\mathcal{R}_{L^{-n}}(\ell^{b_2-b_1})$ in the renormalized nonlinearity leads
to an additional factor $L^{-(b_2-b_1)n}$ as long as $b_2-b_1 \leq 2-p$.
\end{remark}

\begin{lemma}\label{62_l6}
  Let $r\geq3$. For $p \in (0,1/2)$ there exists a $C>0$ such that for all
 $(\alpha_n,w_n) \in X_{L^n}(2,r,2)=B_{L^n}(2,r,2)\times \mathcal{B}_{L^n}(2,r,2)$
with $(\alpha_n,w_n)_{X_{L^n}(2,r,2)}\leq 1$ we have
  \[
    L^{2n}H_n^c(\alpha_n,L^{-(1-p)n} w_n) = s_1 + s_2 + s_3
  \]
  with
\begin{gather}
    \|s_1\|_{B_{L^n}(2,r,p)}  \leq CL^{-(1-p)n}\|\alpha_n\|_{B_{L^n}(2,r,2)}^2, \label{62_6} \\
    \|s_2\|_{B_{L^n}(2,r,1)}  \leq CL^{-(1-p)n}\|\alpha_n\|_{B_{L^n}(2,r,2)} \|w_n\|_{\mathcal{B}_{L^n}(2,r,2)}, \label{62_7}\\
    \|s_3\|_{B_{L^n}(2,r,1)}  \leq CL^{-2(1-p)n}\|w_n\|_{\mathcal{B}_{L^n}(2,r,2)}^2. \label{62_4}
\end{gather}
\end{lemma}

\begin{proof}
  By construction we have
  \begin{equation} \begin{aligned}
& H_n^c(\alpha_n, L^{-(1-p)n}w_n)\\
&= \mathcal{R}_{L^{-n}}\tilde{H}_c(\mathcal{R}_{L^n}\alpha_n, L^{-(1-p)n} \mathcal{R}_{L^n}w_n) \\
&= \mathcal{R}_{L^{-n}}\tilde{E}_c^*\left(\tilde{B}((\mathcal{R}_{L^n}\alpha_n)\phi^1
+ L^{-(1-p)n}\mathcal{R}_{L^n}w_n)\right) - \mathcal{R}_{L^{-n}}\left( \tilde{B}_c(\mathcal{R}_{L^n}\alpha_n)\right)\\
& \quad + \mathcal{R}_{L^{-n}}\tilde{E}_c^*\left(\tilde{G}((\mathcal{R}_{L^n}\alpha_n)\phi^1
+ L^{-(1-p)n}\mathcal{R}_{L^n}w_n)\right),
\end{aligned} \label{62_3}
\end{equation}
 see \eqref{542_al} and \eqref{543_12}. We start with the estimates of the first
two terms on the right-hand side of \eqref{62_3}. Since they are quadratic,
we split them according to the multiplicities of $\alpha_n$ and $w_n$.
\medskip

  \noindent\emph{1. Terms quadratic in $\alpha_n$.}
    In the formal derivation of the Burgers equation in \S \ref{sec_543} we obtained
      \[
        \tilde{E}_c^*\left(\tilde{B}((\mathcal{R}_{L^n}\alpha_n)\phi^1)\right)(\ell)
 - \tilde{B}_c(\mathcal{R}_{L^n}\alpha_n)(\ell) = \beta(\ell) (\mathcal{R}_{L^n}\alpha_n)^{*2}(\ell)
      \]
    with $\beta(\ell) = \mathcal{O}(\ell^2)$, cf.~\eqref{543_res}. Therefore, we have to estimate
    \[
      \mathcal{R}_{L^{-n}}(\beta (\mathcal{R}_{L^n}\alpha_n)^{*2})
 = (\mathcal{R}_{L^{-n}}\beta)\mathcal{R}_{L^{-n}}((\mathcal{R}_{L^n}\alpha_n)^{*2}).
\]
    Applying \eqref{62_2}, Lemma \ref{62_l2}, and Lemma \ref{62_l3} we obtain
\begin{align*}
      \|\mathcal{R}_{L^{-n}}(\beta (\mathcal{R}_{L^n}\alpha_n)^{*2})\|_{B_{L^n}(2,r,p)}
& = L^{-n} \|(\mathcal{R}_{L^{-n}}\beta) (\alpha_n *_{L^n} \alpha_n)\|_{B_{L^n}(2,r,p)} \\
& \leq C L^{-(3-p)n}\|\alpha_n\|_{B_{L^n}(2,r,2)}^2.
 \end{align*}
Thus, the terms considered in this part can be assigned to $s_1$.
Note that here we only used $\beta(\ell) = \mathcal{O}(\ell^{2-p})$ instead
of $\beta(\ell) = \mathcal{O}(\ell^2)$, since otherwise the missing weight in $\ell$ could
not be balanced by the linear semigroup in \eqref{62_1}.
\medskip

