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

Author(s): Yun-guang Lu; Christian Klingenberg
Journal: Proc. Amer. Math. Soc. 132 (2004), 1305-1309.
MSC (2000): Primary 35L65
Posted: December 18, 2003
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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:

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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