\documentclass[reqno]{amsart} \usepackage{graphicx, amssymb} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 24, 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-2007/24\hfil Local ill-posedness] {Local ill-posedness of the 1D Zakharov system} \author[J. Holmer\hfil EJDE-2007/24\hfilneg] {Justin Holmer} \address{Justin Holmer \newline Department of Mathematics\\ University of California, Berkeley, CA 94720-3840, USA} \email{holmer@math.berkeley.edu} \thanks{Submitted March 24, 2006. Published February 12, 2007.} \thanks{Partially supported by an NSF postdoctoral fellowship.} \subjclass[2000]{35Q55, 35Q51, 35R25} \keywords{Zakharov system; Cauchy problem; local well-posedness; \hfill\break\indent local ill-posedness} \begin{abstract} Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system \begin{gather*} i\partial_tu + \Delta u = nu \\ \partial_t^2 n - \Delta n = \Delta |u|^2 \\ u(x,0)=u_0(x), \\ n(x,0)=n_0(x), \quad \partial_tn(x,0)=n_1(x) \end{gather*} where $u=u(x,t)\in \mathbb{C}$, $n=n(x,t)\in \mathbb{R}$, $x\in \mathbb{R}$, and $t\in \mathbb{R}$. The proof was made for any dimension $d$, in the inhomogeneous Sobolev spaces $(u,n)\in H^k(\mathbb{R}^d)\times H^s(\mathbb{R}^d)$ for a range of exponents $k$, $s$ depending on $d$. Here we restrict to dimension $d=1$ and present a few results establishing local ill-posedness for exponent pairs $(k,s)$ outside of the well-posedness regime. The techniques employed are rooted in the work of Bourgain (1993), Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and Christ-Colliander-Tao (2003) applied to the nonlinear Schr\"odinger equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this paper, we examine the one-dimensional Zakharov system (1DZS) $$\label{1DZS} \begin{gathered} i\partial_tu + \partial_x^2 u = nu \\ \partial_t^2 n - \partial_x^2 n = \partial_x^2 |u|^2, \\ u(x,0)=u_0(x), \\ n(x,0)=n_0(x), \quad \partial_tn(x,0)=n_1(x) \end{gathered}$$ where $u=u(x,t)\in \mathbb{C}$, $n=n(x,t)\in \mathbb{R}$, $x\in \mathbb{R}$, and $t\in \mathbb{R}$. Local well-posedness in the inhomogeneous Sobolev spaces $(u,n)\in H^k(\mathbb{R}) \times H^s(\mathbb{R})$ has been obtained by means of the contraction method in the Bourgain space $$\|u\|_{X^S_{k,b_1}} = \Big( \iint_{\xi,\tau} \langle \xi \rangle^{2k} \langle \tau + |\xi|^2 \rangle^{2b_1} |\hat{u}(\xi,\tau)|^2 \, d\xi \; d\tau \Big)^{1/2}$$ by Bourgain-Colliander \cite{BC96} and Ginibre-Tsutsumi-Velo \cite{GTV97}. \footnote{Actually, these papers consider, more generally, the system in dimensions $d=2,3$ and $d\geq 1$, respectively.} In the latter paper, the following result is obtained. \begin{theorem}[{\cite[Prop.\ 1.2]{GTV97}}] \label{T:GTVProp1.2} Problem \eqref{1DZS} is locally well-posed for initial data $(u_0,n_0,n_1) \in H^k\times H^s \times H^{s-1}$ provided that \begin{gather*} k\geq 0 ,\quad s\geq -\tfrac{1}{2};\\ -1\leq s-k < \tfrac{1}{2} ,\quad s\leq 2k-\tfrac{1}{2} \end{gather*} Specifically: \begin{enumerate} \item \emph{Existence}. For all $R>0$, if $\|u_0\|_{H^k}+ \|n_0\|_{H^s} + \|n_1\|_{H^{s-1}}< R$, then there exist $T=T(R)$ and a solution $(u,n)$ to \eqref{1DZS} on $[0,T]$ such that \begin{gather*} \|u\|_{C([0,T]; H_x^k)} \leq c\| u_0\|_{H^k}, \\ \|n\|_{C([0,T]; H_x^s)} + \|\partial_t n\|_{C([0,T]; H_x^{s-1})} \leq c\langle \|u_0\|_{H^k}\rangle ^2 (\|n_0\|_{H^s} + \|n_1\|_{H^{s-1}}) \end{gather*} and $u\in X_{k,b_1}^S$, where $b_1$ is given by Table \ref{TAB:GTV}. \item \emph{Uniqueness}.\footnote{\eqref{1DZS} can be recast as an integral equation in $u$ alone with $W(n_0,n_1)$ solving \eqref{E:110} appearing as a coefficient. Then, $n$ can be expressed in terms of $u$ and $W(n_0,n_1)$, and therefore $n$ need not enter into the uniqueness claim.} This solution is unique among solutions $(u,n)$ such that $$u\in C([0,T]; H_x^k)\cap X^S_{k,b_1}.$$ \item \emph{Uniform continuity of the data-to-solution map}. For a fixed $R>0$, taking $T=T(R)$ as above, the map $(u_0,n_0,n_1) \mapsto (u,n, \partial_t n)$ as a map from the $R$-ball in $H^k\times H^s\times H^{s-1}$ to $C([0,T]; H_x^k) \times C([0,T]; H_x^s) \times C([0,T]; H_x^{s-1})$ is uniformly continuous. \end{enumerate} \end{theorem} The region of local well-posedness in this theorem is depicted in Fig.