\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 78, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/78\hfil 
Generic solvability for the 3-D Navier-Stokes equations]
{Generic solvability for the 3-D Navier-Stokes equations with
nonregular force}

\author[Jihoon Lee\hfil EJDE-2004/78\hfilneg]
{Jihoon Lee} 

\address{Max-Planck institute for mathematics in the sciences\\ 
Insel str. 22-26 04103 Leipzig, Germany}
\email{zhlee@math.snu.ac.kr}

\date{}
\thanks{Submitted February 24, 2004. Published June 4, 2004.}
\thanks{Supported by the post-doctoral Fellowship Program of KOSEF}
\subjclass[2000]{35A05, 76N10}
\keywords{Stochastic Navier-Stokes equations, generic solvability}

\begin{abstract}
 We show that the existence of global strong solutions for the
 Navier-Stokes equations with nonregular force is generically true.
 Similar results for equations without the nonregular force have
 been obtained by  Fursikov \cite{fursikov}.  Our main tools are 
 the Galerkin method and estimates on its solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

We are interested in the generic solvability for the
3-dimensional Navier-Stokes equations with nonregular force
on the periodic domain ${\mathbb{T}}^3\times [0, \infty)$.
\begin{gather}
\frac{\partial u}{\partial t} + (u\cdot \nabla )u +\nabla p
=\nu \Delta u+f+\frac{\partial g}{\partial t}, \label{Navier-Stokes}  \\
    \mathop{\rm div}  u =0 \label{eq-st},\\
      u(x,0)=u_0(x),\label{eq-in}
\end{gather}
where $u$ is the fluid velocity vector field, $p$ is the scalar pressure,
$\nu$ is the positive viscosity constant and $f+\frac{\partial g}{\partial t}$
s the external force. $u_0$ is a given initial data.
The nonregular part is denoted by $\frac{\partial g}{\partial t}$.
We assume $g\in C([0, \infty); V^2)$ which means $g(t)$ is a
continuous function in $V^2$, where the space $V^2$ is defined
below. Since we only consider the periodic domain
${\mathbb{T}}^3=[0, 2\pi]^3$, every function can be regarded as a
periodic vector field with period $2\pi$, i.e., $u(x_1+2\pi, x_2,
x_3)=u(x_1, x_2, x_3)$, etc. For this above Navier-Stokes
equations with nonregular force, the existence of the weak
solution was shown in \cite{Flandoli-Schmalfu}.

Recently, Flandoli
and Romito\cite{Flandoli-Romito} proved the paths of a martingale
suitable weak solution for the Navier-Stokes equations with
nonregular force have a set of singular points of one-dimensional
Hausdorff measure zero. Also the stochastic Navier-Stokes
equations have been intensively studied by many authors (see
\cite{Da Prato-Zabczyk}, \cite{E-Mattingly-Sinai}, \cite{Romito}
and references therein). One of the most important problems in
nonlinear partial differential equations is to show
existence of global strong solution for three-dimensional Navier-Stokes
equations or to construct an example of the finite blow-up of the
solution for the three-dimensional Navier-Stokes equations.
Although, it is still far from being proved the global existence, it
is known to be generically true for
(\ref{Navier-Stokes})--(\ref{eq-in}) without the nonregular part
$\frac{\partial g}{\partial t}$ (see \cite{fursikov} and
\cite{Te}).  In this paper, we show that  generic solvability is
still true even with the nonregular force.

We assume $f$ and $g$ are divergence free vector fields for simplicity.
In the following, we consider the Banach spaces $L^p(0, T; B)$ for any
Banch spaces $B$, i.e. we say $f\in L^p(0, T;B)$ if and only if
$|f|_{L^p(0, T; B)}<\infty$ (we use the same notation for the
Banach space of 3-dimensional vector fields with the Banach space
for scalar valued function for simplicity). We denote
$\cup_{0<T<\infty} L^p(0, T; B)$ is denoted by $L^p_{\rm loc}(0,
\infty; B)$. Let $H^m$ be the usual Sobolev space(see \cite{Te}).
Following the notation in \cite{Te}, we denote the space of  $L^2$
divergence-free vector fields by $H$, the space of $H^m$
divergence free vector fields by $V^m$ ($V^1$ will be denoted by
$V$ for simplicity and convention). We define the projection
operator $P_{\rm div}$ as the projection to the divergence free
vector fields.

