Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

Abstract

The well-posedness for the Cauchy problem of the nonlinear fractional Schrödinger equation

$u_t + i( - \Delta )^\alpha u + i|u|^2 u = 0,(x,t) \in \mathbb{R}^n \times \mathbb{R},\frac{1} {2} < \alpha < 1 $

is considered. The local well-posedness in subcritical space H ^s with s > n/2 -α is obtained. Moreover, the inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

$u_t + (\nu + i)( - \Delta )^\alpha u + i|u|^2 u = 0$

is also considered. It is shown that the solution of the fractional Ginzburg-Landau equation converges to the solution of nonlinear fractional Schrödinger equation in the natural space C([0, T];H)^s) with s > n/2 — α if ν tends to zero.