Global well-posedness for the nonlinear Schr\"{o}dinger equation with derivative in energy space

In this paper, we prove that there exists some small $\varepsilon_*>0$, such that the derivative nonlinear Schr\"{o}dinger equation (DNLS) is global well-posedness in the energy space, provided that the initial data $u_0\in H^1(\mathbb{R})$ satisfies $\|u_0\|_{L^2}<\sqrt{2\pi}+\varepsilon_*$. This result shows us that there are no blow up solutions whose masses slightly exceed $2\pi$, even if their energies are negative. This phenomenon is much different from the behavior of nonlinear Schr\"odinger equation with critical nonlinearity. The technique is a variational argument together with the momentum conservation law. Further, for the DNLS on half-line $\mathbb{R}^+$, we show the blow-up for the solution with negative energy.Comment: To appear in Analysis & PDE. We add some references, and change some expressions in Englis

The Cauchy Problem of the Schr\"odinger-Korteweg-de Vries System

We study the Cauchy problem of the Schr\"odinger-Korteweg-de Vries system. First, we establish the local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show that they are sharp in some well-posedness thresholds. Particularly, we obtain the local well-posedness for the initial data in $H^{-{3/16}+}(\R)\times H^{-{3/4}+}(\R)$ in the resonant case, it is almost the optimal except the endpoint. At last we establish the global well-posedness results in $H^s(\R)\times H^s(\R)$ when $s>\dfrac{1}{2}$ no matter in the resonant case or in the non-resonant case, which improve the results of Pecher (2005).Comment: 38 pages,1 figur

Global well-posedness on the derivative nonlinear Schr\"odinger equation revisited

As a continuation of the previous work \cite{Wu}, we consider the global well-posedness for the derivative nonlinear Schr\"odinger equation. We prove that it is globally well-posed in energy space, provided that the initial data $u_0\in H^1(\mathbb{R})$ with $\|u_0\|_{L^2}< 2\sqrt{\pi}$.Comment: 8 pages. Some typos are correcte

Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension

In this paper, we consider the following nonlinear Klein-Gordon equation \begin{align*} \partial_{tt}u-\Delta u+u=|u|^{p-1}u,\qquad t\in \mathbb{R},\ x\in \mathbb{R}^d, \end{align*} with $1<p< 1+\frac{4}{d}$. The equation has the standing wave solutions $u_\omega=e^{i\omega t}\phi_{\omega}$ with the frequency $\omega\in(-1,1)$, where $\phi_{\omega}$ obeys \begin{align*} -\Delta \phi+(1-\omega^2)\phi-\phi^p=0. \end{align*} It was proved by Shatah (1983), and Shatah, Strauss (1985) that there exists a critical frequency $\omega_c\in (0,1)$ such that the standing waves solution $u_\omega$ is orbitally stable when $\omega_c<|\omega|<1$, and orbitally unstable when $|\omega|<\omega_c$. Further, the critical case $|\omega|=\omega_c$ in the high dimension $d\ge 2$ was considered by Ohta, Todorova (2007), who proved that it is strongly unstable, by using the virial identities and the radial Sobolev inequality. The one dimension problem was left after then. In this paper, we consider the one-dimension problem and prove that it is orbitally unstable when $|\omega|=\omega_c$.Comment: 18 Pages, add a reference and the proof of some lemma

The Cauchy problem for the two dimensional Euler-Poisson system

The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo first constructed a global smooth irrotational solution by using dispersive Klein-Gordon effect. It has been conjectured that same results should hold in the two-dimensional case. In our recent work, we proved the existence of a family of smooth solutions by constructing the wave operators for the 2D system. In this work we completely settle the 2D Cauchy problem.Comment: 56 pages, to appear in JEM

On a quadratic nonlinear Schr\"odinger equation: sharp well-posedness and ill-posedness

We study the initial value problem of the quadratic nonlinear Schr\"odinger equation $$iu_t+u_{xx}=u\bar{u},$$ where u:\R\times \R\to \C. We prove that it's locally well-posed in $H^s(\R)$ when $s\geq -\dfrac{1}{4}$ and ill-posed when $s< -\dfrac{1}{4}$, which improve the previous work in \cite{KPV}. Moreover, we consider the problem in the following space, $$H^{s,a}(\R)={u:\|u\|_{H^{s,a}}\triangleq (\displaystyle\int (|\xi|^s\chi_{\{|\xi|>1\}}+|\xi|^a\chi_{\{|\xi|\leq 1\}})^2|\hat{u}(\xi)|^2 d\xi)^{{1/2}}<\infty}$$ for $s\leq 0, a\geq 0$. We establish the local well-posedness in $H^{s,a}(\R)$ when $s\geq -\dfrac{1}{4}-\dfrac{1}{2}a$ and $a<\dfrac{1}{2}$. Also we prove that it's ill-posed in $H^{s,a}(\R)$ when $s\dfrac{1}{2}$. It remains the cases on the line segment: $a=\dfrac{1}{2}$, $-\dfrac{1}{2}\leq s\leq 0$ open in this paper.Comment: 26 page

Global well-posedness for periodic generalized KdV equation

In this paper, we show the global well-posedness for periodic gKdV equations in the space $H^s(\mathbb{T})$, $s\ge \frac12$ for quartic case, and $s> \frac59$ for quintic case. These improve the previous results of I-team in 2004. In particular, the result is sharp for quintic case. The main approaches are the I-method combining with the resonance decomposition developed by Miao et al in 2010, and a bilinear Strichartz estimate in periodic setting.Comment: Some changes on expression in Englis

Global Attractor for Weakly Damped Forced KdV Equation in Low Regularity on T

In this paper we consider the long time behavior of the weakly damped, forced Korteweg-de Vries equation in the Sololev spaces of the negative indices in the periodic case. We prove that the solutions are uniformly bounded in \dot{H}^s(\T) for $s>-\dfrac{1}{2}$. Moreover, we show that the solution-map possesses a global attractor in \dot{H}^s(\T) for $s>-\dfrac{1}{2}$, which is a compact set in H^{s+3}(\T).Comment: 34 page

Global small solution to the 2D MHD system with a velocity damping term

This paper studies the global well-posedness of the incompressible magnetohydrodynamic (MHD) system with a velocity damping term. We establish the global existence and uniqueness of smooth solutions when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also given.Comment: 27 page

Global well-posedness for the Benjamin equation in low regularity

In this paper we consider the initial value problem of the Benjamin equation $$\partial_{t}u+\nu \H(\partial^2_xu) +\mu\partial_{x}^{3}u+\partial_xu^2=0,$$ where $u:\R\times [0,T]\mapsto \R$, and the constants $\nu,\mu\in \R,\mu\neq0$. We use the I-method to show that it is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>-3/4$. Moreover, we use some argument to obtain a good estimative for the lifetime of the local solution, and employ some multiplier decomposition argument to construct the almost conserved quantities.Comment: 29 page