= Viscous scalar conservation law
{title2=$u_t+F(u)_x=\varepsilon u_{xx}$}
Positive $\varepsilon$ adds diffusion to a one-dimensional <scalar conservation law>. For a sufficiently decaying smooth solution, the divergence structure gives $\frac12\frac d{dt}\|u\|_2^2+\varepsilon\|u_x\|_2^2=0$, so the $L^2$ <energy estimate> is uniform in $\varepsilon$. If $|F'|\leq M$, testing against $-u_{xx}$ also yields $\|u_x(t)\|_2^2\leq e^{M^2t/\varepsilon}\|u_x(0)\|_2^2$. Thin <travelling waves> explain why a uniform <gradient> bound generally fails.
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