A variational approach to the stationary solutions of the Burgers equation (Q2904722)

The authors consider the inhomogeneous Dirichlet problem for the viscous Burgers equation on a bounded interval as the length of the interval tends to infinity. The problem possesses certain Lyapunov functional, whose minimizer is a unique stationary solution. It is studied the behavior of the functional in the limit of infinite interval length. A special attention is paid to the case when the boundary data correspond to a standing wave solution on the whole line. In this case the limiting functional possesses a one-parameter family of minimizers, and the sharp asymptotic cost corresponding to a given shift of the stationary solution is calculated.