A Variational Approach to the Stationary Solutions of the Burgers Equation

Abstract

We consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by Bertini et al. [Comm. Pure Appl. Math., 64 (2011), pp. 649–696], we analyze a Lyapunov functional for such an equation which gives the large deviations asymptotic 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. In particular, 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 a one-parameter family of minimizers and we compute the sharp asymptotic cost corresponding to a given shift of the stationary solution.