Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity |
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Authors: | Keiichi Kato Takayoshi Ogawa |
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Institution: | (1) Department of Mathematics, Science University of Tokyo, Shinjyuku-ku Tokyo 162-8601, Japan (e-mail: kato@ma.kagu.sut.ac.jp) , JP;(2) Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan (e-mail: ogawa@math.kyushu-u.ac.jp, Fax +(81)-92-642-2793) , JP |
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Abstract: | We show that a solution of the Cauchy problem for the KdV equation,
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 () data satisfying the condition
the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac
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 .
Received 22 March 1999 |
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Keywords: | Mathematics Subject Classification (1991):35Q53 |
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