Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation

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 Szegö equation under randomization of initial data.

24 pages. The introduction and Section 4 are expanded. Some statements are made more explicit (e.g. the precise value of s_0.) To appear in Funkcial. Ekvac