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.

参考文献

[1] Antonic N, Lazar M. H-measures and variants applied to parabolic equations. J Math Anal Appl, 2008, 343(1): 207--225

[2] Dacorogna B. Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals. Lecture Notes in Mathematics, Vol 922. Berlin: Springer, 1982

[3] DiPerna R J. Measure-valued solutions to conservation laws. Arch Ration Mech Anal, 1985, 88(3): 223--270

[4] Gerard P. Microlocal defect measures. Comm Partial Differential Equations, 1991, 16(11): 1761--1794

[5] Hille E, Phillips R S. Functional Analysis and Semi-Groups. American Mathematical Society Colloquium Publications, Vol 31.
Providence RI: American Mathematical Society, 1957

[6] Karlsen K H, Risebro N H, Towers J D. L¹ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr K Nor Vidensk Selsk, 2003, (3): 1--49

[7] Karlsen K H, Risebro N H, Towers J D. Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J Numer Anal, 2002, 22(4): 623--664

[8] Karlsen K H, Towers J D. Convergence of the Lax--Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux. Chinese Ann Math Ser B, 2004, 25(3): 287--318

[9] Karlsen K H, Rascle M, Tadmor E. On the existence and compactness of a two-dimensional resonant system of conservation laws.
Commun Math Sci, 2007, 5(2): 253--265

[10] Kruzkov S N. First order quasilinear equations with several independent variables. Mat Sb (N S), 1970, 81(123): 228--255

[11] Ladyzhenskaya O A, Ural'ceva N N. Linear and Quasilinear Elliptic Equations. New York: Academic Press, 1968

[12] Lions P -L, Perthame B, Tadmor E. A kinetic formulation of multidimensional scalar conservation laws and related equations. J Amer Math Soc, 1994, 7(1): 169--191

[13] Panov E Yu. On sequences of measure-valued solutions of a first-order quasilinear equation. Mat Sb, 1994, 185(2): 87--106

[14] Panov E Yu. On the strong precompactness of bounded sets of measure-valued solutions of a first-order quasilinear equation.
Mat Sb, 1995, 186(5): 103--114

[15] Panov E Yu. Property of strong precompactness for bounded sets of measure-valued solutions of a first-order quasilinear equation.
Mat Sb, 1999, 190(3): 427--446

[16] Panov E Yu. Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property. Journal of Mathematical Sciences, 2009, 159(2): 180--228

[17] Panov E Yu. Existence and strong precompactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux. Arch Ration Mech Anal, 2009, doi:10.1007/s00205-009-0217-x

[18] Panov E Yu. On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux.
J Differ Equ, 2009, 247: 2821--2870

[19] Sazhenkov S A. The genuinely nonlinear Graetz--{N}usselt ultraparabolic equation. Sibirsk Mat Zh, 2006, 47(2): 431--454

[20] Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No 30. Princeton NJ: Princeton University Press, 1970

[21] Tadmor E, Tao T. Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear partial differential equations.
Commun Pure Appl Math, 2008, 61: 1--34

[22] Tartar L. Compensated compactness and applications to partial differential equations Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol IV. Boston, Mass: Pitman, 1979: 136--212

[23] Tartar L. H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc Roy Soc Edinburgh Sect A, 1990, 115(3/4): 193--230

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

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Chunxu XU. BOUNDED, COMPACT AND SCHATTEN CLASSES HANKEL OPERATORS ON WEIGHTED FOCK SPACES[J]. 数学物理学报(英文版), 2025, 45(4): 1529-1554.
[2]	Changyu GUO, Changlin XIANG, Gaofeng ZHENG. REFINED CONSERVATION LAW FOR AN EVEN ORDER ELLIPTIC SYSTEM WITH ANTISYMMETRIC POTENTIAL[J]. 数学物理学报(英文版), 2024, 44(6): 2111-2124.
[3]	Zuoling LIU, Hasi WULAN. PRODUCT TYPE OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND BLOCH TYPE SPACES[J]. 数学物理学报(英文版), 2024, 44(4): 1327-1336.
[4]	Zihou ZHANG, Jing ZHOU. THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS[J]. 数学物理学报(英文版), 2024, 44(2): 445-454.
[5]	Yinzheng Sun, Aifang Qu, Hairong Yuan. THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*[J]. 数学物理学报(英文版), 2024, 44(1): 37-77.
[6]	Fei WU, Zejun WANG, Fangqi CHEN. THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1668-1674.
[7]	Fen HE, Zhen WANG, Tingting CHEN. THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1717-1734.
[8]	Qing Guo, Leiga Zhao. POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS^*[J]. 数学物理学报(英文版), 2023, 43(3): 1116-1130.
[9]	Jinlu LI, Zhaoyang YIN, Xiaoping ZHAI. GLOBAL WELL-POSEDNESS FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS[J]. 数学物理学报(英文版), 2022, 42(5): 2131-2148.
[10]	张子厚, 周宇, 刘春燕, 周晶. SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES[J]. 数学物理学报(英文版), 2022, 42(4): 1621-1630.
[11]	杨永富, 琚强昌, 周爽. THE GLOBAL COMBINED QUASI-NEUTRAL AND ZERO-ELECTRON-MASS LIMIT OF NON-ISENTROPIC EULER-POISSON SYSTEMS[J]. 数学物理学报(英文版), 2022, 42(4): 1666-1680.
[12]	黄腾. A COMPACTNESS THEOREM FOR STABLE FLAT $SL(2,\mathbb{C})$ CONNECTIONS ON 3-FOLDS[J]. 数学物理学报(英文版), 2022, 42(3): 1160-1172.
[13]	金建军, 唐树安. GENERALIZED CESÀRO OPERATORS ON DIRICHLET-TYPE SPACES[J]. 数学物理学报(英文版), 2022, 42(1): 212-220.
[14]	Nemat NYAMORADI, Abdolrahman RAZANI. EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES[J]. 数学物理学报(英文版), 2021, 41(4): 1321-1332.
[15]	王晓燕, 杨俊元. DYNAMICS OF A NONLOCAL DISPERSAL FOOT-AND-MOUTH DISEASE MODEL IN A SPATIALLY HETEROGENEOUS ENVIRONMENT[J]. 数学物理学报(英文版), 2021, 41(2): 552-572.