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.

A VARIATIONAL APPROACH TO THE STATIONARY SOLUTIONS OF THE BURGERS EQUATION / BERTINI MALGARINI, Lorenzo; Ponsiglione, Marcello. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - STAMPA. - 44:2(2012), pp. 682-698. [10.1137/100804401]

A VARIATIONAL APPROACH TO THE STATIONARY SOLUTIONS OF THE BURGERS EQUATION

BERTINI MALGARINI, Lorenzo;PONSIGLIONE, Marcello
2012

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.
2012
burgers equation; quasi-potential; development by gamma-convergence; development by g-convergence
01 Pubblicazione su rivista::01a Articolo in rivista
A VARIATIONAL APPROACH TO THE STATIONARY SOLUTIONS OF THE BURGERS EQUATION / BERTINI MALGARINI, Lorenzo; Ponsiglione, Marcello. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - STAMPA. - 44:2(2012), pp. 682-698. [10.1137/100804401]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/487856
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