STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

doi:10.1016/S0252-9602(10)60004-5

数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (6): 1573-1612.doi: 10.1016/S0252-9602(10)60004-5

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Helge Holden, Kenneth H. Karlsen, Darko Mitrovic, Evgueni Yu. Panov

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO--7491 Trondheim, Norway; Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO--0316 Oslo, Norway|Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N--0316 Oslo, Norway|Faculty of Mathematics, University of Montenegro, 81000 Podgorica, Montenegro|Mathematical Analysis Department, Novgorod State University, ul. B. St. Peterburgskaya 41, 173003 Veliky Novgorod, Russia

收稿日期:2009-10-25 出版日期:2009-11-20 发布日期:2009-11-20
基金资助:
This work was supported by the Research Council of Norway through the projects Nonlinear Problems in Mathematical Analysis; Waves In Fluids and Solids; Outstanding Young Investigators Award (KHK), and the Russian Foundation for Basic Research (grant No.~09-01-00490-a) and DFG project No.~436 RUS 113/895/0-1 (EYuP). This article was written as part of the the international research program
on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008--09.

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Helge Holden, Kenneth H. Karlsen, Darko Mitrovic, Evgueni Yu. Panov

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO--7491 Trondheim, Norway; Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO--0316 Oslo, Norway|Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N--0316 Oslo, Norway|Faculty of Mathematics, University of Montenegro, 81000 Podgorica, Montenegro|Mathematical Analysis Department, Novgorod State University, ul. B. St. Peterburgskaya 41, 173003 Veliky Novgorod, Russia

Received:2009-10-25 Online:2009-11-20 Published:2009-11-20
Supported by:
This work was supported by the Research Council of Norway through the projects Nonlinear Problems in Mathematical Analysis; Waves In Fluids and Solids; Outstanding Young Investigators Award (KHK), and the Russian Foundation for Basic Research (grant No.~09-01-00490-a) and DFG project No.~436 RUS 113/895/0-1 (EYuP). This article was written as part of the the international research program
on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008--09.

摘要/Abstract

摘要：

Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measure-valued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.

关键词: degenerate hyperbolic-elliptic equation, degenerate convection-diffusion equation, conservation law, discontinuous flux, approximate solutions, compactness

Abstract:

Key words: degenerate hyperbolic-elliptic equation, degenerate convection-diffusion equation, conservation law, discontinuous flux, approximate solutions, compactness

中图分类号:

35J70

Helge Holden, Kenneth H. Karlsen, Darko Mitrovic, Evgueni Yu. Panov. STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION[J]. 数学物理学报(英文版), 2009, 29(6): 1573-1612.

Helge Holden, Kenneth H. Karlsen, Darko Mitrovic, Evgueni Yu. Panov. STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION[J]. Acta mathematica scientia,Series B, 2009, 29(6): 1573-1612.

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STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

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