Rough solutions for the periodic Korteweg-de Vries equation

TL;DR

This paper introduces rough path theory techniques to analyze low-regularity solutions of the periodic Korteweg-de Vries equation, including convergence of numerical schemes and effects of stochastic forcing.

Contribution

It applies rough path ideas to the KdV equation, providing new insights into low-regularity solutions and stochastic influences.

Findings

Convergence results for Galerkin approximations

Analysis of a modified Euler scheme

Impact of white-noise type random forcing

Abstract

We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain and with initial condition in $\FF L^{\alpha,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.