Communications on Pure and Applied Analysis (CPAA)

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

Pages: 709 - 733, Volume 11, Issue 2, March 2012      doi:10.3934/cpaa.2012.11.709

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Massimiliano Gubinelli - CEREMADE & CNRS (UMR 7534), Université Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France (email)

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.

Keywords:  Dispersive equations, rough paths theory, power series solutions.
Mathematics Subject Classification:  Primary: 35Q53; Secondary: 35D99.

Received: March 2010;      Revised: April 2011;      Available Online: October 2011.