MR2918143 Tadahiro OH Tadahiro OH Remarks on Nonlinear Smoothing under Randomization for the Periodic KdV and the Cubic Szegö Equation Funkcialaj Ekvacioj. Serio Internacia 54 2011 335--365 http://fe.math.kobe-u.ac.jp/FE/FullPapers/54-3/54_335.pdf http://www.ams.org/mathscinet-getitem?mr=MR2918143 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 $H^s(\mathbb{T})$, $s>s_0$ where $s_0 =-\frac{11}{6}+\frac{\sqrt{61}}{6}\thickapprox -0.5316 <-\frac{1}{2}$, 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. Well-posedness, Nonlinear smoothing, KdV, Szegö equation. 35Q53, 35Q55. 54-335 2011 Remarks on Nonlinear Smoothing under Randomization for the Periodic KdV and the Cubic Szegö Equation Tadahiro OH Tadahiro OH
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