\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 78, pp. 1--19.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/78\hfil Rigidity for rough solutions] {On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data} \author[D. Li, X. Zhang\hfil EJDE-2009/78\hfilneg] {Dong Li, Xiaoyi Zhang} % in alphabetical order \address{Dong Li \newline Institute for Advanced Study, Princeton, NJ, 08544, USA} \email{dongli@ias.edu} \address{Xiaoyi Zhang \newline Academy of Mathematics and System Sciences, Beijing, China. \newline Institute for Advanced Study, Princeton, NJ, 08544, USA} \email{xiaoyi@ias.edu} \thanks{Submitted April 15, 2009. Published June 16, 2009.} \subjclass[2000]{35Q55} \keywords{Mass-critical; nonlinear Schrodinger equation} \begin{abstract} For the focusing mass-critical nonlinear Schr\"odinger equation $iu_t+\Delta u=-|u|^{4/d}u$, an important problem is to establish Liouville type results for solutions with ground state mass. Here the ground state is the positive solution to elliptic equation $\Delta Q-Q+Q^{1+\frac 4d}=0$. Previous results in this direction were established in \cite{klvz,lz:2d,merle_duke,weinstein:charact} and they all require $u_0\in H_x^1(\mathbb{R}^d)$. In this paper, we consider the rigidity results for rough initial data $u_0 \in H_x^s(\mathbb{R}^d)$ for any $s>0$. We show that in dimensions $d\ge 4$ and under the radial assumption, the only solution that does not scatter in both time directions (including the finite time blowup case) must be global and coincide with the solitary wave $e^{it}Q$ up to symmetries of the equation. The proof relies on a non-uniform local iteration scheme, the refined estimate involving the $P^{\pm}$ operator and a new smoothing estimate for spherically symmetric solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} \subsection{Background and main results} We consider the focusing mass-critical nonlinear Schr\"odinger equation $$\label{nls} iu_t+\Delta u=-|u|^{4/d}u$$ in dimensions $d\geq 4$; here $u(t,x)$ is a complex-valued function on $\mathbb R\times \mathbb{R}^d$. The name mass critical" refers to the fact that the scaling symmetry $$u(t,x) \mapsto \lambda^{d/2}u(\lambda^{2}t,\lambda x), \quad \forall\lambda>0 \label{scaling}$$ leaves both the equation and the mass invariant. Here the mass is defined as $\mbox{Mass: } M(u(t))=\int_{\mathbb{R}^d} |u(t,x)|^2 dx=M(u_0).$ For the initial value problem of \eqref{nls}, the local theory was established by Cazenave and Weissler in \cite{cwI}. To summarize, for any initial data $u_0\in L_x^2(\mathbb{R}^d)$, they constructed the unique local solution $u(t,x)\in C_t([-T,T];L_x^2)\cap L_{t,x}^{2(d+2)/d}([-T,T]\times\mathbb{R}^d)$. Moreover, when the mass of the initial data is small enough, the solution is global and obeys the global spacetime estimate $\|u\|_{L_{t,x}^{2(d+2)/d}(\mathbb{R}\times\mathbb{R}^d)} \le C(\|u_0\|_{L_x^2}).$ This estimate implies that the solution scatters in both time directions asymptotically: there exist $u_{\pm}\in L_x^2(\mathbb{R}^d)$ such that $\lim_{t\to \pm \infty}\|u(t)-e^{it\Delta}u_{\pm}\|_{L_x^2}=0.$ When the solution has large mass, blowup may occur at finite time. The existence of finite blowup solutions was proved by Glassey \cite{glassey}, basing on the virial argument. On the other hand, the equation \eqref{nls} also admits solitary wave solutions of the form $e^{it}R$, where $R$ solves the elliptic equation $$\Delta R-R+|R|^{4/d}R=0.\label{elliptic}$$ There are infinite many solutions to this equation, but only one positive solution which is spherically symmetric and whose mass is minimal among all these $R's$. This solution is usually called the \begin{definition}[Ground state]\label{ground} \rm The ground state $Q$ refers to the unique positive solution to the equation \eqref{elliptic}. According to \cite{blions,kwong}, $Q$ is spherically symmetric and decays exponentially fast as $|x|\to \infty$. \end{definition} It is believed that the mass of $Q$ is the minimal mass among all the non-scattering solutions. The precise statement of this general belief is the following scattering conjecture: \begin{conjecture} Let $u_0\in L_x^2(\mathbb{R}^d)$ be such that $M(u_0)0$. Let $u_0\in H_x^s(\mathbb{R}^d)$ be spherically symmetric and such that $M(u_0)=M(Q)$. Suppose that the corresponding maximal lifespan solution $u(t,x):(-T_*, T^*)\times \mathbb{R}^d\to \mathbb{C}$ does not scatter in both time directions: $\|u\|_{L_{t,x}^{2(d+2)/d}((-T_*,0]\times\mathbb{R}^d)}= \|u\|_{L_{t,x}^{2(d+2)/d}([0,T^*)\times\mathbb{R}^d)}=\infty.$ Then the solution must be global $T_*=T^*=+\infty.$ And there exist $\lambda_0,\theta_0$ such that $u(t,x)=e^{i\theta_0}e^{i\lambda_0^2t}\lambda_0^{d/2}Q(\lambda_0 x).$ \end{theorem} As expected, the main part of the proof is devoted to upgrading the $H_x^s$ regularity of the solution to $H_x^1$, when the result for $H_x^1$ solution Theorem \ref{h1_result} can be applied. The possibility that we can upgrade the regularity of the solution comes ultimately from the fact that the solution we are considering has the \emph{minimal mass} and does not scatter \emph{in both time directions}. Our strategy for upgrading the regularity is the following: Firstly, since $u$ has the minimal mass and does not scatter on both sides, applying the same argument as in \cite{klvz}, one easily gets that $\|\phi_{>1}\nabla u(t)\|_{L_x^2}\lesssim 1.