= Damped Burgers equation
{title2=$u_t+uu_x+\gamma u=\varepsilon u_{xx}$}
The <Burgers' equation> with constant linear damping has the displayed form. For zero viscosity and $\gamma>0$, <characteristics> are $u=u_0(a)e^{-\gamma t}$ and $x=a+u_0(a)(1-e^{-\gamma t})/\gamma$. A negative initial slope can make this coordinate map cease to be invertible, producing <shock formation by characteristic intersection>. The transformation $u=e^{-\gamma t}v$, $\tau=(1-e^{-\gamma t})/\gamma$ converts the inviscid equation to the ordinary inviscid <Burgers' equation>. Across a thin viscous layer the damping contributes only at lower order to the leading <Rankine-Hugoniot condition>.
Back to article page
