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

We consider self-similar approximations of non-linear hyperbolic systems in one space dimension with Riemann initial data, especially the system ∂tuε+A(uε)∂x uε=εt∂x(B(uε)∂xuε), 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 uε =uε (x/t) which has bounded total variation, uniformly in the diffusion parameter ε>0. In the limit ε→0, the functions uε converge towards a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation ∂tuε+∂ xvε =0, ∂tvε +a2B(u)∂xuε =(f(uε)-vε )/(εt). We also cover the boundary-value problem in a half-space for the same regularizations