\ \ref{F:GTV}. We shall outline the \cite{GTV97} proof of Theorem \ref{T:GTVProp1.2} in \S \ref{S:GTV} since the estimates are needed in the proof of Theorem \ref{T:northip} in \S \ref{S:northip}. \begin{figure}[ht] \label{F:GTV} \begin{center} \includegraphics[width=0.6\textwidth]{fig1} \end{center} \caption{The enclosed gray-shaded strip, which extends infinitely to the upper-right, gives the set of pairs $(k,s)$ for which well-posedness has been established by \cite{GTV97} (see Theorem \ref{T:GTVProp1.2}) for $(u_0,n_0,n_1)\in H^k\times H^s\times H^{s-1}$. Solid lines are included in the well-posedness region, while the dashed line is not. Theorem \ref{T:northip} provides an ill-posedness result of type norm inflation in $n$'' inside the region bounded by the horizontal dotted line $s=-1/2$, the slanted line $s=2k-\frac12$, and the vertical dotted line $k=1$. Theorem \ref{T:deepsouthip} provides an ill-posedness result of type phase decoherence in $u$'' along the solid vertical line extending down from the point $(0,-3/2)$.} \end{figure} Our goal in this paper is to establish local ill-posedness outside of the well-posedness strip, in particular near the optimal corner $k=0$, $s=-1/2$. That is, we consider the region (1) $s>2k-\frac{1}{2}$ (above the strip), and (2) $s<-1/2$ (below the strip). In the first region, the wave data $(n_0,n_1)$ is somewhat smoother than the Schr\"odinger data $u_0$. As a result, the forcing term $\partial_x^2 |u|^2$ of the wave equation, as time evolves, introduces disturbances that are rougher than the wave data, and the wave solution $n$ does not retain its higher initial regularity. This is quantified in Theorem \ref{T:northip} below. In the second region, the Schr\"odinger data $u_0$ is somewhat smoother than the wave data $(n_0,n_1)$. As a result, the forcing term $nu$ of the Schr\"odinger equation introduces disturbances that are rougher than the Schr\"odinger data, and the Schr\"odinger solution $u$ does not retain its higher initial regularity. This is quantified in Theorem \ref{T:deepsouthip} and \ref{T:nearsouthip} below. These simplistic explanations are, at least, accurate for $k>0$. For $k<0$, there are possibly multiple simultaneous causes for breakdown, although we find that our methods still yield information in this setting. We will draw upon and suitably modify techniques developed by Birnir-Kenig-Ponce-Svanstedt-Vega \cite{BKPSV96}, Christ-Colliander-Tao \cite{CCT03b}, and Bourgain \cite{Bou93b}, who addressed ill-posedness issues for the nonlinear Schr\"odinger equation. For a survey of ill-posedness results for nonlinear dispersive equations, see Tzvetkov \cite{T04}. Our first result demonstrates that the boundary line $s\leq 2k-\frac12$ in Theorem \ref{T:GTVProp1.2} is sharp. \begin{theorem} \label{T:northip} Let $02k-\frac12$ or $k\leq 0$ and $s>-1/2$. There exists a sequence $\phi_N\in \mathcal{S}$ such that $\|\phi_N\|_{H^k} \leq 1$ for all $N$ and the corresponding solution $(u_N,n_N)$ to \eqref{1DZS} on $[0,T]$ with initial data $(\phi_N,0,0)$ satisfies \label{E:001} \|n_N(t)\|_{H_x^s} \geq ct N^\alpha \quad \text{for }00$. The time interval$[0,T]$here is independent of$N$. \end{theorem} The form of ill-posedness appearing in Theorem \ref{T:northip} is referred to as norm inflation''. The result is first reduced to the case where$k>0$and$s$is just above the line$s=2k-\frac12$. In this case, Theorem \ref{T:GTVProp1.2} applied with$s=2k-\frac12$(the wave initial data is$0$) provides the existence of a solution$(u_N,n_N)$on a time interval$T$, independent of$N$, with uniform-in-$N$control on$\|u_N\|_{X_{k,b_1}^S}$. The estimates of \cite{GTV97} will enable us to show that$u_N$is comparable to$e^{it\partial_x^2}\phi_N$in a slightly stronger norm than$X_{k,b_1}^S$(on this fixed in$N$time interval) and then Theorem \ref{T:northip} follows from the fact that \eqref{E:001} holds with$n_N=\square^{-1}\partial_x^2|u_N|^2$replaced by$\square^{-1}\partial_x^2|e^{it\partial_x^2}\phi_N|^2$, which can be directly verified.\footnote{$w=\square^{-1}f$is the solution to$\square w =(\partial_t^2-\partial_x^2)w=f$,$w(x,0)=0$,$\partial_t w(x,0)=0$.