Existence of Dafermos profiles for singular shocks
This paper considers the Riemann problem for a strictly hyperbolic, nonlinear system of conservation law \[ \begin{aligned} u_{1t}+(u_1^2-u_2)_x&=0, \quad u_1(x,0)=u_{1L}\text{ if }x<0, \quad u_1(x,0)=u_{1R}\text{ if }x>0, \\ u_{2t}+(\frac{1}{3}u_1^3-u_1)_x&=0, \quad u_2(x,0)=u_{2L}\text{ if }x<0, \quad u_2(x,0)=u_{2R}\text{ if }x>0, \end{aligned} \] where \(u_{iL}\) and \(u_{iR}\in \mathbb{R}\). \textit{B. L. Keyfitz} and \textit{H. C. Kranzer} [Lect. Notes Math. 1402, 185--197 (1989; Zbl 0704.35094), IMA Vol. Math. Appl. 27, 107--125 (1990; Zbl 0718.76071)] produced a unique solution containing {singular shocks}, i.e. allowing discontinuities made of a Heavyside function plus a \(\delta\)-function. They conjectured that such solutions were limits as \(\varepsilon\) goes to zero of self-similar solutions \(u_i^{\varepsilon}(\frac{x}{t})\) of a Dafermos regularization \[ \begin{aligned} u_{1t}+(u_1^2-u_2)_x&=\varepsilon tu_{1xx}, \\ u_{2t}+(\frac{1}{3}u_1^3-u_1)_x&=\varepsilon tu_{2xx}. \end{aligned} \] The main result of the paper gives conditions which ensure existence of such a singular shock which is the limit of solutions of Dafermos regularizations. The proof uses the blowing up approach to geometric singular perturbation problems that lack normal hyperbolicity. It also applies to some generalized Keyfitz-Kranzer system.
scientific article; zbMATH DE number 2117240
scientific article