Global Smoothing for the Periodic KdV Evolution

2011/03/22 by Burak Erdogan, Erdogan, Burak, Nikolaos Tzirakis +1 · 4 citations
Mathematics · #35q53 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35q53

paper · pdf · doi:10.48550/arxiv.1103.4190

24 pages. We demonstrate smoothing for both KdV and mKdV

arxiv created 2011/03/29 · arxiv updated 2011/03/30

Abstract

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for H^s initial data, s>-1/2, and for any s₁<min(3s+1,s+1), the difference of the nonlinear and linear evolutions is in H^s₁ for all times, with at most polynomially growing H^s₁ norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case s≥ 0. Our result and a theorem of Oskolkov for the Airy evolution imply that if one starts with continuous and bounded variation initial data then the solution of KdV (given by the L² theory of Bourgain) is a continuous function of space and time. In addition, we demonstrate smoothing for the modified KdV equation on the torus for s>1/2.

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