Chapter 7 L1-stability of nonlinear waves in scalar conservation laws

Abstract

Abstract We consider various scalar conservation laws with dissipation. The Cauchy problem generates a (nonlinear) semigroup that satisfies a maximum principle, a contraction property with respect to the L 1 -distance, and preserves the total mass (hence the word “conservation law”). A crucial role is played by traveling waves, also called shock profiles, that tend to distinct constants as x · v → ± ∞ (v e S d-1 a direction of propagation). In particular, their stability is a good criterion for the relevance of the underlying inviscid shock waves. The precise question that we address in this article is their asymptotic stability with respect to the L 1 -distance. This topology turns out to be the most natural one in the context of conservation laws because of the properties mentioned above. A related question concerns steady solutions of the IBVP, usually called “boundary layers”. In one space dimension, the stability of shock profiles is, up to some extent, a matter of dynamical systems theory. However, a complete picture needs the more analytical proof that constants are also asymptotically stable, with respect to zero-mass initial disturbances. The latter may or may not hold, depending on the strength of the dissipation involved by a given model. For instance, it does not in the inviscid case. The multidimensional stability of shock waves is significantly more involved, and is still widely open. Numerical approximation by conservative and monotone finite difference schemes provides another kind of dissipative process. We investigate the existence and stability of discrete shock profiles. We fill a gap in Jennings' proof of existence and prove that tails are exponentially small. Throughout this text, we present nine Open Problems that have their own interest.