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Proceedings of the American Mathematical Society

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Viscosity and relaxation approximations for a hyperbolic-elliptic mixed type system


Authors: Yun-guang Lu and Christian Klingenberg
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1305-1309
MSC (2000): Primary 35L65
DOI: https://doi.org/10.1090/S0002-9939-03-07326-X
Published electronically: December 18, 2003
MathSciNet review: 2053334
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Abstract | References | Similar Articles | Additional Information

Abstract: To a given system of conservation laws

\begin{displaymath}\left\{ \begin{array}{l} u_t + f(u,v,h(u,v))_x =0 \\ v_t + g(u,v,h(u,v))_x =0 \\ \end{array}\right. \end{displaymath}

we associate the system

\begin{displaymath}\left\{ \begin{array}{l} u_t + f(u,v,s)_x = \epsilon u_{xx} ... ... {s - h(u,v) \over \tau} = \epsilon s_{xx}, \end{array}\right. \end{displaymath}

which is of mixed type. Under certain conditions, convergence of this latter system for $\epsilon \rightarrow 0$ with $\tau = o(\epsilon)$ is established without the need of stability criteria or hyperbolicity of the left-hand sides of the equations.


References [Enhancements On Off] (What's this?)

  • 1. G. Q. Chen, C. D. Levermore, and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), 787-830. MR 95h:35133
  • 2. R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70. MR 84k:35091
  • 3. A. E. Tzavaras, Materials with internal variables and relaxation to conservation laws, Arch. Rational Mech. Anal. 146 (1999), 129-155. MR 2000i:74004

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Additional Information

Yun-guang Lu
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei and Departamento de Matemáticas Universidad Nacional de Colombia, Bogota
Email: yglu@matematicas.unal.edu.co

Christian Klingenberg
Affiliation: Department of Mathematicas, Würzburg University, Würzburg, 97074, Germany
Email: klingen@mathematik.uni-wuerzburg.de

DOI: https://doi.org/10.1090/S0002-9939-03-07326-X
Received by editor(s): February 10, 2002
Published electronically: December 18, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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