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Multiply the viscous scalar conservation law by $-u_{xx}$, rather than estimating $F^{\prime \prime }$ after differentiating. Integration by parts yieldsThus $d\Vert u_x{\Vert }_2^2/dt\le (M^2/\epsilon )\Vert u_x{\Vert }_2^2$. Applying the Gronwall inequality givesOnly the assumed bound on $F^{\prime }$ is used; a global bound on $F^{\prime \prime }$ is not needed for this energy estimate.

Let $M=\Vert F^{\prime }{\Vert }_{\infty }$. The sharper energy estimate uses the divergence structure of the viscous scalar conservation law. Multiply by $u$ and integrate over the line. With $G(s)={\int }_0^{s}r{F}^{\prime }(r)dr$,because the decay makes $G(u)$ tend to zero at both ends. Integration by parts in the diffusion term therefore givesOne may take $C_0=0$, uniformly in $\epsilon$. This does not require $F(0)=0$.

If a bound explicitly involving $M$ and $\epsilon$ is desired, retaining the transport term and using the Cauchy-Schwarz inequality and the elementary inequality $ab\le (a^2+b^2)/2$ givesThe Gronwall inequality then gives the valid but weaker choice $C_0=M^2/\epsilon$. The exact cancellation explains why its divergence is unnecessary.

Insert the travelling wave into the viscous scalar conservation law. With $s=x-\sigma t$ the equation becomesand one integration givesFor a nonconstant profile, $Q(v(s))$ never vanishes. Indeed, the autonomous ordinary differential equation $v^{\prime }=Q(v)/\epsilon$ has unique local solutions since $Q$ is $C^1$; reaching an equilibrium would force the whole solution to be constant. Separation and $c=v(0)$ therefore giveA different choice of reference point gives $s-s_0$ on the left, expressing the translation freedom of the travelling wave.

The printed formula needs a nonconstant-profile qualification. Constant profiles also solve the PDE, but their denominator vanishes at their constant value, so the separated integral is not defined. They must be included separately as equilibrium solutions of the integrated ordinary differential equation.

This approximation studies the limit of a viscous scalar conservation law as its positive diffusion coefficient tends to zero. Under hypotheses giving compactness and appropriate convergence, the viscous entropy dissipation selects an entropy solution of the inviscid scalar conservation law. Fixed-phase decreasing travelling waves for a strictly convex flux converge to an entropy shock. Translating each profile differently can change or destroy the limit, so phase or initial data must specify the family.