Decay with a rate for noncompactly supported solutions of conservation laws

Abstract:

We show that solutions of the Cauchy problem for systems of two conservation laws decay in the supnorm at a rate that depends only on the ${L^1}$ norm of the initial data. This implies that the dissipation due to increasing entropy dominates the nonlinearities in the problem at a rate depending only on the ${L^1}$ norm of the initial data. Our results apply to any BV initial data satisfying ${u_0}( \pm \infty ) = 0$ and ${\operatorname {Sup}}\{ {u_0}( \cdot )\} \ll 1$. The problem of decay with a rate independent of the support of the initial data is central to the issue of continuous dependence in systems of conservation laws because of the scale invariance of the equations. Indeed, our result implies that the constant state is stable with respect to perturbations in $L_{{\operatorname {loc}}}^1$. This is the first stability result in an ${L^p}$ norm for systems of conservation laws. It is crucial that we estimate decay in the supnorm since the total variation does not decay at a rate independent of the support of the initial data. The main estimate requires an analysis of approximate characteristics for its proof. A general framework is developed for the study of approximate characteristics, and the main estimate is obtained for an arbitrary number of equations.