= Viscous shock
A <viscous shock> is a smooth travelling transition produced when positive <diffusion> resolves a <shock> of a <conservation law>. For $u_t+q(u)_x=\epsilon[D(u)u_x]_x$ with $D>0$, an inner coordinate $X=(x-ct)/\epsilon$ gives $D(U)U'=q(U)-q(u_-)-c(U-u_-)$. 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.
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