Fundamental solutions and singular shocks in scalar conservation laws

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) = u. 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. 2000 Mathematics Subject Classification: 35L60, 35L67.