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We consider self-similar approximations of nonlinear hyperbolic systems inone space dimension with Riemann initial data and general diffusion matrix. Weassume that the matrix of the system is strictly hyperbolic and the diffusionmatrix is close to the identity. No genuine nonlinearity assumption isrequired. We show the existence of a smooth, self-similar solution which hasbounded total variation, uniformly in the diffusion parameter. In thezero-diffusion limit, the solutions converge to a solution of the Riemannproblem associated with the hyperbolic system. A similar result is establishedfor the relaxation approximation and the boundary-value problem in a half-spacefor the same regularizations.

Singular limits for the Riemann problem. General diffusion, relaxation, and boundary conditions