Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions

Abstract

We consider self-similar approximations of non-linear hyperbolic systems in one space dimension with Riemann initial data, especially the system ∂_tu_ε+A(u_ε)∂_x uε=εt∂_x(B(u_ε)∂_xu_ε), with ε>0. We assume that the matrix A(u) is strictly hyperbolic and that the diffusion matrix satisfies |B(u)-Id|«1. No genuine non-linearity assumption is required. We show the existence of a smooth, self-similar solution u_ε =u_ε (x/t) which has bounded total variation, uniformly in the diffusion parameter ε>0. In the limit ε→0, the functions u_ε converge towards a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation ∂_tu^ε+∂ _xv^ε =0, ∂_tv^ε +a²B(u)∂_xu^ε =(f(u^ε)-v^ε )/(_εt). We also cover the boundary-value problem in a half-space for the same regularizations.