\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{ Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 166, pp. 1--22.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/166\hfil Radial defocusing cubic wave equation] {Global well-posedness for the radial defocusing cubic wave equation on $\mathbb{R}^{3}$ and for rough data} \author[T. Roy\hfil EJDE-2007/166\hfilneg] {Tristan Roy} \address{Tristan Roy \newline UCLA Mathematics Department\\ Box 951555 \\ Los Angeles, CA 90095-1555, USA} \email{triroy@math.ucla.edu} \thanks{Submitted August 17, 2007. Published November 30, 2007.} \subjclass[2000]{35Q55} \keywords{Nonlinear Schr\"odinger equation; well-posedness} \begin{abstract} We prove global well-posedness for the radial defocusing cubic wave equation \begin{gather*} \partial_{tt} u - \Delta u = -u^{3} \\ u(0,x) = u_{0}(x) \\ \partial_{t} u(0,x) = u_{1}(x) \end{gather*} with data $( u_{0}, u_{1}) \in H^{s} \times H^{s-1}$, $1 > s >7/10$. The proof relies upon a Morawetz-Strauss-type inequality that allows us to control the growth of an almost conserved quantity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} We shall study the defocusing cubic wave equation on $\mathbb{R}^{3}$ \begin{equation} \begin{gathered} \partial_{tt} u - \Delta u = -u^{3} \\ u(0,x) = u_{0}(x) \\ \partial_{t} u(0,x) = u_{1}(x) \end{gathered} \label{WaveEqRad} \end{equation} We shall focus on the strong solutions of the defocusing cubic wave equation on some interval $[0,T]$ i.e real-valued maps $(u,\partial_{t} u) \in C ([0, T], H^{s}(\mathbb{R}^{3}) ) \times C ( [0, T], H^{s-1} ( \mathbb{R}^{3}) )$ that satisfy for $t \in [0, T] $ the following integral equation \begin{equation} u(t) = \cos(tD) u_{0} + D^{-1} \sin(tD) u_{1} - \int_{0}^{t} D^{-1} \sin ( (t-t') D ) u^{3}(t') \,dt' \end{equation} with $(u_{0},u_{1})$ lying in $H^{s} \times H^{s-1}$. Here $H^{s}$ is the usual inhomogeneous Sobolev space; i.e., $H^{s}$ is the completion of the Schwartz space $\mathcal{S}(\mathbb{R}^{3})$ with respect to the norm \begin{equation} \| f \|_{H^{s}} := \| (1+ D^{s}) f \|_{L^{2}(\mathbb{R}^{3})} \end{equation} where $D$ is the operator defined by \begin{equation} \widehat{Df}(\xi) := |\xi| \hat{f}(\xi) \end{equation} and $\hat{f}$ denotes the Fourier transform \begin{equation} \hat{f}(\xi) := \int_{\mathbb{R}^{3}} f(x) e^{-i x \cdot \xi} \,dx \end{equation} Here $H^{s} \times H^{s-1}$ is the product space of $H^{s}$ and $H^{s-1}$ endowed with the standard norm $ \| (f,g) \|_{H^{s} \times H^{s-1}} := \| f \|_{H^{s}} + \| g \|_{H^{s-1}}$. It is known \cite{lindsogge} that \eqref{WaveEqRad} is locally well-posed in $H^{s}(\mathbb{R}^{3}) \times H^{s-1}(\mathbb{R}^{3})$ for $s \geq \frac{1}{2}$. Moreover if $s > \frac{1}{2}$ the time of local existence only depends on the norm of the initial data $\| (u_{0},u_{1}) \|_{H^{s} \times H^{s-1}}$. Now we turn our attention to the global well-posedness theory of \eqref{WaveEqRad}. In view of the above local well-posedness theory and standard limiting arguments it suffices to establish an a priori bound of the form \begin{equation} \| u(T) \|_{H^{s}} + \| \partial_{t} u (T) \|_{H^{s-1}} \leq C ( s, ( \| u_{0}\|, \| u_{1} \| )_{H^{s} \times H^{s-1}}, T ) \end{equation} for all times $0 < T < \infty $ and all smooth-in-time Schwartz-in-space solutions $(u,\partial_{t} u): [0, T] \times \mathbb{R}^{3} \to \mathbb{R}$, where the right-hand side is a finite quantity depending only on $s$, $ \| u_{0} \|_{H^{s}} $, $\| u_{1} \|_{H^{s-1}}$ and $T$. Therefore in the sequel we shall restrict attention to such smooth solutions. The defocusing cubic wave equation \eqref{WaveEqRad} enjoys the following energy conservation law \begin{equation} E(u(t)) := \frac{1}{2} \int_{\mathbb{R}^{3}} (\partial_{t} u)^{2}(x,t) \,dx + \frac{1}{2} \int_{\mathbb{R}^{3}} | D u (x,t)|^{2} \,dx + \frac{1}{4} \int_{\mathbb{R}^{3}} u^{4}(x,t) \,dx \end{equation} Combining this conservation law to the local well-posedness theory we immediately have global well-posedness for \eqref{WaveEqRad} and for $s=1$. In this paper we are interested in studying global well-posedness for \eqref{WaveEqRad} and for data below the energy norm, i.e $s<1$. It is conjectured that \eqref{WaveEqRad} is globally well-posed in $H^{s} (\mathbb{R}^{3}) \times H^{s-1}(\mathbb{R}^{3}) $ for all $s > \frac{1}{2}$. The global existence for the defocusing cubic wave equation has been the subject of several papers. Let us some mention some results for data lying in a slightly different space than $H^{s} \times H^{s-1}$ i.e $\dot{H}^{s} \times \dot{H}^{s-1}$. Here $\dot{H}^{s}$ is the usual homogeneous Sobolev space i.e the completion of Schwartz functions $\mathcal{S} ( \mathbb{R}^{3} )$ with respect to the norm \begin{equation} \| f \|_{\dot{H}^{s}} = \| D^{s} f \|_{L^{2} ( \mathbb{R}^{3} )} \end{equation} Kenig, Ponce and Vega \cite{kenponcevega} were the first to prove that \eqref{WaveEqRad} is globally well-posed for $1 >s > \frac{3}{4}$. They used the \emph{Fourier truncation method} discovered by Bourgain \cite{bourg}. Gallagher and Planchon \cite{gallagplanch} proposed a different method to prove global well-posedness for $1 >s > \frac{3}{4}$. Bahouri and Jean-Yves Chemin \cite{bahchemin} proved global-wellposedness for \eqref{WaveEqRad} and for $s=\frac{3}{4}$ by using a non linear interpolation method and logarithmic estimates from Klainermann and Tataru \cite{klaintat}. We shall consider global well-posedness for the radial defocusing cubic wave equation i.e global existence for the initial value problem \eqref{WaveEqRad} with radial data. The main result of this paper is the following one \begin{theorem} \label{thm1.1} The radial defocusing cubic wave equation is globally well-posed in $ H^{s} \times H^{s-1} $ for $ 1 > s > \frac{7}{10} $. Moreover if $T$ large then \begin{equation} \| u(T) \|^{2}_{H^{s}} + \| \partial_{t} u(T) \|^{2}_{H^{s-1}} \leq C ( \| u_{0} \|_{H^{s}}, \| u_{1} \|_{H^{s-1}} ) T^{\frac{16s-10}{10s-7}+} \label{LgEstRaduT1} \end{equation} for $ \frac{5}{6} \geq s > \frac{7}{10}$ and \begin{equation} \| u(T) \|^{2}_{H^{s}} + \| \partial_{t} u(T) \|^{2}_{H^{s-1}} \leq C ( \| u_{0} \|_{H^{s}}, \| u_{1} \|_{H^{s-1}} ) T^{\frac{2s}{2s-1}+} \label{LgEstRaduT2} \end{equation} for $ 1 > s > \frac{5}{6}$. Here $C ( \| u_{0} \|_{H^{s}}, \| u_{1} \|_{H^{s-1}} )$ is a constant only depending on $ \| u_{0} \|_{H^{s}} $ and $\| u_{1} \|_{H^{s-1}}$. \label{Thm:GwpRad710} \end{theorem} We set some notation that appear throughout the paper. Given $A,B$ positive number $A \lesssim B$ means that there exists a universal constant $K$ such that $A \leq K B$. We say that $K_{0}$ is the constant determined by the relation $A \lesssim B$ if $K_{0}$ is the smallest $K$ such that $A \leq K B$ is true. We write $A \sim B$ when $A \lesssim B$ and $B \lesssim A$. $A \ll B$ denotes $ A \leq K B$ for some universal constant $K < \frac{1}{100}$ . We also use the notations $A+ = A + \epsilon$, $A-=A - \epsilon$ for some universal constant $0 < \epsilon \ll 1$. Let $\nabla$ denote the gradient operator. If $J$ is an interval then $|J|$ is its size. If $E$ is a set then $\mathop{\rm card}(E)$ is its cardinal. Let $I$ be the following multiplier \begin{equation} \widehat{If}(\xi) := m(\xi) \hat{f}(\xi) \end{equation} where $m(\xi): = \eta ( \frac{\xi}{N} )$, $\eta$ is a smooth, radial, nonincreasing in $|\xi|$ such that \begin{equation} \eta (\xi) := \begin{cases} 1, & |\xi| \leq 1 \\ ( \frac{1}{|\xi|} )^{1-s}, & |\xi| \geq 2 \end{cases} \end{equation} and $N\gg 1$ is a dyadic number playing the role of a parameter to be chosen. We shall abuse the notation and write $m (|\xi|)$ for $m(\xi)$, thus for instance $m(N)=1$. We recall some basic results regarding the defocusing cubic wave equation. Let $\lambda \in \mathbb{R}$ and $u_{\lambda}$ denote the following function \begin{equation} u_{\lambda}(t,x) := \frac{1}{\lambda} u ( \frac{t}{\lambda}, \frac{x}{\lambda} ) \end{equation} If $u$ satisfies \eqref{WaveEqRad} with data $(u_{0},u_{1})$ then $u_{\lambda}$ also satisfies \eqref{WaveEqRad} but with data $ ( \frac{1}{\lambda} u_{0} ( \frac{x}{\lambda} ), \frac{1}{\lambda^{2}} u_{1} ( \frac{x}{\lambda} ) )$. If $u$ satisfies the radial defocusing cubic wave equation then $u$ is radial. Now we recall some standard estimates that we use later in this paper. \begin{proposition}[Strichartz estimates in 3 dimensions \cite{ginebvelo,lindsogge}] Let $m \in [0, 1]$. If $u$ is a strong solution to the IVP problem \begin{equation} \begin{gathered} \partial_{tt} u - \Delta u = F \\ u(0,x) = f(x) \in \dot{H}^{m} \\ \partial_{t} u(0,x) = g(x) \in \dot{H}^{m-1} \\ \end{gathered} \label{LinearWave} \end{equation} then for $0 \leq \tau < \infty$ we have \begin{align*} &\| u \|_{L_{t}^{q}( [0, \tau] ) L_{x}^{r}} + \| u \|_{C ( [0, \tau]; \dot{H}^{m} )} + \| \partial_{t} u \|_{C ( [0, \tau]; \dot{H}^{m-1} )} \\ & \lesssim \| f \|_{\dot{H}^{m}} + \| g \|_{\dot{H}^{m-1}} + \| F \|_{L_{t}^{\tilde{q}} ( [0, \tau] ) L_{x}^{\tilde{r}}} \end{align*} under two assumptions \begin{itemize} \item $(q,r)$ lie in the set $\mathcal{W}$ of \textit{wave-admissible} points; i.e., \begin{equation} \mathcal{W} := \{ (q, r): ( q, r ) \in (2, \infty ] \times [2,\infty), \frac{1}{q}+\frac{1}{r} \leq \frac{1}{2} \} \label{StrCondition1} \end{equation} \item $(\tilde{q}, \tilde{r})$ lie in the dual set $\mathcal{W}'$ of $\mathcal{W}$; i.e., \begin{equation} \mathcal{W}' := \{ (\tilde{q}, \tilde{r}): \frac{1}{\tilde{q}} + \frac{1}{q}=1, \frac{1}{r}+ \frac{1}{\tilde{r}}=1, (q,r) \in \mathcal{W} \} \end{equation} \item $(q,r,\tilde{q}, \tilde{r})$ satisfy the \textit{dimensional analysis} conditions \begin{gather} \frac{1}{q} + \frac{3}{r} = \frac{3}{2} -m, \label{StrCondition2} \\ \frac{1}{\tilde{q}} + \frac{3}{\tilde{r}} -2 = \frac{3}{2} -s\,. \label{StrCondition3} \end{gather} \end{itemize} \end{proposition} We also have the well-known estimate \begin{proposition}[Radial Sobolev inequality] If $u:\mathbb{R}^{3} \to \mathbb{C}$ is radial and smooth, then \begin{equation} |u(x)| \lesssim \frac{\|u\|_{\dot{H}^{1}}}{|x|^{\frac{1}{2}}} \label{SobolevRad} \end{equation} \end{proposition} The Hardy-type inequality is proved in \cite{caz}. \begin{proposition}[Hardy-type inequality] If $1 M} f}(\xi) := \hat{f}(\xi) - \widehat{P_{\leq M} f}(\xi) \end{gather*} Since $\sum_{M \in 2^{\mathbb{Z}}} \psi ( \frac{\xi}{M})=1$ we have \begin{equation} f = \sum_{M \in 2^{\mathbb{Z}}} P_{M} f \end{equation} We conclude this introduction by giving the main ideas of the proof of theorem \ref{Thm:GwpRad710} and explaining how the paper is organized. Following the proof of the global well-posedness for $s=1$ we try to compare for every $T>0$ the relevant quantity $ \| ( u(T), \partial_{t} u(T) ) \|_{H^{s} \times H^{s-1}}$ to the supremum of the energy conservation law $ \sup_{t \in [0, T]} E ( u(t) )$. Unfortunately this strategy does not work if $s<1$ since the energy can be infinite. We get around this difficulty by using the $I$-method designed by Colliander, Keel, Staffilani, H.Takaoka and Tao \cite{almckstt} and successfully applied to prove global well-posedness for semilinear Schr\"odinger equations and for rough data. The idea consists of introducing the following smoothed energy \begin{equation*} E ( Iu(t) ) := \frac{1}{2} \int_{\mathbb{R}^{3}} \big| \partial_{t} I u(x,t) \big|^{2} \,dx + \frac{1}{2} \int_{\mathbb{R}^{3}} | D I u(x,t)|^{2} \,dx + \frac{1}{4} \int_{\mathbb{R}^{3}} |I u(x,t)|^{4} \,dx \end{equation*} We prove in section \ref{sec:RelHsNrj} that $\| ( u(T), \partial_{t} u(T) ) \|^{2}_{H^{s} \times H^{s-1}}$ and the supremum of the smoothed energy on $[0, T]$ are comparable. Therefore we try to estimate the quntity $\sup_{t \in [0,T]}E ( Iu(t) )$ in order to give an upper bound of $ \| ( u(T), \partial_{t} u(T)) \|_{H^{s} \times H^{s-1}} $. For convenience we place the mollified energy at time zero into $[0, \frac{1}{2}]$ by choosing the right scaling factor $\lambda$. This operation shows that we are reduced to estimate $\sup_{t \in [0, \lambda T]} E ( I u_{\lambda}(t) )$. In section \ref{sec:LocalBd} we prove that we can locally control a variable namely $Z(J)$ provided that the interval $J$ satisfies some constraints that give some information about its size. $\sup_{t \in J} E ( I u_{\lambda}(t) ) $ is estimated by the fundamental theorem of calculus. The upper bound depends on the parameter $N$ and the controlled quantity $Z(J)$. This estimate is established in section \ref{sec:AlmCon}. Now we can iterate: the process generates a sequence of intervals $( J_{i} )$ that cover the whole interval $[0, \lambda T]$ and satisfy the same constraints as $J$. We should be able to estimate $\sup_{t \in [0, \lambda T]}E ( Iu_{\lambda}(t) )$ provided that we can control the number of intervals $J_{i}$. This requires the establishment of a long time estimate, the so-called almost Morawetz-Strauss inequality. This estimate is proved in section \ref{sec:AlmMorIneq}. It depends on some remainder integrals that are estimated in section \ref{subsec:RemainderMor}. Combining this inequality to the radial Sobolev inequality (\ref{SobolevRad}) we can give an upper bound of the cardinal of $( J_{i} )$. The proof of theorem \ref{Thm:GwpRad710} is given in section \ref{sec:PfGwRad710}. \section{Proof of global well-posedness for $1 > s > 7/10$} \label{sec:PfGwRad710} In this section we prove the global existence of \eqref{WaveEqRad} for $1 > s > 7/10$. Our proof relies on some intermediate results that we prove in later sections. More precisely we shall show the following results. \begin{proposition}[$H^{s}$ norms and mollified energy estimates] Let $T>0$. Then \begin{equation} \| u(T) \|^{2}_{H^{s}} + \|\partial_{t} u(T)\|^{2}_{H^{s-1}} \lesssim \| u_{0} \|^{2}_{H^{s}} + ( T^{2}+1 )\sup_{t \in [0, T]} E ( Iu(t) ) \label{NrjEstim1} \end{equation} for every $u$. \label{Prop:NrjEst} \end{proposition} \begin{proposition}[Local boundedness] Let $J=[a,b]$ be an interval included in $[0, \infty)$. Assume that $E ( Iu(a) ) \leq 2$ and that $u$ satisfies \eqref{WaveEqRad}. There exist $C_{1}$, $C_{2}$ small and positive constants such that if $J$ satisfies \begin{gather} \| Iu \|_{L_{t}^{6}(J) L_{x}^{6}} \leq \frac{C_{1}}{|J|^{\frac{1}{3}}}, \label{Condition1} \\ |J| \leq C_{2} N^{\frac{1-s}{s-\frac{1}{2}}} \label{Condition2} \end{gather} then $Z(J) \lesssim 1$. %\label{ZRes} \label{prop:LocalBdRad} \end{proposition} \begin{proposition}[Almost conservation law] Let $J=[a,b]$ be an interval included in $[0,\infty)$. Assume that $u$ satisfies \eqref{WaveEqRad}. Then \begin{equation} \big| \sup_{t \in J} E(Iu(t)) - E(Iu(a)) \big| \lesssim \frac{Z^{4} (J)}{N^{1-}} \label{EstNrjRad} \end{equation} \label{prop:EstNrjRad} \end{proposition} \begin{proposition}[Almost Morawetz-Strauss inequality] Let $T \geq 0$. Assume that $u$ satisfies \eqref{WaveEqRad}. Then \begin{equation} \int_{0}^{T} \int_{\mathbb{R}^{3}} \frac{\left| Iu \right|^{4}(t,x)}{|x|} dx \,dt -2 ( E ( I u(0) ) + E ( Iu(T) ) ) \lesssim \left| R_{1}(T)\right| + \left| R_{2}(T) \right|\,. \end{equation} \label{prop:AlmMorawetz} \end{proposition} \begin{proposition}[Estimate of integrals] Let $J$ be an interval included in $[0,\infty)$. Then if $i=1,2$ we have \begin{equation} R_{i}(J) \lesssim \frac{Z^{4}(J)}{N^{1-}} \end{equation} \label{prop:RemainderMor} \end{proposition} For the remainder of this section we show how propositions \ref{prop:LocalBdRad}, \ref{prop:EstNrjRad}, \ref{prop:AlmMorawetz} and \ref{prop:RemainderMor} imply Theorem \ref{Thm:GwpRad710}. Let $T>0$ and $N=N(T)\gg 1$ be a parameter to be chosen later. There are three steps to prove Theorem \ref{Thm:GwpRad710}. \subsection*{(1) Scaling} Let $\lambda \gg 1$ to be chosen later. Then by Plancherel theorem \begin{equation} \begin{aligned} \| D I u_{\lambda}(0) \|^{2}_{L^{2}} & \lesssim \int_{|\xi| \leq 2N} |\xi|^{2} | \widehat{u_{\lambda}}(0,\xi)|^{2} \,d\xi + \int_{|\xi| \geq 2N} |\xi|^{2} \frac{N^{2(1-s)}}{|\xi|^{2(1-s)}} |\widehat{u_{\lambda}}(0, \xi)|^{2} \,d\xi \\ & \lesssim N^{2(1-s)} \| u_{\lambda}(0) \|^{2}_{\dot{H}^{s}} \\ & \lesssim N^{2(1-s)} \lambda^{1-2s} \| u_{0} \|^{2}_{\dot{H}^{s}} \\ & \lesssim N^{2(1-s)} \lambda^{1-2s} \| u_{0} \|^{2}_{H^{s}}, \end{aligned} \label{DIuSc} \end{equation} \begin{equation} \begin{aligned} \| \partial_{t} I u_{\lambda}(0) \|^{2}_{L^{2}} & \lesssim \int_{|\xi| \leq 2N } |\widehat{\partial_{t} u_{\lambda}}(0,\xi)|^{2} \,d\xi + \int_{|\xi| \geq 2N} \frac{N^{2(1-s)}}{|\xi|^{2(1-s)}} |\widehat{\partial_{t} u_{\lambda}}(0,\xi)|^{2} \,d\xi \\ & \lesssim N^{2(1-s)} \| \partial_{t} u_{\lambda}(0) \|^{2}_{H^{s-1}} \\ & \lesssim N^{2(1-s)} \Big( \int_{|\xi| \leq 1} |\widehat{\partial_{t} u_{\lambda}}(0,\xi)|^{2}\,d\xi + \int_{|\xi| \geq 1} |\xi|^{2(s-1)} |\widehat{\partial_{t} u_{\lambda}}(0,\xi)|^{2} \,d\xi \Big) \\ & \lesssim N^{2(1-s)} \Big( \frac{1}{\lambda} \int_{|\xi| \leq \lambda} | \widehat{u_{1}}(\xi) |^{2} \,d\xi + \lambda^{1-2s} \int_{|\xi| \geq \lambda} |\xi|^{2(s-1)} |\widehat{u_{1}}(\xi)|^{2} \,d\xi \Big) \\ & \lesssim N^{2(1-s)} \lambda^{1-2s} \| u_{1} \|^{2}_{H^{s-1}}. \end{aligned} \label{DtIuSc} \end{equation} By homogeneous Sobolev embedding, \begin{equation} \begin{aligned} &\| I u_{\lambda}(0) \|^{2}_{L^{4}}\\ & \lesssim \int_{\mathbb{R}^{3}} |\xi|^{\frac{3}{2}} |\widehat{Iu_{\lambda}}(0,\xi)|^{2} \,d\xi \\ & \lesssim \int_{|\xi| \leq 2N} |\xi|^{\frac{3}{2}} |\widehat{u_{\lambda}}(0,\xi)|^{2} \,d\xi + \int_{|\xi| \geq 2N} |\xi|^{\frac{3}{2}} \frac{N^{2(1-s)}}{|\xi|^{2(1-s)}} |\widehat{u_{\lambda}}(0,\xi)|^{2} \,d\xi \\ & \lesssim \frac{1}{\lambda^{\frac{1}{2}}} \int_{|\xi| \leq 2N \lambda} |\xi|^{\frac{3}{2}} |\widehat{u_{0}}(\xi)|^{2} \,d\xi + N^{2(1-s)} \lambda^{\frac{3}{2}-2s} \int_{|\xi| \geq 2N \lambda} |\xi|^{2s-\frac{1}{2}} |\widehat{u_{0}}(\xi)|^{2} \,d\xi \\ & \lesssim \frac{ \max{( N^{\frac{3}{2}-2s} \lambda^{\frac{3}{2}-2s},1 )}}{\lambda^{\frac{1}{2}}} \| u_{0} \|^{2}_{H^{s}} + N^{\frac{3}{2}-2s} \lambda^{1-2s} \| u_{0} \|^{2}_{H^{s}}. \end{aligned} \end{equation} Hence \begin{equation} \| I u_{\lambda}(0) \|^{4}_{L^{4}} \lesssim N^{2(1-s)} \lambda^{1-2s} \| u_{0} \|^{4}_{H^{s}} \label{IuSc} \end{equation} By (\ref{DIuSc}), (\ref{DtIuSc}) and (\ref{IuSc}) we see that there exists $ C_{0}=C_{0} ( \| u_{0} \|_{H^{s}}, \| u_{1} \|_{H^{s-1}} )$ such that if $\lambda$ satisfies \begin{equation} \lambda = C_{0} N^{\frac{{2(1-s)}}{2s-1}} \label{UpperBdLambda} \end{equation} then \begin{equation} E ( I u_{\lambda}(0) ) \leq \frac{1}{2} \label{InitTruncNrjEst} \end{equation} \subsection*{(2) Boundedness of the mollified energy} Let $F_{T}$ denote the set \begin{align*} F_{T}= \big\{ &T' \in [0, T]: \sup_{t \in [0, \lambda T']} E ( I u_{\lambda}(t) ) \leq 1 \text{ and}\\ &\| I u_{\lambda} \|_{L_{t}^{6} ( [0, \lambda T'] ) L_{x}^{6}} \leq (16 C^{2}_{s})^{\frac{1}{6}} +1 \big\} \end{align*} with $C_{s}$ being the constant determined by $\lesssim$ in (\ref{SobolevRad}) and $\lambda$ satisfying (\ref{UpperBdLambda}). We claim that $F_{T}$ is the whole set $[0, T]$ for $N=N(T) \gg 1$ to be chosen later. Indeed \begin{itemize} \item $F_{T} \neq \emptyset$ since $0 \in F_{T}$ by (\ref{InitTruncNrjEst}). \item $F_{T}$ is closed by continuity and by the dominated convergence theorem \item $F_{T}$ is open. Let $\widetilde{T'} \in F_{T}$. By continuity there exists $\delta > 0$ such that for every $T' \in ( \widetilde{T'} - \delta, \widetilde{T'} + \delta ) \cap [0, T]$ we have \begin{gather} \sup_{t \in [0, \lambda T']} E ( I u_{\lambda} (t) ) \leq 2, \label{HypInducNrj} \\ \| I u_{\lambda} \|_{L_{t}^{6} ( [0, \lambda T'] ) L_{x}^{6}} \leq (16 C_{s}^{2})^{\frac{1}{6}} + 2\,. \label{HypInducLgEst} \end{gather} \end{itemize} We are interested in generating a partition $\{ J_{j}\}$ of $[0, \lambda T']$ such that (\ref{Condition1}) and (\ref{Condition2}) are satisfied for all $J_{j}$. We describe now the algorithm. \noindent \emph{Description of the algorithm}. Let $\mathcal{L}$ be the present list of intervals. Let $L$ be the sum of the lengths of the intervals making up $\mathcal{L}$. Let $n$ be the number of the last interval of $\mathcal{L}$. Initially there is no interval and we start from the time $t=0$. Therefore $\mathcal{L}$ is empty and we assign the value $0$ to $L$ and $n$. Then as long as $L < \lambda T'$ do the following \begin{enumerate} \item consider $f_{L}(\tau)= \| I u_{\lambda} \|_{L_{t}^{6} ( [L, L + \tau] ) L_{x}^{6}} - \frac{C_{1}}{\tau^{\frac{1}{3}}}$, $\tau \geq 0 $ with $C_{1}$ defined in (\ref{Condition1}). \item since $f_{L}$ is continuous, does not decrease and $ f_{L}(\tau) \to -\infty$ as $\tau \to 0$, $\tau \geq 0$ there are two options \begin{itemize} \item $f_{L}$ is always negative on $[0, \lambda T'-L]$: in this case if (\ref{Condition2}) is satisfied by $[L, \lambda T']$ then let $J_{n}:=[L, \lambda T']$. If not let $J_{n}:=[L, L + C_{2} N^{\frac{1-s}{s-\frac{1}{2}}}]$. \item $f_{L}$ has one and only one root on $[0, \lambda T' -L]$: in this case let $\tau_{0}$ be this root. If (\ref{Condition2}) is satisfied by $[L, L+ \tau_{0}]$ then let $J_{n}:= [L, L + \tau_{0}]$. If not let $J_{n} := [L, L + C_{2} N^{\frac{1-s}{s-\frac{1}{2}}}]$. \end{itemize} \item assign the value $L + |J_{n}|$ to $L$. \item assign the value $n+1$ to the variable $n$ \item insert $J_{n}$ into $\mathcal{L}$ so that $\mathcal{L}= ( J_{j} )_{j \in \{1, \dots , n\}}$ \end{enumerate} When we apply this algorithm it is not difficult to see that \begin{itemize} \item $\| I u_{\lambda} \|_{L_{t}^{6} (J_{j}) L_{x}^{6}} = \frac{C_{1}}{|J_{j}|^{\frac{1}{3}}}$ or $|J_{j}|= C_{2} N^{\frac{1-s}{s - \frac{1}{2}}}$ for every $j \in \{1, \dots , \mathop{\rm card}(\mathcal{L}) - 1 \} $ \item $J_{j} \cap J_{k} = \emptyset$ for every $(j,k) \in \{1, \dots , \mathop{\rm card}(\mathcal{L}) \}^{2}$ such that $j \neq k$ \item $\bigcup_{j=1}^{\mathop{\rm card} ( \mathcal{L} )} J_{j}$ is a left-closed interval with left endpoint $0$ and included in $[0,\lambda T']$. Moreover $\bigcup_{j=1}^{\mathop{\rm card} ( \mathcal{L} )} J_{j} = [0, \lambda T']$ if the process is finite. \end{itemize} Let \begin{gather} \mathcal{L}_{1} = \{ J_{j}, J_{j} \in \mathcal{L}, \| I u \|_{L_{t}^{6} (J_{j}) L_{x}^{6}} = \frac{C_{1}}{|J_{j}|^{\frac{1}{3}}} \}, \\ \mathcal{L}_{2} = \{ J_{j}, J_{j} \in \mathcal{L}, |J_{j}|=C_{2} N^{\frac{1-s}{s-\frac{1}{2}}} \} \label{Boundm1} \end{gather} We have $(J_{j})_{j \in \{1, \dots , \mathop{\rm card}(\mathcal{L})-1 \}} \subset \mathcal{L}_{1} \cup \mathcal{L}_{2}$. We claim that $ \mathop{\rm card}(\mathcal{L}_{i}) < \infty $, $i=1,2$. If not let us consider the $m_{1}$, $m_{2}$ first elements of $\mathcal{L}_{1}$, $\mathcal{L}_{2}$ respectively. Then \begin{equation} m_{1} C_{2} N^{\frac{1-s}{s-\frac{1}{2}}} \leq \lambda T' \label{Evalm1} \end{equation} By H\"older inequality and by (\ref{HypInducLgEst}) we have \begin{equation} \begin{aligned} m_{2} & = \sum_{j=1}^{m_{2}} |J_{j}|^{-2/3} |J_{j}|^{2/3} \\ & \leq ( \sum_{j=1}^{m_{2}} \frac{1}{|J_{j}|^{2}} )^{\frac{1}{3}} ( \sum_{j=1}^{m_{2}} |J_{j}| )^{2/3} \\ & \leq \| Iu \|^{2}_{L_{t}^{6} ( [0, \lambda T'] ) L_{x}^{6}} ( \lambda T )^{2/3} \\ & \lesssim ( \lambda T )^{2/3} \end{aligned} \label{Evalm2} \end{equation} Letting $m_{1}$ and $m_{2}$ go to infinity in (\ref{Evalm1}) and (\ref{Evalm2}) we have a contradiction. Therefore $\mathop{\rm card} (\mathcal{L}) < \infty $ and $\bigcup_{j=1}^{\mathop{\rm card} ( \mathcal{L} )} J_{j} = [0, \lambda T']$. Moreover by (\ref{UpperBdLambda}), (\ref{Evalm1}), (\ref{Evalm2}), we have \begin{equation} \mathop{\rm card}(\mathcal{L}) \lesssim ( \lambda T )^{2/3} + \frac{\lambda T}{N^{\frac{1-s}{s- \frac{1}{2}}}} + 1 \lesssim