  \noindent\emph{2. Mixed terms in $\alpha_n, w_n$.}
    The terms in
    \[
      \tilde{E}_c^*\left(\tilde{B}(\tilde{\alpha}\phi^1 + \tilde{w}_s)\right)(\ell)
= \chi(\ell) \int_0^\gamma B_2(\tilde{\alpha}(\ell)\phi^1(\ell) + \tilde{w}_s(\ell))(x) \bar\psi^1_2(\ell,x) {\rm d} x
    \]
    which contain both $\tilde{\alpha}$ and $\tilde{w}_s$ are all of the type
    \[
      N(\tilde{\alpha},\tilde{w}_s)(\ell)
= \chi(\ell)\int_0^\gamma\left[ (\tilde{\alpha}(\cdot)(\partial_x{+}{\rm i}\cdot)^{k_c}\phi^1_i(\cdot,x))
* ((\partial_x{+}{\rm i}\cdot)^{k_s} \tilde{w}_{s,j}(\cdot,x)) \right](\ell)\bar\psi^1_2(\ell,x) {\rm d} x
    \]
    with $k_c, k_s \in \{1,2,3\}$ and $i,j \in \{1,2\}$. Applying \eqref{62_2} yields
\begin{align*}
        & \mathcal{R}_{L^{-n}}N(\mathcal{R}_{L^n}\alpha_n, L^{-(1-p)n}\mathcal{R}_{L^n}w_n) \\
        & = L^{-(1-p)n}\mathcal{R}_{L^{-n}}\Big(\chi\int_0^\gamma
\left[ ((\mathcal{R}_{L^n}\alpha_n)(\partial_x{+}{\rm i}\cdot)^{k_c}\phi^1_i)*((\partial_x{+}{\rm i}
\cdot)^{k_s}(\mathcal{R}_{L^n} w_{n,j})) \right] \bar\psi^1_2 {\rm d} x\Big)\\
        & = L^{-(1-p)n}\mathcal{R}_{L^{-n}}\Big(\chi\int_0^\gamma
\Big[ \mathcal{R}_{L^n}\left(\alpha_n (\partial_x{+}{\rm i}\frac{\cdot}{L^n})^{k_c}\mathcal{R}_{L^{-n}}\phi^1_i\right)\\
&\quad  * \mathcal{R}_{L^n}\left((\partial_x{+}{\rm i}\frac{\cdot}{L^n})^{k_s}w_{n,j}\right) \Big]\bar\psi^1_2
{\rm d} x\Big)\\
        & = L^{-(2-p)n}\int_0^\gamma\left[\left( \alpha_n(\partial_x{+}{\rm i}\frac{\cdot}{L^n})^{k_c}
\mathcal{R}_{L^{-n}}\phi^1_i \right) *_{L^n} \left( (\partial_x{+}{\rm i}\frac{\cdot}{L^n})^{k_s}w_{n,j}
 \right)\right] \mathcal{R}_{L^{-n}}(\chi \bar\psi^1_2) {\rm d} x,
      \end{align*}
where $\chi(\ell)\bar\psi^1_2(\ell) = \mathcal{O}(\ell)$, cf.~\eqref{542_24}. If $u \in B_{L^n}(2,r,1)$
is independent of $x$, we have $\|u\|_{B_{L^n}(2,r,1)} = \|u\|_{B_{L^n}(2,0,1)}$. Thus, we obtain
\begin{align*}
      & \|\mathcal{R}_{L^{-n}}N(\mathcal{R}_{L^n}\alpha_n, L^{-(1-p)n}\mathcal{R}_{L^n}w_n)\|_{B_{L^n}(2,r,1)} \\
      & \leq L^{-(2-p)n}\int_0^\gamma \left\| \left(\left( \alpha_n(\partial_x{+}{\rm i}
\frac{\cdot}{L^n})^{k_c}\mathcal{R}_{L^{-n}}\phi^1_i \right) *_{L^n}
\left( (\partial_x{+}{\rm i}\frac{\cdot}{L^n})^{k_s}w_{n,j} \right)\right) \right\|_{B_{L^n}(2,0,1)} {\rm d} x \\
      & \leq C L^{-(3-p)n} \| \alpha_n(\partial_x{+}{\rm i}\frac{\cdot}{L^n})^{k_c}
\mathcal{R}_{L^{-n}}\phi^1_i  \|_{B_{L^n}(2,0,2)}  \| (\partial_x{+}{\rm i}
\frac{\cdot}{L^n})^{k_s}w_{n,j} \|_{B_{L^n}(2,0,2)} \\
      & \leq C L^{-(3-p)n} \|\alpha_n\|_{B_{L^n}(2,r,2)} \|w_n\|_{\mathcal{B}_{L^n}(2,r,2)}.
 \end{align*}
    Therefore, the mixed terms in $\alpha_n, w_n$ can be assigned to $s_2$.
\medskip

  \noindent\emph{3. Terms quadratic in $w_n$.}
  The estimates for the terms in $\tilde{E}_c^*\big(\tilde{B}(L^{-(1-p)n}\mathcal{R}_{L^n}w_n)\big)$
are the same as for the mixed terms, except that we have an additional factor $L^{-(1-p)n}$
due to the scaling of $w_n$, which yields the estimate for $s_3$ in \eqref{62_4}.
\medskip

  \noindent It remains to estimate the third term in \eqref{62_3}.
We have
  \begin{equation}
    \tilde{E}_c^*\left(\tilde{G}((\mathcal{R}_{L^n}\alpha_n)\phi^1)\right)(\ell)
= \mathcal{O}(\ell^2) (\mathcal{R}_{L^n}\alpha_n)^{*2}(\ell)
+ \sum_{j \geq 3}\mathcal{O}(\ell)(\mathcal{R}_{L^n}\alpha_n)^{*j}(\ell), \label{62_5}
\end{equation}
  where the $\mathcal{O}(\ell^2)$-terms are due to the quadratic terms in the IBL
with a factor $\partial_x q$. Since $(\alpha_n,w_n)_{X_{L^n}(2,r,2)}\leq 1$ we have
$\|\alpha_n\|_{B_{L^n}(2,r,2)}^j \leq \|\alpha_n\|_{B_{L^n}(2,r,2)}^2$
for $j \geq 3$, thus the terms in \eqref{62_5} belong to $s_1$ and can be
estimated as stated in \eqref{62_6}. By the same considerations,
all other terms in $\tilde{E}_c^*\big(\tilde{G}((\mathcal{R}_{L^n}\alpha_n)\phi^1 + L^{-(1-p)n}\mathcal{R}_{L^n}w_n)\big)$
 are absorbed by $s_2$ and $s_3$ and can be estimated as specified in \eqref{62_6}
and \eqref{62_4}.
\end{proof}

To prove estimates for the stable part of the nonlinearity, which in contrast to the
critical part depends on $x$, we have to split $H_n^s$ according to the different
 regularities in space. Moreover, lowering the weight in $\ell$ is not useful in this case.
On the one hand, this is because we do not gain an additional factor $\ell$ by applying
the mode filter $E_s$. On the other hand, the linear semigroup in \eqref{62_8} could
not balance the missing weight without losing powers of $L^{-n}$.

\begin{lemma}\label{62_l4}
  Let $r\geq3$. For $p \in (0,1/2)$ there exists a positive constant $C$ such that
$H_n^s(\alpha_n, L^{-(1-p)n}w_n)$ can be split according to the order of $x$-derivatives in the form
  \[
    L^{(3-p)n} H_n^s(\alpha_n, L^{-(1-p)n}w_n)
= \sum_{j=0}^3 \begin{pmatrix}0\\h_{n,j}^s(\alpha_n, L^{-(1-p)n}w_n)\end{pmatrix},
\]
  where
  \begin{equation} \begin{aligned}
    & \|h_{n,j}^s(\alpha_n, L^{-(1-p)n}w_n)\|_{B_{L^n}(2,r-j,2)}  \\
    & \leq C\Big(L^{(2-p)n} \|\alpha_n\|_{B_{L^n}(2,r,2)}^2
  + L^n\|\alpha_n\|_{B_{L^n}(2,r,2)} \|w_n\|_{\mathcal{B}_{L^n}(2,r,2)}\\
&\quad + L^{pn}\|w_n\|_{\mathcal{B}_{L^n}(2,r,2)}^2\Big)
\end{aligned} \label{62_9}
\end{equation}
  for all $(\alpha_n,w_n)\in X_{L^n}(2,r,2)$ with $(\alpha_n,w_n)_{X_{L^n}(2,r,2)}\leq 1$.
\end{lemma}

