ENTROPY CRITERIA AND STABILITY OF EXTREME SHOCKS: A REMARK ON A PAPER OF LEGER AND VASSEUR.

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable L² dependence on initial data of Lax 1- or n-shock solutions of an n X n system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freistühler and Zum-brun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of exis-tence, at least for small BV or Hs perturbations. [ABSTRACT FROM AUTHOR]