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Action functional and quasi-potential for the Burgers equation in a bounded interval. (English) Zbl 1220.82079

An infinite-dimensional version of the classical Freidlin-Wentzell variational problem for the quasi-potential [M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems. New York etc.: Springer (1984; Zbl 0522.60055)] is described by the viscous Burgers equation \(u_t + f(u)_x = \varepsilon u_{xx}\) on the interval \([0,1]\) with inhomogeneous Dirichlet boundary conditions \(u(t,0) = \rho_0\), \(u(t,1) = \rho_1\) at the end points, where \(u = u(t,x)\) is a scalar function, the flux \(f\) is the function \(f(u) = u(1-u)\), \(\varepsilon > 0\) is the viscosity and the boundary data satisfy \(0 < \rho_0 < \rho_1 < 1\). By analyzing the action functional and the quasi-potential described by nonequilibrium statistical mechanics models with the Burger equation a static variational characterization of the quasi-potential is derived. It is emphasized that in the discussed case there is no uniqueness for the minimizer of the quasi-potential in the inviscid limit \(\varepsilon \downarrow 0\). By a perturbation argument with respect to this limiting case, it is proved that for a class of nonconstant profiles \(\rho\) this phenomenon persists when the viscosity \(\varepsilon\) is small enough. In the context of equilibrium statistical mechanics, the existence of more than a single tangent functional to the quasi-potential is due to the occurence of phase transitions. A nonequilibrium phase transition is then interpreted as the case when the superdifferential of the quasi-potential is not a singleton.

MSC:

82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
49J40 Variational inequalities
35Q53 KdV equations (Korteweg-de Vries equations)

Citations:

Zbl 0522.60055
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