D. Serre
Shock Profiles in Scalar Conservation Laws

Hyperbolic Systems of Balance Laws - Cetraro July 14-21, 2003

The theory of hyperbolic conservation laws is strongly tied with that of dissipative conservation laws. For instance, the limit process as the dissipation coefficient tends to zero, gives an entropy criterion. That correspond to a specific internal structure of shock waves, called Shock Profiles. The existence and the nature of SP is often, but not always, straightforward.
A crucial point is the asymptotic stability of SP under an initial disturbance, as time goes to infinity. When the disturbance is small, a powerfull theory has been developped (see the lectures given by Williams and Zumbrun). For the moment, the case of large disturbances has been solved only in the scalar case, thanks to the L1 contraction of the semi-group. We shall describe the various techniques that are involved in the scalar case.
We shall apply these ideas to several dissipations of distinct natures.

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