Action Functional and Quasi-Potential for the Burgers Equation in a Bounded Interval

Consider the viscous Burgers equation u(t) + f(u)(x) epsilon u(xx) on the interval [1, 1] with the inhomogeneous Dirichlet boundary conditions u(t, 0) = rho(0), u(t, 1) = rho 1. The flux f is the function f(u) = u(1 - u), epsilon > 0 is the viscosity, and the boundary data satisfy 0 < rho(0) < rho(1) < 1. We examine the quasi-potential corresponding to an action functional arising from nonequilibrium statistical mechanical models associated with the above equation. We provide a static variational formula for the quasi-potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi-potential admits more than one minimizer. This phenomenon is interpreted as a nonequilibrium phase transition and corresponds to points where the superdifferential of the quasi-potential is not a singleton. (C) 2010 Wiley Periodicals, Inc.