N^{\frac{4(1-s)}{6s-3}} T^{2/3} + T + 1 \label{BoundCardL} \end{equation} Now by (\ref{InitTruncNrjEst}), (\ref{HypInducNrj}), (\ref{BoundCardL}), proposition \ref{prop:LocalBdRad}, \ref{prop:EstNrjRad}, \ref{prop:AlmMorawetz} and \ref{prop:RemainderMor} we get, after iterating, \begin{equation} \sup_{t \in [0, \lambda T']} E (I u_{\lambda}(t)) - \frac{1}{2} \lesssim \frac{N^{\frac{4(1-s)}{6s-3}} T^{2/3} + T + 1 }{N^{1-}} \label{EstNrjRadIt} \end{equation} and \begin{equation} \begin{aligned} &\int_{0}^{\lambda T'} \int_{\mathbb{R}^{3}} \frac{|Iu_{\lambda}(t,x)|^{4}}{|x|} \,dxdt - 2 ( E ( Iu_{\lambda}(\lambda T') ) + E(Iu_{\lambda}(0)) )\\ & \lesssim \sum_{i=1}^{2} \sum_{j=1}^{\mathop{\rm card}(\mathcal{L}_{i})} R_{i}(J_{j}) \\ & \lesssim \frac{N^{\frac{4(1-s)}{6s-3}} T^{2/3} + T + 1 }{N^{1-}} \end{aligned} \label{EstLgEstRadIt} \end{equation} By (\ref{SobolevRad}), (\ref{HypInducNrj}), (\ref{EstLgEstRadIt}) and the inequality $ (1+x)^{\frac{1}{6}} \leq 1+x$, $x \geq 0$ \begin{equation} \| I u_{\lambda} \|_{L_{t}^{6} ( [0, \lambda T']) L_{x}^{6}} - (16 C_{s}^{2})^{\frac{1}{6}} \lesssim \frac{ N^{\frac{4(1-s)}{6s-3}} T^{2/3} + T + 1 }{N^{1-}} \label{EstLgEstRadIt2} \end{equation} Let $C'$, $C''$ be the constant determined by $\lesssim$ in (\ref{EstNrjRadIt}), (\ref{EstLgEstRadIt2}) respectively. Since $s>\frac{7}{10}$ we can always choose for every $T>0$ a $N=N(T) \gg 1$ such that \begin{gather} \frac{ \max{( C', C'' )} N^{\frac{4(1-s)}{6s-3}} T^{2/3} }{N^{1-}} \leq \frac{1}{6}, \label{ChoiceN1} \\ \frac{\max{( C', C'' )} T }{N^{1-}} \leq \frac{1}{6}, \label{ChoiceN2} \\ \frac{\max{(C', C'')}}{N^{1-}} \leq \frac{1}{6}\,. \label{ChoiceN3} \end{gather} By (\ref{EstNrjRadIt}), (\ref{EstLgEstRadIt2}), (\ref{ChoiceN1}), (\ref{ChoiceN2}) and (\ref{ChoiceN3}) we have $\sup_{t \in [0, \lambda T']} E(I u_{\lambda}(t)) \leq 1 $ and $\| I u_{\lambda} \|_{L_{t}^{6} ( [0, \lambda T'] ) L_{x}^{6}} \leq (16 C_{s}^{2})^{\frac{1}{6}} + 1$. Hence $F_{T}=[0, T]$ with $N=N(T)$ satisfying (\ref{ChoiceN1}), (\ref{ChoiceN2}) and (\ref{ChoiceN3}). \subsection*{(3) Conclusion} Following the $I$-method described in \cite{almckstt} \begin{equation} \sup_{t \in [0, T]} E ( I u(t) ) = \lambda \sup_{t \in [0, \lambda T]} E ( (I u)_{\lambda}(t) ) \lesssim \lambda \sup_{t \in [0, \lambda T]} E (I u_{\lambda}(t)) \lesssim \lambda \label{ComparNrjScaleRad} \end{equation} Combining (\ref{ComparNrjScaleRad}) and proposition \ref{Prop:NrjEst} we have global well-posedness. Now let $T$ be large. If $ \frac{5}{6} \geq s > \frac{7}{10}$ then let $N$ such that \begin{equation} \frac{0.9}{6} \leq \frac{ \max{( C', C'' )} N^{\frac{4(1-s)}{6s-3}} T^{2/3} } {N^{1-}} \leq \frac{1}{6} \label{CondNRad1} \end{equation} Notice that (\ref{ChoiceN2}) and (\ref{ChoiceN3}) are also satisfied. We plug (\ref{CondNRad1}) into (\ref{ComparNrjScaleRad}) and we apply proposition \ref{Prop:NrjEst} to get (\ref{LgEstRaduT1}). If $1 > s > \frac{5}{6}$ then let $N$ such that \begin{equation} \frac{0.9}{6} \leq \frac{ \max{( C', C'' )} T } {N^{1-}} \leq \frac{1}{6} \label{CondNRad2} \end{equation} Notice that (\ref{ChoiceN1}) and (\ref{ChoiceN3}) are also satisfied. We plug (\ref{CondNRad2}) into (\ref{ComparNrjScaleRad}) and we apply proposition \ref{Prop:NrjEst} to get (\ref{LgEstRaduT2}). \section{Proof of the $H^{s}$ norms and mollified energy estimates} \label{sec:RelHsNrj} In this section we are interested in proving proposition \ref{Prop:NrjEst}. By Plancherel theorem \begin{equation*} \| u(T) \|^{2}_{H^{s}} \lesssim \| P_{\leq 1} u(T) \|^{2}_{H^{s}} + \int_{1\leq |\xi| \leq 2N} |\xi|^{2s} |\hat{u}(T,\xi)|^{2} d\xi + \int_{|\xi| \geq 2N}|\xi|^{2s} \left| \hat{u}(T,\xi) \right|^{2} \,d\xi \end{equation*} But \begin{equation} \begin{aligned} \int_{1 \leq |\xi| \leq 2N} |\xi|^{2s} \left| \hat{u}(T,\xi) \right|^{2} \,d\xi & \leq \int_{|\xi| \leq 2N} |\xi|^{2} \left| \hat{u}(T,\xi) \right|^{2} \,d\xi \\ & \lesssim \int_{\mathbb{R}^{3}} \left| D I u (T,x) \right|^{2} \,dx \\ & \lesssim E ( Iu(T) ) \end{aligned} \label{HighFrequEIu1} \end{equation} \begin{equation} \begin{aligned} \int_{|\xi| \geq 2N} |\xi|^{2s} |\hat{u}(T,\xi)|^{2} \,d\xi & \leq \int_{|\xi| \geq 2N} |\xi|^{2} \frac{N^{2(1-s)}}{|\xi|^{2(1-s)}} \left| \hat{u}(T,\xi) \right|^{2} \,d\xi \\ & \lesssim \int_{\mathbb{R}^{3}} \left| D I u (T,x) \right|^{2} \,dx \\ & \lesssim E ( Iu(T) ) \end{aligned} \label{HighFreqEIu2} \end{equation} and by the fundamental theorem of calculus and Minkowski inequality \begin{equation} \begin{aligned} \| P_{\leq 1} u(T) \|_{H^{s}} & \lesssim \| P_{\leq 1} u_{0} \|_{H^{s}}+ \int_{0}^{T} \| P_{\leq 1} \partial_{t} u(t) \|_{H^{s}} \,dt \\ & \lesssim \| u_{0} \|_{H^{s}} + T \sup_{t \in [0, T]} \| \partial_{t} I u(t) \|_{L^{2}} \end{aligned} \end{equation} which implies that \begin{equation} \| P_{\leq 1} u(T) \|^{2}_{H^{s}} \lesssim \| u_{0} \|^{2}_{H^{s}} + T^{2} \sup_{t \in [0, T]} E ( Iu(t) ) \label{LowFrequEIu} \end{equation} We also have \begin{equation} \| \partial_{t} u(T) \|^{2}_{H^{s-1}} \lesssim E ( Iu(T)) \label{DerivuEIu} \end{equation} Combining (\ref{HighFrequEIu1}), (\ref{HighFreqEIu2}), (\ref{LowFrequEIu}) and (\ref{DerivuEIu}) we get (\ref{NrjEstim1}). \section{Proof of the local boundedness estimate} \label{sec:LocalBd} We are interested in proving proposition \ref{prop:LocalBdRad} in this section. In what follows we also assume that $J=[0, \tau]$: the reader can check after reading the proof that the other cases can be reduced to that one. Before starting the proof let us state the following lemma. \begin{lemma}[Strichartz estimates with derivative] Let $m \in [0, 1]$ and $ 0 \leq \tau < \infty$. If $u$ satisfies the IVP problem \begin{equation} \begin{gathered} \Box u = F \\ u(t=0) = f \\ \partial_{t} u(t=0) = g \end{gathered} \label{EqWaveGen} \end{equation} then we have the $m$-Strichartz estimate with derivative \begin{equation} \| u \|_{L_{t}^{q}( [0, \tau] ) L_{x}^{r}} + \| \partial_{t} D^{-1} u \|_{L_{t}^{q} ( [0, \tau] ) L_{x}^{r}} \lesssim \| f \|_{\dot{H}^{m}} + \| g \|_{\dot{H}^{m-1}} + \| F \|_{L_{t}^{\tilde{q}} ( [0, \tau] ) L_{x}^{\tilde{r}}} \label{StrEstDeriv} \end{equation} for $(q,r) \in \mathcal{W} $, $(\tilde{q},\tilde{r}) \in \widetilde{\mathcal{W}}$ and $(q,r,\tilde{q},\tilde{r})$ satisfying the gap condition \begin{equation} \frac{1}{q} + \frac{3}{r} = \frac{3}{2} -m = \frac{1}{\tilde{q}} + \frac{3}{\tilde{r}} -2 \end{equation} \label{lem:StrEstDeriv} \end{lemma} We postpone the proof of lemma \ref{lem:StrEstDeriv} to subsection \ref{subsec:PfLemStrDer}. Assuming that is true we now show how lemma \ref{lem:StrEstDeriv} implies proposition \ref{prop:LocalBdRad}. Multiplying the $m$-Strichartz estimate with derivative (\ref{StrEstDeriv}) by $D^{1-m}I$ we get \begin{equation} \begin{aligned} Z_{m,s}(\tau) & \lesssim \| D I u_{0} \|_{L^{2}} + \| I u_{1} \|_{L^{2}} + \| D^{1-m} I F \| _{L_{t}^{\tilde{q}}([0,\tau]) L_{x}^{\tilde{r}} } \\ & \lesssim 1 + \| D^{1-m} I F \|_{L_{t}^{\tilde{q}}([0, \tau]) L_{x}^{\tilde{r}} } \end{aligned} \label{ZEst} \end{equation} The remainder of proof is divided into three steps. \subsection*{First Step} First we assume that $m \leq s$. Notice that the point $(\frac{1}{1-s},6)$ is $s$-wave admissible. In this case we get from the fractional Leibnitz rule the H\"older in time and the H\"older in space inequalities \begin{equation} \begin{aligned} Z_{m,s}(\tau) & \lesssim 1 + \| D^{1-m} I (uuu) \|_{L_{t}^{1}([0, \tau]) L_{x}^{\frac{6}{5-2m}}} \\ & \lesssim 1+ \| D^{1-m} I u \|_{L_{t}^{\infty}([0, \tau]) L_{x}^{\frac{6}{3-2m}}} \| u \|_{L_{t}^{2} ( [0,\tau]) L_{x}^{6}}^{2} \\ & \lesssim 1+ Z_{m,s} (\tau) \Big( \tau^{\frac{1}{3}} \| P_{\leq N} u \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \tau^{s-\frac{1}{2}} \| P_{>N} u \|_{L_{t}^{\frac{1}{1-s}} ([0,\tau]) L_{x}^{6}} \Big)^{2} \\ & \lesssim 1 + Z_{m,s}(\tau) \Big( \tau^{\frac{1}{3}} \| Iu \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \tau^{s-\frac{1}{2}} \frac{ \| D^{1-s} Iu \|_{L_{t}^{\frac{1}{1-s}}( [0, \tau] ) L_{x}^{6}}}{N^{1-s}} \Big)^{2} \\ & \lesssim 1 + Z_{m,s}(\tau) \Big( \tau^{\frac{1}{3}} \| Iu \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \tau^{s-\frac{1}{2}} \frac{Z_{s,s}(\tau)}{N^{1-s}} \Big)^{2} \end{aligned} \label{IndZms} \end{equation} Assume $m=s$. Then if we apply a continuity argument to (\ref{IndZms}) we get, from the inequalities (\ref{Condition1}) and (\ref{Condition2}), \begin{equation} Z_{s,s}(\tau) \lesssim 1 \label{BdZss} \end{equation} Now assume $ms$. By (\ref{IndZms}), (\ref{BdZss}), (\ref{BdZms}), (\ref{Condition1}) and (\ref{Condition2}) we have \begin{equation} \| D^{1-r} I (uuu) \|_{L_{t}^{1} ([0,\tau]) L_{x}^{\frac{6}{5-2r}}} \lesssim Z_{r,s}(\tau) ( \tau^{\frac{1}{3}} \| Iu \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \frac{ \tau^{s-\frac{1}{2}} Z_{s,s}(\tau)}{N^{1-s}} )^{2} \lesssim 1 \label{BdFcTerm} \end{equation} for $r \leq s$. The inequality \begin{equation} \| D^{1-m} I (uuu) \|_{L_{t}^{1} ([0,\tau]) L_{x}^{\frac{6}{5-2m}}} \lesssim \| D^{1-r} I (uuu) \|_{L_{t}^{1} ([0,\tau]) L_{x}^{\frac{6}{5-2r}}} \label{SobIneqm} \end{equation} follows from the application of Sobolev homogeneous embedding. We get from (\ref{ZEst}), (\ref{BdFcTerm}) and (\ref{SobIneqm}) \begin{equation} \begin{aligned} Z_{m,s}(\tau) & \lesssim 1 + \| D^{1-m} I (uuu) \|_{L_{t}^{1} ( [ 0, \tau] ) L_{x}^{\frac{6}{5-2m}}} \\ & \lesssim 1 + \| D^{1-r} I (uuu) \|_{L_{t}^{1} ( [ 0, \tau] ) L_{x}^{\frac{6}{5-2r}}} \lesssim 1 \end{aligned} \end{equation} \subsection{Proof of lemma \ref{lem:StrEstDeriv}} \label{subsec:PfLemStrDer} By decomposition it suffices to prove that $u_{l}^{1}(t)=e^{ \pm itD}f$, $u_{l}^{2}(t)=\frac{e^{\pm itD}}{D}g$ and $u_{n}(t)= \int_{0}^{t} D^{-1} \sin { ( (t-t')D ) } F \,dt'$ satisfy (\ref{StrEstDeriv}). We have $ \partial_{t} u_{l}^{1}(t)= \pm i D e^{ \pm itD} f $ and $\partial_{t} u_{l}^{2} = \pm e^{\pm it D} g$. We know from the Strichartz estimates that \begin{equation} \| D^{-1} \partial_{t} u_{l}^{1} \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r} } \lesssim \| e^{\pm it D} f \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} \lesssim \| f \|_{\dot{H}^{m}} \label{Lin1} \end{equation} and \begin{equation} \| D^{-1} \partial_{t} u_{l}^{2} \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} = \| e^{\pm itD} D^{-1} g \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} \lesssim \| D^{-1} g \|_{\dot{H}^{m}} \lesssim \| g \|_{\dot{H}^{m-1}} \label{Lin2} \end{equation} We also have \begin{equation} D^{-1} \partial_{t} u_{n}(t) = \int_{0}^{t} \cos{ ( (t-t')D ) } F(t{'}) \,dt' \end{equation} and by the Strichartz estimates \begin{equation} \begin{aligned} \| D^{-1} \partial_{t} u_{n} \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} & \lesssim \| \int_{0}^{t} D^{-1} e^{ i(t-t') D } F(t') dt' \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} \\ &\quad+ \| \int_{0}^{t} D^{-1} e^{ -i(t-t') D } F(t') dt' \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}}\\ &\lesssim \| F \|_{L_{t}^{\tilde{q}} ([0,\tau]) L_{x}^{\tilde{r}}} \end{aligned} \label{ResNlin} \end{equation} Inequality (\ref{StrEstDeriv}) follows from (\ref{Lin1}), (\ref{Lin2}) and (\ref{ResNlin}). \section{Proof of almost conservation law} \label{sec:AlmCon} Now we prove proposition \ref{prop:EstNrjRad}. In what follows we also assume that $J=[0,\tau]$: the reader can check after reading the proof that the other cases can be reduced to that one. Let $\tau_{0} \in J$. It suffices to prove \begin{equation} | E ( Iu(\tau_{0}) ) - E ( Iu(0) ) | \lesssim \frac{Z^{4}(\tau)}{N^{1-}} \end{equation} In what follows we also assume that $\tau_{0}= \tau$: the reader can check after reading the proof that the other cases can be reduced to this one. The Plancherel formula and the fundamental theorem of calculus yield \begin{align*} & E ( Iu(\tau) ) - E ( Iu(0) ) \\ & = \int_{0}^{\tau} \int_{\xi_{1} + \dots + \xi_{4} = 0} \mu(\xi_{2},\xi_{3},\xi_{4}) \widehat{\partial_{t} I u}(t,\xi_{1}) \widehat{I u}(t,\xi_{2}) \widehat{I u} (t,\xi_{3}) \widehat{I u} (t,\xi_{4}) \,d\xi_{2} d\xi_{3} d \xi_{4} dt \end{align*} with \begin{equation} \mu(\xi_{2},\xi_{3},\xi_{4}) = 1 - \frac{m(\xi_{2}+ \xi_{3} + \xi_{4})}{m(\xi_{2}) m(\xi_{3}) m(\xi_{4})} \label{DfnMu} \end{equation} It is left to prove that \begin{equation} \begin{aligned} &\big| \int_{0}^{\tau} \int_{\xi_{1} + \dots + \xi_{4} = 0} \mu(\xi_{2},\xi_{3},\xi_{4}) \widehat{\partial_{t} I u}(t,\xi_{1}) \hat{I u}(t, \xi_{2}) \widehat{I u }(t,\xi_{3}) \widehat{I u} (t, \xi_{4}) \,d\xi_{2}d\xi_{3}d \xi_{4} \,dt \big| \\ & \lesssim \frac{Z^{4}(\tau)}{N^{1-}} \end{aligned} \label{AlmConsToProve} \end{equation} We perform a Paley-Littlewood decomposition to prove (\ref{AlmConsToProve}). Let $u_{i}=P_{N_{i}} u$ with $i \in \{1,\dots , 4 \}$ and let \begin{equation} \begin{aligned} X &= \Big| \int_{0}^{\tau} \int_{\xi_{1}+\dots +\xi_{4}=0} \mu(\xi_{2},\xi_{3},\xi_{4}) \widehat{\partial_{t} I u_{1}}(t, \xi_{1}) \widehat{I u_{2}}(t,\xi_{2}) \\ &\quad \times \widehat{I u_{3}}(t,\xi_{3}) \widehat{I u_{4}} (t,\xi_{4}) \,d\xi_{2}d\xi_{3}d \xi_{4}dt \Big| \end{aligned} \label{DfnXAlmRad} \end{equation} There are different cases resulting from this Paley-Littlewood analysis and we describe now the strategy to estimate (\ref{AlmConsToProve}). We suggest that the reader at first ignores the second and third steps of the description and the $N_{j}^{\pm}$ appearing in the study of these cases to solve the summation issue. \noindent\emph{Description of the strategy} \subsection*{(1)} We follow \cite{morckstt} to estimate $X$. First we recall the following Coifman-Meyer theorem \cite{coifmey2}, p179 for a class of multilinear operators \begin{theorem}[Coifman Meyer multiplier theorem] Consider an infinitely differentiable symbol $\sigma: \mathbb{R}^{nk} \to \mathbb{C}$ so that for all $\alpha \in N^{nk}$ there exists $c(\alpha)$ such that for all $ \xi= ( \xi_{1},\dots ,\xi_{k} ) \in \mathbb{R}^{nk}$ \begin{equation} | \partial_{\xi}^{\alpha} \sigma (\xi) | \leq \frac{c(\alpha)} { ( 1+ |\xi| )^{| \alpha|}} \label{MeyMultBound} \end{equation} Let $\Lambda_{\sigma}$ be the multilinear operator \begin{equation} \Lambda_{\sigma} (f_{1},\dots ,f_{k})(x) = \int_{\mathbb{R}^{nk}} e^{ix \cdot (\xi_{1}+\dots +\xi_{k})} \sigma(\xi_{1},\dots ,\xi_{k}) \widehat{f_{1}}(\xi_{1})\dots \widehat{f_{k}}(\xi_{k}) \,d \xi_{1}\dots d \xi_{k} \label{ThmCoifMey} \end{equation} Assume that $q_{j} \in (1,\infty)$, $j \in \{1, \dots , k \}$ are such that $\frac{1}{q}=\frac{1}{q_{1}}+\dots +\frac{1}{q_{k}} \leq 1$. Then there is a constant $C=C ( q_{j},n,k,c(\alpha) )$ so that for all Schwarz class functions $f_{1},\dots ,f_{k}$ \begin{equation} \| \Lambda_{\sigma}(f_{1},\dots ,f_{k}) \|_{L^{q}(\mathbb{R}^{n})} \leq C \| f_{1} \|_{L^{q_{1}} (\mathbb{R}^{n})}\dots \| f_{k} \|_{L^{q_{k}} (\mathbb{R}^{n})} \end{equation} \label{thm:CoifMeyer} \end{theorem} Then we proceed as follows. We seek a pointwise bound on the symbol \begin{equation} \left| \mu(\xi_{2},\xi_{3},\xi_{4}) \right| \leq B ( N_{2},N_{3},N_{4} ) \label{DfnB} \end{equation} We factor $B=B(N_{2},N_{3},N_{4})$ out of the right side of (\ref{DfnXAlmRad}) and we are left to evaluate \begin{equation*} B \int_{0}^{\tau} \int_{\mathbb{R}^{3}} \widehat{\Lambda_{\frac{\mu}{B}} ( \partial_{t} I u_{1}(t), I u_{2}(t), I u_{3}(t) )} (\xi_{4}) \widehat {I u_{4}}(t,\xi_{4}) \,d \xi_{4} \,dt \end{equation*} We notice that the multiplier $\frac{\mu}{B}$ satisfy the bound (\ref{MeyMultBound}) and by the Plancherel theorem, H\"older inequality, theorem \ref{thm:CoifMeyer} and Bernstein inequalities we have \begin{equation} \begin{aligned} X & \lesssim B \| \partial_{t} I u_{1} \|_{L_{t}^{p_{1}} ([0, \tau]) L_{x}^{q_{1}} } \| I u_{2} \|_{L_{t}^{p_{2}} ([0, \tau]) L_{x}^{q_{2}}}\dots \| I