$ This means that away from the origin, the solution is regular uniformly in time, thus it suffices for us to examine the solution near the origin. There we carefully design a local iteration scheme enabling us to go from $H_x^t$ to $H_x^{t+\epsilon}$ for any $t<1$ and an $\epsilon$ increasing in $t$. After finite many times of iteration, we get the desired $H_x^1$ regularity. Here by "local", we mean that the scheme is designed to upgrade the regularity of the solution at some fixed time $t$, for example $t=0$, not uniformly in time. More precisely, the quantity we will look at is $$\|\phi_{\le 1} P_N u_0\|_{L_x^2}, \ N\ge 1.\label{small1}$$ (Not that the piece $\|\phi_{>1} P_N u_0\|_{L_x^2}$ already gives us $N^{-1-\epsilon(d)}$ decay following the argument in \cite{klvz}, which is already very good). Now we split \eqref{small1} into two parts by introducing a spatial cutoff \begin{gather} \| \phi_{\le N^{-1-\gamma}} P_N u_0 \|_{L_x^2},\label{first}\\ \| \phi_{N^{-1-\gamma}<\cdot\le 1} P_N u_0 \|_{L_x^2}.\label{second} \end{gather} By H\"older and Bernstein, the first quantity gives us the bound: $N^{-s-\frac d2\gamma}$ which is good for the iteration. To estimate the second piece, we project it into the incoming and outgoing wave, for the incoming wave, we use the Duhamel formula backward in time; for the outgoing wave, we use the Duhamel formula forward in time. The assumption that the solution does not scatter on both sides forbids the scattering wave, for which there is no hope to upgrade the regularity, to participate in the estimates. The first issue when we estimate these two pieces comes from the fact that in \eqref{second}, the spatial cutoff and the frequency cutoff does not obey the scaling like in \cite{ktv:2d}, \cite{kvz:blowup}, there we have good estimates for $P_N^{\pm}$ with a natural spatial cutoff $\phi_{>\frac 1N}$. Indeed, when approaching the origin, the operator $P^{\pm}$ have strong singularities. To get around this problem, we refine the estimates for the operator $P_N^{\pm}$ with spatial cutoff of the form $\phi_{N^{-1-\gamma}<\cdot\le 1}$. It turns out that there will be a loss of $N$ to some power related to $\gamma$. This loss of power is not too harmful for us if we make a judicious choice of $\gamma$ and other relevant parameters in the iteration scheme. We give the detailed discussion on the properties of the operators in Section 3. Having the operator estimate in hand, we then estimate the contribution from the in-out wave by chopping the $t$-integration into different pieces. Since the stationary phase point moves with time $t$ at speed $N$, the contribution from the large time piece is presumably fine due to the decay property of radial functions. It turns out that the most troublesome term is the following local piece $$\label{trouble} \Big\|\phi_{N^{-1-\gamma}<\cdot\le 1}\int_0^{\frac 1{N^{2-\sigma}}} P_N^+ e^{-i\tau\Delta} \phi_{\le 1}F(u)(\tau)d\tau\Big\|_{L_x^2}.$$ Here $0<\sigma<2$ is a small constant to be specified later in the proof. One observation from the expression \eqref{trouble} is that it is spatially localized, which suggests that the additional regularity should come from some sort of smoothing estimate. The classical smoothing estimate \cite{sjolin,kpv:smoothing93,constantin_saut} asserts that the linear propagation gain half derivative locally. This is crucial to study the NLS containing first order derivatives. In our setting, since we are considering the spherically symmetric functions, we develop the following global smoothing estimate: $$\label{smoothing} \||x|^{(d-1)/2}|\nabla|^{1/2} e^{it\Delta}u_0\|_{L_x^{\infty}L_t^2(\mathbb{R}^d\times\mathbb{R})}\lesssim \|u_0\|_{L_x^2}.$$ Using the dual form of this estimate, we can successfully control the term \eqref{trouble} and close the argument. The proof of the estimate \eqref{smoothing} can be found in Section 3. We make two remarks here. First of all, it is worth pointing out that the strategy here for upgrading the regularity is quite different from the one in \cite{ktv:2d, kvz:blowup}. There the argument relies on the fact that the solution is almost periodic modulo scaling, and the solution is uniformly flat. Namely, there exists $N(t)>0$ such that $\{N(t)^{-\frac d2}u(t,\frac x{N(t)})\}$ is precompact in $L_x^2(\mathbb{R}^d)$ \emph{and $N(t)\le 1$}. In our setting, the solution also enjoys such compactness, but there is no a priori control on $N(t)$. Actually, the most difficult case is that $N(t)$ can fluctuate out of any control. Secondly, like in \cite{klvz}, our proof needs the assumption $d\ge 4$ since the nonlinearity $|u|^{4/d} u$ can easily be controlled without knowing other information than $M(u)$ being finite. It is