• 제목/요약/키워드: Kato

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ANALYTIC SMOOTHING EFFECT AND SINGLE POINT SINGULARITY FOR THE NONLINEAR SCHRODINGER EQUATIONS

  • Kato, Keiichi;Ogawa, Takayoshi
    • 대한수학회지
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    • 제37권6호
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    • pp.1071-1084
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    • 2000
  • We show that a weak solution of the Cauchy problem for he nonlinear Schrodinger equation, {i∂(sub)t u + ∂$^2$(sub)x u = f(u,u), t∈(-T,T), x∈R, u(0,x) = ø(x).} in the negative solbolev space H(sup)s has a smoothing effect up to real analyticity if the initial data only have a single point singularity such as the Dirac delta measure. It is shown that for H(sup)s (R)(s>-3/4) data satisfying the condition (※Equations, See Full-text) the solution is analytic in both space and time variable. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [18] and previous work by Kato-Ogawa [12]. We give an improved new argument in the regularity argument.

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