## Abstract

We show that a solution of the Cauchy problem for the KdV equation, {∂_{t}υ + ∂^{3}_{x}υ + ∂_{x}(υ^{2}) = 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 H^{s}(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 language | English |
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Pages (from-to) | 577-608 |

Number of pages | 32 |

Journal | Mathematische Annalen |

Volume | 316 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2000 Mar |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)