certainly an interesting problem to prove the theorem in lower dimensions. \section{Preliminaries} \subsection{Some notation} We write $X \lesssim Y$ or $Y \gtrsim X$ to indicate $X \leq CY$ for some constant $C>0$. We use $O(Y)$ to denote any quantity $X$ such that $|X| \lesssim Y$. We use the notation $X \sim Y$ whenever $X \lesssim Y \lesssim X$. The fact that these constants depend upon the dimension $d$ will be suppressed. If $C$ depends upon some additional parameters, we will indicate this with subscripts; for example, $X \lesssim_u Y$ denotes the assertion that $X \leq C_u Y$ for some $C_u$ depending on $u$. We denote by $X\pm$ any quantity of the form $X\pm \epsilon$ for any $\epsilon>0$. We use the Japanese bracket' convention $\langle x \rangle := (1 +|x|^2)^{1/2}$. We write $L^q_t L^r_{x}$ to denote the Banach space with norm $$\| u \|_{L^q_t L^r_x(\mathbb{R} \times \mathbb{R}^d)} := \Bigl(\int_\mathbb{R} \Bigl(\int_{\mathbb{R}^d} |u(t,x)|^r\ dx\Bigr)^{q/r}\ dt\Bigr)^{1/q},$$ with the usual modifications when $q$ or $r$ are equal to infinity, or when the domain $\mathbb{R} \times \mathbb{R}^d$ is replaced by a smaller region of spacetime such as $I \times \mathbb{R}^d$. When $q=r$ we abbreviate $L^q_t L^q_x$ as $L^q_{t,x}$. Throughout this paper, we will use $\phi\in C^\infty(\mathbb{R}^d)$ be a radial bump function supported in the ball $\{ x \in \mathbb{R}^d: |x| \leq \frac{25} {24} \}$ and equal to one on the ball $\{ x \in \mathbb{R}^d: |x| \leq 1 \}$. For any constant $C>0$, we denote $\phi_{\le C}(x):= \phi \bigl( \tfrac{x}{C}\bigr)$ and $\phi_{> C}:=1-\phi_{\le C}$. \subsection{Basic harmonic analysis}\label{ss:basic} For each number $N > 0$, we define the Fourier multipliers \begin{gather*} \widehat{P_{\leq N} f}(\xi) := \phi_{\leq N}(\xi) \hat f(\xi)\\ \widehat{P_{> N} f}(\xi) := \phi_{> N}(\xi) \hat f(\xi)\\ \widehat{P_N f}(\xi) := (\phi_{\leq N} - \phi_{\leq N/2})(\xi) \hat f(\xi) \end{gather*} and similarly $P_{0$. Then \begin{gather*} \bigl\| \phi_{> R} \nabla P_{\le N} \phi_{\le\frac R2} f \bigr\|_{L_x^p} \lesssim_m N^{1-m} R^{-m} \|f\|_{L_x^p} \\ \bigl\| \phi_{> R} P_{\leq N} \phi_{\le\frac R2} f \bigr\|_{L_x^p} \lesssim _m N^{-m} R^{-m} \|f\|_{L_x^p} \end{gather*} for any $1\le p\le \infty$ and $m\geq 0$. \end{lemma} \begin{proof} We will only prove the first inequality; the second follows similarly. It is not hard to obtain kernel estimates for the operator $\phi_{> R}\nabla P_{\le N}\phi_{\le\frac R2}$. Indeed, an exercise in non-stationary phase shows $\bigl|\phi_{> R}\nabla P_{\le N}\phi_{\le R/2}(x,y)\bigr| \lesssim N^{d+1-2k} |x-y|^{-2k}\phi_{|x-y|>\frac R2}$ for any $k\geq 0$. An application of Young's inequality yields the claim. \end{proof} Similar estimates hold when the roles of the frequency and physical spaces are interchanged. The proof is easiest when working on $L_x^2$, which is the case we will need; nevertheless, the following statement holds on $L_x^p$ for any $1\leq p\leq \infty$. \begin{lemma}[Mismatch estimates in frequency space] \label{L:mismatch_fre} For $R>0$ and $N,M>0$ such that $\max\{N,M\}\geq 4\min\{N,M\}$, \begin{gather*} \bigl\| P_N \phi_{\le{R}} P_M f \bigr\|_{L_x^2} \lesssim_m \max\{N,M\}^{-m} R^{-m} \|f\|_{L_x^2} \\ \bigl\| P_N \phi_{\le {R}} \nabla P_M f \bigr\|_{L_x^2} \lesssim_m M \max\{N,M\}^{-m} R^{-m} \|f\|_{L_x^2}. \end{gather*} for any $m\geq 0$. The same estimates hold if we replace $\phi_{\le R}$ by $\phi_{>R}$. \end{lemma} \begin{proof} The first claim follows from Plancherel's Theorem and Lemma~\ref{L:mismatch_real} and its adjoint. To obtain the second claim from this, we write $$P_N \phi_{\le {R}} \nabla P_M = P_N \phi_{\le {R}} P_M \nabla \tilde P_M$$ and note that $\|\nabla \tilde P_M\|_{L_x^2\to L_x^2}\lesssim M$. \end{proof} We will need the following radial Sobolev embedding to exploit the decay property of a radial function. For the proof and the more complete version, one refers to see \cite{tvz:hd}. \begin{lemma}[Radial Sobolev embedding, \cite{tvz:hd}] \label{L:radial_embed} Let dimension $d\ge 2$. Let $s>0$, $\alpha>0$, $10} \rho^{(d-1)/2} |h(\rho)| \le C_1<\infty, where$C_1$is a constant depending only on the dimension$d$. By Plancherel, one can show that for any one-dimensional function$F, we have $$\label{eq1b} \big\| \int_0^\infty e^{-itk^2} F(k) dk \big\|_{L_t^2}^2 = \frac 12 \int_0^\infty |F(k)|^2 \frac {dk} k.