 Note that $\{ \vec{e}_i e^{i k\cdot x}\, |\, i=1, 2,3, k\in {\mathbb{Z}}^3\}$,
 where $\vec{e}_i$ is an $i$-th standard unit vector, is a complete
orthonormal basis for $L^2$. Hence projection operator, $P_{\rm div}$, is defined as
\[
P_{\rm div} (\vec{a} e^{ik\cdot x})
= \Big(\vec{a} -\frac{k\otimes k}{|k|} \cdot \vec{a}\Big) e^{ik\cdot x}.
\]
Let $\alpha_i(k)= | P_{\rm div}(\vec{e}_i e^{ik\cdot x}) |_{L^2}$.
Hence
$K=\big\{\frac{1}{\alpha_i(k)} P_{\rm div} (\vec{e}_i e^{ik\cdot x}):
k\in {\mathbb{Z}}^3$, $i=1,2,3\big\}$
is a complete orthonormal basis for $H$.
 Define $B(u, v)=-P_{\rm div}(u\cdot \nabla )v$, $\Lambda^2 u=-P_{\rm div}\Delta u$,
 where $P_{\rm div}$ is the $L^2$ projection operator as above.
 By projecting (\ref{Navier-Stokes})--(\ref{eq-st}) to the divergence-free vector
 field, we obtain
\begin{equation}\label{main-equation}
\frac{du}{dt} +\nu \Lambda^2 u(t)-B(u, u)=f(t)+\frac{dg(t)}{dt}.
\end{equation}

 For the three-dimensional Navier-Stokes equations with regular force,
 Fursikov\cite{fursikov} and Temam \cite{Te} proved that for any initial data
 $u_0 \in V$,  there exists a set $F$ which is included in $L^2(0,T;H)$
  and dense in $L^q(0, T; V')$ with $1\leq q <\frac43$
  ($V'$ is the dual space of $V$),
 such that for every external force $f\in F$, the equations have a unique
 strong solution $u$.
Using methods similar to those developed in \cite{Flandoli-Schmalfu},
\cite{fursikov} and \cite{Te}, we obtain the following generic existence
and uniqueness of the Navier-Stokes
equations with nonregular force.

\begin{theorem}\label{main theorem}
Assume that the initial data is $u_0\in V$. We also assume that
$f\in L^2_{\rm loc}(0, \infty; H)$. There exist $f_m \in L^2_{\rm loc}(0, \infty; H)$
 satisfying $f_m\to f$ in $L^q_{\rm loc}(0,\infty; L^{6/5})$ for all
$q$ satisfying $1 \leq q <\frac43$ such that (\ref{main-equation}) corresponding
to $u_0$ and $f_m$ possesses a unique strong solution in
$L^{\infty}_{\rm loc}(0, \infty; V)\cap L^2_{\rm loc}(0, \infty; V^2)$.
\end{theorem}

We remark that since $L^{6/5}\subset V'$, Theorem \ref{main theorem} can
be regarded as a slight generalization of the  results in
\cite{Te}.



