Well-posedness of scalar conservation laws with singular sources

We consider scalar conservation laws with nonlinear singular sources with a concentration effect at the origin. We assume that the flux A is not degenerated and we study whether it is possible to define a well-posed limit problem. We prove that when A is strictly monotonic then the limit problem is well-defined and has a unique solution. The definition of this limit problem involves a layer which is shown to be very stable. But when A is not monotonic this problem can be unstable. Indeed we can construct two sequences of approximate solutions which converge to two different functions although their initial values coincide in the limit.

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