$$ Now by \eqref{eq0}, \eqref{eq1a}, \eqref{eq1b}, we obtain \begin{align*} \big\| |x|^{(d-1)/2} |\nabla|^{1/2} e^{it\Delta} f\big\|_{L_t^2}^2 & = \frac 12 \int_0^\infty |\hat f(k)|^2 \cdot |x|^{d-1} \cdot k^{2(d-1)} \cdot |h(k|x|)|^2 dk \\ & \le \frac 12 \int_0^\infty |\hat f(k)|^2 k^{d-1} dk \cdot \Big(\sup_{\rho>0} \rho^{\frac {d-1}2} |h(\rho)| \Big)^2 \\ & \le C_1^2 \|f\|_2^2. \end{align*} The lemma is proved. \end{proof} \subsection{The in-out decomposition and refined operator estimates} We will need an incoming/outgoing decomposition; we will use the one developed in \cite{ktv:2d, kvz:blowup}. As there, we define operatorsP^{\pm}$by $[P^{\pm} f](r) :=\tfrac12 f(r)\pm \tfrac{i}{\pi} \int_0^\infty \frac{r^{2-d}\,f(\rho)\,\rho^{d-1}\,d\rho}{r^2-\rho^2},$ where the radial function$f: \mathbb{R}^d\to \mathbb{C}$is written as a function of radius only. We will refer to$P^+$is the projection onto outgoing spherical waves; however, it is not a true projection as it is neither idempotent nor self-adjoint. Similarly,$P^-$plays the role of a projection onto incoming spherical waves; its kernel is the complex conjugate of the kernel of$P^+$as required by time-reversal symmetry. \subsection{The two-dimensional case} For$N>0$let$P_N^{\pm}$denote the product$P^{\pm}P_N$where$P_N$is the Littlewood-Paley projection. We record the following properties of$P^{\pm}$from \cite{ktv:2d, kvz:blowup}: \begin{proposition}[Properties of$P^\pm$, \cite{ktv:2d, kvz:blowup}] \label{P:P properties}\leavevmode \begin{itemize} \item[(i)]$P^+ + P^- $represents the projection from$L^2$onto$L^2_{\rm rad}$. \item[(ii)]$P^{\pm}$are bounded on$L^2(\mathbb{R}^2)$. \item[(iii)] For$|x|\gtrsim N^{-1}$and$t\gtrsim N^{-2}$, the integral kernel obeys $\bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \begin{cases} (|x||y|)^{-1/2}|t|^{-1/2} & |y|-|x|\sim Nt \\[1ex] \frac{N^2}{(N|x|)^{1/2}\langle N|y|\rangle^{1/2}} \bigl\langle N^2t + N|x| - N|y| \bigr\rangle^{-m} &\text{otherwise}\end{cases}$ for all$m\geq 0$. \item[(iv)] For$|x|\gtrsim N^{-1}$and$|t|\lesssim N^{-2}$, the integral kernel obeys \begin{equation*} \bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac{N^2}{(N|x|)^{1/2}\langle N|y|\rangle^{1/2}} \bigl\langle N|x| - N|y| \bigr\rangle^{-m} \end{equation*} for any$m\geq 0$. \end{itemize} \end{proposition} For a proof of the above proposition, see \cite{ktv:2d, kvz:blowup}. We will also need the following Proposition concerning the properties of$P^{\pm}$in the small$x$regime (i.e.$|x| \lesssim N^{-1}$) where Bessel functions have logarithmic singularities. More precisely, we have the following result. \begin{proposition}[Properties of$P^\pm$, small$x$regime, \cite{lz:2d}] \label{Psmall_properties} Let dimension$d=2$. \begin{itemize} \item[(i)] For$t \gtrsim N^{-2}$,$N^{-3} \lesssim |x| \lesssim N^{-1}$,$|y|\ll Nt$or$|y| \gg Nt$, the integral kernel satisfies $\bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac {N^2 \log N}{\langle N|y| \rangle^{1/2} } \langle N^2 t + N|y| \rangle^{-m}, \quad \forall\, m\ge 0.$ \item[(ii)] For$t \gtrsim N^{-2}$,$N^{-3} \lesssim |x| \lesssim N^{-1}$,$|y|\sim Nt$, the integral kernel satisfies $\bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac {N^2 \log N}{\langle N|y| \rangle^{1/2} }.$ \end{itemize} \end{proposition} For a proof of the above proposition, see \cite{lz:2d}. \subsection{The case$d\ge 3$} The next lemma allows us to bound the operator$P_N^{\pm}$slightly below the$|x| \sim 1/N$barrier, i.e. in the regime$\frac 1 {N^{1+\gamma}} \le |x| \le \frac 1N$for some$\gamma>0$. The price to pay is a polynomial growth factor in$N$. \begin{lemma} \label{L:bdd} Let the dimension$d\ge 3$. Fix$N \gtrsim 1$and$\gamma>0$. For any spherically symmetric function$f\in L_x^2(\mathbb{R}^d)$, $\bigl\|P^\pm P_{ N} f \bigr\|_{L^2_x(\frac 1 {N^{1+\gamma} } \le |x| \le \frac 1N)} \lesssim \begin{cases} N^{\frac{(d-4)\gamma}2} \cdot \bigl\| f \bigr\|_{L^2_x(\mathbb{R}^d)}, &\text{if d\ge 5}, \\ \langle \log N \rangle^{1/2} \cdot \bigl\| f \bigr\|_{L^2_x(\mathbb{R}^d)}, &\text{if d=4}, \\ \bigl\| f \bigr\|_{L^2_x(\mathbb{R}^d)}, &\text{if d=3}, \end{cases}$ where the implied constant depends only on$\gamma$and$d$. Here$\langle \cdot \rangle$is the Japanese bracket. \end{lemma} \begin{proof} We shall only prove the inequality for$P^+$. The result for$P^{-}$is similar (or one can use the fact$P^++P^-$acts as an identity on$L^2_{\rm rad}(\mathbb{R}^d)$). By the definition of$P^{+}$, we have $$\bigl\|P^\pm P_{N} f \bigr\|_{L^2_x(\frac 1 {N^{1+\gamma} } \le |x| \le \frac 1 N)}^2 = \int_{\frac 1 {N^{1+\gamma}}}^{\frac 1 N} \Big| \int_0^\infty H^{(1)}_{\frac {d-2} 2} (kr) \hat f(k) k^{d/2} \psi(\tfrac k N) dk \Big|^2 r dr. \label{tmp_1047_a}$$ Since$k\sim N$,$\frac 1 {N^{1+\gamma} } \le r \le \frac 1 N$,$\frac 1 {N^{\gamma}} \le kr \lesssim 1, we have $\big| H^{(1)}_{(d-2)/2} (kr) \big| \lesssim (kr)^{-(d-2)/2}.$ Therefore, by Cauchy-Schwartz, we obtain \begin{aligned} \text{RHS of \eqref{tmp_1047_a} } & \lesssim \int_{\frac 1 {N^{1+\gamma}}}^{\frac 1 N} r^{3-d} dr \cdot N^{2-d} \cdot \int_0^\infty |\hat f(k)|^2 k^{d-1} dk \cdot \int_0^\infty |\psi(\frac kN)|^2 k dk \\ & \lesssim \int_{\frac 1 {N^{1+\gamma}}}^{\frac 1N} r^{3-d} dr \cdot N^{4-d} \cdot \|f\|_{L_x^2(\mathbb{R}^d)}^2. \end{aligned} \label{tmp_1047_b} Now ifd \ge 5$, then $\int_{\frac 1{N^{1+\gamma}}}^{\frac 1 N} r^{3-d} dr \lesssim N^{(1+\gamma)(d-4)}$ and RHS of \eqref{tmp_1047_b}$\lesssim N^{(d-4)\gamma} \|f\|_{L_x^2(\mathbb{R}^d)}^2$. If$d=4$, then $\int_{\frac 1{N^{1+\gamma}}}^{\frac 1 N} r^{3-d} dr \lesssim \log N$ and RHS of \eqref{tmp_1047_b}$\lesssim (\log N)\cdot \|f\|_{L_x^2(\mathbb{R}^d)}^2$. If$d=3$, then clearly $\text{RHS of \eqref{tmp_1047_b} } \lesssim \|f\|_{L_x^2(\mathbb{R}^d)}^2.