Dissipative interface coupling of conservation laws

Abstract

International audienceWe give a brief account on the theory of $L^1$-contractive solvers of the model conservation law with discontinuous flux: \begin{equation*}\label{eq:1D-model} \!\!\leqno(MP)\;\; u_t + (\mathfrak{f}(x,u))_x=0, \quad \mathfrak {f}(x,\cdot)= f^l(\cdot)\char_{x0}, \end{equation*} constructed in the work \cite{AKR-ARMA} of K.H.~Karlsen, N.H.~Risebro and the author. We discuss the modifications that can be used for extending our approach to the multi-dimensional setting and curved flux discontinuity hypersurfaces; the vanishing viscosity case (see \cite{AKR-NHM}) is presented as an illustration. Applications to a road traffic with point constraint and to a coupled particle-fluid interaction model, coming from the joint works \cite{AGS} with P.~Goatin, N.~Seguin and \cite{AS,ALST} with F.~Lagoutiére, N.~Seguin, T.~Takahashi, are presented