Conservation laws I: viscosity solutions

YAN, JIN; CHENG, ZHIXIN; TAO, MING

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Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.41 no.1 Bogotá Jan./June 2007

Conservation laws I: viscosity solutions

Leyes de conservación I: soluciones de viscosidad

JIN YAN¹, ZHIXIN CHENG², MING TAO³

¹University of Science and Technology of China, Department of Mathematics, Hefei 230026, China.
E-mail: yanjin@mail.ustc.edu.cn
²University of Science and Technology of China, Department of Mathematics, Hefei 230026, China.
E-mail: czx@mail.ustc.edu.cn
³University of Science and Technology of China, Department of Mathematics, Hefei 230026, China.
E-mail: mtao@mail.ustc.edu.cn

Abstract

In this paper we use the Brouwer-Schauder's fixed point theorem to obtain the existence of local smooth viscosity solutions of the Cauchy problem for the parabolic system

with the bounded measurable initial data

Then based on the local existence and the maximum principle, we get the existence of global smooth solutions for two special systems, one related to the hyperbolic system of quadratic flux and the other related to the LeRoux system.

Key words: Hyperbolic conservation laws, viscosity solution, Cauchy problem, a priori estimate.

2000 Mathematics Subject Classification. Primary: 35B40. Secondary: 35L65.

Resumen

En este artículo usamos el teorema de punto fijo de Brouwer-Schauder para obtener la existencia de soluciones locales de viscosidad suave al problema de Cauchy para el sistema parabólico

con data inicial medible acotada

Luego, basados en la existencia local y el principio del máximo, obtenemos la existencia de soluciones globales suaves para dos sistemas especiales , uno relacionado con el sistema parabólico de flujo cuadrático y el otro relacionado con el sistema LeRoux.

Palabras clave: Leyes de conservación hiperbólica, soluciones de viscosidad, problema de Cauchy, estimación a priori.

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References

[1] Y. G. LU, Hyperbolic conservation laws and the compensated compactess method, Vol. 128, Chapman and Hall, New York, 2002. [ Links ]

[2] Y. G. LU, Convergence of the viscosity method for a nonstrictly hyperbolic system, Acta Math. Sci. 12 (1992), 349-360. [ Links ]

[3] A. Y. LEROUX, Numerical stability for some equations of gas dynamics, Mathematics of Computation 37 (1981), 435-446. [ Links ]

[4] J. SMOLLER, Shock waves and reaction-diffusion equations, Springer-Verlag, Berlin-Heidelberg-New York, 1983. [ Links ]

(Recibido en enero de 2007. Aceptado en marzo de 2007)