\begin{proof}
  The proof works along similar lines as for the critical part. Again, it is sufficient
to estimate the quadratic terms. The appropriate estimates for higher order terms follow
a fortiori since each convolution yields an additional factor $L^{-n}$.
Moreover, every $w_n$ gives a factor $L^{-(1-p)n}$. We only have to pay attention
to the different regularities in space. Since in \eqref{540_1} the highest
derivative $\partial_x^3 h$ occurs only linearly (i.e., the IBL  is quasilinear),
the second component of $H_n^s$ maps to $B_{L^n}(2,r-j,2), \, j\in\{0,3\}$ due
to Lemma \ref{62_l3}.  Inequality  \eqref{62_9} then follows by counting the
respective powers of $L^{-n}$.
\end{proof}


\begin{remark} \rm
  At first view, estimate \eqref{62_9} for the stable part seems worse than those
for the critical part in Lemma \ref{62_l4}, since the powers of $L$ in the
coefficients do not converge to zero for $n \to \infty$. However, applying
the linear semigroup in \eqref{62_vdk2} yields an additional factor $L^{-2n}$,
which in \S \ref{sec_63} allows to prove that the stable component decays
polynomially for $t \to \infty$ . Furthermore, the nonlinearity $h_{n,3}^s$
only lies in $B_{L^n}(2,r-3,2)$. But since in the second component the phase
space is $B_{L^n}(2,r-1,2)$, this can be smoothed out by the linear semigroup.
\end{remark}


\subsection{Splitting, iteration, and conclusion}\label{sec_63}

The result of the formal calculation in \S \ref{sec_543} was
\begin{equation}
  \partial_t \tilde{\alpha}(t,\ell) = -c_2\ell^2\tilde{\alpha}(t,\ell) + {\rm i} d\ell\chi(\ell)\tilde{\alpha}^{*2}(t,\ell)
 + \text{h.o.t.} \label{63_1}
\end{equation}
Since $\tilde{\alpha}$ is independent of $x$, \eqref{63_1} is reminiscent of the Fourier
 transform $\partial_t \hat{v} = -c_2\ell^2 \hat{v} +{\rm i} d\ell\hat{v}^{*2} + \text{h.o.t.}$
of the Burgers equation. According to \S\ref{sec_522}, the higher order perturbations
are asymptotically irrelevant, and the renormalized solution $t^{1/2}v(t,t^{1/2}x)$
converges for $t\to\infty$ towards $f_z(x)$, see \eqref{522_2}. In Fourier space,
 this corresponds to $\hat{v}(t,t^{-1/2}\ell) \to \hat{f}_z(\ell)$. If we consider the initial
condition $v(1,x) = v_0(x)$, the parameter $z$ is given by
$\ln(z+1) = \frac{d}{c_2}\int_\mathbb{R} v_0(x) \,{\rm d} x = 2\pi\frac{d}{c_2}\hat{v}_0(0)$.

Transferring this result to \eqref{63_1}, we expect the rescaled critical
component $\tilde{\alpha}(t,t^{-1/2}\ell)$ to converge towards $\hat{f}_{z_0}(\ell)$ for $t \to\infty$, where
\begin{equation}
\ln(z_0+1) = 2\pi\frac{d}{c_2}\tilde{\alpha}(1,0). \label{63_20}
\end{equation}
Thus, for fixed times $t=L^{2n}$, $n \in \mathbb{N}$, the renormalized
solution $\alpha_n(1,\ell)=\tilde{\alpha}(L^{2n},L^{-n}\ell)$ is expected to converge
towards $\hat{f}_{z_0}(\ell)$ for $n \to \infty$.

\subsection*{Splitting}
The formal considerations above give reason to split $\alpha_n$ into
\[
  \alpha_n(t,\ell) = \alpha_n^{(z)}(t,\ell) + L^{-(1-p)n}\gamma_n(t,\ell)
\]
with the Fourier transformed profile
\[
  \alpha_n^{(z)}(t,\ell) := \chi(L^{-n}\ell)\hat{v}_{z_0}(t,\ell)
 = \chi(L^{-n}\ell) \hat{f}_{z_0}(t^{1/2}\ell).
\]
Then, according to \eqref{61_1}, the correction term $\gamma_n$ satisfies
\begin{equation} \begin{aligned}
  \partial_t \gamma_n
&=  L^{2n}\mu_1(L^{-n}\cdot)\gamma_n + L^{(3-p)n}
\Big( B_n^c(\alpha_n)-B_n^c(\alpha_n^{(z)}) \\
&\quad + H_n^c(\alpha_n,L^{-n(1-p)}w_n) \Big)
   + L^{(1-p)n}{\rm Res}_n,
\end{aligned} \label{63_2}
\end{equation}
where ${\rm Res}_n := -\partial_t\alpha_n^{(z)} + L^{2n}\mu_1(L^{-n}\cdot)\alpha_n^{(z)}
+ L^{2n}B_n^c(\alpha_n^{(z)})$.

\begin{lemma} \label{63_l2}
  Let $|z_0| <1$. Then there exists a $C>0$ such that
  \[
    \sup_{t\in[L^{-2},1]} \|{\rm Res}_n\|_{B_{L^n}(2,r,2)} \leq C L^{-n} |z_0|.
 \]
\end{lemma}

\begin{proof}
  By construction we have $L^{2n}B_n^c(\alpha_n^{(z)})(\ell) = {\rm i} d\ell \chi(L^{-n}\ell)(\alpha_n^{(z)}*\alpha_n^{(z)})(\ell)$,
while the renormalization of the largest eigenvalue reads
\[
L^{2n}\mu_1(L^{-n}\ell) = -c_2\ell^2 + \mathcal{O}(L^{-n}\ell^3).
\]
As $\hat{v}_{z_0}$ is an exact solution of
$\partial_t \hat{v}_{z_0}(t,\ell) = -c_2\ell^2 \hat{v}_{z_0}(t,\ell) + {\rm i} d\ell(\hat{v}_{z_0}*\hat{v}_{z_0})(t,\ell)$,
 we obtain
\begin{align*}
    {\rm Res}_n(\ell)
& = {\rm i} d\ell \chi(L^{-n}\ell)\left((\alpha_n^{(z)}*\alpha_n^{(z)})(\ell) - (\hat{v}_{z_0}*\hat{v}_{z_0})(\ell) \right) \\
& = {\rm i} d\ell \chi(L^{-n}\ell)\int_{-L^n/2}^{L^n/2}\left( \chi\left(\frac{\ell-m}{L^n}\right)
\chi\left(\frac{m}{L^n}\right)-1\right) \hat{v}_{z_0}(\ell-m)\hat{v}_{z_0}(m) \,{\rm d} m.
\end{align*}
  This can be estimated in $B_{L^n}(2,r,2)$ by $CL^{-n}|z_0|$ since the first factor
in the integral is zero for both $\ell-m$ and $m$ small, and since $\hat{v}_{z_0}$ is a
smooth and exponentially decaying function.
\end{proof}

