Hydrodynamic limit of a B.G.K. like model on domains with boundaries and analysis of kinetic boundary conditions for scalar multidimensional conservation laws

TL;DR

This paper investigates the hydrodynamic limit of a kinetic model with boundaries, establishing existence results for scalar conservation laws with kinetic boundary conditions and extending analysis to include force effects and source terms.

Contribution

It provides new existence results for scalar conservation laws with kinetic boundary conditions using BV theory and extends the analysis to models with forces and source terms.

Findings

Existence of solutions for scalar conservation laws with kinetic boundary conditions.

Extension of hydrodynamic limit analysis to models with forces and source terms.

Application of BV_{loc} theory to boundary value problems in kinetic models.

Abstract

In this paper we study the hydrodynamic limit of a B.G.K. like kinetic model on domains with boundaries via $BV_{loc}$ theory. We obtain as a consequence existence results for scalar multidimensional conservation laws with kinetic boundary conditions. We require that the initial and boundary data satisfy the optimal assumptions that they all belong to $L^1\cap L^\infty$ with the additional regularity assumptions that the initial data are in $BV_{loc}$. We also extend our hydrodynamic analysis to the case of a generalized kinetic model to account for forces effects and we obtain as a consequence the existence theory for conservation laws with source terms and kinetic boundary conditions.