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Guixiang Xu

Publications and source records attributed to Guixiang Xu.

At least 37 records · Page 2Linked to original sources
arXiv2017

Global center stable manifold for the defocusing energy critical wave equation with potential

In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(ϕ, 0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(ϕ,0)$. It is constructed in a first step locally around any solution scattering to $ϕ$, which might be very far away from $ϕ$ in the $\dot{H}^1\times L^2(\mathbb{R}^3)$ norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends our previous result [18] to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from [3] to construct a local center stable manifold near any solution that scatters to $(ϕ, 0)$. This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle [14], which is used to show that solutions that start off but near the local manifolds associated with $ϕ$ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state $ϕ$.

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arXiv2017

Solitary waves for nonlinear Schrödinger equation with derivative

In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesn't enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4ω>c^2$ and the critical parameters $4ω=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters $4ω=c^2, c\leq 0$ and the supercritical parameters $4ω 0$ or $4ω>c^2$. On one hand, different with the scattering result for the $L^2$-critical NLS in \cite{Dod:NLS_sct}, the scattering result of (DNLS) doesn't hold for initial data in $\mathcal{K}^+_{ω,c}$ because of the existence of infinity many small solitary/traveling waves in $\mathcal{K}^+_{ω,c},$ with $4ω=c^2, c>0$ or $4ω>c^2$. On the other hand, our global result improves the global result in \cite{Wu-DNLS, Wu-DNLS2} (see Corollary \ref{cor:gwp}).

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arXiv2017

Stability of the traveling waves for the derivative Schrödinger equation in the energy space

In this paper, we continue the study of the dynamics of the traveling waves for nonlinear Schrödinger equation with derivative (DNLS) in the energy space. Under some technical assumptions on the speed of each traveling wave, the stability of the sum of two traveling waves for DNLS is obtained in the energy space by Martel-Merle-Tsai's analytic approach in \cite{MartelMT:Stab:gKdV, MartelMT:Stab:NLS}. As a by-product, we also give an alternative proof of the stability of the single traveling wave in the energy space in \cite{ColinOhta-DNLS}, where Colin and Ohta made use of the concentration-compactness argument.

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arXiv2017

Stability of the sum of two solitary waves for (gDNLS) in the energy space

In this paper, we continue the study in \cite{MiaoTX:DNLS:Stab}. We use the perturbation argument, modulational analysis and the energy argument in \cite{MartelMT:Stab:gKdV, MartelMT:Stab:NLS} to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schrödinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case $σ>1$ we considered corresponds to the $L^2$-supercritical case.

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arXiv2016

The dynamics of the NLS with the combined terms in five and higher dimensions

In this paper, we continue the study in \cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schrödinger equation with the energy below the threshold $m$ in the energy space $H^1(\R^d)$, iu_t + Δu = -|u|^{4/(d-2)}u + |u|^{4/(d-1)}u, \; d\geq 5. \tag{CNLS} The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + Δu = -|u|^{4/(d-2)}u$. Compared with the argument in \cite{MiaoWZ:NLS:3d Combined}, the new ingredient is that we use the double duhamel formula in \cite{Kiv:Clay Lecture, TaoVZ:NLS:mass compact} to lower the regularity of the critical element in $L^{\infty}_tH^1_x$ to $L^{\infty}\dot H^{-ε}_x$ for some $ε>0$ in five and higher dimensions and obtain the compactness of the critical element in $L^2_x$, which is used to control the spatial center function $x(t)$ of the critical element and furthermore used to defeat the critical element in the reductive argument.

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arXiv2015

Generic and non-generic behavior of solutions to the defocusing energy critical wave equation with potential in the radial case

In this paper, we continue our study [16] on the long time dynamics of radial solutions to defocusing energy critical wave equation with a trapping radial potential in 3 + 1 dimensions. For generic radial potentials (in the topological sense) there are only finitely many steady states which might be either stable or unstable. We first observe that there can be stable excited states (i.e., a steady state which is not the ground state) if the potential is large and attractive, although all small excited states are unstable. We prove that the set of initial data for which solutions scatter to any one unstable excited state forms a finite co-dimensional connected $C^1$ manifold in energy space. This amounts to the construction of the global path-connected, and unique, center-stable manifold associated with, but not necessarily close to, any unstable steady state. In particular, the set of data for which solutions scatter to unstable states has empty interior in the energy space, and generic radial solutions scatter to one of the stable steady states. Our main tools are (i) near any given finite energy radial initial data $(u_0,u_1)$ for which the solution $u(t)$ scatters to some unstable steady state $ϕ$ we construct a $C^1$ manifold containing $(u_0,u_1)$ with the property that any solution starting on the manifold scatters to $ϕ$; moreover, any solution remaining near the manifold for all positive times lies on the manifold (ii) an exterior energy inequality from [9,10,16]. The latter is used to obtain a result in the spirit of the one-pass theorem [22], albeit with completely different techniques

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arXiv2014

Long time dynamics of defocusing energy critical 3 + 1 dimensional wave equation with potential in the radial case

Using channel of energy inequalities developed by T.Duyckaerts, C.Kenig and F.Merle, we prove that, modulo a free radiation, any finite energy radial solution to the defocusing energy critical wave equation with radial potential in 3 + 1 dimensions converges to the set of steady states as time goes to infinity. For generic potentials we prove there are only finitely many steady states, and in this case modulo some free radiation the solution converges to one steady state as time goes to infinity.