\section{Proof of main Theorem}

For the proof of Theorem \ref{main theorem}, we use the methods developed
in \cite{Flandoli-Schmalfu}, \cite{fursikov}, and \cite{Te}.
First, we consider the following Galerkin approximation of the system
(\ref{Navier-Stokes})--(\ref{eq-st}).
\begin{equation} \label{Galerkin1}
\begin{gathered}
\frac{d u_m}{d t} + \nu \Lambda^2 u_m -P_{m} B(u_m, u_m)
=P_m f+ P_m \frac{d g}{d t},\\
 u_m(0)=P_m u_0.
\end{gathered}
\end{equation}
The  projection onto the space spanned by
$\{ \frac{1}{\alpha_i(k)}P_{\rm div}(\vec{e}_ie^{ik\cdot x})\,|\, |k|\leq m\}$
is denoted by $P_m$.
Note that (\ref{Galerkin1}) is equivalent to the integral equation
\begin{equation} \label{eqn1}
\begin{aligned}
& u_m(t)+\nu \int_0^t \Lambda^2 u_m(s) ds-\int_0^t P_m B(u_m(s), u_m(s)) ds\\
&= u_m(0)+ \int_0^t P_m f(s) ds +P_m g(t).
\end{aligned}
\end{equation}
Using contraction mapping argument, we can show, local in time, existence of
solution $u_m$ for (\ref{eqn1}). Following the argument given
in \cite{Flandoli-Schmalfu}, we consider the following auxiliary equation for
given $z_0\in V$.
\begin{equation}\label{auxiliary integral equation}
z_m(t)+ \nu\int_0^t \Lambda^2 z_m(s) ds = P_m z_0+P_m g(t), \quad t\geq 0.
\end{equation}
This equation has a unique solution, which is continuous with values in $V$ and
global in time. We show $z_m$ converges in
$C([0, \infty); V)\cap L^2_{\rm loc}(0, \infty; V^2)$.
(\ref{auxiliary integral equation}) is equivalent to
\[
z_m(t)= e^{-\nu t \Lambda^2}P_m z_0 +P_m g(t)-\nu\int_0^t \Lambda^2 e^{-\nu (t-s) \Lambda^2}P_m g(s) ds.
\]
Note that for $c(\gamma)=\max_{x \geq 0} x^{2\gamma}e^{-x}$, we have
\begin{equation}\label{Linear norm}
\| \Lambda^{2\gamma} e^{-t \Lambda^2}\|_{L(H)}
\leq c(\gamma) \frac{e^{-\frac{t}{2}\lambda_1}}{t^{\gamma}},
\end{equation}
where $\| F\|_{L(H)}= \sup_{\|f\|_{H}\leq 1}\| F(f)\|_{H}$ and
$\lambda_1$ is the smallest eigenvalue of $\Lambda^2$. Hence
\[
\Lambda z_m(t)=e^{-\nu t\Lambda^2} \Lambda P_m z_0 +\Lambda P_m g -\nu\int_0^t \Lambda^{2-2\epsilon} e^{-\nu (t-s)\Lambda^2} \Lambda^{1+2\epsilon}P_m g(s) ds
\]
is a continuous function in $H$. Since $z_m\in C([0, \infty); V)$, it follows that
$z_m$ converges in $C([0, \infty); V)$. We let $z$ be the limit of $z_m$.
Set
\[
\rho_m(t)= e^{-\nu t \Lambda^2} P_m z_0 -\int_0^t \nu \Lambda^2 e^{-\nu (t-s)
\Lambda^2}P_m g(s) ds.
\]
Then $\rho_m$ satisfies the linear equation
\[
\frac{d\rho_m}{dt} +\nu \Lambda^2 \rho_m = -\nu\Lambda^2 P_m g, \rho_m(0)=P_mz_0.
