On the physical and the self-similar viscous approximation of a boundary Riemann problem

We deal with the viscous approximation of a system of conservation laws in one space dimension and we focus on initial-boundary value problems. It is known that, in general, different viscous approximation provide different limits because of boundary layer phenomena. We focus on Riemann-type data and we discuss a uniqueness criterion for distributional solutions which applies to both the non characteristic and the boundary characteristic case. As an application, one gets that the limits of the physical viscous approximation ∂tU ε + ∂x ˆ F (Uε) ̃ = ε∂x ˆ B(Uε)∂xU ε ̃ and of the self-similar viscous approximation ∂tU ε + ∂x ˆ F (Uε) ̃ = εt∂x ˆ B(Uε)∂xU ε ̃ introduced by Dafermos et al. are expected to coincide.