 Viscous shock

A viscous shock is a smooth travelling transition produced when positive diffusion resolves a shock of a conservation law. For $u_t+q(u{)}_x=ϵ[D(u)u_x{]}_x$ with $D>0$, an inner coordinate $X=(x-ct)/ϵ$ gives $D(U)U^{\prime }=q(U)-q(u_{-})-c(U-u_{-})$. The endpoint conditions impose the Rankine-Hugoniot speed. If $q$ is strictly concave, the right side is positive between its roots, so the admissible profile increases from its rear state to its front state. This supplies the entropy selection directly from a regularizing mechanism.