\]
It follows that
\[
\frac{d}{dt} |\Lambda \rho_m|_{L^2}^2 + \nu | \Lambda^2 \rho_m|_{L^2}^2 \leq \nu | \Lambda^2 P_m g|_{L^2}^2.
\]
Therefore, integrating we have
\[
\nu \int_0^t |\Lambda^2 \rho_m(s)|_{L^2}^2 ds
\leq |\Lambda P_m z_0|_{L^2}^2 +\nu \int_0^t |\Lambda^2 P_m g(s)|_{L^2}^2 ds.
\]
By taking subsequence, $\rho_m $ converges in $L^2{\rm loc} (0, \infty;V^2)$.
Let $\rho$ be the limit of $\rho_m$.
Since we have $g\in L^2_{\rm loc} (0, \infty; V^2)$, we have $z_m=\rho_m +P_mg$
converges to $z:=\rho+g$ in $L^2{\rm loc} (0,\infty; V^2)$.
Now define $\tilde{u}_m= u_m-z_m$. Let $\tilde{u}_m$ satisfy the equations
\begin{gather}
 \frac{d\tilde{u}_m}{dt} +\nu \Lambda^2 \tilde{u}_m-P_m B(\tilde{u}_m+z_m,
 \tilde{u}_m+z_m)=P_m f, t\in[0, \infty),\\
\tilde{u}_m(0)=u_m(0)-P_m z_0.
 \end{gather}
In \cite{Flandoli-Schmalfu}, the boundedness of $\tilde{u}_m$ in
$L^{\infty}(0, T; L^2)\cap L^2(0, T; V)$ is shown, i.e.,
\begin{align*}
& |\tilde{u}_m(t)|_{L^2}^2 +\nu \int_0^t |\Lambda \tilde{u}_m(s)|_{L^2}^2 ds \\
& \leq |\tilde{u}_m(0)|_{L^2}^2 + C\int_0^t |\tilde{u}_m|_{L^2}^2 |z_m|_{L^4}^8
+|z_m|_{L^4}^4+|z_m|_{L^2}^2 +|P_mf|_{V'}^2 ds.
\end{align*}
Thus $\nu \Lambda^2 \tilde{u}_m\in L^2_{\rm loc} (0, \infty; V')
\subset L^{4/3}_{\rm loc} (0, \infty; V')$. Note that
\begin{gather*}
\| \tilde{u}_m \cdot \nabla \tilde{u}_m \|_{V'}
  \leq  C \| \tilde{u}_m \|_{L^2}^{1/2} \| \tilde{u}_m \|_{V'}^{3/2},\\
\| \tilde{u}_m \cdot \nabla z_m\|_{V'}
  \leq  C \| \tilde{u}_m \|_{L^2} \| z_m \|_{V}^{1/2} \|  z_m \|_{V^2}^{1/2},\\
\| \tilde{z}_m \cdot \nabla \tilde{u}_m \|_{V'}
  \leq  C \| z_m \|_{L^2}^{1/2} \| z_m \|_{V}^{1/2} \| \tilde{u}_m \|_{V},\\
\| z_m \cdot \nabla z_m \|_{V'} \leq  C \| z_m \|_{L^2}^{1/2} \|z_m \|_{V}^{3/2}.
\end{gather*}
 Hence, we have
$P_m B(\tilde{u}_m+z_m, \tilde{u}_m+z_m)\in L^{4/3}_{\rm loc} (0, \infty; V')$.
 Thus we conclude that $\frac{\partial \tilde{u}_m}{\partial t}$ is bounded in
 $L^{4/3}_{\rm loc}(0, \infty; V')$.
Since $\{ \tilde{u}_m \}$ is bounded in $W^{1, \frac43}{\rm loc}([0,\infty); V')$,
we have ${\tilde{u}}_m$ converges strongly in $L^2{\rm loc}(0,\infty ; H)$.
 Thus there exists $u$ such that $u_m$ converges to $u$ in
$L^2(0, T; V^{1-\epsilon})$ and
$ L^{\frac{1}{\epsilon}} (0, T; H)$ for any small $\epsilon>0$.
 Since $u_m$ is a finite Galerkin approximation, we have
$|u_m|_{V^m} \leq C(m)|u_m|_{L^2}$.
 Hence $u_m$ is in $L^{\infty} (0,T; V)\cap L^2(0, T;  V^2)$.
Thus we have showed the following lemma.

