Rough solutions of the fifth-order KdV equations

Zihua Guo, Chulkwang Kwak, Soonsik Kwon

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy{∂tu+∂x5u+c1∂xu∂x2u+c2u∂x3u=0,x,t∈R,u(0,x)=u0(x),u0∈Hs(R). We prove a priori bound of solutions for Hs(R) with s≥54 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-30u2∂xu+20∂xu∂x2u+10u∂x3u=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
StatePublished - 1 Dec 2013

Keywords

  • Fifth-order KdV equation
  • KdV hierarchy
  • Local well-posedness
  • X space

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