Coupling Two Scalar Conservation Laws via Dafermos’ Self-Similar Regularization

Abstract

We are interested in the problem of coupling two scalar conservation laws with distinct flux-functions. This problem arises, for instance, in modeling fluid flows in media with discontinuous porosity and has important possible applications in the numerical computation of a singular pressure drop. This problem is also well-known to exhibit several technical difficulties due to the presence of nonconservative terms and to the resonant behavior of the system of equations. We present here a global approach consisting of two scalar problems in a half-space coupled through an algebraic jump relation. We view this problem as a 2 × 2 system of conservation laws, and introduce a viscous regularization à la Dafermos. We establish that this approximation converges as the viscosity tends to zero and we analyze the structure of the entropy solutions constructed in this way.