} The proof is given in \S\ref{S:northip}. Our second theorem demonstrates lack of uniform continuity of the data-to-solution map, for any$T>0$, as a map from the unit ball in$H^k\times H^s\times H^{s-1}$to$C([0,T]; H^k)\times C([0,T]; H^s) \times C([0,T]; H^{s-1})$for$k=0$and any$s< -\frac{3}{2}$. We first show that if one issue is ignored, we can, in a manner similar to \cite{BKPSV96}, make use of an explicit soliton class to demonstrate that for any$T>0$there are two waves, close in amplitude on all of$[0,T]$, initially of the same phase but that slide completely out of phase by time$T$. This form of ill-posedness is termed phase decoherence''. The soliton class for \eqref{1DZS} that we use appears in \cite{Guo89} \cite{Wu94}. The ignored issue'' pertains to low frequencies of$n_0(x)$, and can be resolved by invoking the method of \cite{CCT03b} to construct a near soliton'' class offering more flexibility than the exact explicit soliton class in the selection of$n_0(x)$. This is, however, not straightforward since \eqref{1DZS} lacks scaling and Galilean invariance, which was used to manufacture the solution class in \cite{CCT03b}. \begin{theorem} \label{T:deepsouthip} Suppose$k=0$,$s<-\frac{3}{2}$. Fix any$T>0$and$\delta>0$. Then there is a pair of Schwartz class initial data tuples$(u_{0},n_{0},0)$and$(\tilde{u}_{0}, \tilde{n}_{0}, 0)$giving rise to solutions$(u,n)$and$(\tilde{u}, \tilde{n})$on$[0,T]$such that the data is of unit size $$\|u_0\|_{H^k}, \|n_0\|_{H^s} \sim 1, \quad \|\tilde{u}_0\|_{H^k}, \|\tilde{n}_0\|_{H^s} \sim 1$$ and initially close $$\| u_0-\tilde{u}_0 \|_{H^k} + \|n_0-\tilde{n}_0\|_{H^s} \leq \delta$$ but the solutions become well-separated by time$T$in the Schr\"odinger variable $$\| u(\cdot, t)-\tilde{u}(\cdot, t) \|_{L_{[0,T]}^\infty H_x^k} \sim 1 \,.$$ \end{theorem} We expect that this result can be extended to all$k\in \mathbb{R}$and$s<-3/2$, although preliminary efforts were abandoned since the computations became very lengthy and technical. The proof of Theorem \ref{T:deepsouthip} appears in \S \ref{S:CCT}. Our final theorem employs a method of Bourgain \cite{Bou93b}. \begin{theorem} \label{T:nearsouthip} For any$T>0$, the data-to-solution map, as a map from the unit ball in$H^k\times H^{s}\times H^{s-1}$to$C([0,T]; H^k)\times C([0,T]; H^s) \times C([0,T]; H^{s-1})$fails to be$C^2$for$k\in \mathbb{R}$and$s<-1/2$. \end{theorem} This is a weaker form of ill-posedness than the phase decoherence of Theorem \ref{T:deepsouthip}, although it covers the full region below the well-posedness boundary$s=-1/2$of \cite{GTV97}. The proof is given in \S\ref{S:Bourgain}. \section{The local theory} \label{S:GTV} We outline and review the local well-posedness argument in \cite{GTV97} since the estimates will be needed in the proofs of Theorems \ref{T:northip}, \ref{T:nearsouthip}. Let$[ U(t) u_0 ]\hat{}(\xi) = e^{-it\xi^2}\hat{u}_0(\xi)$and $$U \ast_R f(\cdot, t) = \int_0^t U(t-t')f(t') \,dt'$$ denote the Schr\"odinger group and Duhamel operators, respectively. Define the Schr\"odinger Bourgain spaces$X^S_{k,\alpha}$,$Y^S_k$by the norms $$\label{E:112} \begin{gathered} \|z\|_{X^S_{k,\alpha}} = \Big( \iint_{\xi,\tau} \langle \xi \rangle^{2k} \langle \tau + |\xi|^2 \rangle^{2\alpha} |\hat{z}(\xi,\tau)|^2 \, d\xi \, d\tau \Big)^{1/2} \\ \|z\|_{Y^S_k} = \Big( \int_{\xi} \langle \xi \rangle^{2k} \Big(\int_\tau \langle \tau + |\xi|^2 \rangle^{-1} |\hat{z}(\xi,\tau)| \, d\tau\Big)^2 \, d\xi \Big)^{1/2} \,. \end{gathered}$$ Consider an initial wave data pair$(n_0, n_1)$. Split$n_1 = n_{1L}+n_{1H}$into low and high frequencies\footnote{This decomposition is needed, for otherwise the estimate in Lemma \ref{L:Group}\eqref{I:Groupwave} would have to be modified to have$\|n_1\|_{H^s}$in place of$\|n_1\|_{H^{s-1}}$on the right-hand side}, and set$\hat{\nu}(\xi) = \frac{\hat{n}_{1H}(\xi)}{i\xi}$, so that$\partial_x \nu = n_{1H}$. Let \begin{gather*} W_+(n_0,n_1)(x,t) = \tfrac{1}{2}n_0(x-t) -\tfrac{1}{2}\nu(x-t) + \tfrac{1}{2} \int_{x-t}^x n_{1L}(y)\, dy\\ W_-(n_0,n_1)(x,t) = \tfrac{1}{2}n_0(x+t) +\tfrac{1}{2}\nu(x+t) + \tfrac{1}{2} \int_x^{x+t} n_{1L}(y)\, dy \end{gather*} so that \begin{gather*} (\partial_t\pm\partial_x)W_\pm(n_0,n_1)(x,t) = \tfrac{1}{2}n_{1L}(x) \\ W_\pm(n_0,n_1)(x,0)=\tfrac{1}{2}n_0(x)\mp \tfrac{1}{2}\nu(x) \,. \end{gather*} By setting$n = W_+(n_0,n_1)+W_-(n_0,n_1)$, we obtain a solution to the linear homogeneous problem $$\label{E:110} \begin{gathered} \partial_t^2 n - \partial_x^2 n = 0 \quad t,x\in \mathbb{R}\\ n(x,0) = n_0(x), \; \partial_t n(x,0) = n_1(x) \quad n=n(t,x)\in \mathbb{R} \end{gathered}$$ Let $$\label{E:101} W_\pm\ast_R f (x,t) = \tfrac{1}{2}\int_0^t f(x\mp s, t-s) \ ds$$ so that $$(\partial_t \pm \partial_x)W_\pm\ast_R f(x,t) = \tfrac{1}{2}f(x,t), \quad W_\pm f(x,0)= 0, \quad \partial_tW_\pm f(x,0) = \tfrac{1}{2}f(x,0) \,.