Well-posedness for the supercritical gKdV equation

TL;DR

This paper establishes local and global well-posedness results for the supercritical generalized KdV equation in critical Besov and Sobolev spaces, advancing understanding of solution behavior in supercritical regimes.

Contribution

It proves local well-posedness in critical Besov spaces and extends results to inhomogeneous Sobolev spaces, also providing global well-posedness for small initial data.

Findings

Local well-posedness in homogeneous Besov space $\, ext{dot} B^{s_p,2}_{ ext{infty}}$

Extension to inhomogeneous Sobolev space $H^{s_p}$

Global well-posedness for small initial data

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 $5\leq p\in\R$. We prove a local well-posedness result in the homogeneous Besov space $\dot B^{s_p,2}_{\infty}(\mathbb{R})$, where $s_p=\frac12-\frac{2}{p-1}$ 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.