Rough solutions of the fifth-order KdV equations

TL;DR

This paper establishes local and global well-posedness results for the fifth-order KdV equation using short time X^{s,b} spaces and weighted techniques, advancing understanding of high-order dispersive PDEs.

Contribution

It introduces a novel application of short time X^{s,b} spaces and weighted methods to analyze the fifth-order KdV equation, achieving well-posedness results.

Findings

A priori bounds for solutions in H^s with s >= 5/4.

Local well-posedness in H^s for s >= 2.

Global well-posedness in H^2 due to conservation laws.

Abstract

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u u_{x} = 0 x,t \in \R We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and the local well-posedness for s >= 2. The method is a short time X^{s,b} space, which is first developed by Ionescu-Kenig-Tataru in the context of the KP-I equation. In addition, we use a weight on localized X^{s,b} structures to reduce the contribution of high-low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we have the fifth-order equation in the KdV hierarchy is globally well-posed in the energy space H^2.