Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws (Q1578963)

The authors consider a scalar conservation law \(u_t+ f(u)_x= 0\) with a weakly genuinely nonlinear flux, i.e. \(f\) is not affine on any nontrivial interval. The main result states that weak solutions \(u\in L^\infty\), that satisfy the entropy condition in the sense of distributions only for \(t> 0\), are in fact weakly continuous in \(L^1\) as \(t\) tends to zero. This implies in particular uniqueness. The proof is based on properties of divergence-measure fields. As an application of the main result it is shown that for a weakly genuinely nonlinear flux the initial layer of the semilinear relaxation approximation collapses to the line \(t= 0\).