$ The lemma is proved. \end{proof} In the next lemma we shall give bounds of some integrals needed later in the kernel estimates. To fix notations, we assume$\tilde g_1$,$\tilde g_2$are one-dimensional functions such that \label{eq_1236a} \big| \frac{d^m \tilde g_i (r)} { d r^m} \big| \lesssim 1, \quad \forall 00$. Let $N\gtrsim 1$ be a dyadic number. \begin{itemize} \item[(i)] For $t \gtrsim N^{-2}$, $\frac 1 {N^{1+\gamma}} \lesssim |x| \lesssim N^{-1}$, $|y|\ll Nt$ or $|y| \gg Nt$, the integral kernel satisfies $\bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac {N^{(1+\gamma)(d-2)+2}}{\langle N|y| \rangle^{1/2} } \langle N^2 t + N|y| \rangle^{-m}, \quad \forall\, m\ge 0.$ \item[(ii)] For $t \gtrsim N^{-2}$, $\frac 1 {N^{1+\gamma}} \lesssim |x| \lesssim N^{-1}$, $|y|\sim Nt$, the integral kernel satisfies $\bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac {N^{(1+\gamma)(d-2)+2} \log N}{\langle N|y| \rangle^{1/2} }.$ \end{itemize} \end{proposition} \begin{proof} We shall only provide the proof for $P_N^{+} e^{-it \Delta}$ since the other kernel is its complex conjugate. The first claim is an exercise in stationary phase. By Fourier transform we have the following formula for the kernel \label{kernel_highD} \begin{aligned} &[P^+_N e^{-it\Delta}](x,y)\\ &= \tfrac12 \bigl(|x||y|\bigr)^{-(d-2)/2} \int_0^\infty H^{(1)}_{(d-2)/2}(k |x|) J_{(d-2)/2}(k |y|) e^{itk^2} \psi\bigl(\tfrac kN\bigr)\,k\,dk \end{aligned} where $\psi$ is the multiplier function from the Littlewood--Paley projection. First note that $$\label{eq_tmp_1210_1} H^{(1)}_{(d-2)/2} (r) = J_{(d-2)/2} (r) + iY_{(d-2)/2}(r).$$ Since $k\sim N$, $\frac 1{N^{1+\gamma}} \lesssim |x| \lesssim \frac 1N$, we have $r=k|x|$ satisfies $\frac 1{N^{\gamma}} \lesssim r \lesssim 1$. Now using the expansion $J_{(d-2)/2} (r) = \sum_{m=0}^\infty \frac {(-1)^m} {m! \Gamma(m+\tfrac d2)} \cdot (\tfrac r 2)^{2m+\frac {d-2}2},$ we can write $$\label{eq_tmp_1210_2} r^{-(d-2)/2} J_{(d-2)/2} (r) = \tilde g_1(r),$$ where $\big| \frac {\partial^m \tilde g_1(r)} {\partial r^m} \big| \lesssim 1, \quad \forall\, m \ge0, \, r\lesssim 1.$ Here the factor $r^{-(d-2)/2}$ in \eqref{eq_tmp_1210_2} is needed since the dimension $d$ may possibly be a odd integer. To treat the function $Y_{\frac {d-2} 2}$ in the regime $\frac 1 {N^{\gamma}} \lesssim r \lesssim 1$, we discuss two cases. If the dimension $d$ is even, then we use the series \begin{align*} Y_{(d-2)/2} (r) &= - \frac{ (\tfrac r 2)^{-(d-2)/2}}{\pi} \sum_{k=0}^{\frac{d-4}2} \frac{(\tfrac {d-4}2 -k)!}{k!} \cdot \big( \frac 14 r^2 \big)^k + \frac 2 {\pi} \log (\tfrac r 2) J_{(d-2)/2} (r) \\ & \quad - \frac{ (\tfrac r 2)^{\frac{d-2}2}}{\pi} \sum_{k=0}^\infty \bigl( \psi_0(k+1) + \psi_0(n+k+1) \bigr) \cdot \frac {(-\tfrac 14 r^2)^k} {k! (\tfrac{d-2}2 +k)!}, \end{align*} where $\psi_0$ is the digamma function defined by $\psi_0(n) = - \gamma_0 + \sum_{k=1}^{n-1} \frac 1k,$ and $\gamma_0$ is the Euler-Masheroni constant. It follows easily that $$\label{eq_tmp_1210_3a} Y_{(d-2)/2} (r) = r^{-(d-2)/2} \tilde g_2(r) + \log r \cdot r^{\frac{d-2}2} \tilde g_3(r) +r^{\frac{d-2}2} \cdot\tilde g_4(r),$$ where $\big| \frac{\partial^m \tilde g_j (r)}{\partial r^m} \big| \lesssim 1, \quad \forall\, m\ge0,\, r\lesssim 1,\; j=2,3,4.$ Now if the dimension $d$ is odd, then we use the formula $Y_{\frac {d-2} 2}(r) = Y_{\frac {d-3}2 +\frac 12} (r) = - \frac {2 \cdot (\tfrac r 2)^{\frac{d-2}2}} {\sqrt{ \pi \cdot (\tfrac{d-3}2)!}} \cdot \Big( 1+\frac{d^2}{dr^2} \Big)^{\frac{d-3}2} \big( \frac {\cos r}{r} \big).$ It follows that we can write $$\label{eq_tmp_1210_3b} r^{-(d-2)/2} \cdot Y_{(d-2)/2} (r) = r^{-(d-2)} \tilde g_5(r),$$ where $\big| \frac{\partial^m \tilde g_5 (r)}{\partial r^m} \big| \lesssim 1, \quad \forall\, m\ge0,\, r\lesssim 1.$ Next we also use the following information about Bessel functions in the regime $r\gtrsim 1$: $$\label{eq_tmp_1210_4} J_{(d-2)/2}(r) = \frac{a(r) e^{ir}}{\langle r\rangle^{1/2}} + \frac{\bar a(r) e^{-ir}}{\langle r\rangle^{1/2}}\,,$$ where $a(r)$ obeys the symbol estimates $\Bigr| \frac{\partial^m a(r)}{\partial r^m} \Bigr| \lesssim \langle r \rangle^{-m} \quad \text{for all m\geq0, r\gtrsim 1}$ Finally substitute \eqref{eq_tmp_1210_1}, \eqref{eq_tmp_1210_2}, \eqref{eq_tmp_1210_3a} (when $d$ is even), \eqref{eq_tmp_1210_3b} (when $d$ is odd), \eqref{eq_tmp_1210_4} into \eqref{kernel_highD}. Consider three regimes of $y$: $1/N \lesssim |y| \ll N|t|$, $|y|\lesssim 1/N$, $|y| \gg N|t|$ and use different asymptotics of the Bessel function in these regimes. Note also that the singular part of the Hankel function near $r=0$ adds only a power of $N$ due to our lower bound on $x$. It is then easy to see that a stationary phase point can only occur when $|y| \sim Nt$. Since we assume $|y|\ll Nt$ or $|y| \gg Nt$, integrating by parts and using Lemma \ref{lem_tmp12} yield the first claim. The second claim follows from a trivial $L^1$ estimate. We omit the details. \end{proof} \section{The proof of Theorem \ref{main_thm}} We first explain why it suffices for us