Uniqueness of nonnegative weak solution to $u^ple(-Delta)^frac{alpha}{2}u$ on $mathbb R^N$

This note shows that under $(p,alpha, N)in (1,infty)times(0,2)timesmathbb Z_+$ the fractional order differential inequality $$ (dagger)quad u^p le (-Delta)^{frac{alpha}{2}} uquadhbox{in}quadmathbb R^{N} $$ has the property that if $Nlealpha$ then a nonnegative solution to $(dagger)$ is unique, and if $N>alpha$ then the uniqueness of a nonnegative weak solution to $(dagger)$ occurs when and only when $ple N/(N-alpha)$, thereby innovatively generalizing Gidas-Sprucks result for $u^p+Delta ule 0$ in $R^N$ discovered in cite{GS}.

Periodic Cubic Hyperbolic Schrodinger equation on $T^2$

We consider the cubic Hyperbolic Schrodinger equation eqref{eq:nls} on torus $T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that eqref{eq:nls} is analytic locally well-posed in in $H^s(T^2)$ with $s>1/2$, meanwhile, the ill-posedness in $H^s(T^2)$ for $s<1/2$ is also obtained. The main difficulty comes from estimating the number of representations of an integer as a difference of squares.

Periodic nonlinear Schrodinger equation in critical $H^s(T^n)$ spaces

In this paper we prove some multi-linear Strichartz estimates for solutions to the linear Schrodinger equations on torus $T^n$. Then we apply it to get some local well-posed results for nonlinear Schrodinger equation in critical $H^{s}(T^n)$ spaces. As by-products, the energy critical global well-posed results and energy subcritical global well-posed results with small initial data are also obtained.

On the Well-posedness of the Schrodinger-Korteweg-de Vries system

We prove that the Cauchy problem for the Schrodinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(R)times H^{-{3/4}}(R)$. The new ingredient is that we use the $bar{F}^s$ type space, introduced by the first author in cite{G}, to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares.

Remark on Well-posedness of Quadratic Schrodinger equation with nonlinearity $ubar u$ in $H^{-1/4}(R)$

In this remark, we give another approach to the local well-posedness of quadratic Schrodinger equation with nonlinearity $ubar u$ in $H^{-1/4}$, which was already proved by Kishimoto cite{kis}. Our resolution space is $l^1$-analogue of $X^{s,b}$ space with low frequency part in a weaker space $L^{infty}_{t}L^2_x$. Such type spaces was developed by Guo. cite{G} to deal the KdV endpoint $H^{-3/4}$ regularity.

Global well-posedness and scattering for Derivative Schr{o}dinger equation

In this paper we study the Cauchy problem for the elliptic and non-elliptic derivative nonlinear Schrodinger equations in higher spatial dimensions ($ngeq 2$) and some global well-posedness results with small initial data in critical Besov spaces $B^s_{2,1}$ are obtained. As by-products, the scattering results with small initial data are also obtained.

Local well-posedness for hyperbolic-elliptic Ishimori equation

In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of Ionescu and Kenig cite{IK} and cite{IK2} to approach the problem in a perturbative way.