Fundamental solutions and singular shocks in scalar conservation laws. (English) Zbl 1057.35002

Summary: We study the existence and non-existence of fundamental solutions for the scalar conservation laws \(u_tf(u)_x=0,\) 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 \(f(u)=u^m\). 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.


35A08 Fundamental solutions to PDEs
35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
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