Propagation properties for scalar conservation laws

Abstract

We study the propagation of an initial disturbance u(0)(x) of an equilibrium state s epsilon R for the scalar conservation law u(t) + phi(u)(x) = 0 in (0, + infinity) x R. We give a necessary and sufficient condition on phi for the following propagation property: if support of (u(0)(.) - s) is compact then the support of (u(0)(t,.) - s) is also compact for t epsilon [0, T-0), for some T-0 epsilon (0, + infinity]. The proofs are based on the study of suitable associated Riemann problems.