Rough solutions of the fifth-order KdV equations

Zihua Guo, Chulkwang Kwak, Soonsik Kwon

Research output: Contribution to journalArticleResearchpeer-review

25 Citations (Scopus)

Abstract

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy

{∂tu+∂x5u+c1xu∂x2u+c2ux3u=0,  x, t ∈ R,

u(0,x)=u0(x),  u0Hs(R). 

We prove a priori bound of solutions for Hs(R) with s ≥ 5/4 and the local well-posedness for s ≥ 2. The method is a short time Xs,b space, which was first developed by Ionescu, Kenig and Tataru [13] in the context of the KP-I equation. In addition, we use a weight on localized Xs,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 obtain that the fifth-order equation in the KdV hierarchy,

tu-∂x5u-30u2xu+20∂xux2u+10ux3u=0 

is globally well-posed in the energy space H2.

Original languageEnglish
Pages (from-to)2791-2829
Number of pages39
JournalJournal of Functional Analysis
Volume265
Issue number11
DOIs
Publication statusPublished - 1 Dec 2013
Externally publishedYes

Keywords

  • Fifth-order KdV equation
  • KdV hierarchy
  • Local well-posedness
  • Xs,b space

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