Well-posedness for the supercritical gKdV equation

Abstract

In this paper we consider the supercritical generalized Korteweg-de Vries equation ${\partial }_t\psi +{\partial }_{xxx}\psi +{\partial }_x(|\psi {|}^{p-1}\psi )=0$, where $5\le p\in \mathbb{R}$. We prove a local well-posedness result in the homogeneous Besov space ${\dot{B}}_{\infty }^{s_p,2}(\mathbb{R})$, where $s_p=\frac{1}{2}-\frac{2}{p-1}$ is the scaling critical index. In particular local well-posedness in the smaller inhomogeneous Sobolev space $H^{s_p}(\mathbb{R})$ can be proved similarly. As a byproduct a global well-posedness result for small initial data is also obtained.

20 pages