Calculus of Variations and Geometric Measure Theory

L. Bertini - M. Ponsiglione

A variational approach to the stationary solutions of Burgers equation

created by ponsiglio on 03 Aug 2010


Submitted Paper

Inserted: 3 aug 2010

Year: 2010


Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced in Bertini-De Sole-Gabrielli-Jona-Lasinio-Landim, 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 one-parameter family of minimizers and we analyze the so-called development by Gamma-convergence; this amounts to compute the sharp asymptotic cost corresponding to a given shift of the stationary solution.

Keywords: burgers equation, variational convergence, Quasi-potential