\begin{lemma} \label{lemma1}
If $u_0 \in H$, then $u_m$ converges in $L^2(0, T;V^{1-\epsilon})$ and
$L^\frac{1}{\epsilon}(0, T; H)$ for any small $\epsilon >0$ as
$m\to \infty$.
The sequence  $u_m$ is bounded in  $L^{\infty} (0, T;H) \cap L^2(0,  T;V)$.
 Furthermore, $u_m$ is in
$L^{\infty}_{\rm loc}(0, \infty; V)\cap L^2_{\rm loc}(0, \infty; V^2)$.
\end{lemma}

To proceed further, we consider the  linear  equations
\begin{equation} \label{linear equation1}
\begin{gathered}
 \frac{\partial v_m}{\partial t}+\nu \Lambda^2 v_m
=(I-P_m)f+(I-P_m) \frac{\partial g}{\partial t}, \\
 v_m (0) =(I-P_m) u_0,
\end{gathered}
\end{equation}
where $I-P_m$ is the projection onto the space spanned by
$\{ \frac{1}{\alpha_i(k)}P_{\rm div}(\vec{e}_ie^{ik\cdot x})\,|\, |k|> m\}$.

\begin{lemma}\label{lemma3}
If $u_0 \in H$, then there exist a unique solution $v_m$ of
\eqref{linear equation1} in the space
$L^2(0, T; V)\cap L^{\infty}(0, T; H)$ and
$v_m\to 0$ in $ L^2(0, T; V)\cap L^{\infty}(0, T; H)$ as
$m\to \infty$. Furthermore, if $u_0\in V$, then
$v_m \in L^2(0, T; V^2)\cap L^{\infty}(0, T; V)$ and
$v_m\to 0$ in $L^2(0, T; V^2)\cap L^{\infty}(0, T; V)$ as
$m\to \infty$.
\end{lemma}

\begin{proof} Since (\ref{linear equation1}) is a simple linear dissipative system,
the existence and uniqueness are immediate consequence of the standard results.
Similarly to the proof of Lemma \ref{lemma1},
we can prove $v_m\in L^2(0, T; V)\cap L^{\infty}(0, T; H)$ and $v_m\to 0$
in $L^2(0, T; V)\cap L^{\infty}(0,T; H)$.
We only provide the proof of the second claim of this lemma.
Equation (\ref{linear equation1}) is equivalent to the integral equation
\begin{align*}
v_m(t)=& e^{-\nu t \Lambda^2} (I-P_m) u_0 +(I-P_m)g(t)
+\int_0^t e^{-\nu (t-s)\Lambda^2} (I-P_m) f(s) \,ds\\
&- \int_0^t \nu \Lambda^2 e^{-\nu (t-s) \Lambda^2}(I-P_m) g(s)\, ds.
\end{align*}
 Since
\begin{align*}
\Lambda v_m(t)=& e^{-\nu t\Lambda^2} \Lambda (I-P_m) u_0 +\Lambda (I-P_m) g(t)
+\int_0^t \Lambda e^{-\nu (t-s) \Lambda^2} (I-P_m) f(s)\, ds\\
&-\int_0^t \nu \Lambda^{2(1-\epsilon)} e^{-\nu (t-s) \Lambda^2}
 \Lambda^{1+2\epsilon} (I-P_m) g(s) \,ds
\end{align*}
is a continuous function in $H$ (see (\ref{Linear norm})), we have
$v_m\in C([0,T); V)$. Set
\begin{align*}
h_m(t)=& e^{-\nu t\Lambda^2} (I-P_m) u_0
-\int_0^t \nu \Lambda^2 e^{-\nu (t-s) \Lambda^2} (I-P_m)g(s)\, ds\\
& + \int_0^t e^{-\nu (t-s) \Lambda^2} (I-P_m) f(s)\, ds,
\end{align*}
i.e., $v_m(t)= (I-P_m)g(t)+h_m(t)$. It follows that $h_m$ satisfies the equation
\[
\frac{dh_m}{dt} +\nu \Lambda^2 h_m= -\nu \Lambda^2 (I-P_m) g+ (I-P_m)f.
\]
Taking inner product with $\Lambda^2 h_m$ in $L^2$ produces
\[
\frac12 \frac{d}{dt} |\Lambda h_m|_{L^2}^2 +\nu |\Lambda^2 h_m(t)|_{L^2}^2
\leq \frac{\nu}{2} |\Lambda^2 h_m|_{L^2}^2 + C|\Lambda^2(I-P_m) g|_{L^2}^2 +C|(I-P_m) f|_{L^2}^2.
\]
Integrating over $[0, T)$, we obtain
\begin{align*}
&|\Lambda^2 h_m(t)|_{L^2}^2 +\nu \int_0^T |\Lambda^2 h_m(t)|_{L^2}^2 dt \\
&\leq |\Lambda (I-P_m) u_0|_{L^2}^2 +C\int_0^T |\Lambda^2(I-P_m) g(t)|_{L^2}^2 dt
 +C\int_0^T |(I-P_m)f(t)|_{L^2}^2 dt.
\end{align*}
Thus  $h_m\in L^2(0, T; V^2)$. Since $g\in L^2(0, T; V^2)$, we obtain
$v_m\in L^2(0, T; V^2)$.
Lebesgue's dominated convergence Theorem produces $v_m\to 0$ in
$L^2(0, T; V^2) \cap L^{\infty}(0, T; V)$ as $m\to \infty$.
For the uniqueness, if there exists two solutions $v_m^1$ and $v_m^2$ of
(\ref{linear equation1}), then we denote by $\rho(t)=v_m^1(t)-v_m^2(t)$.
We have the following deterministic equations.
\[
\frac{d \rho}{dt} +\nu \Lambda^2 \rho=0,\quad  \rho(0)=0.
\]
Thus we have $\rho(t)=0$,
which completes the proof. \end{proof}

