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Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szeg\"o equation

Authors
  • Oh, Tadahiro
Type
Preprint
Publication Date
Aug 17, 2011
Submission Date
Jul 13, 2010
Source
arXiv
License
Yellow
External links

Abstract

We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on H^s(T), s > s_0 where s_0 = -11/6 + \sqrt{61}/6 \thickapprox -0.5316 < -1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the integration formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szeg\"o equation under randomization of initial data.

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