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.