$$ It follows that if we set$n=W_-\ast_R f - W_+\ast_R f$, then we obtain a solution to the linear inhomogeneous problem \begin{gather*} \partial_t^2 n - \partial_x^2 n = \partial_x f \quad t,x\in \mathbb{R}\\ n(x,0) = 0, \; \partial_t n(x,0) = 0 \quad n(x,t)\in \mathbb{R} \end{gather*} Define the one-dimensional reduced wave Bourgain spaces$X^{W\pm}_{s,\alpha}$,$Y^{W\pm}_s$as $$\label{E:113} \begin{gathered} \|z\|_{X^{W_\pm}_{s,\alpha}} = \Big( \iint_{\xi,\tau} \langle \xi \rangle^{2s} \langle \tau \pm \xi \rangle^{2\alpha} |\hat{z}(\xi,\tau)|^2 \, d\xi \; d\tau \Big)^{1/2} \\ \|z\|_{Y^{W_\pm}_s} = \Big( \int_\xi \langle \xi \rangle^{2s} \Big( \int_{\tau} \langle \tau \pm \xi \rangle^{-1} |\hat{z}(\xi,\tau)| \, d\tau \Big)^2 d\xi \Big)^{1/2}\,. \end{gathered}$$ Let$\psi(t)=1$on$[-1,1]$and$\psi(t)=0$outside of$[-2,2]$. Let$\psi_T(t)= \psi(t/T)$, which will serve as a time cutoff for the Bourgain space estimates. For clarity, we write$\psi_1(t) = \psi(t)$. We can now recast \eqref{1DZS} as $$\label{E:Zakharovreduced} \begin{gathered} i\partial_tu + \partial_x^2 u = (n_++n_-)u \quad x\in \mathbb{R}, t\in \mathbb{R}\\ (\partial_t\pm \partial_x)n_\pm = \mp \tfrac{1}{2}\partial_x |u|^2 +\tfrac{1}{2}n_{1L} \end{gathered}$$ where$n=n_++n_-$, which has the integral equation formulation \begin{gather*} u(t) = U(t)u_0 -iU\ast_R[(n_++n_-)u](t) \\ n_\pm(t) = W_\pm(t)(n_0,n_1) \mp W_\pm\ast_R(\partial_x|u|^2)(t)\,. \end{gather*} \begin{lemma}[Group estimates] \label{L:Group} \quad \begin{enumerate} \item \label{I:GroupSch} \emph{Schr\"odinger}.$\|\psi_1(t) U(t)u_0\|_{X_{k,b_1}^S} \lesssim \|u_0\|_{H^k}$. \item \label{I:Groupwave}\emph{1-d Wave}.$\|\psi_1(t) W_\pm(t)(n_0,n_1)\|_{X^{W_\pm}_{s,b}} \lesssim \|n_0\|_{H_x^{s}} + \|n_1\|_{H_x^{s-1}} $. \end{enumerate} \end{lemma} \begin{lemma}[Duhamel estimates] \label{L:Duhamel} Suppose$T\leq 1$. \begin{enumerate} \item \label{I:DuhamelSch}\emph{Schr\"odinger}. If$0\leq c_1<\frac{1}{2}$,$0\leq b_1$,$b_1+c_1\leq 1$, then$\| \psi_T U\ast_R f \|_{X_{k,b_1}^S} \lesssim T^{1-b_1-c_1}\|f\|_{X_{k,-c_1}^S}$. \\ If$0\leq b_1\leq \frac{1}{2}$, then$\| \psi_T U\ast_R f \|_{X_{k,b_1}^S} \lesssim T^{\frac{1}{2}-b_1}(\|f\|_{X_{k,-\frac{1}{2}}^S\cap Y^S_k})$. \\$\| U\ast_R f\|_{C(\mathbb{R}_t; H_x^k)} \lesssim \|f\|_{Y^S_k}$. \item \label{I:Duhamelwave}\emph{1-d Wave}. If$0\leq c<\frac{1}{2}$,$0\leq b$,$b+c\leq 1$, then$\| \psi_T W_\pm \ast_R f \|_{X_{s,b}^{W\pm}} \lesssim T^{1-b-c}\|f\|_{X_{s,-c}^{W\pm}}$. \\ If$0\leq b\leq \frac{1}{2}$, then$\| \psi_T W_\pm \ast_R f \|_{X_{s,b}^{W\pm}} \lesssim T^{\frac{1}{2}-b}(\|f\|_{X_{s,-\frac{1}{2}}^{W\pm}\cap Y^{W\pm}_s})$.\\$\|W_\pm \ast_R f\|_{C(\mathbb{R}_t; H_x^s)} \lesssim \|f\|_{Y^{W\pm}_s}$. \end{enumerate} \end{lemma} \begin{lemma}[{\cite[Lemma 4.3/4.5]{GTV97}}] \label{L:GTV4.3} Let$k,s,b,c_1,b_1$satisfy \begin{gather*} s\geq -\tfrac{1}{2}, \quad k\geq 0, \quad s-k \geq -1 ,\\ b,c_1, b_1 > \tfrac{1}{4},\quad b+c_1 > \tfrac{3}{4},\quad b+b_1 >\tfrac{3}{4}, \\ s-k \geq -2c_1 \end{gather*} Then $$\|n_\pm u \|_{X^S_{k,-c_1}\cap Y^S_k} \lesssim \|n_\pm \|_{X^{W\pm}_{s,b}} \|u\|_{X^S_{k,b_1}}\,.$$ \end{lemma} \begin{lemma}[{\cite[Lemma 4.4/4.6]{GTV97}}] \label{L:GTV4.4} Let$k,s,c,b_1$satisfy \begin{gather*} s-2k \leq -\tfrac{1}{2},\quad k \geq 0, \quad s-k < \tfrac{1}{2}, \\ c,b_1 > \tfrac{1}{4},\quad c+b_1 >\tfrac{3}{4}, \\ s-k \leq 2b_1-1, \quad s-k < 2c-\tfrac{1}{2} \end{gather*} Then $$\| \partial_x (u_1 \bar{u}_2) \|_{X^{W\pm}_{s,-c}\cap Y^{W\pm}_s} \lesssim \|u_1\|_{X^S_{k,b_1}}\|u_2\|_{X^S_{k,b_1}} \,.$$ \end{lemma} To obtain Theorem \ref{T:GTVProp1.2}, fix$0\frac{1}{2}$and$c_1,c<\frac{1}{2}$for all cases except$s-k=-1$.} \label{TAB:GTV} \end{table} Consider first the case$s-k>-1$. We note from Table \ref{TAB:GTV} that$b_1, b>\frac{1}{2}$, and thus we have the Sobolev imbeddings $$\label{E:107} \begin{gathered} \| u\|_{C(\mathbb{R}_t; H_x^k)} \lesssim \|u\|_{X^S_{k,b_1}} \\ \|n_\pm\|_{C(\mathbb{R}_t; H_x^s)} \lesssim \|n_\pm \|_{X^{W\pm}_{s,b}}\,. \end{gathered}$$ Also, $$\partial_t n(x,t) = \partial_t(n_++n_-)(x,t) = \partial_x(-n_++n_-)(x,t) + n_{1L}(x)$$ and thus $$\label{E:108} \| \partial_t n \|_{C(\mathbb{R}_t; H_x^{s-1})} \lesssim \|n_\pm\|_{X^{W\pm}_{s,b}} + \|n_1\|_{H^{s-1}} \,.