to show that such two way non-scattering solution with minimal mass must be regular: $u_0\in H_x^1$. Indeed, if $u_0\in H_x^1$ and the corresponding solution blows up at finite time, according to Merle's result \cite{merle_duke}, we know it must scatter one way which contradicts our assumption. Then the solution must be global, here a direct application of Theorem \ref{h1_result} immediately yields the coincidence of the solution with $SW$ up to symmetries. Since the following proof of upgrading the regularity works for all two-way non-scattering solutions, for the sake of simplicity, we assume the solution is global. The discussion of the finite time blowup solutions is only notationally more complicated. To begin with, we recall the following result. The proof of this result is implicitly contained in \cite{klvz}. \begin{lemma}[Regularity of solutions away from the origin, \cite{klvz}]\label{weak_com} Let $d\ge 4$. Let $u_0\in L_x^2(\mathbb{R}^d)$ be spherically symmetric and $M(u_0)=M(Q)$. Let $u(t,x)$ be the corresponding solution such that it does not scatter in both time directions: $\|u\|_{L_{t,x}^{\frac {2(d+2)}d}((-\infty,0]\times\mathbb{R}^d)} =\|u\|_{L_{t,x}^{\frac {2(d+2)}d}([0,\infty)\times\mathbb{R}^d)}=\infty.$ Then there exists $\epsilon=\epsilon(d)>0$ such that $\|\phi_{>1}P_N u(t)\|_{L_x^2}\lesssim N^{-1-\epsilon}, \ \forall N\ge 1, \ t\in \mathbb{R}.$ In particular, $\|\phi_{>1}\nabla u(t)\|_{L_x^2}\lesssim 1, \ \forall t\in \mathbb{R}.$ \end{lemma} Now we use this information to upgrade the regularity of the initial data. To this end, we seek for the refined decay estimate for single frequency $P_N u_0$ with $N\ge 1$. Let $\gamma>0$ be a small parameter to be chosen later, we use triangle inequality to bound \begin{align} \|P_N u_0\|_{L_x^2} &\lesssim \|\phi_{\le N^{-1-\gamma}} P_N u_0\|_{L_x^2}\label{low_cut}\\ &\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1} P_N u_0\|_{L_x^2}\label{mid_cut}\\ &\quad +\|\phi_{>1}P_N u_0\|_{L_x^2}.\label{hi_cut} \end{align} First of all, Lemma \ref{weak_com} yields that $\eqref{hi_cut}\lesssim N^{-1-\epsilon}$. Next, using H\"older and Bernstein, \eqref{low_cut} can be controlled rather easily: $\eqref{hi_cut} \lesssim N^{\frac d2(-1-\gamma)}\|P_N u_0\|_{L_x^{\infty}} \lesssim N^{-s-\frac d2\gamma}\|u_0\|_{H_x^s}\lesssim N^{-s-\frac d2\gamma}.$ The task now is to estimate \eqref{mid_cut}, for which we will use the in-out decomposition and the improved Duhamel formula as we explain now. Since the solution $u$ does not scatter in both time directions and has minimal mass, according to \cite{compact, kvz:blowup} \footnote{The first reference established the improved Duhamel formula for minimal-mass non-scattering solution in which the scattering wave vanish when the $t$ approaches the maximal life time. The second one identifies $M(Q)$ as the minimal mass within all the spherically symmetric solutions.} we have \begin{align} u(t)&=\lim_{T\to \infty} -i\int_t^T e^{i(t-s)\Delta}F(u(s))ds\label{duh_for}\\ &=\lim_{T\to -\infty}i\int_T^t e^{i(t-s)\Delta}F(u(s))ds,\label{duh_bak} \end{align} where $F(u)=|u|^{4/d}u$ and the limit is understood in the weak $L_x^2$ sense. Using the in-out decomposition and \eqref{duh_for}, \eqref{duh_bak}, we estimate \begin{align} \eqref{mid_cut} &\le \|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ u_0\|_{L_x^2} +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^- u_0\|_{L_x^2}\notag\\ &\lesssim \|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+\int_0^{\infty} e^{-i\tau\Delta} F(u(\tau))d\tau \|_{L_x^2}\label{6}\\ &\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^-\int_0^{\infty} e^{i\tau\Delta} F(u(-\tau)) d\tau\| _{L_x^2}.\label{7} \end{align} Expression \eqref{6} and \eqref{7} will give the same contribution so we only need to estimate one of them. By splitting into different time pieces and introducing spatial cutoffs, we estimate \eqref{6} as follows \begin{align} \eqref{6} &\lesssim \|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int_{\frac 1N}^\infty e^{-i\tau\Delta} \phi_{>\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.1}\\ &\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int_{\frac 1N}^\infty e^{-i\tau\Delta} \phi_{\le\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.2}\\ &\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int^{\frac 1N}_{\frac 1{N^{2-\sigma}}} e^{-i\tau\Delta} \phi_{>\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.3}\\ &\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int^{\frac 1N}_{\frac 1{N^{2-\sigma}}} e^{-i\tau\Delta} \phi_{\le\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.4}\\ &\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int_{0}^{\frac 1{N^{2-\sigma}}} e^{-i\tau\Delta} F(u(\tau))d\tau\|_{L_x^2},\label{6.5} \end{align} where $0<\sigma<2$ is a small constant to be fixed later. We first look at \eqref{6.2}, \eqref{6.4} where the desired decay in $N$ comes from the kernel estimate Lemma \ref{Psmall_properties}. Let $A=(1+\gamma)(d-2)+2$, then for any $\tau>\frac 