u_{4} \|_{L_{t}^{p_{4}} ([0, \tau]) L_{x}^{q_{4}}} \\ & \lesssim B N_{1}^{m_{1}} N_{2}^{m_{2}-1}\dots N_{4}^{m_{4}-1} \| \partial_{t} D^{-m_{1}} I u_{1} \|_{L_{t}^{p_{1}} ([0, \tau]) L_{x}^{q_{1}} } \\ &\quad\times \| D^{1-m_{2}} I u_{2} \|_{L_{t}^{p_{2}} ([0, \tau]) L_{x}^{q_{2}}}\dots \| D^{1-m_{4}} I u_{4} \|_{L_{t}^{p_{4}} ([0, \tau]) L_{x}^{q_{4}}} \\ & \lesssim B N_{1}^{m_{1}} N_{2}^{m_{2}-1}\dots N_{4}^{m_{4}-1} Z^{4}(\tau) \end{aligned} \label{AlmconsRadpj} \end{equation} with $(p_{j},q_{j})$ such that $p_{j} \in [1, \infty]$ and $q_{j} \in (1, \infty)$ for $j=\{1, \dots , 4 \}$, $\sum_{j=1}^{4} \frac{1}{p_{j}}= 1$, $\sum_{j=1}^{4} \frac{1}{q_{j}} =1$, $(p_{j},q_{j})$ $m_{j}$-wave admissible for some $m_{j}'s$ such that $0 \leq m_{j} < 1$ and $\frac{1}{p_{j}} + \frac{1}{q_{j}} =\frac{1}{2}$. In other words $(p_{j},q_{j}) = ( \frac{2}{m_{j}}, \frac{2}{1-m_{j}} )$. \subsection*{(2)} The series must be summable. Therefore in some cases we might create $N_{k}^{\pm}$ for some $k's$ by considering slight variations $(p_{k} \pm , q_{k} \pm) \in [1, \infty] \times (1, \infty)$ of $(p_{k},q_{k})$ that are $m_{k} \pm$ - wave admissible and such that $\frac{1}{p_{k} \pm} + \frac{1}{q_{k} \pm} =\frac{1}{2}$. For instance if we create slight variations $(p_{2}+,q_{2}-)$, $(p_{4}-,q_{4}+)$ of $(p_{2},q_{2})$, $(p_{4},q_{4})$ respectively we have \begin{equation} \begin{gathered} \| I u_{2} \|_{L_{t}^{p_{2}+} L_{x}^{q_{2}-}} \lesssim N_{2}^{-} N_{2}^{m_{2}-1} \| D^{1-(m_{2}-)} I u_{2} \|_{L_{t}^{p_{2}+} L_{x}^{q_{2}-}} \\ \| I u_{4} \|_{L_{t}^{p_{4}-} L_{x}^{q_{4}+}} \lesssim N_{4}^{+} N_{4}^{m_{4}-1} \| D^{1-(m_{4}+)} I u_{4} \|_{L_{t}^{p_{4}-} L_{x}^{q_{4}+}} \end{gathered} \label{Ex1DirectCreat} \end{equation} and (\ref{AlmconsRadpj}) becomes \begin{equation} X \lesssim B N_{2}^{-} N_{4}^{+} N_{1}^{m_{1}} N_{2}^{m_{2}-1}\dots N_{4}^{m_{4}-1} Z^{4}(\tau) \end{equation} \subsection*{(3)} When we deal with low frequencies, i.e $N_{k}<1$ for some $k \in \{1,\dots ,4 \}$ we might consider generating $N_{k}^{+}$ by creating a variation $(2+, \infty-)$ of $(2,\infty)$. Such a task cannot be directly performed since we unfortunately have \begin{equation} \| I u_{k} \|_{L_{t}^{2+} L_{x}^{\infty -}} \lesssim N_{k}^{-} \| D^{1-(1-)} I u_{k} \|_{L_{t}^{2+} L_{x}^{\infty -}} \lesssim N_{k}^{-} Z(\tau) \label{Ex2DirectCreat} \end{equation} But we can indirectly create $N_{k}^{+}$ by appropriately using H\"older in time inequality. Indeed if $\epsilon > 0$, $\epsilon' > 0 $ and $ \epsilon'' > 0 $ are such that $\frac{\epsilon}{2}= \frac{\epsilon'}{2} - \frac{\epsilon''}{3}$ we get from Bernstein inequalities, H\"older in time inequality and Sobolev homogeneous embedding \begin{equation} \begin{aligned} \| I u_{k} \|_{L_{t}^{\frac{2}{1- \epsilon}}(([0,\tau])) L_{x}^{\frac{2}{\epsilon}}} & \lesssim N_{k}^{\epsilon'} \| D^{-\epsilon'} I u_{k} \|_{L_{t}^{\frac{2}{1- \epsilon}}(([0,\tau])) L_{x}^{\frac{2}{\epsilon}}} \\ & \lesssim N_{k}^{\epsilon'} \tau^{\frac{\epsilon' -\epsilon}{2}} \| D^{-\epsilon'} I u_{k} \|_{L_{t}^{\frac{2}{1- \epsilon'}}(([0,\tau])) L_{x}^{\frac{2}{\epsilon}}} \\ & \lesssim N_{k}^{\epsilon'} \tau^{\frac{\epsilon' -\epsilon}{2}} \| D^{-\epsilon'+ \epsilon''} I u_{k} \|_{L_{t}^{\frac{2}{1- \epsilon'}}(([0,\tau])) L_{x}^{\frac{2}{\epsilon'}}} \\ & \lesssim N_{k}^{\epsilon'' - \epsilon'} \tau^{\frac{\epsilon' -\epsilon}{2}} \| D^{ 1- (1- \epsilon')} I u_{k} \|_{L_{t}^{\frac{2}{1- \epsilon'}}(([0,\tau])) L_{x}^{\frac{2}{\epsilon'}}} \\ & \lesssim N_{k}^{\epsilon'' - \epsilon'} \tau^{\frac{\epsilon' -\epsilon}{2}} Z(\tau) \end{aligned} \label{IndirectStepsQuad} \end{equation} We would like $\epsilon'' > \epsilon'$. A quick computation show that it suffices that $ \epsilon' > 3 \epsilon $. Letting $\epsilon'= 5 \epsilon$ we get \begin{equation} \| I u_{k} \|_{L_{t}^{\frac{2}{1- \epsilon}}(([0,\tau])) L_{x}^{\frac{2}{\epsilon}}} \lesssim N_{k}^{\epsilon} \tau^{ 2 \epsilon } Z(\tau) \end{equation} Now if we choose $\epsilon > 0 $ so small that $ |\tau|^{2 \epsilon} \leq 2 $ we eventually get \begin{equation} \| I u_{k} \|_{L_{t}^{2+}(([0,\tau])) L_{x}^{\infty -}} \lesssim N_{k}^{+} Z(\tau) \label{IndirectRes} \end{equation} For the remainder of the paper we say that we directly create $N_{k}^{\pm}$ if we directly use Bernstein inequality like in (\ref{Ex1DirectCreat}) or (\ref{Ex2DirectCreat}) and we say that we indirectly create $N_{k}^{+}$ if we also use H\"older in time inequality to get (\ref{IndirectRes}). This completes the general description of the strategy. \smallskip Let us get back to the proof. By symmetry we may assume that $N_{2} \geq N_{3} \geq N_{4}$. There are several cases. \subsection*{Case 1: $N \gg N_{2} \geq N_{3}$} In this case $X=0$ since $\mu=0$. \subsection*{Case 2: $N_{2} \gtrsim N \gg N_{3}$} In this case we have \begin{equation} | \mu(\xi_{2},\dots ,\xi_{4}) | \lesssim \frac{|\nabla m(\xi_{2})| |\xi_{3} + \xi_{4}|}{m(\xi_{2})} \lesssim \frac{N_{3}}{N_{2}} \label{EstMultRad1} \end{equation} We also get $N_{1} \sim N_{2}$ from the convolution constraint $\xi_{1}+\dots + \xi_{4}=0$. We assume that $N_{4} \geq 1$. By (\ref{EstMultRad1}) and by the Bernstein inequalities we have \begin{align*} X & \lesssim \frac{N_{3}}{N_{2}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{3}}{N_{2}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creating $N_{1}^{+}$ and $N_{4}^{-}$. If $N_{4} < 1$ the proof is similar except that we indirectly create $N_{4}^{+}$ to get $X \lesssim \frac{N_{2}^{-} N_{4}^{+}}{N^{1-}} Z^{4}(\tau) $. This makes the summation possible. We get (\ref{AlmConsToProve}) after summation. \subsection*{Case 3: $ N_{3} \gtrsim N \gg N_{4}$} In this case we have \begin{equation} \left| \mu(\xi_{2},\dots,\xi_{4}) \right| \lesssim \frac{m(\xi_{1})}{m(\xi_{2}) m(\xi_{3}) m(\xi_{4})} \label{EstMultRad2} \end{equation} There are two subcases: \subsection*{Case 3.a: $N_{1} \sim N_{2}$} We assume that $N_{4} \geq 1$. By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{3}^{1-s}}{N^{1-s}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creating $N_{1}^{+}$ and $N_{4}^{-}$. If $N_{4} < 1$ the proof is similar except that we indirectly create $N_{4}^{+}$. We get (\ref{AlmConsToProve}) after summation. \subsection*{Case 3.b: $N_{1} \ll N_{2}$} In this case by the convolution constraint $\xi_{1}+\dots +\xi_{4}=0$ we have $N_{2} \sim N_{3}$. There are two subcases \subsection*{Case 3.b.1: $N_{1} \ll N$} We assume that $N_{1} \geq 1$ and $N_{4} \geq 1$. By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \\ & \quad\times \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{1}^{-} N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creating $N_{1}^{+}$ and $N_{4}^{-}$. If $N_{1} < 1$ and $N_{4} < 1$ the proof is similar except that we indirectly create $N_{4}^{+}$ and we substitute $N_{1}^{-}$ for $N_{1}^{+}$. The proof for the other cases \footnote{i.e $N_{1} \geq 1$, $N_{4} \leq 1$ or $N_{1} \leq 1$, $N_{4} \geq 1$} is a slight variant to that for the case $N_{1} \geq 1$, $N_{4} \geq 1$ and that for the case $N_{1} < 1$, $N_{4}<1$. Details are left to the reader. We get (\ref{AlmConsToProve}) after summation. \subsection*{Case 3.b.2: $N_{1} \gg N$} We assume that $N_{4} \geq 1$. By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{2}^{2(1-s)}}{N^{1-s} N_{1}^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \\ & \quad\times \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{2}^{2(1-s)}}{N^{1-s} N_{1}^{1-s}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creating $N_{1}^{+}$ and $N_{4}^{-}$. If $N_{4} < 1$ the proof is similar except that we indirectly create $N_{4}^{+}$. We get (\ref{AlmConsToProve}) after summation. % N_{1}\ll N_{2} N_{1} \lesssim N % N_{3} \gtrsim N \gg N_{4} N_{1} \sim N_{2} \subsection*{Case 4: $N_{4} \gtrsim N$} There are two subcases. \subsection*{Case 4.a: $N_{1} \sim