A variational approach to the stationary solutions of Burgers equation
Abstract
Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by BertiniDe SoleGabrielliJonaLasinioLandim C, we analyze a Lyapunov functional for such equation which gives the large deviations asymptotics of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functional, whose minimizer is given by the unique stationary solution, as the length of the interval diverges. We focus on boundary data corresponding to a standing wave solution to the Burgers equation in the whole line. In this case, the limiting functional has in fact a oneparameter family of minimizers and we analyze the socalled development by Gammaconvergence; this amounts to compute the sharp asymptotic cost corresponding to a given shift of the stationary solution.
 Publication:

arXiv eprints
 Pub Date:
 August 2010
 DOI:
 10.48550/arXiv.1008.0550
 arXiv:
 arXiv:1008.0550
 Bibcode:
 2010arXiv1008.0550B
 Keywords:

 Mathematics  Probability