$$ Similar estimates apply to differences of solutions. Consider now the case$s-k=-1$, where it is necessary to take$b_1<\frac{1}{2}$. We return to \eqref{E:102} and estimate directly using Lemma \ref{L:Duhamel} to obtain $$\|u\|_{C(\mathbb{R}_t; H_x^k)} \lesssim \|u_0\|_{H^k} + \|n_\pm u \|_{Y^S_k}$$ and by Lemma \ref{L:GTV4.3}, $$\|n_\pm u \|_{Y^S_k} \lesssim \|n_\pm \|_{X^{W\pm}_{k,b}} \|u\|_{X^S_{s,b_1}}$$ where$b_1$,$b$are as specified in the Table \ref{TAB:GTV}, and the right-hand side is appropriately bounded by \eqref{E:104}, \eqref{E:105}. The bounds in \eqref{E:107}, \eqref{E:108} apply in this case since$b>\frac{1}{2}$. We further note that we can re-estimate$u$in$X^S_{k,\frac{1}{2}}$in \eqref{E:102} to obtain $$\label{E:106} \|u\|_{X^S_{k,\frac{1}{2}}} \lesssim \|u_0\|_{H^k} + (\|n_0\|_{H^s}+\|n_1\|_{H^{s-1}} + \|u_0\|_{H^k}^2 )\|u_0\|_{H^k} \,.$$ \section{Wave norm-inflation for$s> 2k-\frac12$} \label{S:northip} Here we prove Theorem \ref{T:northip}. In Steps 1--3, the result will be established for$02k-\frac12$but with$s$\textit{near}$2k-\frac12$. In Steps 4--5, the general case of the theorem is reduced to the case considered in Steps 1--3. \begin{proof} Let$00$is not chosen too large. This says that$u_N(t)$is well-approximated by the linear flow$\psi_1(t)U(t)\phi_N$in the \textit{stronger} norm$X_{k+\sigma}^S. We now prove \eqref{E:201}. From \eqref{E:202}, \begin{align*} &\|u_N - \psi_1 U\phi_N\|_{X^S_{k+\sigma,b_1}} \\ &\leq \begin{cases} \| (W_\pm \ast_R \partial_x |u_N|^2) \cdot u_N \|_{X^S_{k+\sigma,-c_1}} & \text{if }0\leq k+\sigma \leq \frac{1}{2} \\ \| (W_\pm \ast_R \partial_x |u_N|^2) \cdot u_N \|_{X^S_{k+\sigma,-c_1} \cap Y^S_{k+\sigma}} & \text{if }\frac{1}{2}\leq k+\sigma \leq \frac{5}{2} \end{cases} \end{align*} forb_1as defined above and $$c_1 = \begin{cases} \frac{1}{4} + \frac{k+\sigma}{2} & \text{if }00 such that \label{E:212} s \leq \begin{cases} 2(k+\sigma) - \frac{1}{2} & \text{if }02k-\frac12. Let s' be such that s'\leq s and s' meets the restrictions outlined in Step 3 with s replaced by s'. Then by Steps 1--3 (with s replaced by s')$$ \|n_N(t)\|_{H^s} \geq \|n_N(t)\|_{H^{s'}} \geq tN^{s'-(2k-\frac{1}{2})} \quad \text{for }N\geq t^{-1} $$so we can take \alpha=s'-(2k-\frac{1}{2}) in the statement of the theorem. \noindent\textbf{Step 5}. Next, suppose k<0 and s>-1/2. By the reasoning of Step 4, it suffices to restrict to s<\frac32. Set 02k''-1/2. Clearly \|u_N(t)\|_{H^k} \leq \|u_N(t)\|_{H^{k''}}, so we can just appeal to the conclusion of Steps 1--4 applied with k replaced by k''. \end{proof} \section{A preliminary analysis for s\leq -\frac{3}{2}} Let f(x) = \sqrt 2 \operatorname{sech}(x), which is the unique positive ground state solution to $$\label{E:501} -f + \partial_x^2 f + |f|^2 f =0$$ Let f_\lambda(x) = \lambda f(\lambda x) and set \begin{gather*} u_{\lambda,N}(x,t) = e^{it(\lambda^2 - N^2)}e^{iNx} \sqrt{1-4N^2} f_\lambda (x-2Nt) \\ n_{\lambda,N}(x,t) = -|f_\lambda (x-2Nt)|^2 \end{gather*} From \eqref{E:501}, it follows that (u_{\lambda,N},n_{\lambda,N}) solves \eqref{1DZS} for all \lambda\in \mathbb{R} and -\frac{1}{2}0, \delta>0. Then there exist M(\delta) sufficiently large and N(\delta)<\frac{1}{2} sufficiently close to \frac{1}{2} so that if$$ \lambda_1=M, \quad \lambda_2=\sqrt{M^2+\frac{\pi}{2T}} then the solutions are of unit size on [0,T], $$\label{E:504} \begin{gathered} \|u_{\lambda_j,N}(\cdot, t)\|_{L_x^2} \sim 1 \\ \|n_{\lambda_j,N}(\cdot, t)\|_{H^s(|\xi| \geq M)} \sim 1, \quad \|\partial_t n_{\lambda_j,N}(\cdot, t)\|_{H^{s-1}(|\xi| \geq M)} \sim 1 \end{gathered}$$ and are initially close $$\label{E:502} \| u_{\lambda_2,N}(\cdot, 0) -u_{\lambda_1,N}(\cdot, 0) \|_{L^2} \leq \delta$$ $$\label{E:503} \begin{gathered} \| n_{\lambda_2,N}(\cdot, 0) -n_{\lambda_1,N}(\cdot, 0) \|_{H^s(|\xi|\geq M)} \leq \delta\\ \| \partial_t n_{\lambda_2,N}(\cdot, 0) -\partial_t n_{\lambda_1,N}(\cdot, 0) \|_{H^{s-1}(|\xi|\geq M)} \leq \delta \end{gathered}$$ but become fully separated in the u-variable by time