Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

Keiichi Kato, Takayoshi Ogawa

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)

Abstract

We show that a solution of the Cauchy problem for the KdV equation, {∂tυ + ∂3xυ + ∂x2) = 0, t ∈ (-T, T), x ∈ ℝ, υ(0, x) = φ(x). has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for Hs(ℝ) (s > -3/4) data satisfying the condition (equation presented) the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac δ measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator 3t∂t + x∂x.

Original languageEnglish
Pages (from-to)577-608
Number of pages32
JournalMathematische Annalen
Volume316
Issue number3
DOIs
Publication statusPublished - 2000 Mar
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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