Metastability for scalar conservation laws in a bounded domain

¹ Dipartimento di Matematica “G. Castelnuovo”, Sapienza – Università di Roma, P.le Aldo Moro, 2 - 00185 Roma ( ITALY),
mascia@mat.uniroma1.it
² Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche (associated in the framework of the program “Intracellular Signalling”)
³ Ecole Normale Supérieure, Département de Mathématiques et Applications, 45 rue d’Ulm - 75005 Paris ( France),
strani@dma.ens.fr

Abstract

The initial-boundary-value problem for a viscous scalar conservation law in a bounded interval I = ( − ℓ,ℓ) is considered, with emphasis on the metastable dynamics, whereby the time-dependent solution develops internal transition layers that approach their steady state in an asymptotically exponentially long time interval as the viscosity coefficient ε> 0 goes to zero. We describe such behavior by deriving an ODE for the position ξ of the internal interface. The main tool of our analysis is the construction of a one-parameter family of approximate stationary solutions {U^εε(.;ξ)}_ξ∈I, parametrized by the location of the shock layer ξ, to be considered as an approximate invariant manifold for the problem. By using the properties of the linearized operator at U^ε, we estimate the size of the layer location.