Next we study the evolution of $\gamma_n$ at the fixed wave number $\ell=0$.
Due to the definition of the critical mode filter $\tilde{E}_c^*$ we obtain
$\tilde{\alpha}(t,0) = \langle \tilde{w}(t,0,\cdot), \psi^1(0,\cdot) \rangle$
where $\tilde{w}={\rm e}^{-{\rm i}\ell c_1 t}(\tilde{h}, \tilde{q})^\top$, see \eqref{542_17} and \eqref{542_1}.
Since according to \eqref{542_24} we have $\psi^1(0,x)=(c_0,0)^\top$, we obtain
\begin{align*}
  \tilde{\alpha}(t,0) & = c_0\int_0^\gamma \tilde{h}(t,0,x) \,{\rm d} x
 = c_0 \int_0^\gamma \sum_{j \in \mathbb{Z}} {\rm e}^{{\rm i} j k_0 x} \mathcal{F} h(t,k_0j) \,{\rm d} x\\
 &= c_0\gamma \mathcal{F} h(t,0)
   = \frac{1}{2\pi} c_0\gamma \int_\mathbb{R} h(t,x) \,{\rm d} x.
\end{align*}
The perturbation's mass $\int_\mathbb{R} h \, {\rm d} x$ is conserved in the IBL,
cf.~\S\ref{sec_540}. Thus, we have $\tilde{\alpha}(t,0) = \tilde{\alpha}(1,0)$ for all $t\geq 1$, which yields
\begin{align*}
  L^{-(1-p)n}\gamma_n(t,0)
& = \alpha_n(t,0) - \alpha_n^{(z)}(t,0)
     = \tilde{\alpha}(L^{2n}t,0) - \hat{f}_{z_0}(0)\\
& = \tilde{\alpha}(1,0) - \frac{c_2}{2\pi d} \ln(z_0+1)
     = 0
\end{align*}
for all $t \in [L^{-2},1]$, cf.~\eqref{63_20}. The following lemma shows a
contraction property of the rescaled linear semigroup when acting
on the remainder $\gamma_n$ with $\gamma_n(0)=0$, and explains
why we require some regularity in $\ell$ in the spaces $B(m,r,b)$.

\begin{lemma}\label{63_l1}
  Let $g\in H^2(2)$ with $g(0) = 0$. Then
\[
\| {\rm e}^{(1-L^{-2})L^{2n}\mu_1(L^{-n}\cdot)}\mathcal{R}_{1/L}g \|_{H^2(2)} \leq CL^{-1}\|g\|_{H^2(2)}.
\]
\end{lemma}

\begin{proof}
  We state here only the estimates for the $L^2(2)$-norm, as the additional
factor $L^{-1}$ coming from $\frac{{\rm d}}{{\rm d}\ell}(\mathcal{R}_{1/L}g)(\ell) = L^{-1}g'(L^{-1}\ell)$
leads to easier estimates in case of derivatives. Since $g(0) = 0$, we have
for a $\tilde\ell\in [0, L^{-1}\ell]$
  \[
    |g(L^{-1}\ell)|  = L^{-1}\ell g'(\tilde\ell)
       \leq L^{-1}\ell \|g\|_{C^1}  \leq C L^{-1}\ell \|g\|_{H^2(2)}
 \]
  due to standard Sobolev embedding. Thus, we obtain
\begin{align*}
&\| {\rm e}^{(1-L^{-2})L^{2n}\mu_1(L^{-2n}\cdot)}\mathcal{R}_{1/L}g \|_{L^2(2)}^2\\
&= \int_\mathbb{R} {\rm e}^{2(1-L^{-2})L^{2n}\mu_1(L^{-n}\ell)} (g(L^{-1}\ell))^2(1+\ell)^2 \,{\rm d} \ell \\
& \leq CL^{-2}\|g\|_{H^2(2)}^2 \int_\mathbb{R} {\rm e}^{2(1-L^{-2})L^{2n}
 \mu_1(L^{-n}\ell)}\ell^2 (1+\ell)^2 \,{\rm d} \ell,
\end{align*}
  where the integral can be estimated independently of $n$ since
$L^{2n}\mu_1(L^{-n}\ell) = -c_2\ell^2 + \mathcal{O}(L^{-n}\ell^3)$.
\end{proof}

Let
\begin{gather*}
  g_{n,c}(\ell)  := \gamma_n(1,\ell), \quad \rho_{n,c} := \| g_{n,c} \|_{B_{L^n}(2,r,2)}, \\
  g_{n,s}(\ell,x)  := w_n(1,\ell,x),  \quad \rho_{n,s} := \| g_{n,s} \|_{\mathcal{B}_{L^n}(2,r,2)},\\
  \rho_n := \|\alpha_n(1)\|_{B_{L^n}(2,r,2)} + \|w_n(1)\|_{\mathcal{B}_{L^n}(2,r,2)}.
\end{gather*}
Hence, $\rho_n \leq L^{-(1-p)n}\rho_{n,c} + \|\alpha_n^{(z)}(1)\|_{H^2(2)} + \rho_{n,s}$,
and to prove Theorem \ref{main_th3} we will show
that both $\rho_{n,c}$ and $\rho_{n,s}$ are bounded for
$n\to \infty$.



