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With diffusion absent, the characteristic equations for a first-order quasilinear equation are $du/dt=0$, $dx/dt=q^{\prime }(u)$. A characteristic curve labelled by its starting point $s$ isThis determines a smooth single-valued solution while $s\mapsto x$ is one-to-one. Its derivative isAll characteristic curves move right because $q^{\prime }>0$. Because $q^{\prime \prime }<0$, increasing portions of the initial profile have slower characteristic curves ahead than behind, so they steepen. The first positive gradient catastrophe occurs atThis is a minimum when attained. The numerator at such a point is positive and the denominator decreases to zero from above, so $u_x\to +\infty$.

Resolve the thin transition with $X=(x-x_s(t))/ϵ$ and $u\simeq U(X)$. The leading balance is $(-c+q^{\prime }(U))U^{\prime }=(D(U)U^{\prime }{)}^{\prime }$, where $c={\dot{x}}_s$. Integrating once and matching to the state behind givesMatching the state ahead requires $F(u_{+})=0$ and gives the Rankine-Hugoniot speedThe strictly concave graph of $F$ has zeros at the two states and lies above its chord, hence is strictly positive between them. Since $D>0$, $U^{\prime }>0$ through the layer. A heteroclinic profile increasing with $X$ can therefore connect the rear state at $X=-\infty$ to the front state at $X=+\infty$ only when $\boxed{u_{-}<u_{+}}$. The concavity also gives $q^{\prime }(u_{-})>c>q^{\prime }(u_{+})$: characteristic curves enter the viscous shock from both sides. Reversing the states would demand a decreasing profile, contradicting the sign of $F/D$.

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.