Fundamental solutions and singular shocks in scalar conservation laws

Résumé

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.