Burgers' equation

Remarks

The inviscid equation is the scalar conservation law with flux $F(u) = u^2/2$, which is uniformly convex. For smooth $u$ the two forms agree, but for discontinuous weak solutions only the conservation form is meaningful, and it fixes the shock speed $(u_l + u_r)/2$. By the method of characteristics, a smooth solution with initial value $g$ satisfies $u = g(x - ut)$; if $g$ is somewhere decreasing, the solution ceases to be smooth at time $1/\max(-g')$.

Example: $u(x, t) = x/(t + 1)$ solves the inviscid equation, and also the viscous one for every $\nu$. Check: $\partial_t u = -x/(t + 1)^2$, $u\, \partial_x u = x/(t + 1)^2$, and $\partial_x^2 u = 0$.

Non-example: $u(x, t) = e^{-t} \sin x$ solves the linear heat equation $u_t = u_{xx}$ but not Burgers' equation with $\nu = 1$. Indeed for it $\partial_t u - \partial_x^2 u = 0$, while $u\, \partial_x u = e^{-2t} \sin x \cos x$ is not identically zero.