\subsection*{Proof of Theorem \ref{main_th3}}
  Taking $L_0$ and $C_1$ from Theorem \ref{61_th1}, we assume the initial condition
 $(\alpha_0, w_0)$ to be small enough to fulfill
  \begin{equation}
    \rho_0 \leq L^{-m_0-1}, \quad L^{-m_0} \leq C_1 L^{-5/2},
  \end{equation}
  where $L>L_0$ and $m_0 \in \mathbb{N}$ are specified later. In particular, this yields
  \begin{equation}
    |z_0| \leq C \|\alpha_0^{(z)}(1,\cdot)\|_{H^2(2)} \leq C L^{-m_0-1}. \label{63_17}
\end{equation}
By \eqref{63_2} we have
\begin{equation}
\begin{aligned}
    \gamma_n(t,\ell)
 &=  {\rm e}^{(t-L^{-2})L^{2n}\mu_1(L^{-n}\ell)}\gamma_{n}(L^{-2},\ell) \\
      &\quad + L^{(3-p)n}\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\mu_1(L^{-n}\ell)}
\left( B_n^c(\alpha_n(s))(\ell)-B_n^c(\alpha_n^{(z)}(s))(\ell) \right) {\rm d} s \\
      &\quad + L^{(3-p)n}\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\mu_1(L^{-n}\ell)}
 H_n^c(\alpha_n(s),L^{-(1-p)n}w_n(s))(\ell) \,{\rm d} s \\
      &\quad + L^{(1-p)n}\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\mu_1(L^{-n}\ell)}
 {\rm Res}_n(s,\ell) \,{\rm d} s,
\end{aligned}\label{63_3}
\end{equation}
while $w_n$ is obtained from \eqref{62_vdk2}. In order to apply an iteration scheme,
we now assume
  \begin{equation}
    \rho_{n-1} \leq L^{-m_0}, \quad \rho_{n-1,c} \leq L^{-m_0}, \label{63_5}
  \end{equation}
  which is obviously true for $n=1$. Since $\rho_{n-1}\leq C_1L^{-5/2}$,
Theorem \ref{61_th1} implies
 \begin{gather*}
 \|(\alpha_n,w_n)\|_{\mathcal{X}_{L^n}^{r+1}((1/L^2,1),2,2)} \leq C L^{5/2} \rho_{n-1},\\
\|(\alpha_n,w_n)\|_{\mathcal{X}_{L^n}^{r+2}((1/2,1),2,2)} \leq C L^{5/2} \rho_{n-1}
 \end{gather*}
  for a $C>0$. Due to Corollary \ref{71_c1}, these estimates yield
\begin{gather}
    \sup_{t\in[L^{-2},1]}\|\alpha_n\|_{B_{L^n}(2,r,2)}  \leq C L^{5/2} \rho_{n-1}, \label{63_9} \\
    \sup_{t\in[L^{-2},1]}\|w_n\|_{\mathcal{B}_{L^n}(2,r,2)}  \leq C L^{5/2} \rho_{n-1}, \\
 \sup_{t\in[1/2,1]}\|w_n\|_{\mathcal{B}_{L^n}(2,r+1,2)} \leq C L^{5/2} \rho_{n-1}. \label{63_10}
\end{gather}

  First, we show an a-priori estimate for
$\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}$ by estimating \eqref{63_3}.
We start with the first term on the right-hand side of \eqref{63_3}.
The initial condition \eqref{61_6} yields
\begin{align*}
    \gamma_n(L^{-2},\ell) & = L^{(1-p)n}
\left(\alpha_{n-1}(1,L^{-1}\ell) - \chi(L^{-n}\ell)\hat{f}_{z_0}(L^{-1}\ell)\right)  \\
      & = L^{1-p}\gamma_{n-1}(1,L^{-1}\ell) + L^{(1-p)n}
\left( \chi(L^{-(n-1)}\ell) - \chi(L^{-n}\ell) \right) \hat{f}_{z_0}(L^{-1}\ell).
\end{align*}
  Since $\chi(L^{-(n-1)}\ell) - \chi(L^{-n}\ell) = 0$ for $|\ell| \leq L^{n-1}r_\chi$
and since $\hat{f}_{z_0}$ decays exponentially, we obtain
  \begin{equation} \begin{aligned}
    \|\gamma_n(L^{-2},\cdot)\|_{B_{L^n}(2,r,2)}
& \leq CL^{1-p}L^{5/2}\|\gamma_{n-1}(1,\cdot)\|_{B_{L^{n-1}}(2,r,2)} + CL^{-1}|z_0|  \\
& \leq CL^{7/2-p}\rho_{n-1,c} +  CL^{-1}|z_0|,
\end{aligned}\label{63_7}
\end{equation}
  where the factor $L^{5/2}$ comes from the different scalings of $\gamma_{n-1}$
and $\gamma_n$, see \eqref{61_7}. Next we estimate the first integral in \eqref{63_3}.
Due to the definitions of $\tilde{B}_c$ in \eqref{542_22} and of
the convolution $*_{L^n}$ in \eqref{62_2} we have
$ L^{2n}B_n^c(\alpha_n)(\ell) = {\rm i} d \ell\chi(\ell)(\alpha_n *_{L^n} \alpha_n)(\ell)$,
  and thus
\begin{align*}
 &L^{(3-p)n}\left(B_n^c(\alpha_n) - B_n^c(\alpha_n^{(z)})\right)(\ell)\\
 &= L^{(1-p)n} {\rm i} d \ell \chi(\ell)
\left(L^{-2(1-p)n}\gamma_n *_{L^n}\gamma_n
+ L^{-(1-p)n}\gamma_n*_{L^n}\alpha_n^{(z)}\right)(\ell).
\end{align*}
  Therefore,
\begin{align*}
    & L^{(3-p)n}\sup_{t\in [L^{-2},1]}\|B_n^c(\alpha_n) - B_n^c(\alpha_n^{(z)})\|_{B_{L^n}(2,r,1)} \\
    & \leq CL^{-(1-p)n}\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}^2
+ C|z_0|\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}.
\end{align*}
  The missing weight in $\ell$ can be balanced by the linear semigroup in Lemma \ref{62_l1},
which gives
  \begin{equation} \label{63_8}
\begin{aligned}
    & L^{(3-p)n}\sup_{t\in [L^{-2},1]}
\Big\|\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\mu_1(L^{-n}\cdot)}
\left( B_n^c(\alpha_n(s))-B_n^c(\alpha_n^{(z)}(s)) \right) {\rm d} s \Big\|_{B_{L^n}(2,r,2)} \\
    & \leq L^{(3-p)n}\sup_{t\in [L^{-2},1]}\|B_n^c(\alpha_n)
- B_n^c(\alpha_n^{(z)})\|_{B_{L^n}(2,r,1)} \sup_{t\in [L^{-2},1]} \int_{L^{-2}}^t (t-s)^{-1/2} \,{\rm d} s \\
    & \leq CL^{-(1-p)n}\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}^2
+ C|z_0|\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}.
\end{aligned}
\end{equation}
 Similarly, we can estimate the second integral in \eqref{63_3} by applying
 Lemma \ref{62_l6} and the estimates for $\alpha_n, w_n$ in \eqref{63_9}, \eqref{63_10}, which gives
  \[
    L^{(3-p)n}\sup_{t\in [L^{-2},1]}\|H_n^c(\alpha_n(t),L^{-(1-p)n}w_n(t))\|_{B_{L^n}(2,r,p)}
\leq C (L^{5/2}\rho_{n-1})^2.
\]
  Using the properties of the linear semigroup in Lemma \ref{62_l1} yields
  \begin{equation} \begin{aligned}
    & L^{(3-p)n}\sup_{t\in [L^{-2},1]}
\Big\|\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\mu_1(L^{-n}\cdot)}
H_n^c(\alpha_n(s),L^{-(1-p)n}w_n(s)) \,{\rm d} s \Big\|_{B_{L^n}(2,r,2)}\\
&\leq CL^5\rho_{n-1}^2.
\end{aligned} \label{63_11}
\end{equation}
  Finally, by Lemma \ref{63_l2}, we obtain
  \begin{equation}\label{63_12}
    L^{(1-p)n}\sup_{t\in [L^{-2},1]}
\Big\|\int_{L^{-2}}^t {\rm e}^{(t-s)L^{2n}\mu_1(L^{-n}\ell)} {\rm Res}_n(s,\ell) \,{\rm d} s \Big\|_{B_{L^n}(2,r,2)}
 \leq C L^{-pn}|z_0|.
\end{equation}
  Combining \eqref{63_3} and \eqref{63_7}-\eqref{63_12}, we achieve
  \begin{equation} \begin{aligned}
&\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}\\
& \leq CL^{7/2-p}\rho_{n-1,c} +  CL^{-1}|z_0|
      + CL^{-(1-p)n}\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}^2  \\
&\quad + C|z_0|\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}
       + CL^5\rho_{n-1}^2 + C L^{-pn}|z_0|.
\end{aligned} \label{63_4}
\end{equation}
  By choosing $m_0$ large enough we obtain $C|z_0| \leq CL^{-m_0-1}\leq 1/3$, and thus
  \begin{align*}
&L^{-(1-p)n}\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)} \\
& \leq \sup_{t\in [L^{-2},1]}\|\alpha_n(t,\cdot) - \alpha_n^{(z)}(t,\cdot) \|_{B_{L^n}(2,r,2)} \\
&\leq C(L^{5/2}\rho_{n-1} + |z_0|)
\leq CL^{5/2}L^{-m_0} + CL^{-m_0-1}  \leq 1/(3C).
\end{align*}
  Finally, \eqref{63_4} yields the a-priori estimate
  \begin{equation} \begin{aligned}
& 1/3\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}\\
&\leq CL^{7/2-p}\rho_{n-1,c} +  CL^{-1}|z_0| + CL^5\rho_{n-1}^2 + C L^{-pn}|z_0|.
\end{aligned} \label{63_6}
\end{equation}

