1. A Theory of L 1-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux
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Kenneth H. Karlsen, Boris Andreianov, Nils Henrik Risebro, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Center of Mathematics for Applications [Oslo] (CMA), Department of Mathematics [Oslo], Faculty of Mathematics and Natural Sciences [Oslo], University of Oslo (UiO)-University of Oslo (UiO)-Faculty of Mathematics and Natural Sciences [Oslo], University of Oslo (UiO)-University of Oslo (UiO), and This paper was written as part of the research program on Nonlinear Partial Dierential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo, which took place during the academic year 2008-09.
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Pure mathematics ,Admissibility ,Adapted entropy ,01 natural sciences ,Mathematics - Analysis of PDEs ,Mathematics (miscellaneous) ,Boundary trace ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Discontinuous flux ,Germ ,Uniqueness ,Mathematics - Numerical Analysis ,0101 mathematics ,Scalar conservation law ,Mathematics ,Conservation law ,Finite volume method ,Mechanical Engineering ,010102 general mathematics ,Primary 35L65 ,Secondary 35R05 ,Numerical Analysis (math.NA) ,010101 applied mathematics ,Convergence of numerical approximations ,Uniqueness criteria ,Finite volume scheme ,Piecewise ,Dissipative system ,Adapted viscosity ,Entropy solution ,Vanishing viscosity ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Analysis ,Analysis of PDEs (math.AP) - Abstract
International audience; We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: (CL) u_t + f(x; u)_x = 0; f(x; u) = f^l(u), x < 0 and f(x;u)= f^r(u); x > 0; where the fluxes f^l and f^r 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], Buerger 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) = c^l 1l_{x0}. 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 specic attention to the "vanishing viscosity" germ, which is a way to express the "Gamma-condition" of Diehl [32]. 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 [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 specic viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.
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