Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions
Kayyunnapara Thomas Joseph, Philippe G. LeFloch · Comptes Rendus Mathématique · 2006
We consider self-similar approximations of non-linear hyperbolic systems in one space dimension with Riemann initial data, especially the system ∂ t u ε + A ( u ε ) ∂ x u ε = ε t ∂ x ( B ( u ε ) ∂ x u 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 ∂ t u ε + ∂ x v ε = 0 , ∂ t v ε + a 2 B ( u ) ∂ x u ε = ( f ( u ε ) − v ε ) / ( ε t ) . We also cover the boundary-value problem in a half-space for the same regularizations.