\subsection*{Iteration}
  To conclude $\rho_{n}\leq L^{-m_0}$ and $\rho_{n,c}\leq L^{-m_0}$
from assumption \eqref{63_5}, the first term on the right-hand side of \eqref{63_6}
is not yet small enough. Thus, we have to use
\eqref{63_3} once more for the fixed time $t=1$. In this case, as $\gamma_{n-1}(t,0)=0$,
 we can apply Lemma \ref{63_l1}. In contrast to \eqref{63_7}, we achieve
\begin{align*}
&\|{\rm e}^{(1-L^{-2})L^{2n}\mu_1(L^{-n}\cdot)}\gamma_{n}(L^{-2},\cdot)\|_{B_{L^n}(2,r,2)} \\
& \leq CL^{-p} \|\gamma_{n-1}(1,\cdot)\|_{B_{L^{n-1}}(2,r,2)} + CL^{(1-p)n-1}\|\\
&\quad\times \left( \chi(L^{-(n-2)}\cdot) - \chi(L^{-(n-1)}\cdot) \right) \hat{f}_{z_0} \|_{H^2(2)}  \\
& \leq C L^{-p} \rho_{n-1,c} + CL^{-1} \|\hat{f}_{z_0}\|_{H^2(2)}
   \leq C L^{-p} \rho_{n-1,c} + CL^{-1} |z_0|.
\end{align*}
  Similar to \eqref{63_4}, we obtain
  \begin{equation} \begin{aligned}
    \rho_{n,c}
&\leq  CL^{-p}\rho_{n-1,c} + CL^{-1}|z_0| + CL^{-(1-p)n}\sup_{t\in [L^{-2},1]}
 \|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}^2  \\
&\quad + C|z_0|\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}
     + CL^5\rho_{n-1}^2 + C L^{-pn}|z_0|.
\end{aligned} \label{63_13}
\end{equation}
  From \eqref{63_6} we obtain
\begin{align*}
&\sup_{t\in [L^{-2},1]}\|\gamma_n(t,\cdot)\|_{B_{L^n}(2,r,2)}\\
&\leq C\left( L^{7/2-p}L^{-m_0} + L^{-1}L^{-m_0-1} + L^{5-2m_0} + L^{-pn}L^{-m_0-1} \right)
 \leq C L^{7/2-p-m_0},
\end{align*}
  and plugging this estimate into \eqref{63_13} yields
 \begin{align*}
    \rho_{n,c} &\leq  C\Big(L^{-p}L^{-m_0} + L^{-1}L^{-m_0-1} + L^{-(1-p)n}L^{7-2p-2m_0}
 + L^{-m_0-1}L^{7/2-p-m_0} \\
      & \quad+ L^5L^{-2m_0} + L^{-pn}L^{-m_0-1}\Big).
\end{align*}
  Choosing now $L>L_0$ such that $C \leq L^p/18$, we obtain
  \[
    \rho_{n,c} \leq \frac{1}{18} \left( L^{-m_0} + L^{-m_0-1} + L^{7-2m_0} + L^{3/2-2m_0}
+ L^{5-2m_0} + L^{-m_0-1} \right).
\]
  Choosing finally $m_0 > 7$ such that $L^{7-m_0} \leq 1$ leads to
  \begin{equation}
    \rho_{n,c} \leq \frac{1}{3} L^{-m_0}. \label{63_16}
 \end{equation}