 From Lemma \ref{lemma1}, we have $u_m$ is in
 $L^{\infty} (0,T;V) \cap L^2(0,  T;V^2)$.
Now for every $m$, we consider also the solution $v_m$ of the linearized problems
(\ref{linear equation1})
We then set $w_m = u_m +v_m$ and observe $w_m$ satisfies
$w_m \in L^2(0, T; V^2)\cap L^{\infty} (0, T; V)$ if $u_0 \in V$.
By adding two equations, we have the following Navier-Stokes equations with
nonregular force
\begin{equation}\label{main equation}
\frac{d w_m}{d t} +\nu \Lambda^2 w_m - B(w_m, w_m) = f_m +\frac{d g}{d t},
\end{equation}
where $f_m= f-B(v_m, v_m)  - B(v_m, u_m) - B(u_m, v_m)  -(I-P_m) B(u_m, u_m)$.
Let $\tilde{w}_m$ be another solution of (\ref{main equation}). Then
by letting $\tilde{\rho}_m= w_m-\tilde{w}_m$, we have
\[
\frac{d \tilde{\rho}_m}{d t} +\nu \Lambda^2 \tilde{\rho}_m = B(\tilde{\rho}_m, w_m)+B(\tilde{w}_m, \tilde{\rho}_m).
\]
Hence we have
\begin{align*}
\frac{d}{dt} | \tilde{\rho}_m|_{L^2}^2
& \leq  -2\nu |\Lambda \tilde{\rho}_m|_{L^2}^2
  +C |\tilde{\rho}_m|_{L^6} |\nabla w_m|_{L^3} |\tilde{\rho}_m|_{L^2}\\
& \leq  -\nu |\Lambda \tilde{\rho}_m|_{L^2}^2
  +C| w_m |_{V^2}|w_m|_{V} | \tilde{\rho}_m|_{L^2}^2.
\end{align*}
Using Gronwall's inequality, we have
\[
|\tilde{\rho}_m (t)|_{L^2}^2 \leq |\tilde{\rho}_m(0)|_{L^2}^2
\exp \Big(C\int_0^t |w_m|_{V^2} |w_m|_{V} ds  \Big).
\]
Since $w_m \in L^2(0, T; V^2)\cap L^{\infty}(0, T; V)$, it is clear that $w_m$
is the unique solution in
$L^2_{\rm loc}(0, \infty; V^2)\cap L^{\infty}_{\rm loc}(0, \infty; V)$.
Hence for the proof of Theorem \ref{main theorem}, it is sufficient to
show that $f_m$ converges to $f$ in $L^q(0, T; L^{6/5})$ with
$1\leq q <\frac43$.

