Yongsheng Li — Page 3

arXiv2009

Global well-posedness for the Benjamin equation in low regularity

In this paper we consider the initial value problem of the Benjamin equation $$ \partial_{t}u+ν\H(\partial^2_xu) +μ\partial_{x}^{3}u+\partial_xu^2=0, $$ where $u:\R\times [0,T]\mapsto \R$, and the constants $ν,μ\in \R,μ\neq0$. 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.

math.AP↗

arXiv2009

On a quadratic nonlinear Schrödinger equation: sharp well-posedness and ill-posedness

We study the initial value problem of the quadratic nonlinear Schrödinger 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, $$ H^{s,a}(\R)={u:\|u\|_{H^{s,a}}\triangleq (\displaystyle\int (|ξ|^sχ_{\{|ξ|>1\}}+|ξ|^aχ_{\{|ξ|\leq 1\}})^2|\hat{u}(ξ)|^2 dξ)^{1/2}<\infty} $$ 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 $s<-\dfrac{1}{4}-\dfrac{1}{2}a$ or $a>\dfrac{1}{2}$. It remains the cases on the line segment: $a=\dfrac{1}{2}$, $-\dfrac{1}{2}\leq s\leq 0$ open in this paper.

math.AP↗