Rough solutions of the fifth-order KdV equations

Zihua Guo, Chulkwang Kwak, Soonsik Kwon

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28 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

Bibliographical note

Funding Information:
We are grateful to Didier Pilod for pointing out an error in the first draft, and to the anonymous referee for careful reading and improving the clearness of this paper. Z.G. is partially supported by NNSF of China (Nos. 11001003 , 11371037 ). S.K. is partially supported by TJ Park Science Fellowship and NRF (Korea) grant 2010-0024017 .

Keywords

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

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