\begin{proof}[Proof of Theorem \ref{main theorem}]
For the remaining of this proof, we use only the weaker assumption
$u_0\in H$ instead of $u_0\in V$. Since we have $v_m\to 0$ in
$L^2(0, T; V)\cap L^{\infty} (0, T; H)$ as $m\to \infty$, it is clear that
$B(v_m, v_m) \to 0$ in $L^q(0, T; L^{6/5})$ by the  inequalities
\begin{align*}
 \int_{0}^T | B(v_m, v_m)|_{L^{6/5}}^q dt
& \leq  C\int_{0}^T |v_m|_{L^{3}}^q \, |v_m |_{V}^q dt \\
& \leq  C \int_0^T |v_m |_{L^2}^{q/2} |v_m|_{V}^{\frac32 q}dt \\
& \leq  C\Big(\int_0^T |v_m|_{L^2}^{\frac{2q}{4-3q}}dt \Big)^{\frac{4-3q}{4}}
  \Big( \int_0^T |v_m|_{V}^2  dt  \Big)^{3q/4}\\
& \leq  CT^{\frac{4-3q}{4}}|v_m |_{L^{\infty}(0, T; H)}^{\frac{4q}{4-3q}}
 |v_m |_{L^2(0, T; V)}^{3q/2}.
\end{align*}
It is well known from the interpolation inequality that
\[
\quad | B(u_m, v_m)|_{L^{6/5}} \leq  C | u_m |_{L^6} |\nabla v_m|_{L^{3/2}}
\leq  C|\nabla u_m |_{L^2} | v_m |_{L^2}^{1/2} |v_m |_{V}^{1/2}.
\]
Then
\begin{align*}
& \int_{0}^T | B(u_m, v_m )|_{L^{6/5}}^q dt\\
& \leq C\int_0^T | \nabla u_m|_{L^2}^q |v_m |_{L^2}^{q/2} |v_m|_{V}^{q/2}dt\\
& \leq  C \Big(\int_{0}^T |\nabla u_m|_{L^2}^2 dt   \Big)^{q/2}
 \Big( \int_{0}^T |v_m|_{L^2}^{\frac{2q}{4-3q}}dt   \Big)^{\frac{4-3q}{4}}
 \Big(\int_{0}^T |v_m |_{V}^2 dt  \Big)^{q/4}\\
& \leq  C T^{\frac{4-3q}{4}}|u_m |_{L^2 (0, T; V)}^q |v_m|_{L^{\infty}(0,T; H)}^{q/2}
v_m|_{L^2(0, T; V)}^{q/2} \to  0\,.
\end{align*}
Similarly, we have
\begin{align*}
\int_{0}^T | B(v_m, u_m)|_{L^{6/5}}^q dt
& \leq  \int_{0}^T |v_m|_{L^3}^q |\nabla u_m |_{L^2}^q dt \\
& \leq   C T^{\frac{4-3q}{4}}|u_m |_{L^2 (0, T; V)}^q |v_m|_{L^{\infty}
(0,T; H)}^{q/2}|v_m|_{L^2(0, T; V)}^{q/2}  \to  0\,.
\end{align*}
To complete the proof, it is sufficient to show that
\[
(I-P_m) B(u_m, u_m) \to  0 \quad\mbox{in } L^q(0, T;  L^{6/5})\mbox{ as } m\to
 \infty.
\]
First we recall that $u_m$ converges to its limit $u$ from Lemma \ref{lemma1}.
We rewrite $u$ as an expansion by the complete orthonormal basis $K$, i.e.,
\[
u=\sum_{k\in{\mathbb{Z}}^3\; i=1,2,3} u^i_k \frac{1}{\alpha_i(k)} P_{\rm div}
(\vec{e}_i e^{ik\cdot x})=:
\sum_{k\in {\mathbb{Z}}^3} u_k e_k(x),
\]
 where $u_k^i$ is the corresponding coefficient, and for
 simplicity of notation we introduced the right-hand-side.
Then we have
\begin{equation} \label{identity}
\begin{aligned}
(I-P_m) B(u, u)& = (I-P_m)P_{\rm div} ((u\cdot \nabla)u) \\
&= (I-P_m) P_{\rm div} \sum_{k'\in{\mathbb{Z}}^3} ( \sum_{k\in {\mathbb{Z}}^3}
u_k e_k(x)\cdot k')u_{k'} e_{k'}(x) \\
&= P_{\rm div} \Big(  \big(\sum_{|k|\geq [\frac{m}{2}]} u_k e_k(x)\cdot \nabla\big)
\sum_{k'}^* u_{k'} e_{k'}(x) \Big)\\
&\quad + P_{\rm div} \Big( \big(\sum_{k}^* u_k e_k(x) \cdot \nabla\big)
\sum_{|k'| \geq [\frac{m}{2}]} u_{k'} e_{k'}(x)  \Big),
\end{aligned}
\end{equation}
where $[a]$ denotes the largest integer less than or equal to $a$,
 and $\sum_{h}^* $ denotes the summation over all $h\in{\mathbb{Z}}^3$
satisfying $|h+j|>m$ when $|j|\geq [\frac{m}{2}]$.