1{N^2}$, $m>0$, \begin{align*} \Big| \Bigl( \phi_{N^{-1-\gamma}<\cdot\le 1} P_N^+ e^{-i\tau\Delta} \phi_{\le \frac{N\tau}2} \Bigr) (x,y)\Big| &\lesssim_m N^A\langle N^2\tau+N|x|+N|y|\rangle^{-2m}\\ &\lesssim_m N^A|N^2\tau|^{-m}\langle N|x-y|\rangle^{-m}. \end{align*} Using this and Young's inequality, \eqref{6.2}, \eqref{6.4} can be bounded as follows \begin{align*} \eqref{6.2} &\lesssim_m N^A \int_{\frac 1N}^{\infty} |N^2\tau|^{-m}\|\langle N|\cdot|\rangle* F(u(\tau))\|_{L_x^2} d\tau\\ &\lesssim_m N^A N^{-2m}\int_{\frac 1N}^{\infty} \tau^{-m} d\tau \|\langle N|\cdot|\rangle\| _{L_x^{\frac d{d-2}}}\|F(u)\|_{L_t^{\infty}L_x^{\frac{2d}{d+4}}}\\ &\lesssim_m N^{-m+1+A-d}\|u\|_{L_t^{\infty}L_x^2}^{1+\frac 4d}\\ &\lesssim_m N^{-m+1+A-d}. \end{align*} Thus, by taking $m$ large enough depending on $d$, $\eqref{6.2}\lesssim N^{-10}.$ Expression \eqref{6.4} can be estimated in a similar way: \begin{align*} \eqref{6.4} &\lesssim_m N^A\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N}|N^2\tau|^{-m} \|\langle N|\cdot|\rangle* F(u(\tau))\|_{L_x^2} d\tau\\ &\lesssim_m N^{-2m+A+2-d}\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N} \tau^{-m}d\tau \\ &\lesssim_m N^{A-(m-1)\sigma-d}. \end{align*} For \eqref{6.1} and \eqref{6.3}, we will use the weighted Strichartz estimate Lemma \ref{L:wes}. In what follows we shall only present the details for $d\ge 5$. The case $d=4$ is similar and will be omitted. In dimension $d\ge 5$, from the $L_x^2$-boundedness of the operator $\phi_{>\frac 1{N^{1+\gamma}}}P_N^+$ Lemma \ref{L:bdd}, Lemma \ref{L:wes} and Lemma \ref{L:strichartz}, we have \begin{align} \eqref{6.1} &\lesssim N^{\frac{\gamma}2(d-4)} \Big\|\int_{\frac 1N}^{\infty}e^{-i\tau\Delta}\tilde{P}_N \phi_{>\frac{N\tau}2}F(u(\tau))d\tau\Big\|_{L_x^2}\notag\\ &\lesssim N^{\frac{\gamma}2(d-4)}\Big\|\int_{\frac 1N}^{\infty} e^{-i\tau\Delta}\tilde{P}_N\phi_{>\frac {N\tau}2}F(u\phi_{>\frac{N\tau}4})(\tau)d\tau\Big\|_{L_x^2}\notag\\ &\lesssim N^{\frac{\gamma}2(d-4)}\Big(\Big\|\int_{\frac 1N}^{\infty} e^{-i\tau\Delta}\tilde{P}_N\phi_{>\frac{N\tau}2}P_{\le N/8}F(u\phi_{>\frac{N\tau}4})(\tau)d\tau\Big\|_{L_x^2}\notag\\ &\quad+\Big\|\int_{\frac 1N}^{\infty}e^{-i\tau\Delta}\tilde{P}_N\phi_{>\frac{N\tau}2}P_{>N/8} F(u\phi_{>\frac{N\tau}4})(\tau)d\tau\Big\|_{L_x^2}\Big)\notag\\ &\lesssim N^{\frac{\gamma}2(d-4)}\Big(\|\tilde{P}_N \phi_{>\frac{N\tau}2}P_{\le N/8}F(u\phi_{>\frac{N\tau}4}) \|_{L_{\tau}^1L_x^2([\frac 1N,\infty)\times\mathbb{R}^d)}\label{8}\\ &\quad+\|(N\tau)^{-\frac{2(d-1)}d}P_{>N/8}F(u\phi_{>\frac{N\tau}4})\|_{L_{\tau}^{\frac d{d-1}} L_x^{\frac{2d}{d+4}}([\frac 1N,\infty)\times\mathbb{R}^d)}\Big).\label{9} \end{align} Using the mismatch estimate Lemma \ref{L:mismatch_fre}, we can bound \eqref{8} as \begin{align*} \eqref{8} &\lesssim \|(N^2\tau)^{-11}P_{\le N} F(u\phi_{>\frac{N\tau}4})\|_{L_{\tau}^1L_x^2 ([\frac 1N,\infty)\times\mathbb{R}^d)}\\ &\lesssim N^{-20}\|\tau^{-11} F(u\phi_{>\frac{N\tau}4})\|_{L_{\tau}^1L_x^{\frac{2d}{d+4}} ([\frac 1N,\infty)\times\mathbb{R}^d)}\\ &\lesssim N^{-20}\|\tau^{-11}\|_{L_{\tau}^1([\frac 1N,\infty))}\lesssim N^{-10}. \end{align*} For \eqref{9}, we use Bernstein estimate and Lemma \ref{weak_com} to get \begin{align*} \eqref{9} &\lesssim N^{-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}N^{-1}\| \nabla F(u\phi_{>\frac{N\tau}4}) \|_{L_x^{\frac{2d}{d+4}}}\|_{L_{\tau}^{\frac d{d-1}}([\frac 1N,\infty))}\\ &\lesssim N^{-1-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}\|u \phi_{>\frac{N\tau}4}\|_{L_x^2}^{4/d} \|\nabla (u\phi_{>\frac{N\tau}4})\|_{L_x^2}\|_{L_{\tau}^{\frac d{d-1}} ([\frac 1N,\infty))}\\ &\lesssim N^{-1-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}\|_{L_{\tau} ^{d/(d-1)}(\frac 1N,\infty)}\\ &\lesssim N^{-1-\frac {d-1}d}. \end{align*} Therefore, summarizing the two pieces together we have $\eqref{6.1}\lesssim N^{\frac{\gamma}2(d-4)}N^{-1-\frac{d-1}d}.$ Now we look at the piece \eqref{6.3} where the uniform kinetic energy estimate Lemma \ref{weak_com} is no longer available. Instead, we will use the fact $u_0\in H_x^s$, therefore locally we have the bound $$\|u\|_{L_{\tau}^{\infty}H_x^s([0,1]\times\mathbb{R}^d)}\lesssim_{u_0} 1.\label{local_hs}$$ Using this information, Lemma \ref{L:bdd}, Lemma \ref{L:mismatch_fre} and Lemma \ref{L:wes}, we control \eqref{6.3} as \begin{align*} \eqref{6.3} &\lesssim N^{\frac{\gamma}2(d-4)}\Big\|\tilde{P}_N\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N} e^{-i\tau\Delta}\phi_{>\frac{N\tau}2}F(u(\tau))d\tau\Big\|_{L_x^2}\\ &\lesssim N^{\frac{\gamma}2(d-4)}\Big(\Big\|\tilde{P}_N\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N} e^{-i\tau\Delta}\phi_{>\frac{N\tau}2}P_{\le\frac N8}F(u(\tau))d\tau\Big\|_{L_x^2}\\ &\quad +\Big\|\tilde{P}_N\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N}e^{-i\tau\Delta} \phi_{>\frac{N\tau}2} P_{>N/8}F(u(\tau))d\tau\Big\|_{L_x^2}\Big)\\ &\lesssim N^{\frac{\gamma}2(d-4)}\Big(\|\tilde{P}_N\phi_{>\frac{N\tau}2}P_{\le N/8}F(u)\| _{L_{\tau}^1L_x^2([\frac 1{N^{2-\sigma}},\frac 1N]\times\mathbb{R}^d)}\\ &\quad+\||N\tau|^{-\frac{2(d-1)}d}P_{>\frac N8}F(u(\tau))\|_{L_{\tau}^{\frac