TY - JOUR

T1 - Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

AU - Kato, Keiichi

AU - Ogawa, Takayoshi

PY - 2000/3

Y1 - 2000/3

N2 - We show that a solution of the Cauchy problem for the KdV equation, {∂tυ + ∂3xυ + ∂x(υ2) = 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.

AB - We show that a solution of the Cauchy problem for the KdV equation, {∂tυ + ∂3xυ + ∂x(υ2) = 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.

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U2 - 10.1007/s002080050345

DO - 10.1007/s002080050345

M3 - Article

AN - SCOPUS:0034384756

SN - 0025-5831

VL - 316

SP - 577

EP - 608

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3

ER -