Using the identity (\ref{identity}), we obtain
\begin{align*}
|(I-P_m) B(u, u)|
& \leq  C |\sum_{|k|\geq [\frac{m}{2}]} u_k e_k(x) |_{L^3} | \nabla u |_{L^2}
+ C|u|_{L^6} |\nabla \sum_{|k'| \geq \frac{m}{2}} u_{k'} e_{k'}(x) |_{L^{3/2}}\\
&\leq C| \sum_{|k| \geq [\frac{m}{2}]} u_k e_k(x) |_{L^2}^{1/2}
|\nabla u|_{L^2}^{3/2}.
\end{align*}
We obtain that for any $q<4/3$,
\begin{align*}
&\int_{0}^T | (I-P_m)B(u, u)|_{L^{6/5}}^q dt\\
& \leq  C  \int_{0}^T |\nabla u|_{L^2}^{\frac{3q}{2}}
|\sum_{|k|\geq [\frac{m}{2}]} u_k e_k(x)|_{L^2}^{q/2} dt\\
& \leq  C\Big(\int_{0}^T |\nabla u|_{L^2}^2dt \Big)^{3q/4}
\Big( \int_{0}^T |\sum_{|k|\geq [\frac{m}{2}]}u_k e_k(x)|_{L^2}^{\frac{2q}{4-3q}}
dt \Big)^{\frac{4-3q}{4}} .
\end{align*}
Since $K$ is a complete orthonormal basis for $H$, we have
\[
(I-P_m) B(u, u) \to  0 \quad\mbox{in } L^q(0, T; L^{6/5})\mbox{ as } m\to \infty.
\]
Thus it only remains to prove that $ B(u_m, u_m)- B(u, u) \to  0$ in
$L^q(0, T; L^{6/5})$. From Lemma \ref{lemma1}, we have
\[
u_m \to u \quad \mbox{ in } L^2(0, T; V^{1-\epsilon})\cap L^{\frac{1}{\epsilon}}
(0, T; H)\mbox{ for any } \epsilon >0.
\]
We complete the proof by showing that
\[
B(u_m-u,u_m), B(u,u_m-u)\to  0 \quad\mbox{in } L^q(0, T; L^{6/5})
{\mbox{ for all }} q< 4/3.
\]
By the interpolation inequality, we have for $\epsilon <1/2$,
\begin{gather*}
|B(u_m-u,u_m)|_{L^{6/5}}\leq C |u_m -u|_{L^2}^{\frac{1-2\epsilon}{2(1-\epsilon)}}
 |u_m-u|_{V^{1-\epsilon}}^{\frac{1}{2(1-\epsilon)}}|\nabla u_m|_{L^2},\\
|B(u, u_m-u)|_{L^{6/5}} \leq C|u|_{L^6} |u_m-u|_{L^2}^{1/2} |u_m-u|_{V}^{1/2}.
\end{gather*}
Setting $r=\frac{2q(1-2\epsilon)}{4-3q-2\epsilon(2-q)}$, we have
 \[
\frac{q}{2}+ \frac{q}{4(1-\epsilon)} +\frac{q(1-2\epsilon)}{2r(1-\epsilon)}=1.
  \]
By  H\"{o}lder's inequality and Lemma \ref{lemma1}, we obtain
\begin{align*}
&\int_{0}^T |B(u_m-u, u_m)|_{L^{6/5}}^q dt\\
&\leq C\Big( \int_{0}^T|u_m-u|_{L^2}^r dt
 \Big)^{\frac{q(1-2\epsilon)}{2r(1-\epsilon)}}
 \Big( \int_{0}^T |u_m-u|_{V^{1-\epsilon}}^2dt \Big)^{\frac{q}{4(1-\epsilon)}}
 \Big(\int_{0}^T |\nabla u_m|_{L^2}^2dt \Big)^{q/2}
\end{align*}
which approaches zero as $m$ approaches $\infty$.
Again using H\"{o}lder's inequality(note that
$\frac{q}{2}+\frac{q}{4} +\frac{4-3q}{4}=1$), we have
\begin{align*}
&\int_{0}^T |B(u, u_m-u)|_{L^{6/5}}^q dt\\
&\leq C\Big( \int_{0}^T |u|_{V}^2 dt\Big)^{q/2}
\Big( \int_{0}^T |u_m-u|_{L^2}^{\frac{2q}{4-3q}}dt \Big)^{\frac{4-3q}{4}}
\Big(\int_{0}^T |u_m-u|_{V}^2 dt \Big)^{q/4}
 \to 0\,.
\end{align*}
This completes the proof of Theorem \ref{main theorem}.
\end{proof}

\subsection*{Acknowledgements}
The author would like to thank Professor D. Chae for the many helpful
suggestions on the generic solvability, and Professor Ya Sinai for his helpful
discussions. The author also wants to thank the anonymous referee for
his/her suggestions and corrections.

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\end{document}