Nonclassical shocks and the Cauchy problem for nonconvex conservation laws
The authors consider the Cauchy problem for a conservation law with nonconvex flux-function (taken to be cubic, for simplicity) \[ \partial_tu+\partial_xu^3=0, \quad u(x,0)=u_0(x), \quad u(x,t)\in \mathbb{R},\;x\in \mathbb{R},\;t>0. \] The Riemann problem for this law can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality of the form \[ \partial_tU(u)+\partial_xF(u)\leq 0, \] where \(U\) is strictly convex, and \(F'(u)=3U'(u)u^2\), and kinetic relation, which is required for nonclassical shocks only and determines them uniquely. The limiting solution \(u=\lim_{\varepsilon\rightarrow 0}u_\varepsilon\) given by \[ \partial_tu_\varepsilon+\partial_xu^3_\varepsilon= \varepsilon\partial_{xx}u_\varepsilon+A\varepsilon^2\partial_{xxx}u_\varepsilon \] is referred to as the admissible nonclassical solution. The authors use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variation must be modified accordingly. The authors prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality.
scientific article; zbMATH DE number 1271352
scientific article