d{d-1}} L_x^{\frac{2d}{d+4}}([\frac 1{N^{2-\sigma}},\frac 1N]\times\mathbb{R}^d)}\Big)\\ &\lesssim_m N^{\frac{\gamma}2(d-4)} \Big(\||N^2\tau|^{-m}\|P_{\le N/8}F(u)\|_{L_x^2}\|_{L_{\tau}^1([\frac 1{N^{2-\sigma}},\frac 1N])}\\ &\quad+\||N\tau|^{-\frac{2(d-1)}d}N^{-s}\||\nabla|^s P_{> N/8}F(u)\|_{L_x^{\frac{2d}{d+4}}} \|_{L_{\tau}^{\frac d{d-1}}([\frac 1{N^{2-\sigma}},\frac 1N])}\\ &\lesssim_{m,u_0} N^{\frac{\gamma}2(d-4)}(N^{2-2m}\| \tau^{-m}\|_{L_{\tau}^1([\frac 1{N^{2-\sigma}},\frac 1N])} \\ &\quad +N^{-s-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}\|_{L_{\tau}^{\frac d{d-1}}([\frac 1{N^{2-\sigma}}, \frac 1N])}\Big)\\ &\lesssim_{m,u_0} N^{\frac{\gamma}2(d-4)}(N^{-\sigma(m-1)}+N^{-s-\frac{d-1}d\sigma}). \end{align*} Finally, we give the estimate of \eqref{6.5} which can firstly be trivially bounded as \begin{align} \eqref{6.5} &\lesssim \|\phi_{N^{-1-\gamma}<\cdot\le 1}\int_0^{\frac 1{N^{2-\sigma}}} P_N^+ e^{-i\tau\Delta}\phi_{>1}\tilde{P}_N F(u(\tau))d\tau \|_{L_x^2}\label{6.6}\\ &\quad+\|\phi_{N^{-1-\gamma}<\cdot\le 1}\int_0^{\frac 1{N^{2-\sigma}}} P_N^+ e^{-i\tau\Delta}\phi_{\le 1}\tilde{P}_N F(u(\tau)) d\tau\|_{L_x^2}. \label{6.7} \end{align} For \eqref{6.6}, we use the $L_x^2$ boundedness Lemma \ref{L:bdd}, weighted Strichartz Lemma \ref{L:wes}, Bernstein and local estimate \eqref{local_hs} to get \begin{align*} \eqref{6.6}&\lesssim N^{\frac \gamma 2(d-4)}\|\int_0^{\frac 1{N^{2-\sigma}}}e^{-i\tau\Delta}\phi_{>1}\tilde{P}_N F(u(\tau))d\tau\|_{L_x^2}\\ &\lesssim N^{\frac{\gamma}2(d-4)}\|\tilde{P}_N F(u)\|_{L_{\tau}^{\frac d{d-1}}L_x^{\frac{2d}{d+4}}([0, \frac 1{N^{2-\sigma}}]\times\mathbb{R}^d)}\\ &\lesssim N^{\frac{\gamma}2(d-4)}N^{-(2-\sigma)\frac{d-1}d}N^{-s} \||\nabla|^s F(u)\|_{L_{\tau}^{\infty}L_x^{\frac{2d}{d+4}}}\\ &\lesssim_{u_0} N^{\frac{\gamma}2(d-4)}N^{-\frac{(2-\sigma)(d-1)}d}N^{-s}. \end{align*} To estimate \eqref{6.7}, we will use the duality of the smoothing estimate \eqref{smoothing} as follows: $$\|\int_{\mathbb{R}} e^{-i\tau\Delta} |\nabla|^{1/2} f(\tau)d\tau\|_{L_x^2}\lesssim \||x|^{-\frac{d-1}2}f\|_{L_x^1L_t^2(\mathbb{R}^d\times\mathbb{R})}.\label{dual}$$ Let $\eta>0$ be a tiny number to be chosen later, using Lemma \ref{L:bdd} and \eqref{dual} we have \begin{align*} \eqref{6.7}&\lesssim N^{\frac{\gamma}2(d-4)}N^{-1/2}\|\int_{\mathbb{R}} e^{-i\tau\Delta}|\nabla|^{1/2}(\phi_{\le 1}\tilde{P}_N F(u(\tau))\chi_{0<\tau\le\frac 1{N^{2-\sigma}}}) d\tau\|_{L_x^2}\\ &\lesssim N^{\frac{\gamma}2(d-4)}N^{-1/2}\||x|^{-\frac{d-1} 2}\phi_{\le 1}\tilde{P}_N F(u)\| _{L_x^1L_{\tau}^2(\mathbb{R}^d\times[0,\frac 1{N^{2-\sigma}}])}\\ &\lesssim N^{\frac{\gamma}2(d-4)}N^{-1/2}\||x|^{\frac 12-\eta}\tilde{P}_N F(u)\|_{L_{\tau,x}^2([0, \frac 1{N^{2-\sigma}}]\times\mathbb{R}^d)}\||x|^{-\frac d2+\eta}\phi_{\le 1}\|_{L_x^2}.\\ &\lesssim N^{\frac{\gamma}2(d-4)-\frac 12-\frac{2-\sigma}2}\||x|^{\frac 12-\eta}\tilde{P}_N F(u)\|_{L_{\tau}^{\infty}L_x^2([0,\frac 1{N^{2-\sigma}}]\times\mathbb{R}^d)}. \end{align*} Now, using the radial Sobolev embedding Lemma \ref{L:radial_embed}, Bernstein, and \eqref{local_hs}, we bound the $F(u)$ term as \begin{align*} \||x|^{\frac 12-\eta}\tilde{P}_N F(u)\|_{L_x^2}&\lesssim \||\nabla|^{\eta}\tilde{P}_N F(u)\|_ {L_x^{\frac{2d}{d+1}}}\\ &\lesssim N^{\eta}N^{d(\frac{d^2+4d-8s}{2d^2}-\frac{d+1}{2d})}\|\tilde{P}_N F(u)\|_{L_x^{\frac{2d^2} {d^2+4d-8s}}}\\ &\lesssim N^{\eta+\frac{3d-8s}{2d}}N^{-s}\||\nabla|^s u\|_{L_x^2}\|u\|_{L_x^{\frac{2d}{d-2s}}}^{4/d}\\ &\lesssim_{u_0} N^{\eta+\frac 32-\frac{4s}d-s}. \end{align*} Plugging in this estimate back to the estimate of \eqref{6.7} we have $\eqref{6.7}\lesssim_{u_0} N^{\frac{\gamma}2(d-4)+\eta +\frac{\sigma}2-\frac{4s}d-s}.$ Combining the estimate for \eqref{6.6} and \eqref{6.7} together gives the final estimate of \eqref{6.5}: $\eqref{6.5}\lesssim_{u_0} N^{\frac{\gamma}2(d-4)}(N^{-\frac{(2-\sigma)(d-1)}d -s} +N^{\eta+\frac{\sigma}2-\frac{4s}d-s}).$ Now adding all the estimate for \eqref{6.1} through \eqref{6.5}, we finish estimating the term \eqref{6}. This together with the estimate for \eqref{low_cut} and \eqref{hi_cut} finally gives that \begin{align*} \|P_N u_0\|_{L_x^2} &\lesssim_{m,u_0} N^{-1-\epsilon}+N^{-s-\frac d2\gamma} +N^{(1+\gamma)(d-2)+2-d-\sigma(m-1)} \\ &\quad+N^{\frac{\gamma} 2(d-4)}(N^{-\frac{d-1}d\sigma-s}+N^{-\sigma(m-1)} +N^{-\frac{d-1}d(2-\sigma) -s} +N^{\eta+\frac{\sigma}2-\frac{4s}d-s}). \end{align*} For any $s>0$, choosing $\sigma=\frac s{100d}$, $\eta=\frac s{1000d}$, $\gamma=\frac s{1000d^2}$, $m=1+\frac{200d}s$, we finally obtain $\|P_N u_0\|_{L_x^2}\lesssim_{u_0}N^{-1-\epsilon}+N^{-s-\frac s{2000d}}.$ It is easy to see that after finite many times of iteration we obtain $\|P_N u_0\|_{L_x^2}\lesssim_{u_0} N^{-1-},\ \forall N\ge 1.$ Therefore, $u_0\in H_x^1$. The proof of Theorem \ref{main_thm} is complete. \subsection*{Acknowledgements} Both authors were supported by the National Science Foundation under agreement No. DMS-0635607 and start-up funding from the Mathematics Department of University of Iowa. 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