  Next, we estimate the stable part $\rho_{n,s}$. From
\eqref{62_vdk2} we obtain
  \begin{equation} \begin{aligned}
    g_{n,s}(\ell,x)
&=  L^{1-p}{\rm e}^{(1-L^{-2})L^{2n}\tilde{A}_s(L^{-n}\ell)}g_{n-1,s}(L^{-1}\ell,x)  \\
& + L^{(3-p)n}\int_{L^{-2}}^1 {\rm e}^{(1-s)L^{2n}\tilde{A}_s(L^{-n}\ell)}
H_n^s(\alpha_n(s),L^{-(1-p)n}w_n(s))(\ell,x) \,{\rm d} s.
\end{aligned} \label{63_18}
\end{equation}
  Applying Lemma \ref{62_l1} we can estimate the first term on the right-hand side
of \eqref{63_18} as
  \begin{align*}
&L^{1-p}\|{\rm e}^{(1-L^{-2})L^{2n}\tilde{A}_s(L^{-n}\cdot)}
 \mathcal{R}_{L^{-1}}g_{n-1,s}\|_{\mathcal{B}_{L^n}(2,r,2)} \\
& \leq L^{1-p}{\rm e}^{-\sigma_1(1-L^{-2})L^{2n}}\|g_{n-1,s}\|_{\mathcal{B}_{L^{n-1}}(2,r,2)} \\
& \leq CL^{-1}\rho_{n-1,s}.
\end{align*}
To estimate the integral in \eqref{63_18} we use Lemma \ref{62_l4}, which gives
 \begin{align*}
    & L^{(3-p)n}\Big\| \int_{L^{-2}}^1 {\rm e}^{(1-s)L^{2n}\tilde{A}_s(L^{-n}\cdot)}
 H_n^s(\alpha_n(s),L^{-(1-p)n}w_n(s)) \,{\rm d} s \Big\|_{\mathcal{B}_{L^n}(2,r,2)} \\
    & \leq \sum_{j=0}^3 \Big\| \int_{L^{-2}}^1 {\rm e}^{(1-s)L^{2n}\tilde{A}_s(L^{-n}\cdot)}
\begin{pmatrix}0\\h_{n,j}^s(\alpha_n, L^{-(1-p)n}w_n)\end{pmatrix}
 {\rm d} s \Big\|_{\mathcal{B}_{L^n}(2,r,2)} \\
    & =: \sum_{j=0}^3 \|I_j\|_{\mathcal{B}_{L^{n}}(2,r,2)}.
\end{align*}
  For the integrals $I_0$ and $I_1$, the smoothing properties of the linear semigroup
are not required since $\mathcal{B}_{L}(m,r,b) := B_{L}(m,r,b)\times B_{L}(m,r-1,b)$; i.e.,
the regularity needed for the second component is only $r-1$. Applying Lemma \ref{62_l1}
and Lemma \ref{62_l5} we achieve
\begin{align*}
    \|I_{j}\|_{\mathcal{B}_{L^n}(2,r,2)}
& \leq C \int_{L^{-2}}^1 {\rm e}^{-\sigma_1(1-s)L^{2n}}\|h_{n,j}^s\|_{B_{L^n}(2,r-j,2)} \,{\rm d} s \\
& \leq C L^{(2-p)n}(L^{5/2}\rho_{n-1})^2 \int_{L^{-2}}^1 {\rm e}^{-\sigma_1(1-s)L^{2n}} \,{\rm d} s
 \leq C L^{-pn} L^5\rho_{n-1}^2
\end{align*}
  for $j \in \{0,1\}$. For the estimate of $I_2$, the linear semigroup has to smooth
 out one $x$-derivative, which yields
  \begin{align*}
\|I_{2}\|_{\mathcal{B}_{L^n}(2,r,2)}
   &\leq C\int_{L^{-2}}^1 {\rm e}^{-\sigma_1(1-s)L^{2n}}(1+L^{-n}(1{-}s)^{-1/2})\\
&\quad\times \|h_{n,2}^s(\alpha_n(s),L^{-(1-p)n}w_n(s)) \|_{B_{L^n}(2,r-2,2)} \,{\rm d} s \\
    & \leq CL^{(2-p)n}(L^{5/2}\rho_{n-1})^2\int_{L^{-2}}^1
{\rm e}^{-\sigma_1(1-s)L^{2n}}(1+L^{-n}(1{-}s)^{-1/2})\,{\rm d} s  \\
&\leq CL^{-pn}L^5\rho_{n-1}^2.
\end{align*}
Treating $I_3$ the same way would lead to factor $(1-s)^{-1}$ and therefore
to a non-integrable singularity. Thus, we split $I_3$ into
  \[
    I_3 = \int_{L^{-2}}^{1/2} \ldots {\rm d} s + \int_{1/2}^{1} \ldots {\rm d} s.
  \]
  While on the first time interval $[{L^{-2}},{1/2}]$ the singularity does not occur,
we can use the higher regularity of $w_n$ on the second time interval as
stated in \eqref{63_10}. Note again that $\alpha_n$ is independent of $x$,
thus the value of $r$ plays no role in the spaces for $\alpha_n$. We obtain
\begin{align*}
\|I_{3}\|_{\mathcal{B}_{L^n}(2,r,2)}
&\leq C\int_{L^{-2}}^{1/2} {\rm e}^{-\sigma_1(1{-}s)L^{2n}}(1{+}L^{-n}(1{-}s)^{-1}) \\
&\quad\times \|h_{n,3}^s(\alpha_n(s),L^{-(1-p)n}w_n(s)) \|_{B_{L^n}(2,r-3,2)} \,{\rm d} s  \\
&\quad + C\int_{1/2}^{1} {\rm e}^{-\sigma_1(1{-}s)L^{2n}}(1{+}L^{-n}(1{-}s)^{-1/2}) \\
&\quad\times \|h_{n,3}^s(\alpha_n(s),L^{-(1-p)n}w_n(s)) \|_{B_{L^n}(2,r-2,2)} \,{\rm d} s  \\
&\leq CL^{-pn}L^5 \rho_{n-1}^2.
\end{align*}
Collecting the estimates for $I_j$ gives
\begin{equation} \begin{aligned}
\rho_{n,s}
& \leq CL^{-1}\rho_{n-1} + CL^{-pn}L^5 \rho_{n-1}^2\\
&\leq CL^{-1}L^{-m_0} + CL^{-p}L^5L^{-2m_0} \leq \frac{1}{3} L^{-m_0}.
\end{aligned} \label{63_15}
\end{equation}
  Combining \eqref{63_17}, \eqref{63_16}, and \eqref{63_15}, we finally obtain
  \[
    \rho_n  = \|\alpha_n(1)\|_{B_{L^n}(2,r,2)} + \|w_n(1)\|_{\mathcal{B}_{L^n}(2,r,2)}
\leq L^{-(1-p)n}\rho_{n,c} + C|z_0| + \rho_{n,s} \leq L^{-m_0}.
\]

