\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 $\mathbb{R}^{3}$ $$\begin{gathered} \partial_{tt} u - \Delta u = -u^{3} \\ u(0,x) = u_{0}(x) \\ \partial_{t} u(0,x) = u_{1}(x) \end{gathered} \label{WaveEqRad}$$ 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 $$u(t) = \cos(tD) u_{0} + D^{-1} \sin(tD) u_{1} - \int_{0}^{t} D^{-1} \sin ( (t-t') D ) u^{3}(t') \,dt'$$ 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 $$\| f \|_{H^{s}} := \| (1+ D^{s}) f \|_{L^{2}(\mathbb{R}^{3})}$$ where $D$ is the operator defined by $$\widehat{Df}(\xi) := |\xi| \hat{f}(\xi)$$ and $\hat{f}$ denotes the Fourier transform $$\hat{f}(\xi) := \int_{\mathbb{R}^{3}} f(x) e^{-i x \cdot \xi} \,dx$$ 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 $$\| u(T) \|_{H^{s}} + \| \partial_{t} u (T) \|_{H^{s-1}} \leq C ( s, ( \| u_{0}\|, \| u_{1} \| )_{H^{s} \times H^{s-1}}, T )$$ 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 $$E(u(t)) := \frac{1}{2} \int_{\mathbb{R}^{3}} (\partial_{t} u)^{2}(x,t) \,dx + \frac{1}{2} \int_{\mathbb{R}^{3}} | D u (x,t)|^{2} \,dx + \frac{1}{4} \int_{\mathbb{R}^{3}} u^{4}(x,t) \,dx$$ 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 $$\| f \|_{\dot{H}^{s}} = \| D^{s} f \|_{L^{2} ( \mathbb{R}^{3} )}$$ 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 $$\| u(T) \|^{2}_{H^{s}} + \| \partial_{t} u(T) \|^{2}_{H^{s-1}} \leq C ( \| u_{0} \|_{H^{s}}, \| u_{1} \|_{H^{s-1}} ) T^{\frac{16s-10}{10s-7}+} \label{LgEstRaduT1}$$ for $\frac{5}{6} \geq s > \frac{7}{10}$ and $$\| u(T) \|^{2}_{H^{s}} + \| \partial_{t} u(T) \|^{2}_{H^{s-1}} \leq C ( \| u_{0} \|_{H^{s}}, \| u_{1} \|_{H^{s-1}} ) T^{\frac{2s}{2s-1}+} \label{LgEstRaduT2}$$ 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 $$\widehat{If}(\xi) := m(\xi) \hat{f}(\xi)$$ where $m(\xi): = \eta ( \frac{\xi}{N} )$, $\eta$ is a smooth, radial, nonincreasing in $|\xi|$ such that $$\eta (\xi) := \begin{cases} 1, & |\xi| \leq 1 \\ ( \frac{1}{|\xi|} )^{1-s}, & |\xi| \geq 2 \end{cases}$$ 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 $$u_{\lambda}(t,x) := \frac{1}{\lambda} u ( \frac{t}{\lambda}, \frac{x}{\lambda} )$$ 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 $$\begin{gathered} \partial_{tt} u - \Delta u = F \\ u(0,x) = f(x) \in \dot{H}^{m} \\ \partial_{t} u(0,x) = g(x) \in \dot{H}^{m-1} \\ \end{gathered} \label{LinearWave}$$ 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., $$\mathcal{W} := \{ (q, r): ( q, r ) \in (2, \infty ] \times [2,\infty), \frac{1}{q}+\frac{1}{r} \leq \frac{1}{2} \} \label{StrCondition1}$$ \item $(\tilde{q}, \tilde{r})$ lie in the dual set $\mathcal{W}'$ of $\mathcal{W}$; i.e., $$\mathcal{W}' := \{ (\tilde{q}, \tilde{r}): \frac{1}{\tilde{q}} + \frac{1}{q}=1, \frac{1}{r}+ \frac{1}{\tilde{r}}=1, (q,r) \in \mathcal{W} \}$$ \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 $$|u(x)| \lesssim \frac{\|u\|_{\dot{H}^{1}}}{|x|^{\frac{1}{2}}} \label{SobolevRad}$$ \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 $$f = \sum_{M \in 2^{\mathbb{Z}}} P_{M} f$$ 