Differentiability With Respect to Initial Data for a Scalar Conservation Law

Abstract

. We linearize a scalar conservation law around an entropy initial datum. The resulting equation is a linear conservation law with discontinuous coefficient, solved in the context of duality solutions, for which existence and uniqueness hold. We interpret these solutions as weak derivatives with respect to the initial data for the nonlinear equation. 1. Introduction Consider the one-dimensional scalar conservation law @ t u + @ x f(u) = 0; 0 ! t ! T ; x 2 R; (1) where f is a C 1 convex function, provided with entropy admissible initial data u ffi 2 L 1 (R). Kruzkov's results [4] assert that the entropy solution u to (1) lies in L 1 (]0; T [\ThetaR) " C(0; T ; L 1 loc (R)), and that the following contraction property holds: if u (resp. v) corresponds to the initial data u ffi (resp. v ffi ), then for all R ? 0 and any t ? 0 Z jxjR ju(t; x) \Gamma v(t; x)j dx Z jxjR+Mt ju ffi (x) \Gamma v ffi (x)j dx; (2) where M = maxfjf 0 (s)j; jsj max(ku ffi kL 1 ; kv..