N_{2}$} By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} \frac{N_{4}^{1-s}}{N^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{2} \|_{L_{t}^{4+} ([0,\tau]) L_{x}^{4-}} \| I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ &\quad\times \| I u_{4} \|_{L_{t}^{4-} ([0,\tau]) L_{x}^{4+}} \\ & \lesssim N_{2}^{-} N_{4}^{+} \frac{N_{3}^{1-s}}{N^{1-s}} \frac{N_{4}^{1-s}} {N^{1-s}} N_{1}^{\frac{1}{2}} \frac{1}{N_{2}^{\frac{1}{2}}} \frac{1}{N_{3}^{\frac{1}{2}}} \frac{1}{N_{4}^{\frac{1}{2}}} \| \partial_{t} D^{- \frac{1}{2} } I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ &\quad\times \| D^{1- ( \frac{1}{2} - ) } I u_{2} \|_{L_{t}^{4+} ([0,\tau]) L_{x}^{4-}} \| D^{1-\frac{1}{2}} I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1- ( \frac{1}{2} + ) } I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{2}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creating $N_{2}^{-}$ and $N_{4}^{+}$. We get (\ref{AlmConsToProve}) after summation. \subsection*{Case 4.b: $N_{1} \ll N_{2}$} In this case we have $N_{2} \sim N_{3}$. There are two subcases \subsection*{Case 4.b.1: $N_{1} \gtrsim N$} By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} \frac{N^{1-s}}{N_{1}^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{2} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ &\quad\times \| I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} \frac{N^{1-s}}{N_{1}^{1-s}} N_{1}^{\frac{1}{2}} \frac{1}{N_{2}^{\frac{1}{2}}} \frac{1}{N_{3}^{\frac{1}{2}}} \frac{1}{N_{4}^{\frac{1}{2}}} \| \partial_{t} D^{- \frac{1}{2} } I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ &\quad\times \| D^{1- \frac{1}{2} } I u_{2} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1-\frac{1}{2}} I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1- \frac{1}{2} } I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{2}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} We get (\ref{AlmConsToProve}) after summation. \subsection*{Case 4.b.2: $N_{1} \ll N$} We assume that $N_{1} \geq 1$. We have \begin{align*} X & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{2} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \quad\times \| I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} N_{1}^{\frac{1}{2}} \frac{1}{N_{2}^{\frac{1}{2}}} \frac{1}{N_{3}^{\frac{1}{2}}} \frac{1}{N_{4}^{\frac{1}{2}}} \| \partial_{t} D^{- \frac{1}{2} } I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1- \frac{1}{2} } I u_{2} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \quad\times \| D^{1-\frac{1}{2}} I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1- \frac{1}{2} } I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{1}^{-} N_{2}^{-} }{N^{1-}} Z^{4}(\tau) \end{align*} If $N_{1} <1$ the proof is similar except that we create $N_{1}^{+}$ instead of $N_{1}^{-}$. We get (\ref{AlmConsToProve}) after summation. \section{Proof of Almost Morawetz-Strauss inequality} \label{sec:AlmMorIneq} We prove proposition \ref{prop:AlmMorawetz} in this section. The proof is divided into two steps \subsection*{First Step: Morawetz-Strauss inequality} We recall the proof of this inequality in \cite{mor,morstr}. We have the identity \begin{equation} \begin{aligned} &\big( \frac{x.\nabla u}{|x|} + \frac{u}{|x|} \big) ( u_{tt} - \triangle u + u^{3} )\\ &= \partial_{t} ( \frac{1}{|x|} ( x.\nabla u + u ) \partial_{t}u ) + \mathop{\rm div} \Big[ \frac{1}{|x|} \Big( -\frac{1}{2} ( \partial_{t} u )^{2} -( x. \nabla u) \nabla u \\ &\quad +\frac{1}{2}|\nabla u|^{2}x -u \nabla u -\frac{u^{2}}{2 |x|^{2}} x + \frac{1}{4} u^{4} x \Big) \Big] + \frac{1}{|x|} ( |\nabla u|^{2} -\frac{( x. \nabla u)^{2}}{|x|^{2}} ) + \frac{u^{4}}{2 |x|} \end{aligned} \label{Identity} \end{equation} and since $u$ satisfies \eqref{WaveEqRad} we have, after integration, \begin{align*} &2 \pi \int_{0}^{T} u^{2}(t,0) dt + \int_{0}^{T} \int_{\mathbb{R}^{3}} \frac{u^{4}(t,x)}{2 |x|} dx \,dt \\ & = - \int_{\mathbb{R}^{3}} ( \frac{\nabla u (T,x).x}{|x|} + \frac{u(T,x)}{|x|} ) \partial_{t} u (T,x) dx \\ &\quad + \int_{\mathbb{R}^{3}} ( \frac{\nabla u (0,x).x}{|x|} + \frac{u(0,x)}{|x|} ) \partial_{t} u (0,x) dx \end{align*} Now we apply the basic inequality $|ab| \leq \frac{|a|^{2}}{2} + \frac{|b|^{2}}{2}$ to the right hand side of the integral and we get \begin{equation} \begin{aligned} \int_{0}^{T} \int_{\mathbb{R}^{3}} \frac{u^{4}(t,x)}{2 |x|} dx \,dt & \leq \frac{1}{2} \int_{\mathbb{R}^{3}} ( \frac{\nabla u (T,x).x}{|x|} + \frac{u(T,x)}{|x|} )^{2} + (\partial_{t} u)^{2}(T,x) dx \\ &\quad + \frac{1}{2} \int_{\mathbb{R}^{3}} ( \frac{\nabla u (0,x).x}{|x|} + \frac{u(0,x)}{|x|} )^{2} + (\partial_{t} u)^{2}(0,x) dx \end{aligned} \label{MorCauch} \end{equation} We also notice that \begin{equation} \big( \frac{\nabla u.x}{|x|} + \frac{u}{|x|} \big)^{2} = \frac{ ( \nabla u.x )^{2}}{|x|^{2}} + \mathop{\rm div}( \frac{u^{2}}{|x|^{2}} x ) \leq | \nabla u |^{2} + \mathop{\rm div}( \frac{u^{2}}{|x|^{2}} x ) \label{PtwiseEquality} \end{equation} We substitute (\ref{PtwiseEquality}) into (\ref{MorCauch}). We get the Morawetz-Strauss's inequality \begin{equation} \int_{0}^{T} \int_{ \mathbb{R}^{3}} \frac{u^{4}(t,x)}{ |x|} dx \,dt \leq 2 ( E(u(T))+ E(u(0)) ) \end{equation} \subsection*{Second Step} Almost Morawetz-Strauss's inequality. We substitute $u$ for $Iu$ in (\ref{Identity}) and we proceed similarly. We get \begin{equation} \begin{aligned} &\int_{0}^{T} \int_{\mathbb{R}^{3}} \frac{|Iu |^{4}(t,x)}{ |x|} dx \,dt -2 ( E ( Iu(T) ) + E ( Iu(0) ) )\\ & \leq |R_{1}(T) + R_{2}(T)| \\ & \leq |R_{1}(T)| + |R_{2}(T)| \end{aligned} \label{IntermMor} \end{equation} \section{Proof of the integral estimates} \label{subsec:RemainderMor} We are interested in proving proposition \ref{prop:RemainderMor} in this section. In what follows we also assume that $J=[0, \tau]$; the reader can check after reading the proof that the other cases can be reduced to that one. Plancherel formula yields \begin{align*} R_{1}(\tau) & = \int_{0}^{\tau} \int_{\xi_{1}+\dots + \xi_{4}=0} \mu(\xi_{2}, \xi_{3}, \xi_{4}) \widehat{\frac{\nabla I u.x}{|x|}} (t,\xi_{1}) \\ &\quad\times \widehat{I u} (t,\xi_{2}) \widehat{I u}(t,\xi_{3}) \widehat{I u} (t,\xi_{4}) d \xi_{2}\dots d \xi_{4} \, dt \end{align*} and \[ R_{2}(\tau) = \int_{0}^{\tau} \int_{\xi_{1}+\dots +\xi_{4}} \mu(\xi_{2},\xi_{3},\xi_{4}) \widehat{\frac{Iu}{|x|}} (t,\xi_{1}) \widehat{I u} (t,\xi_{2}) \widehat{I u}(t,\xi_{3}) \widehat{I u} (t,\xi_{4}) d \xi_{2}\dots d \xi_{4} \, dt \] with $\mu$ defined in (\ref{DfnMu}). It suffices to prove \begin{equation} \begin{aligned} &\Big| \int_{0}^{\tau} \int_{\xi_{1}+\dots + \xi_{4}=0} \mu(\xi_{2},\xi_{3},\xi_{4}) \widehat{\frac{\nabla I u.x}{|x|}} (t,\xi_{1}) \widehat{I u} (t,\xi_{2}) \widehat{I u}(t,\xi_{3}) \widehat{I u} (t,\xi_{4}) d \xi_{2}\dots d \xi_{4} \, dt \Big| \\ & \lesssim \frac{Z^{4}(\tau)}{N^{1-}} \end{aligned} \label{R1Four} \end{equation} and \begin{equation} \begin{aligned} &\Big| \int_{0}^{\tau} \int_{\xi_{1}+\dots + \xi_{4}=0} \mu(\xi_{2},\xi_{3},\xi_{4}) \widehat{\frac{I u}{|x|}} (t,\xi_{1}) \widehat{I u} (t,\xi_{2}) \widehat{I u}(t,\xi_{3}) \widehat{I u} (t,\xi_{4}) d \xi_{2}\dots d \xi_{4} \, dt \Big| \\ &\lesssim \frac{Z^{4}(\tau)}{N^{1-}} \end{aligned} \label{R2Four} \end{equation} We perform a Paley-Littlewood decomposition to prove (\ref{R1Four}) and (\ref{R2Four}). Let $u_{i}:=P_{N_{i}} u$, $i \in \{2, \dots \,4\}$, $(\frac{\nabla I u \cdot x}{|x|})_{1}:= P_{N_{1}} (\frac{\nabla I u \cdot x}{|x|}) $ and $ (\frac{I u}{|x|})_{1} := P_{N_{1}} (\frac{Iu}{|x|})$. \begin{align*} X_{1} & = \Big| \int_{0}^{\tau} \int_{\xi_{1}+\dots + \xi_{4}=0} \mu(\xi_{2},\xi_{3},\xi_{4}) \\ &\quad \times \widehat{(\frac{\nabla I u.x}{|x|} )_{1}} (t,\xi_{1}) \widehat{I u_{2}} (t,\xi_{2}) \widehat{I u_{3}}(t,\xi_{3}) \widehat{I u_{4}} (t,\xi_{4}) d \xi_{2}\dots d \xi_{4} \, dt \Big| \end{align*} and \begin{align*} X_{2} & = \Big| \int_{0}^{\tau} \int_{\xi_{1}+\dots + \xi_{4}=0} \mu (\xi_{2}, \xi_{3}, \xi_{4}) \widehat{(\frac{I