A Theory of L-dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: ut + f(x, u)x = 0, f(x, u) = ( f (u), x < 0, f(u), x > 0, (CL) where the fluxes f , f are mainly assumed to be continuous. Developing the ideas of a number of preceding works (Baiti and Jenssen [14], Audusse and Perthame [12], Garavello et al. [35], Adimurthi et al. [3], Bürger et al. [21]), we claim that the whole admissibility issue is reduced to the selection of a family of “elementary solutions”, which are piecewise constant weak solutions of the form c(x) = c1l{x<0} + c 1l{x>0}. We refer to such a family as a “germ”. It is well known that (CL) admits many different L1 contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the “vanishing viscosity” germ, which is a way to express the “Γ-condition” of Diehl [32]. For any given germ, we formulate “germbased” admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} (in the spirit of Vol’pert [80]) and in the form of a family of global entropy inequalities (following Kruzhkov [50] and Carrillo [22]). We characterize those germs that lead to the L1-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities “adapted” to the choice of a germ), or for specific germ-adapted finite volume schemes. Date: April 23, 2010. 2000 Mathematics Subject Classification. Primary 35L65; Secondary 35R05.