A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications

Abstract

Abstract. We state a kinetic formulation of weak entropy solutions of a general multidimensional scalar conservation law with initial and boundary conditions. We first associate with any weak entropy solution an entropy defect measure; the analysis of this measure at the boundary of the domain relies on the study of weak entropy sub-and supersolutions and implies the introduction of the notion of sided boundary defect measures. As a first application, we prove that any weak entropy subsolution of the initial-boundary value problem is bounded above by any weak entropy supersolution (comparison theorem). We next study a Bhatnagar-Gross-Krook-like kinetic model that approximates the scalar conservation law. We prove that such a model converges by adapting the proof of the comparison theorem.

On the Dirichlet problem for first order quasilinear equations on a manifold

“…for short). Using the approaches of papers [9,13,16], below we introduce the notions of generalized entropy sub-solutions (g.e.sub-s.) and generalized entropy super-solutions (g.e.super-s.) of the Dirichlet problem (0.1), (0.2). Letμ = (μ, μ b ) be a pair of smooth measures on M and S, respectively.…”

“…Similarly to [9], we define notions of a kinetic sub-and super-solution (k.sub-s. and k.super-s. for short).…”

“…By similar reasons we prove that (3.16) implies (3.14). Using the Kruzhkov method of doubling variable, we can establish (as in [9]) the following result.…”

“…In BGK-like approximations usually applied in kinetic models (see [9,12,23]) the corresponding operators do not satisfy this property.…”

“…In the present paper we will follow Otto's formulation. To prove our main results we extend the kinetic approach developed for initial-boundary value problems by Imbert and Vovelle in [9].…”