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arXiv2011

The dynamics of the 3D radial NLS with the combined terms

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schrödinger equation (NLS) with the combined terms iu_t + Δu = -|u|^4u + |u|^2u \tag{CNLS} in the energy space $H^1(\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + Δu = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the scaling parameter, then apply it into the scattering theory. Our result shows that the defocusing, $\dot H^1$-subcritical perturbation $|u|^2u$ does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.

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arXiv2011

Dynamics for the focusing, energy-critical nonlinear Hartree equation

In \cite{LiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har}, the dynamics of the solutions for the focusing energy-critical Hartree equation have been classified when $E(u_0)<E(W)$, where $W$ is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the threshold energy. Our arguments closely follow those in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution, DuyRouden:NLS:ThresholdSolution, LiZh:NLS, LiZh:NLW}. The new ingredient is that we show that the positive solution of the nonlocal elliptic equation in $L^{\frac{2d}{d-2}}(\R^d)$ is regular and unique by the moving plane method in its global form, which plays an important role in the spectral theory of the linearized operator and the dynamics behavior of the threshold solution.

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arXiv2010

Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\R)$

In this paper, we consider the Cauchy problem of the cubic nonlinear Schrödinger equation with derivative in $H^s(\R)$. This equation was known to be the local well-posedness for $s\geq \frac12$ (Takaoka,1999), ill-posedness for $s<\frac12$ (Biagioni and Linares, 2001, etc.) and global well-posedness for $s>\frac12$ (I-team, 2002). In this paper, we show that it is global well-posedness in $H^{1/2(\R)$. The main approach is the third generation I-method combined with some additional resonant decomposition technique. The resonant decomposition is applied to control the singularity coming from the resonant interaction.

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arXiv2009

The low regularity global solutions for the critical generalized KdV equation

We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>6/13$. Of course, we require that the mass is strictly less than that of the ground state in the focusing case. The main approach is the "I-method" together with the multilinear correction analysis. Moreover, we use some "partially refined" argument to lower the upper control of the multiplier in the resonant interactions. The result improves the previous works of Fonseca, Linares, Ponce (2003) and Farah (2009).

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arXiv2009

Scattering for the focusing ${\dot H}^{1/2}$-critical Hartree equation with radial data

We investigate the focusing $\dot H^{1/2}$-critical nonlinear Schrödinger equation (NLS) of Hartree type $i\partial_t u + Δu = -(|\cdot|^{-3} \ast |u|^2)u$ with $\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if the maximal life-span solution obeys $\sup_{t}\big\||\nabla|^{1/2}u\big\|_2 < \frac{\sqrt{6}}{3} \big\||\nabla|^{1/2}Q\big\|_2$, where $Q$ is the positive radial solution to the elliptic equation(\ref{e14}). Then the solution is global and scatters.

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arXiv2009

Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbb{R}^{1+n}$

Using the same induction on energy argument in both frequency space and spatial space simultaneously as in \cite{CKSTT07}, \cite{RyV05} and \cite{Vi05}, we obtain global well-posedness and scattering of energy solutions of defocusing energy-critical nonlinear Hartree equation in $\mathbb{R}\times \mathbb{R}^n$($n\geq 5$), which removes the radial assumption on the data in \cite{MiXZ07a}. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition \ref{freqdelocaimplystbound} and Corollary \ref{frequencylocalization}) of the minimal energy blow up solutions, which can not be obtained from the classical long time perturbation and bilinear estimate and that we obtain the spatial concentration of minimal energy blow up solution after proving that $L^{\frac{2n}{n-2}}_x$-norm of minimal energy blow up solutions is bounded from below, the $L^{\frac{2n}{n-2}}_x$-norm is larger than the potential energy.

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arXiv2008

Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order $iu_t+Δ^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys $\displaystyle\sup_{t\in I}\|Δu(t)\|_{2}<\|ΔW\|_{2}$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters.

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arXiv2008

On the blow up phenomenon for the $L^2$-critical focusing Hartree equation in $\Bbb R^4$

We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we also analyze the mass concentration phenomenon of such blow up solutions.

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arXiv2008

Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $\mathbb{R}^d$

We prove the global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical (that is, $2<γ<3$) Hartree equation with low regularity data in $\mathbb{R}^d$, $d\geq 3$. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space $H^s\big(\mathbb{R}^d\big)$ with $s>4(γ-2)/(3γ-4)$, which also scatters in both time directions. This improves the result in \cite{ChHKY}, where the global well-posedness was established for any $s>\max\big(1/2,4(γ-2)/(3γ-4)\big)$. The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution $Iu$, instead of an interaction Morawetz estimate for the solution $u$, and that we make careful analysis of the monotonicity property of the multiplier $m(ξ)\cdot < ξ>^p$. As a byproduct of our proof, we obtain that the $H^s$ norm of the solution obeys the uniform-in-time bounds.

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