= Viscous Burgers equation
{title2=$u_t+uu_x=\nu u_{xx}$}
The <viscous Burgers equation> combines a nonlinear <transport equation> with <diffusion> with constant $\nu>0$. On the infinite line, a bounded <initial condition> gives a <smooth function> solving the equation for every positive time; a jump in the <initial condition> is smoothed rather than retained as a discontinuity. Small $\nu$ can nevertheless produce a steep layer, as the <Viscous Burgers travelling shock> illustrates. The <Cole-Hopf transformation> $u=-2\nu\,\partial_x\log\phi$, with positive $\phi$, converts this equation to the <heat equation> $\phi_t=\nu\phi_{xx}$.<Asymptotic and Perturbation Methods by Victor Ivrii/Burgers equation. 1>[§9.1, Problem set-up, (9.1.1)–(9.1.2); Cole-Hopf transform, (9.1.11)–(9.1.13)]{ref}
Positive <diffusion> also explains the admissibility of the inviscid limit. For the <entropy flux for a scalar conservation law> associated with $\eta(u)=u^2/2$, direct multiplication by $u$ gives
$$
\partial_t\frac{u^2}{2}+\partial_x\frac{u^3}{3}=\nu\partial_{xx}\frac{u^2}{2}-\nu u_x^2.
$$
The last term is nonpositive. With convergence sufficient to pass to the limit, the <vanishing viscosity approximation> therefore retains an entropy inequality rather than just the <weak solution> equation. For general bounded data, the full <Kruzhkov entropy inequalities> specify the <entropy solution>; the quadratic identity is a useful illustration of dissipation.<Asymptotic and Perturbation Methods by Victor Ivrii/Burgers equation. 1>[§9.1, Case epsilon=0 as a limit, identity following (9.1.10)]{ref}
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