A variational approach to the stationary solutions of Burgers equation

TL;DR

This paper investigates the large deviations and asymptotic behavior of stationary solutions to the viscous Burgers equation on bounded intervals, using a variational framework and Gamma-convergence analysis.

Contribution

It extends the variational approach to analyze stationary solutions and their large deviations for Burgers equation with inhomogeneous boundary conditions, including boundary data related to standing waves.

Findings

Identifies the minimizer of the Lyapunov functional as the stationary solution.

Analyzes the asymptotic behavior of the energy functional as the interval length diverges.

Computes the sharp asymptotic cost for shifts of stationary solutions via Gamma-convergence.

Abstract

Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by Bertini-De Sole-Gabrielli-Jona-Lasinio-Landim 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 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.