Driving force and kinetic relations for scalar conservation laws

Abstract

This paper is concerned with the circumstances under which the dissipative character of a one-dimensional scalar conservation law may be described by a formalism strictly analogous to that arising naturally in the dynamics of nonlinearly elastic materials. It is shown that this occurs if and only if the entropy density, entropy flux pair associated with the conservation law takes a particular form. We compare the admissibility condition associated with this special entropy with other admissibility criteria such as those of Lax, Oleinik and regularization theory. Using the special entropy, we consider the Riemann problem for an example in which genuine nonlinearity fails and a kinetic relation is needed to determine a unique solution.

Additional Information

© 2007 World Scientific Publishing Company. Received 5 Feb. 2006; Accepted 3 Mar. 2006. Communicated by P. G. LeFloch. I am pleased to acknowledge many helpful conversations on the general subject of this work with Rohan Abeyaratne and Phoebus Rosakis. Special thanks must go to Rohan, who read the manuscript and made a number of valuable criticisms and suggestions. Talks by Andrea Bertozzi and Michael Shearer about their joint work on thin films have stimulated my interest in scalar conservation laws. Research for the present paper was begun while the author was at the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, in 2003; the hospitality and financial support of the Institute are gratefully acknowledged. I also wish to thank Philippe LeFloch for the invitation to participate in the program on Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics at the Newton Institute.

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Driving force and kinetic relations for scalar conservation laws

Abstract

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