A Theory of L 1 -Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

TLDR: In this paper, a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux is proposed, where the fluxes fl, fr are mainly assumed to be continuous.

Abstract: We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: $$ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x 0, \end{array} \right\quad\quad\quad (\rm CL) $$ where the fluxes fl, fr are mainly assumed to be continuous Developing the ideas of a number of preceding works (Baiti and Jenssen in J Differ Equ 140(1):161–185, 1997; Towers in SIAM J Numer Anal 38(2):681–698, 2000; Towers in SIAM J Numer Anal 39(4):1197–1218, 2001; Towers et al in Skr K Nor Vidensk Selsk 3:1–49, 2003; Adimurthi et al in J Math Kyoto University 43(1):27–70, 2003; Adimurthi et al in J Hyperbolic Differ Equ 2(4):783–837, 2005; Audusse and Perthame in Proc Roy Soc Edinburgh A 135(2):253–265, 2005; Garavello et al in Netw Heterog Media 2:159–179, 2007; Burger et al in SIAM J Numer Anal 47:1684–1712, 2009), 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 $$ c(x)=c^l11_{\left\{{x 0}\right\}} $$ We refer to such a family as a “germ” It is well known that (CL) admits many different L1 contractive semigroups, some of which reflect 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 of expressing the “Γ-condition” of Diehl (J Hyperbolic Differ Equ 6(1):127–159, 2009) For any given germ, we formulate “germ-based” admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} [in the spirit of Vol’pert (Math USSR Sbornik 2(2):225–267, 1967)] and in the form of a family of global entropy inequalities [following Kruzhkov (Math USSR Sbornik 10(2):217–243, 1970) and Carrillo (Arch Ration Mech Anal 147(4):269–361, 1999)] 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