u}{|x|} )_{1}} (t,\xi_{1}) \\ &\quad\times \widehat{I u_{2}} (t,\xi_{2}) \widehat{I u_{3}}(t,\xi_{3}) \widehat{I u_{4}} (t,\xi_{4}) d \xi_{2}\dots d \xi_{4} \, dt \Big| \end{align*} Notice that by Bernstein inequality, H\"older inequality, Plancherel theorem and (\ref{HardyIneq}) we have \begin{equation} \begin{aligned} \big\| \big(\frac{\nabla I u \cdot x}{|x|}\big)_{1} \Big\|_{ L_{t}^{\infty -} ([0,\tau]) L_{t}^{2+}} & \lesssim N_{1}^{+} \Big\| \frac{\nabla I u \cdot x}{|x|} \Big\|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \lesssim N_{1}^{+} \| \nabla I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \lesssim N_{1}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \end{aligned} \label{1Mor} \end{equation} and \begin{equation} \big\| (\frac{ I u}{|x|})_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} \lesssim N_{1}^{+} \big\| \frac{Iu}{|x|} \big\|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \lesssim N_{1}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \label{2Mor} \end{equation} If $p_{j} \in [1, \infty]$ and $q_{j} \in (1, \infty)$, $j \in \{2, \dots , 4\}$ such that $ \frac{1}{(\infty-)} + \sum_{j=2}^{4} \frac{1}{p_{j}} = 1$, $ \frac{1}{(2+)} + \sum_{j=2}^{4} \frac{1}{q_{j}} =1$, $(p_{j},q_{j})$ -$m_{j}$ wave admissible for some $m_{j}^{'} \, s$ such that $0 \leq m_{j} < 1$ and $\frac{1}{p_{j}} + \frac{1}{q_{j}}=\frac{1}{2}$ then we have by the methodology explained in the proof of Proposition \ref{prop:EstNrjRad} \begin{align*} X_{1} & \lesssim B (N_{2},N_{3},N_{4}) \big\| (\frac{\nabla Iu \cdot x}{|x|})_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}}\\ &\quad\times \| I u_{2} \|_{L_{t}^{p_{2}} ([0,\tau]) L_{x}^{q_{2}}} \dots \| I u_{4} \|_{L_{t}^{p_{4}} ([0,\tau]) L_{x}^{q_{4}}} \end{align*} and \[ X_{2} \lesssim B (N_{2},N_{3},N_{4}) \big\| ( \frac{I u}{|x|})_{1} \big\|_{L_{t}^{\infty -} ([0,\tau]) L_{x}^{2+}} \| I u_{2} \|_{L_{t}^{p_{2}} ([0,\tau]) L_{x}^{q_{2}}} \dots \| I u_{4} \|_{L_{t}^{p_{4}} ([0,\tau]) L_{x}^{q_{4}}} \] By symmetry we can assume that $N_{2} \geq N_{3} \geq N_{4}$. There are different cases \subsection*{Case 1: $N >> N_{2} \geq N_{3}$} In this case $X_{1}=0$ and $X_{2}=0$ since $\mu=0$. \subsection*{Case 2: $N_{2} \gtrsim N >> N_{3}$} By (\ref{IndirectRes}), (\ref{EstMultRad1}), (\ref{1Mor}) and (\ref{2Mor}) we have \begin{align*} X_{1} & \lesssim \frac{N_{3}}{N_{2}} \big\| (\frac{\nabla Iu.x}{|x|})_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} \| I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ &\quad\times \| I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim \frac{N_{3}}{N_{2}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ &\quad\times \| D^{1-(1-)} I u_{3}\|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{ N_{2}^{--} N_{4}^{+}}{N^{1-}} Z^{4}(\tau) \end{align*} and \begin{align*} X_{2} & \lesssim \frac{N_{3}}{N_{2}} \big\| (\frac{Iu}{|x|} )_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} \| I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ &\quad\times \| I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim \frac{N_{3}}{N_{2}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \quad \times \| D^{1-(1-)} I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{ N_{2}^{--} N_{4}^{+}}{N^{1-}} Z^{4}(\tau) \end{align*} \subsection*{Case 3: $N_{3} \gtrsim N >> N_{4}$} There are two subcases \subsection*{Case 3.a: $N_{1} \sim N_{2}$} By (\ref{IndirectRes}), (\ref{EstMultRad2}) and (\ref{1Mor}) \begin{equation} \begin{aligned} X_{1} & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} \big\| ( \frac{\nabla Iu.x}{|x|})_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} \| I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| I u_{3} \|_{L_{t}^{2+} ([0,\tau])L_{x}^{\infty-}} \\ &\quad\times \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}}\\ & \quad \times \| D^{1-(1-)} I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{ N_{2}^{--} N_{4}^{+}}{N^{1-}} Z^{4}(\tau) \end{aligned} \label{Case3a} \end{equation} Similarly we get $X_{2} \lesssim \frac{ N_{2}^{--} N_{4}^{+}}{N^{1-}} Z^{4}(\tau)$ after substituting $X_{1}$, $ \| (\frac{\nabla Iu.x}{|x|})_{1} \|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} $ for $X_{2}$, $ \|(\frac{Iu}{|x|})_{1} \|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} $ respectively in (\ref{Case3a}). \subsection*{Case 3.b: $N_{1} << N_{2}$} There are two subcases \subsection*{Case 3.b.1: $N_{1} << N$} \begin{align*} X_{1} & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \big\| ( \frac{\nabla Iu.x}{|x|})_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} \| I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ &\quad \times \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \quad \times \|D^{1-(1-)} I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{ N_{1}^{+} N_{2}^{---} N_{4}^{+}}{N^{1-}} Z^{4}(\tau) \end{align*} Similarly $ X_{2} \lesssim \frac{ N_{1}^{+} N_{2}^{---} N_{4}^{+}}{N^{1-}} Z^{4}(\tau)$. \subsection*{Case 3.b.2: $N_{1} \gtrsim N$} \begin{align*} X_{1} & \lesssim \frac{N_{2}^{2(1-s)}}{N^{1-s} N_{1}^{1-s}} \big\| (\frac{\nabla Iu.x}{|x|})_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} \| I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ &\quad\times \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim \frac{N_{2}^{2(1-s)}}{N^{1-s} N_{1}^{1-s}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \quad \times \| D^{1-(1-)} I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{ N_{2}^{--} N_{4}^{+}}{N^{1-}} Z^{4}(\tau) \end{align*} Similarly $X_{2} \lesssim \frac{ N_{2}^{--} N_{4}^{+}}{N^{1-}} Z^{4}(\tau)$. \subsection*{Case 4: $N_{4} \gtrsim N$} There are two subcases \subsection*{Case 4.a: $N_{1} \sim N_{2}$} \begin{align*} X_{1} & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} \frac{N_{4}^{1-s}}{N^{1-s}} \big\| (\frac{ \nabla Iu.x}{|x|} )_{1} \big\|_{L_{t}^{\infty -} ([0,\tau]) L_{x}^{2+}} \| I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ &\quad \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} \frac{N_{4}^{1-s}}{N^{1-s}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \quad \times \| D^{1-(1-)} I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-}} {N^{1-}} Z^{4}(\tau) \end{align*} Similarly $X_{2} \lesssim \frac{N_{2}^{-}}{N^{1-}} Z^{4}(\tau)$. \subsection*{Case 4.b: $N_{1}<< N_{2}$} There are two subcases \subsection*{Case 4.b.1: $N_{1} \gtrsim N$} We have \begin{align*} X_{1} & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} \frac{N^{1-s}}{N_{1}^{1-s}} \big\| (\frac{ \nabla Iu.x}{|x|})_{1} \big\|_{L_{t}^{\infty -} ([0,\tau]) L_{x}^{2+}} \left\| I u_{2} \right \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ &\quad\times \| I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \|I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} \frac{N^{1-s}}{N_{1}^{1-s}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \quad \times \|D^{1-(1-)} I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \|D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} Similarly $X_{2} \lesssim \frac{N_{2}^{-}}{N^{1-}} Z^{4}(\tau)$. \subsection*{Case 4.b.2: $N_{1} << N$} We have \begin{align*} X_{1} & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} \big\| (\frac{\nabla Iu.x}{|x|} )_{1} \big\|_{L_{t}^{\infty -} ([0,\tau]) L_{x}^{2+}} \left\| I u_{2} \right \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}}\\ &\quad \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} N_{1}^{+} \frac{1}{N_{2}} N_{3}^{+} N_{4}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \| D I u_{2} \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \\ & \quad \times \| D^{1-(1-)} I u_{3} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{1}^{+} N_{2}^{--}}{N^{1-}} Z^{4}(\tau) \end{align*} Similarly $X_{2} \lesssim \frac{N_{1}^{+} N_{2}^{--}}{N^{1-}} Z^{4}(\tau)$. 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