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$% E. Chasseigne, Fundamental solutions and singular shocks in scalar conservation laws, Rev. Mat. Complut. 16 (2003), 443–463.%$

Fundamental Solutions and Singular Shocks in Scalar Conservation Laws

Emmanuel CHASSEIGNE
Laboratoire de Mathématiques
Université de Tours — France
echasseigne@univ-tours.fr

Received: October 8, 2002
 Accepted: February 5, 2003
ABSTRACT

We study the existence and non-existence of fundamental solutions for the scalar conservation laws

ut+ f(u)x = 0,

related to convexity assumptions on f. We also study the limits of those solutions as the initial mass goes to infinity. We especially prove the existence of so-called Friendly Giants and Infinite Shock Solutions according to the convexity of f, which generalize the explicit power case f(u) = um. We introduce an extended notion of solution and entropy criterion to allow infinite shocks in the theory, and the initial data also has to be understood in a generalized sense, since locally infinite measures appear.

Key words: conservation laws, fundamental solutions, singular solutions.
2000 Mathematics Subject Classification: 35L60, 35L67.