Global Smoothing for the Periodic KdV Evolution

TL;DR

This paper proves that solutions to the periodic KdV equation exhibit smoothing effects, with the difference between nonlinear and linear evolutions gaining regularity over time, extending to modified KdV and equations with potentials.

Contribution

It establishes new smoothing results for the periodic KdV and mKdV equations, including cases with time-dependent potentials and initial data of low regularity.

Findings

Difference between nonlinear and linear evolutions lies in higher regularity spaces.

Solutions with continuous and bounded variation initial data are continuous in space and time.

Smoothing effect demonstrated for modified KdV on the torus for s > 1/2.

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 $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$ for all times, with at most polynomially growing $H^{s_1}$ norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case $s\geq 0$. 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 $L^2$ 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 $s>1/2$.