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 $$\| u(T) \|^{2}_{H^{s}} + \|\partial_{t} u(T)\|^{2}_{H^{s-1}} \lesssim \| u_{0} \|^{2}_{H^{s}} + ( T^{2}+1 )\sup_{t \in [0, T]} E ( Iu(t) ) \label{NrjEstim1}$$ 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 $$\big| \sup_{t \in J} E(Iu(t)) - E(Iu(a)) \big| \lesssim \frac{Z^{4} (J)}{N^{1-}} \label{EstNrjRad}$$ \label{prop:EstNrjRad} \end{proposition} \begin{proposition}[Almost Morawetz-Strauss inequality] Let$T \geq 0$. Assume that$u$satisfies \eqref{WaveEqRad}. Then $$\int_{0}^{T} \int_{\mathbb{R}^{3}} \frac{\left| Iu \right|^{4}(t,x)}{|x|} dx \,dt -2 ( E ( I u(0) ) + E ( Iu(T) ) ) \lesssim \left| R_{1}(T)\right| + \left| R_{2}(T) \right|\,.$$ \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 $$R_{i}(J) \lesssim \frac{Z^{4}(J)}{N^{1-}}$$ \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 1to be chosen later. Then by Plancherel theorem \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} \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} By homogeneous Sobolev embedding, \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} Hence $$\| I u_{\lambda}(0) \|^{4}_{L^{4}} \lesssim N^{2(1-s)} \lambda^{1-2s} \| u_{0} \|^{4}_{H^{s}} \label{IuSc}$$ 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 $$\lambda = C_{0} N^{\frac{{2(1-s)}}{2s-1}} \label{UpperBdLambda}$$ then $$E ( I u_{\lambda}(0) ) \leq \frac{1}{2} \label{InitTruncNrjEst}$$ \subsection*{(2) Boundedness of the mollified energy} Let$F_{T}denote the set \begin{align*} F_{T}= \big\{ &T' \in [0, T]: \sup_{t \in [0, \lambda T']} E ( I u_{\lambda}(t) ) \leq 1 \text{ and}\\ &\| I u_{\lambda} \|_{L_{t}^{6} ( [0, \lambda T'] ) L_{x}^{6}} \leq (16 C^{2}_{s})^{\frac{1}{6}} +1 \big\} \end{align*} withC_{s}$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 $$m_{1} C_{2} N^{\frac{1-s}{s-\frac{1}{2}}} \leq \lambda T' \label{Evalm1}$$ By H\"older inequality and by (\ref{HypInducLgEst}) we have \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} Lettingm_{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 $$\mathop{\rm card}(\mathcal{L}) \lesssim ( \lambda T )^{2/3} + \frac{\lambda T}{N^{\frac{1-s}{s- \frac{1}{2}}}} + 1 \lesssim N^{\frac{4(1-s)}{6s-3}} T^{2/3} + T + 1 \label{BoundCardL}$$ 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, $$\sup_{t \in [0, \lambda T']} E (I u_{\lambda}(t)) - \frac{1}{2} \lesssim \frac{N^{\frac{4(1-s)}{6s-3}} T^{2/3} + T + 1 }{N^{1-}} \label{EstNrjRadIt}$$ and \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} By (\ref{SobolevRad}), (\ref{HypInducNrj}), (\ref{EstLgEstRadIt}) and the inequality (1+x)^{\frac{1}{6}} \leq 1+x$,$x \geq 0$$$\| I u_{\lambda} \|_{L_{t}^{6} ( [0, \lambda T']) L_{x}^{6}} - (16 C_{s}^{2})^{\frac{1}{6}} \lesssim \frac{ N^{\frac{4(1-s)}{6s-3}} T^{2/3} + T + 1 }{N^{1-}} \label{EstLgEstRadIt2}$$ 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} $$\sup_{t \in [0, T]} E ( I u(t) ) = \lambda \sup_{t \in [0, \lambda T]} E ( (I u)_{\lambda}(t) ) \lesssim \lambda \sup_{t \in [0, \lambda T]} E (I u_{\lambda}(t)) \lesssim \lambda \label{ComparNrjScaleRad}$$ 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 $$\frac{0.9}{6} \leq \frac{ \max{( C', C'' )} N^{\frac{4(1-s)}{6s-3}} T^{2/3} } {N^{1-}} \leq \frac{1}{6} \label{CondNRad1}$$ 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 $$\frac{0.9}{6} \leq \frac{ \max{( C', C'' )} T } {N^{1-}} \leq \frac{1}{6} \label{CondNRad2}$$ 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{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} \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} and by the fundamental theorem of calculus and Minkowski inequality \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} which implies that $$\| P_{\leq 1} u(T) \|^{2}_{H^{s}} \lesssim \| u_{0} \|^{2}_{H^{s}} + T^{2} \sup_{t \in [0, T]} E ( Iu(t) ) \label{LowFrequEIu}$$ We also have $$\| \partial_{t} u(T) \|^{2}_{H^{s-1}} \lesssim E ( Iu(T)) \label{DerivuEIu}$$ 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 thatJ=[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 $$\begin{gathered} \Box u = F \\ u(t=0) = f \\ \partial_{t} u(t=0) = g \end{gathered} \label{EqWaveGen}$$ then we have the$m$-Strichartz estimate with derivative $$\| u \|_{L_{t}^{q}( [0, \tau] ) L_{x}^{r}} + \| \partial_{t} D^{-1} u \|_{L_{t}^{q} ( [0, \tau] ) L_{x}^{r}} \lesssim \| f \|_{\dot{H}^{m}} + \| g \|_{\dot{H}^{m-1}} + \| F \|_{L_{t}^{\tilde{q}} ( [0, \tau] ) L_{x}^{\tilde{r}}} \label{StrEstDeriv}$$ 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 $$\frac{1}{q} + \frac{3}{r} = \frac{3}{2} -m = \frac{1}{\tilde{q}} + \frac{3}{\tilde{r}} -2$$ \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$D^{1-m}Iwe get \begin{aligned} Z_{m,s}(\tau) & \lesssim \| D I u_{0} \|_{L^{2}} + \| I u_{1} \|_{L^{2}} + \| D^{1-m} I F \| _{L_{t}^{\tilde{q}}([0,\tau]) L_{x}^{\tilde{r}} } \\ & \lesssim 1 + \| D^{1-m} I F \|_{L_{t}^{\tilde{q}}([0, \tau]) L_{x}^{\tilde{r}} } \end{aligned} \label{ZEst} The remainder of proof is divided into three steps. \subsection*{First Step} First we assume thatm \leq s$. Notice that the point$(\frac{1}{1-s},6)$is$s-wave admissible. In this case we get from the fractional Leibnitz rule the H\"older in time and the H\"older in space inequalities \begin{aligned} Z_{m,s}(\tau) & \lesssim 1 + \| D^{1-m} I (uuu) \|_{L_{t}^{1}([0, \tau]) L_{x}^{\frac{6}{5-2m}}} \\ & \lesssim 1+ \| D^{1-m} I u \|_{L_{t}^{\infty}([0, \tau]) L_{x}^{\frac{6}{3-2m}}} \| u \|_{L_{t}^{2} ( [0,\tau]) L_{x}^{6}}^{2} \\ & \lesssim 1+ Z_{m,s} (\tau) \Big( \tau^{\frac{1}{3}} \| P_{\leq N} u \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \tau^{s-\frac{1}{2}} \| P_{>N} u \|_{L_{t}^{\frac{1}{1-s}} ([0,\tau]) L_{x}^{6}} \Big)^{2} \\ & \lesssim 1 + Z_{m,s}(\tau) \Big( \tau^{\frac{1}{3}} \| Iu \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \tau^{s-\frac{1}{2}} \frac{ \| D^{1-s} Iu \|_{L_{t}^{\frac{1}{1-s}}( [0, \tau] ) L_{x}^{6}}}{N^{1-s}} \Big)^{2} \\ & \lesssim 1 + Z_{m,s}(\tau) \Big( \tau^{\frac{1}{3}} \| Iu \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \tau^{s-\frac{1}{2}} \frac{Z_{s,s}(\tau)}{N^{1-s}} \Big)^{2} \end{aligned} \label{IndZms} Assumem=s$. Then if we apply a continuity argument to (\ref{IndZms}) we get, from the inequalities (\ref{Condition1}) and (\ref{Condition2}), $$Z_{s,s}(\tau) \lesssim 1 \label{BdZss}$$ Now assume$ms$. By (\ref{IndZms}), (\ref{BdZss}), (\ref{BdZms}), (\ref{Condition1}) and (\ref{Condition2}) we have $$\| D^{1-r} I (uuu) \|_{L_{t}^{1} ([0,\tau]) L_{x}^{\frac{6}{5-2r}}} \lesssim Z_{r,s}(\tau) ( \tau^{\frac{1}{3}} \| Iu \|_{L_{t}^{6} ([0,\tau]) L_{x}^{6}} + \frac{ \tau^{s-\frac{1}{2}} Z_{s,s}(\tau)}{N^{1-s}} )^{2} \lesssim 1 \label{BdFcTerm}$$ for$r \leq s. The inequality $$\| D^{1-m} I (uuu) \|_{L_{t}^{1} ([0,\tau]) L_{x}^{\frac{6}{5-2m}}} \lesssim \| D^{1-r} I (uuu) \|_{L_{t}^{1} ([0,\tau]) L_{x}^{\frac{6}{5-2r}}} \label{SobIneqm}$$ follows from the application of Sobolev homogeneous embedding. We get from (\ref{ZEst}), (\ref{BdFcTerm}) and (\ref{SobIneqm}) \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} \subsection{Proof of lemma \ref{lem:StrEstDeriv}} \label{subsec:PfLemStrDer} By decomposition it suffices to prove thatu_{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 $$\| D^{-1} \partial_{t} u_{l}^{1} \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r} } \lesssim \| e^{\pm it D} f \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} \lesssim \| f \|_{\dot{H}^{m}} \label{Lin1}$$ and $$\| D^{-1} \partial_{t} u_{l}^{2} \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} = \| e^{\pm itD} D^{-1} g \|_{L_{t}^{q} ([0,\tau]) L_{x}^{r}} \lesssim \| D^{-1} g \|_{\dot{H}^{m}} \lesssim \| g \|_{\dot{H}^{m-1}} \label{Lin2}$$ We also have $$D^{-1} \partial_{t} u_{n}(t) = \int_{0}^{t} \cos{ ( (t-t')D ) } F(t{'}) \,dt'$$ and by the Strichartz estimates \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} 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 thatJ=[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 $$| E ( Iu(\tau_{0}) ) - E ( Iu(0) ) | \lesssim \frac{Z^{4}(\tau)}{N^{1-}}$$ 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 $$\mu(\xi_{2},\xi_{3},\xi_{4}) = 1 - \frac{m(\xi_{2}+ \xi_{3} + \xi_{4})}{m(\xi_{2}) m(\xi_{3}) m(\xi_{4})} \label{DfnMu}$$ It is left to prove that \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} We perform a Paley-Littlewood decomposition to prove (\ref{AlmConsToProve}). Letu_{i}=P_{N_{i}} u$with$i \in \{1,\dots , 4 \}and let \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} 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 theN_{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}$$$| \partial_{\xi}^{\alpha} \sigma (\xi) | \leq \frac{c(\alpha)} { ( 1+ |\xi| )^{| \alpha|}} \label{MeyMultBound}$$ Let$\Lambda_{\sigma}$be the multilinear operator $$\Lambda_{\sigma} (f_{1},\dots ,f_{k})(x) = \int_{\mathbb{R}^{nk}} e^{ix \cdot (\xi_{1}+\dots +\xi_{k})} \sigma(\xi_{1},\dots ,\xi_{k}) \widehat{f_{1}}(\xi_{1})\dots \widehat{f_{k}}(\xi_{k}) \,d \xi_{1}\dots d \xi_{k} \label{ThmCoifMey}$$ 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}$$$\| \Lambda_{\sigma}(f_{1},\dots ,f_{k}) \|_{L^{q}(\mathbb{R}^{n})} \leq C \| f_{1} \|_{L^{q_{1}} (\mathbb{R}^{n})}\dots \| f_{k} \|_{L^{q_{k}} (\mathbb{R}^{n})}$$ \label{thm:CoifMeyer} \end{theorem} Then we proceed as follows. We seek a pointwise bound on the symbol $$\left| \mu(\xi_{2},\xi_{3},\xi_{4}) \right| \leq B ( N_{2},N_{3},N_{4} ) \label{DfnB}$$ 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$\frac{\mu}{B}satisfy the bound (\ref{MeyMultBound}) and by the Plancherel theorem, H\"older inequality, theorem \ref{thm:CoifMeyer} and Bernstein inequalities we have \begin{aligned} X & \lesssim B \| \partial_{t} I u_{1} \|_{L_{t}^{p_{1}} ([0, \tau]) L_{x}^{q_{1}} } \| I u_{2} \|_{L_{t}^{p_{2}} ([0, \tau]) L_{x}^{q_{2}}}\dots \| I u_{4} \|_{L_{t}^{p_{4}} ([0, \tau]) L_{x}^{q_{4}}} \\ & \lesssim B N_{1}^{m_{1}} N_{2}^{m_{2}-1}\dots N_{4}^{m_{4}-1} \| \partial_{t} D^{-m_{1}} I u_{1} \|_{L_{t}^{p_{1}} ([0, \tau]) L_{x}^{q_{1}} } \\ &\quad\times \| D^{1-m_{2}} I u_{2} \|_{L_{t}^{p_{2}} ([0, \tau]) L_{x}^{q_{2}}}\dots \| D^{1-m_{4}} I u_{4} \|_{L_{t}^{p_{4}} ([0, \tau]) L_{x}^{q_{4}}} \\ & \lesssim B N_{1}^{m_{1}} N_{2}^{m_{2}-1}\dots N_{4}^{m_{4}-1} Z^{4}(\tau) \end{aligned} \label{AlmconsRadpj} with(p_{j},q_{j})$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 $$\begin{gathered} \| I u_{2} \|_{L_{t}^{p_{2}+} L_{x}^{q_{2}-}} \lesssim N_{2}^{-} N_{2}^{m_{2}-1} \| D^{1-(m_{2}-)} I u_{2} \|_{L_{t}^{p_{2}+} L_{x}^{q_{2}-}} \\ \| I u_{4} \|_{L_{t}^{p_{4}-} L_{x}^{q_{4}+}} \lesssim N_{4}^{+} N_{4}^{m_{4}-1} \| D^{1-(m_{4}+)} I u_{4} \|_{L_{t}^{p_{4}-} L_{x}^{q_{4}+}} \end{gathered} \label{Ex1DirectCreat}$$ and (\ref{AlmconsRadpj}) becomes $$X \lesssim B N_{2}^{-} N_{4}^{+} N_{1}^{m_{1}} N_{2}^{m_{2}-1}\dots N_{4}^{m_{4}-1} Z^{4}(\tau)$$ \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 $$\| I u_{k} \|_{L_{t}^{2+} L_{x}^{\infty -}} \lesssim N_{k}^{-} \| D^{1-(1-)} I u_{k} \|_{L_{t}^{2+} L_{x}^{\infty -}} \lesssim N_{k}^{-} Z(\tau) \label{Ex2DirectCreat}$$ 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{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} We would like\epsilon'' > \epsilon'$. A quick computation show that it suffices that$ \epsilon' > 3 \epsilon $. Letting$\epsilon'= 5 \epsilon$we get $$\| I u_{k} \|_{L_{t}^{\frac{2}{1- \epsilon}}(([0,\tau])) L_{x}^{\frac{2}{\epsilon}}} \lesssim N_{k}^{\epsilon} \tau^{ 2 \epsilon } Z(\tau)$$ Now if we choose$\epsilon > 0 $so small that$ |\tau|^{2 \epsilon} \leq 2 $we eventually get $$\| I u_{k} \|_{L_{t}^{2+}(([0,\tau])) L_{x}^{\infty -}} \lesssim N_{k}^{+} Z(\tau) \label{IndirectRes}$$ 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 $$| \mu(\xi_{2},\dots ,\xi_{4}) | \lesssim \frac{|\nabla m(\xi_{2})| |\xi_{3} + \xi_{4}|}{m(\xi_{2})} \lesssim \frac{N_{3}}{N_{2}} \label{EstMultRad1}$$ We also get$N_{1} \sim N_{2}$from the convolution constraint$\xi_{1}+\dots + \xi_{4}=0$. We assume that$N_{4} \geq 1. By (\ref{EstMultRad1}) and by the Bernstein inequalities we have \begin{align*} X & \lesssim \frac{N_{3}}{N_{2}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{3}}{N_{2}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creatingN_{1}^{+}$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 $$\left| \mu(\xi_{2},\dots,\xi_{4}) \right| \lesssim \frac{m(\xi_{1})}{m(\xi_{2}) m(\xi_{3}) m(\xi_{4})} \label{EstMultRad2}$$ There are two subcases: \subsection*{Case 3.a:$N_{1} \sim N_{2}$} We assume that$N_{4} \geq 1. By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{3}^{1-s}}{N^{1-s}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creatingN_{1}^{+}$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$N_{4} \geq 1. By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \\ & \quad\times \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{1}^{-} N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creatingN_{1}^{+}$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$N_{4} \geq 1. By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{2}^{2(1-s)}}{N^{1-s} N_{1}^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \| I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \\ & \quad\times \| I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty -}} \\ & \lesssim N_{1}^{+} N_{4}^{-} \frac{N_{2}^{2(1-s)}}{N^{1-s} N_{1}^{1-s}} N_{1}^{\frac{1}{3}} N_{2}^{-2/3} N_{3}^{-2/3} \| \partial_{t} D^{- ( \frac{1}{3}+ ) } I u_{1} \|_{L_{t}^{6-} ([0,\tau]) L_{x}^{3+}} \\ &\quad\times \| D^{1- \frac{1}{3} } I u_{2} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-\frac{1}{3}} I u_{3} \|_{L_{t}^{6} ([0,\tau]) L_{x}^{3}} \| D^{1-(1-)} I u_{4} \|_{L_{t}^{2+} ([0,\tau]) L_{x}^{\infty-}} \\ & \lesssim \frac{N_{2}^{-} N_{4}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creatingN_{1}^{+}$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:$N_{1} \sim N_{2}} By (\ref{EstMultRad2}) we have \begin{align*} X & \lesssim \frac{N_{3}^{1-s}}{N^{1-s}} \frac{N_{4}^{1-s}}{N^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{2} \|_{L_{t}^{4+} ([0,\tau]) L_{x}^{4-}} \| I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ &\quad\times \| I u_{4} \|_{L_{t}^{4-} ([0,\tau]) L_{x}^{4+}} \\ & \lesssim N_{2}^{-} N_{4}^{+} \frac{N_{3}^{1-s}}{N^{1-s}} \frac{N_{4}^{1-s}} {N^{1-s}} N_{1}^{\frac{1}{2}} \frac{1}{N_{2}^{\frac{1}{2}}} \frac{1}{N_{3}^{\frac{1}{2}}} \frac{1}{N_{4}^{\frac{1}{2}}} \| \partial_{t} D^{- \frac{1}{2} } I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ &\quad\times \| D^{1- ( \frac{1}{2} - ) } I u_{2} \|_{L_{t}^{4+} ([0,\tau]) L_{x}^{4-}} \| D^{1-\frac{1}{2}} I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1- ( \frac{1}{2} + ) } I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{2}^{-}}{N^{1-}} Z^{4}(\tau) \end{align*} after directly creatingN_{2}^{-}$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$N_{1} \geq 1. We have \begin{align*} X & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} \| \partial_{t} I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{2} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \quad\times \| I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{2}^{2(1-s)}}{N^{2(1-s)}} \frac{N_{4}^{1-s}}{N^{1-s}} N_{1}^{\frac{1}{2}} \frac{1}{N_{2}^{\frac{1}{2}}} \frac{1}{N_{3}^{\frac{1}{2}}} \frac{1}{N_{4}^{\frac{1}{2}}} \| \partial_{t} D^{- \frac{1}{2} } I u_{1} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1- \frac{1}{2} } I u_{2} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \quad\times \| D^{1-\frac{1}{2}} I u_{3} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \| D^{1- \frac{1}{2} } I u_{4} \|_{L_{t}^{4} ([0,\tau]) L_{x}^{4}} \\ & \lesssim \frac{N_{1}^{-} N_{2}^{-} }{N^{1-}} Z^{4}(\tau) \end{align*} IfN_{1} <1$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{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} and sinceusatisfies \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{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} We also notice that $$\big( \frac{\nabla u.x}{|x|} + \frac{u}{|x|} \big)^{2} = \frac{ ( \nabla u.x )^{2}}{|x|^{2}} + \mathop{\rm div}( \frac{u^{2}}{|x|^{2}} x ) \leq | \nabla u |^{2} + \mathop{\rm div}( \frac{u^{2}}{|x|^{2}} x ) \label{PtwiseEquality}$$ We substitute (\ref{PtwiseEquality}) into (\ref{MorCauch}). We get the Morawetz-Strauss's inequality $$\int_{0}^{T} \int_{ \mathbb{R}^{3}} \frac{u^{4}(t,x)}{ |x|} dx \,dt \leq 2 ( E(u(T))+ E(u(0)) )$$ \subsection*{Second Step} Almost Morawetz-Strauss's inequality. We substituteu$for$Iuin (\ref{Identity}) and we proceed similarly. We get \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} \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 thatJ=[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\mudefined in (\ref{DfnMu}). It suffices to prove \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} and \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} We perform a Paley-Littlewood decomposition to prove (\ref{R1Four}) and (\ref{R2Four}). Letu_{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{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} and $$\big\| (\frac{ I u}{|x|})_{1} \big\|_{L_{t}^{\infty-} ([0,\tau]) L_{x}^{2+}} \lesssim N_{1}^{+} \big\| \frac{Iu}{|x|} \big\|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \lesssim N_{1}^{+} \| D I u \|_{L_{t}^{\infty} ([0,\tau]) L_{x}^{2}} \label{2Mor}$$ Ifp_{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 thatN_{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{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} Similarly we getX_{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*} SimilarlyX_{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*} SimilarlyX_{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*} SimilarlyX_{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*} SimilarlyX_{2} \lesssim \frac{N_{1}^{+} N_{2}^{--}}{N^{1-}} Z^{4}(\tau)$. 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