T, $$\label{E:505} \| u_{\lambda_2,N}(\cdot, T) -u_{\lambda_1,N}(\cdot, T) \|_{L^2} \sim 1$$ \end{proposition} \begin{proof} We will select M=M(\delta) sufficiently large later. Take 0 \leq N < \frac{1}{2} sufficiently close to \frac{1}{2} so that (1-2N)^{1/2}M^{1/2}=1. Then since N\sim \frac{1}{2} we have \sqrt{1-4N^2} \sim (1-2N)^{1/2} and noting that \lambda_1=M and (1-2N)^{1/2}M^{1/2} =1 gives \begin{align*} \|u_{\lambda_2,N}(\cdot,0)- u_{\lambda_1,N}(\cdot, 0)\|_{L^2} & = (1-2N)^{1/2} \big\| \hat f\big( \frac{\xi}{\lambda_2} \big) - \hat f \big( \frac{\xi}{\lambda_1} \big) \big\|_{L_\xi^2} \\ &= \big\| \hat f\big( \frac{\lambda_1 \xi}{\lambda_2} \big) - \hat f(\xi) \big\|_{L_\xi^2} \end{align*} Take M sufficiently large so that \lambda_1/\lambda_2 is sufficiently close to 1 in order to make the above expression \leq \delta. Thus \eqref{E:502} is established. Next, we establish \eqref{E:503}. By the change of variable \xi \mapsto \lambda_1 \xi \begin{align*} & \| n_{\lambda_2,N}(\cdot, 0) - n_{\lambda_1,N}(\cdot, 0) \|_{H^s(|\xi|\geq M)}^2 \\ &= \lambda_1^{3+2s} \int_{|\xi| \geq 1} \Big| \frac{\lambda_2}{\lambda_1}( f^2)\hat{} \big( \frac{\xi\lambda_1}{\lambda_2} \big) - (f^2)\hat{}(\xi) \Big|^2 |\xi|^{2s} \, d\xi \end{align*} Since s\leq -\frac{3}{2} we have \lambda_1^{3+2s} \leq 1 and the above difference is made \leq \delta by again taking M sufficiently large. Also \begin{align*} & \| \partial_t n_{\lambda_2,N}(\cdot, 0) - \partial_t n_{\lambda_1,N}(\cdot, 0) \|_{H^{s-1}(|\xi|\geq M)}^2 \\ &= N^2\lambda_1^{3+2s} \int_{|\xi| \geq 1} \Big| \frac{\lambda_2^2}{\lambda_1^2}( f^2\,')\hat{} \big( \frac{\xi\lambda_1}{\lambda_2} \big) - (f^2\,')\hat{}(\xi) \Big|^2 |\xi|^{2(s-1)} \, d\xi \end{align*} (Here, the notation ' indicates the derivative). Since s\leq -\frac{3}{2} we have \lambda_1^{3+2s} \leq 1 and the above difference is made \leq \delta by again taking M sufficiently large. The statements \eqref{E:504} are proved by similar change of variable calculations. The need for the restrictions to |\xi|\geq M in \eqref{E:503} is clear from these calculations. In fact, one can show that for s<-\frac12, we have \|n_{\lambda, N}(\cdot, 0)\|_{H^s} \sim \lambda as \lambda \to +\infty due to the |\xi|\leq \lambda frequency contribution. Now we establish \eqref{E:505}. The key observation here is that while \lambda_2-\lambda_1 is very small (as M\to +\infty), \lambda_2^2-\lambda_1^2 is of fixed size \pi/(2T) and thus e^{iT(\lambda_2^2-\lambda_1^2)} = i is purely imaginary. Now \|u_2(\cdot, T) - u_1(\cdot, T)\|_{L^2}^2 = \|u_2(\cdot, T)\|_{L^2}^2 + \|u_1(\cdot, T)\|_{L^2}^2 - 2\operatorname{Re} \int_x u_2(x,T) \overline{u_1(x,T)} \, dx $$but the last term on the right-hand side is$$ -2\operatorname{Re} e^{iT(\lambda_2^2-\lambda_1^2)}(1-4N^2) \int_x \lambda_2f(\lambda_2x) \lambda_1f(\lambda_1x) \, dx =0 $$which, combined with \eqref{E:504} gives \eqref{E:505}. \end{proof} \section{Schr\"odinger phase decoherence for s<-3/2} \label{S:CCT} Here, we remove the shortcoming of Proposition \ref{P:deepsouthippartial} (high frequency truncated norms H^s(|\xi|\geq M), H^{s-1}(|\xi|\geq M) used instead of H^s, H^{s-1}) and prove Theorem \ref{T:deepsouthip}. The soliton class employed in the proof of Proposition \ref{P:deepsouthippartial} involved assigning$$ n(x,t) = -\lambda^2 |f|^2(\lambda(x-2tN)) $$and thus \hat{n}(\xi,t) = -\lambda (|f|^2)\hat{}(\xi/\lambda) e^{-2itN\xi}. Replace |f|^2 in the definition of n by g defined by \hat{g}(\xi) = (|f|^2)\hat{}(\xi) \chi_{|\xi|\geq 1}(\xi) (i.e.\ the restriction to frequencies \geq 1) and set$$ \tilde{n}(x,t) = -\lambda^2 g(\lambda (x-2tN)) $$Then$$ \| \tilde n(\cdot, t)\|_{H^s} + \| \partial_t \tilde n(\cdot, t)\|_{H^{s-1}} \leq 1, \quad \text{as }\lambda \to +\infty $$Unfortunately, (u,\tilde n) is no longer a solution to \eqref{1DZS}. We shall thus adapt the method of Christ-Colliander-Tao \cite{CCT03b} to construct a near soliton'' class that grants more flexibility in the selection of the wave initial data. The method proceeds by solving a small dispersion approximation'' to the equation, and by introducing scaling and phase translation parameters, building the near soliton'' class. The main new obstacle, in comparison to the work of \cite{CCT03b} applied to the nonlinear Schr\"odinger equation, is that \eqref{1DZS} does not possess scaling nor the Galilean (phase shift) identity. We thus need to carry out the small dispersion approximation for a \textit{modified} Zakharov system with the property that when scaling and phase shift operations are performed, the modified Zakharov system is converted into the true Zakharov system. Consider fixed initial data (n_0,u_0) (to be defined later). \noindent\textbf{Step 1.