Existence theory for nonclassical entropy solutions of scalar conservation laws (Q1764406)

A general existence theory for the Cauchy problem for a one space dimensional conservation law: \[ \partial_ t u + \partial_ x f(u) =0, \quad u\in \mathbb R^ N,\;x\in \mathbb R,\;t>0\,, \] is presented. The flux function \(f\) is assumed to be nonconvex in this paper. The aim is to establish existence of solutions having the so-called nonclassical shocks. Notice that the classical entropy solution satisfies the Oleinik entropy inequalities and the total variation \(TV(u(.,t))\) satisfies: \(TV(u(.,t))\leq TV(u(.,0))\). The main difficulty for the solutions having nonclassical shocks lies in the fact that their total variation may increase. In this paper the authors introduce and investigate a generalized total variation functional \(V=V(u)\) for which \(V(u(.,t))\leq V(u(.,0))\) even when nonclassical shocks are involved.