\subsection*{Conclusion}
  So far we have shown that if $\rho_0 \leq L^{-m_0-1}$,
then $\rho_{n,c}$, $\rho_{n,s}$, and $\rho_n$ stay smaller than $L^{-m_0}$
for all $n \in \mathbb{N}$. In order to prove \eqref{55_1}, we estimate
  \[
    (\ell,x) \mapsto \tilde{w}(t,t^{-1/2}\ell,x) - \chi(t^{-1/2}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x)
  \]
  at the discrete times $t = L^{2n}$, $n \in \mathbb{N}$; i.e.,
  \begin{equation}
 \begin{aligned}
    & \tilde{w}(L^{2n},L^{-n}\ell,x) - \chi(L^{-n}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x) \\
    & = \tilde{\alpha}(L^{2n},L^{-n}\ell)\phi^1(L^{-n}\ell,x) + \tilde{w}_s(L^{2n},L^{-n}\ell,x)
- \chi(L^{-n}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x) \\
    & = \alpha_n(1,\ell)\phi^1(L^{-n}\ell,x) + L^{-(1-p)n}w_n(1,\ell,x)
- \chi(L^{-n}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x) \\
    & = \left( \alpha_n(1,\ell) - \chi(L^{-n}\ell)\hat{f}_{z_0}(\ell) \right) \phi^1(0,x)
+ \alpha_n(1,\ell)\left( \phi^1(L^{-n}\ell,x) - \phi^1(0,x) \right)\\
&\quad + L^{-(1-p)n}w_n(1,\ell,x) \\
    & = L^{-(1-p)n}\gamma_n(1,\ell) \phi^1(0,x) + \alpha_n(1,\ell)\left( \phi^1(L^{-n}\ell,x)
- \phi^1(0,x) \right) \\
&\quad + L^{-(1-p)n}w_n(1,\ell,x).
\end{aligned} \label{63_21}
\end{equation}
  Taking the $\mathcal{B}_{\sqrt{t}}(2,r,2)$-norm at $t=L^{2n}$, the first term
on the right-hand side of \eqref{63_21} yields
  \begin{align*}
  &   \|(\ell,x)\mapsto \gamma_n(1,\ell)\phi^1(0,x)\|_{\mathcal{B}_{L^n}(2,r,2)}\\
   & = \Big( \sum_{j=0}^2 \int_{-1/2k_0L^n}^{1/2k_0L^n}(1+\ell^2)^2
 |\partial_\ell^j\gamma_n(1,\ell)|^2\| \phi^1(0,\cdot) \|_{H^r(I_\gamma)}^2 \,{\rm d}\ell \Big)^{1/2} \\
 & \leq C \| \gamma_n(1,\cdot) \|_{B_{L^n}(2,r,2)} \leq CL^{-m_0}.
\end{align*}
  The second term in \eqref{63_21} is estimated as follows. We have
  \begin{equation} \begin{aligned}
    & \|(\ell,x)\mapsto \alpha_n(1,\ell)\left( \phi^1(L^{-n}\ell,x) - \phi^1(0,x) \right)
 \|_{\mathcal{B}_{L^n}(2,r,1)} \\
 & = \Big( \sum_{j=0}^2 \int_{-1/2k_0L^n}^{1/2k_0L^n} (1+\ell^2)
\big\| \frac{{\rm d}^j}{{\rm d}\ell^j} [ \alpha_n(1,\ell)\left( \phi^1(L^{-n}\ell,\cdot)
- \phi^1(0,\cdot) \right) ]\big\|_{H^r(I_\gamma)}^2 \,{\rm d}\ell \Big)^{1/2}.
\end{aligned}\label{63_22}
\end{equation}
  We now have to distinguish between terms in which the eigenvector $\phi^1$
is differentiated with respect to $\ell$ and those in which only $\alpha_n$
is differentiated. For the first group, we obtain estimates of the type
\begin{align*}
    & \int_{-1/2k_0L^n}^{1/2k_0L^n} (1+\ell^2) |\partial_\ell^{j_1}\alpha_n(1,\ell)|^2
 \Big\| \frac{{\rm d}^{j_2}}{{\rm d}\ell^{j_2}}\phi^1(L^{-n}\ell,\cdot)
 \Big\|_{H^r(I_\gamma)}^2 \,{\rm d}\ell \\
    & \leq L^{-2n} \int_{-1/2k_0L^n}^{1/2k_0L^n} (1+\ell^2) |\partial_\ell^{j_1}\alpha_n(1,\ell)|^2
\Big\| \frac{{\rm d}^{j_2{-}1}}{{\rm d}\ell^{j_2{-}1}}(\partial_\ell\phi^1(L^{-n}\ell,\cdot))
  \Big\|_{H^r(I_\gamma)}^2 \,{\rm d}\ell \\
    & \leq CL^{-2n} \|\alpha_n(1,\cdot)\|_{B_{L^n}(2,r,2)}^2,
\end{align*}
  where $j_2$ is at least one. For the terms without a derivative of $\phi^1$
we can use Taylor expansion, which leads to
  \[
    \phi^1(L^{-n}\ell,x) - \phi^1(0,x) = L^{-n}\ell \partial_\ell\phi^1(\tilde{\ell}(\ell),x),
  \]
  where $|\tilde{\ell}(\ell)|\leq L^{-n}|\ell|$. Thus, in \eqref{63_22} there also occur terms
of the type
  \begin{align*}
    & \int_{-1/2k_0L^n}^{1/2k_0L^n} (1+\ell^2) \| \partial_\ell^{j}\alpha_n(1,\ell)
 L^{-n}\ell \partial_\ell\phi^1(\tilde{\ell}(\ell),\cdot) \|_{H^r(I_\gamma)}^2 \,{\rm d}\ell \\
    & \leq L^{-2n} \int_{-1/2k_0L^n}^{1/2k_0L^n} (1+\ell^2)^2
|\partial_\ell^{j}\alpha_n(1,\ell)|^2 \|\partial_\ell\phi^1(\tilde{\ell}(\ell),\cdot)
\|_{H^r(I_\gamma)}^2 \,{\rm d}\ell\\
&\leq CL^{-2n} \|\alpha_n(1,\cdot)\|_{B_{L^n}(2,r,2)}^2.
\end{align*}
  Note that the additional factor $\ell$ in this estimate is the reason why we
have to lower the weight in $\ell$ from $\mathcal{B}_{\sqrt{t}}(2,r,2)$ to $\mathcal{B}_{\sqrt{t}}(2,r,1)$
in Theorem \ref{main_th3}. Altogether, we obtain
\begin{align*}
  \|(\ell,x)\mapsto \alpha_n(1,\ell)\left( \phi^1(L^{-n}\ell,x)
  - \phi^1(0,x) \right)\|_{\mathcal{B}_{L^n}(2,r,1)}
 & \leq CL^{-n} \|\alpha_n(1,\cdot)\|_{B_{L^n}(2,r,2)}\\
 & \leq CL^{-m_0}L^{-n}.
\end{align*}
  The $\mathcal{B}_{L^n}(2,r,2)$-norm of third term on the right-hand side of \eqref{63_21}
can be easily estimated by $L^{-(1-p)n}\rho_{n,s}$. Combining all these estimates,
we obtain
\[
    \|(\ell,x) \mapsto \tilde{w}(L^{2n},L^{-n}\ell,x)
- \chi(L^{-n}\ell)\hat{f}_{z_0}(\ell)\phi^1(0,x) \|_{\mathcal{B}_{L^n}(2,r,1)} \leq C L^{-(1-p)n}.
\]
  This is \eqref{55_1} for $t=L^{2n}$, and the local existence
Theorem \ref{61_th1} yields the result for all $t \in [L^{2n},L^{2(n+1)}]$.


\subsection*{Acknowledgements}
This work was supported by the Deutsche Forschungsgemeinschaft
DFG under grant Schn 520/6.


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\end{document}