} The solution to the small dispersion approximation$$i\partial_t v = n_0(x) v$$with v(x,0)=u_0(x) is$$ v(x,t) = e^{-itn_0(x)}u_0(x) $$\noindent\textbf{Step 2.} For parameters \lambda \gg 1, 0<\nu \ll 1, -\frac{1}{2}0 sufficiently small so that$$ \| u_2(x,t) - u_1(x,t) \|_{L_x^2} = \| u^{(\lambda_2,\nu,N)}( x, \lambda_2^2t ) - u^{(\lambda_1,\nu,N)}(x,\lambda_1^2 t) \|_{L_x^2} + \mathcal{O}(\delta) $$By the results of Step 1 and 3, again taking \nu=\nu(\delta) sufficiently small, if 0 \leq t \leq |\ln \nu|/M^2, then $$\label{E:609} \| u_2(x,t) - u_1(x,t) \|_{L_x^2} = \| (e^{i(\lambda_2^2-\lambda_1^2)t \, n_0(x)} - 1)u_0(x) \|_{L_x^2} + \mathcal{O}(\delta)$$ Here, the first condition of \eqref{E:606} is met, since the T appearing there is our \lambda_1^2t \sim \lambda_2^2t \lesssim |\ln v|. We see trivially from \eqref{E:609} that$$ \| u_2(x,0) - u_1(x,0) \|_{L_x^2} \lesssim \delta $$But by \eqref{E:612} and \eqref{E:609} and the choice of u_0(x) and n_0(x),$$ \| u_2(x,T) - u_1(x,T) \|_{L_{[0,T]}^\infty L_x^2} = \mathcal{O}(1) $$We further note that$$ T= \frac{|\ln \nu|}{M^2} \leq |\ln \nu| \nu^4 \to 0 \quad \text{as }\nu \to 0 $$and therefore we can accommodate an arbitrarily small preselected time, T as in the statement of the theorem. \section{The Schr\"odinger flow map is not C^2 for s<-1/2} \label{S:Bourgain} In this section, we give the proof of Theorem \ref{T:nearsouthip}. For fixed H^\infty data (u_0,n_0,n_1), to be specified later, and a parameter \gamma \in \mathbb{R}. We consider the initial data $(u\big|_{t=0},n\big|_{t=0},\partial_tn\big|_{t=0}) =(\gamma u_0, \gamma n_0, \gamma n_1)$ and the corresponding 1DZS solutions (u,n)=(u_\gamma,n_\gamma). Clearly $$\label{E:701} u\big|_{\gamma=0}=0, \quad \partial_xu\big|_{\gamma=0}=0, \quad \partial_x^2u\big|_{\gamma=0}=0, \quad n\big|_{\gamma=0}=0, \quad \partial_xn\big|_{\gamma=0}=0$$ The solution, written in integral equation form, is: \begin{gather*} u(t) = U(t)(\gamma u_0) -i \int_0^t U(t-t')[(un)(t')]\, dt' \\ n(t) = W(t)(\gamma n_0,\gamma n_1) \pm \tfrac{1}{2} \int_0^t \partial_x |u|^2(x\pm s,t-s) \, ds \end{gather*} from which it follows that $$\label{E:702} \begin{gathered} \partial_\gamma u(t) = U(t)u_0 -i \int_0^t U(t-t')[(\partial_\gamma u \, n + u \,\partial_\gamma n)(t')]\, dt' \\ \partial_\gamma n(t) = W(t)(n_0,n_1) \pm \tfrac{1}{2}\int_0^t \partial_x (\partial_\gamma u \, \bar{u} + u \overline{\partial_\gamma u})(x\pm s,t-s) \, ds \end{gathered}$$ By \eqref{E:701}, $$\label{E:703} \partial_\gamma u\big|_{\gamma=0} = Uu_0, \quad \partial_\gamma n\big|_{\gamma=0}=W(n_0,n_1)$$ By applying \partial_x to \eqref{E:702} and again appealing to \eqref{E:701}, we get $$\label{E:704} \partial_x \partial_\gamma u\big|_{\gamma=0} =\partial_xUu_0, \quad \partial_\gamma\partial_x n\big|_{\gamma=0} =\partial_xW(n_0,n_1)$$ By applying \partial_\gamma to \eqref{E:702}, we obtain \begin{gather*} \partial_\gamma^2 u(t) = -i \int_0^t U(t-t')[(\partial_\gamma^2 u \, n +2\partial_\gamma u \, \partial_\gamma n + u \,\partial_\gamma^2 n)(t')]\, dt' \\ \partial_\gamma^2 n(t) = \pm \int_0^t \partial_x (\partial_\gamma^2 u \, \bar{u} +2|\partial_\gamma u|^2+ u \overline{\partial_\gamma^2 u}) (x\pm s,t-s) \, ds \end{gather*} from which we find, together with \eqref{E:703}\eqref{E:704}, that \begin{gather*} \partial_\gamma^2 u \big|_{\gamma=0}(t) = -2i \int_0^t U(t-t')[ Uu_0(t') \; W(n_0,n_1)(t')]\, dt'\\ \partial_\gamma^2 n \big|_{\gamma=0}(t) = \pm \int_0^t \partial_x|U(t')u_0|^2(x\mp s, t-s) \, ds \end{gather*} Let X=H^k\times H^{s}\times H^{s-1}, Y=H^k\times H^s. Fix t>0, and let F:X\to Y be the solution map F(u_0,n_0,n_1) = (u(t),n(t)). Let G:\mathbb{R}\to X be given by G(\gamma)=(\gamma u_0,\gamma n_0, \gamma n_1). Let H(\gamma)=F\circ G(\gamma) so that H:\mathbb{R} \to Y. Then (here \mathcal{L}(A;B) denotes a linear map A\to B)$$ \underbrace{DH(\gamma)}_{\mathcal{L}(\mathbb{R};Y)} = \underbrace{DF(G(\gamma))}_{\mathcal{L}(X;Y)} \circ \underbrace{DG(\gamma)}_{\mathcal{L}(\mathbb{R};X)} $$Also$$ D^2H(\gamma)=\underbrace{D^2F(G(\gamma))}_{\mathcal{B}(X\times X;Y)} \circ(\underbrace{DG(\gamma)}_{\mathcal{L}(\mathbb{R};X)}, \underbrace{DG(\gamma)}_{\mathcal{L}(\mathbb{R};X)}) +\underbrace{DF(G(\gamma))}_{\mathcal{L}(X;Y)}\circ \underbrace{D^2G(\gamma)}_{\mathcal{B}(\mathbb{R}\times \mathbb{R}; X)} $$and since D^2G(\gamma)=0,$$ D^2H(\gamma)(\alpha_1,\alpha_2)=D^2F(G(\gamma))((\alpha_1 u_0, \alpha_1 n_0, \alpha_1 n_1), \;(\alpha_2 u_0, \alpha_2 n_0, \alpha_2 n_1)) $$Hence$$ D^2F(0)((u_0,n_0,n_1),(u_0,n_0,n_1)) = D^2H(0)(1,1) = (\partial_\gamma^2 u\big|_{\gamma=0}(t), \partial_\gamma^2 n\big|_{\gamma=0}(t)) $$We now note how to prescribe an appropriate sequence (u_{N,0},n_{N,0},n_{N,1}) (indexed by N) to show that D^2F(0)\in \mathcal{B}(X\times X; Y) is not a bounded (continuous) bilinear map in the two cases (1) s<-1/2 and (2) s>2k-\frac12. \begin{itemize} \item If s<-1/2, \begin{gather*} \hat{u}_{N,0}(\xi) = N^{\frac{1}{2}-k}\chi_{[-N-\frac{1}{N},-N]}(\xi),\\ \hat{n}_{N,0}(\xi) = N^{\frac{1}{2}-s}\chi_{[2N-1,2N-1+\frac{1}{N}]} (\xi) + N^{\frac{1}{2}-s}\chi_{[-2N+1-\frac{1}{N},-2N+1]}(\xi) \end{gather*} and n_{N,1}=0, then (u_{N,0},n_{N,0},n_{N,1}) is a sequence such that $\|(u_{N,0},n_{N,0},n_{N,1})\|_X \sim 1,$ but$$ \|\partial_\gamma^2u\big|_{\gamma=0}(t)\|_{H^k} \geq c(t) N^{-s-\frac{1}{2}}. $$We note that the second term in the definition of \hat n_{N,0}(\xi) is included solely to make n_0(x) real. \item If s>2k-\frac12 and we set$$ \hat{u}_{N,0}(\xi) = N^{\frac{1}{2}-k}(\chi_{[-N-\frac{1}{N},-N]}(\xi) + \chi_{[N+1,N+1+\frac{1}{N}]}(\xi)) $$and n_{N,0}=0, n_{N,1}=0, then (u_{N,0},n_{N,0},n_{N,1}) is a sequence such that \|(u_{N,0},n_{N,0},n_{N,1})\|_X \sim 1 but$$ \|\partial_\gamma^2n\big|_{\gamma=0}(t)\|_{H^s} \geq c(t) N^{s-(2k-\frac{1}{2})} \end{itemize} The second was considered in \S \ref{S:northip} as part of the proof of Theorem \ref{T:northip}, and thus reproduces a weaker version of that result. We now carry out a proof of the first case to establish Theorem \ref{T:nearsouthip}. Since $$\label{E:705} [\partial_\gamma^2u\Big|_{\gamma=0}(t)]\hat{}(\xi) = \int_0^t e^{-i(t-t')\xi^2}[Uu_{N,0}(t')W(n_{N,0},0)(t')]\hat{}(\xi) \, dt'$$ we need to examine \begin{align*} & [Uu_{N,0}(t')W(n_{N,0},0)(t')]\hat{}(\xi) \\ &= \int_{\xi_1} e^{-it'\xi_1^2}\hat{u}_{N,0}(\xi_1) \cos(t'(\xi-\xi_1)) \hat n_{N,0}(\xi-\xi_1) \, d\xi_1 \\ &= (e^{-it'N^2}\cos(t'(2N-1))+ \mathcal{O}(t')) \int_{\xi_1} \hat{u}_{N,0}(\xi_1) \hat n_{N,0}(\xi-\xi_1) \, d\xi_1 \end{align*} by the support properties of u_{N,0} and n_{N,0}. Directly evaluating the convolution gives (e^{-it'(N-1)^2}+e^{-it'(N^2+2N-1)}+\mathcal{O}(t')) N^{1-k-s}g(\xi) where g(\xi)=g_1(\xi)+g_2(\xi) consists of two triangular step functions, each of height 1/N and width 2/N, centered at N-1 and -3N+1, respectively. Specifically, \begin{gather*} g_1(\xi) =\begin{cases} \tfrac{1}{N} - |\xi-(N-1)| & \text{if }|\xi-(N-1)| \leq \tfrac{1}{N} \\ 0 & \text{otherwise} \end{cases} \\ g_2(\xi) =\begin{cases} \tfrac{1}{N} - |\xi-(-3N+1)| & \text{if }|\xi-(-3N+1)| \leq \tfrac{1}{N}\\ 0 & \text{otherwise} \end{cases} \end{gather*} We have by the support properties of g_1(\xi) and g_2(\xi) and \eqref{E:705}, \begin{align*} & [\partial_\gamma^2 u\Big|_{\gamma=0}(t)]\hat{}(\xi)\\ &=+N^{1-k-s} g_1(\xi) \int_0^t e^{-i(t-t')(N-1)^2}(e^{-it'(N-1)^2} + e^{-it'(N^2+2N-1)}) \, dt' \\ &\quad +N^{1-k-s} g_2(\xi)\int_0^t e^{-i(t-t')(-3N+1)^2}(e^{-it'(N-1)^2} + e^{-it'(N^2+2N-1)}) \, dt'\\ &\quad +N^{1-k-s}g(\xi)\mathcal{O}(t^2) \end{align*} Evaluating each component separately gives [\partial_\gamma^2 u\Big|_{\gamma=0}(t)]\hat{}(\xi) = N^{1-k-s}g_1(\xi)e^{-it(N-1)^2}t + N^{1-k-s}g(\xi)(\mathcal{O}(t^2) + \mathcal{O}(N^{-1})) $$Thus, provided t is chosen small and N sufficiently large, the first term is pointwise dominant, giving$$ \|\partial_\gamma^2 u\big|_{\gamma=0}(t)\|_{H^k} \geq tN^{-\frac{1}{2}-s}\$ completing the proof. \subsection*{Acknowledgments} I would like to thank Jim Colliander for his clear explanation of how to construct counterexamples to bilinear estimates and for other helpful discussion on this topic. 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