Gauss kernel method for generalized solutions to conservation laws in heterogeneous media

Abstract

Gauss kernel method and Duhamel's integral principle are used to obtain solutions in the sense of Colombeau generalized functions to scalar conservation laws with flux explicitly dependent on space variable, and to the corresponding parabolic approximation. In homogeneous case, the existence and uniqueness of generalized solution are obtained in [M. Oberguggenberger and Y.-G. Wang, Generalized solutions to conservation laws, J. Anal. Appl. 13(1) (1994), pp. 7–18]. This paper presents generalization of results from [16] for wider class of problems.

Keywords:

• conservation law
• space-dependent flux
• generalized solution
• Duhamel's principle

AMS Subject Classifications :

• 35D05
• 35L65
• 46F30
• 33E30
• 44A20

Acknowledgements

The author is grateful to Stevan Pilipović and Darko Mitrović for numerous discussions, and to the referee for the useful comments. The author also thanks the organizers of GF2009. The research is supported by the Ministry of Science and Technological Development, Republic of Serbia, project 144016.