Rough solutions for the periodic Korteweg--de~Vries equation

Massimiliano Gubinelli

doi:10.3934/cpaa.2012.11.709

Article Contents

2012, Volume 11, Issue 2: 709-733. Doi: 10.3934/cpaa.2012.11.709

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Rough solutions for the periodic Korteweg--de~Vries equation

Massimiliano Gubinelli^1,

1.
CEREMADE & CNRS (UMR 7534), Université Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16

Received: February 28, 2010

Revised: March 31, 2011

Published: February 2012

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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 $F L^{\alpha,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35D99.

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