I have the following boundary value problem (for my applied finite elements course):
\begin{equation} \begin{cases} \mathbf{v}\cdot\nabla u - \nabla \cdot \left[ D(u) \nabla u\right] = 0 & \mathrm{in}\, \Omega, \\ u = g(x,y) &\mathrm{on}\,\partial\Omega. \end{cases} \end{equation} Here $g(x,y)$ and $D(u)$ are given functions and the velocity field $\mathbf{v}$ is assumed to be given and constant over the domain. It is stated (without proof) that one can expect problems with the existence of a solution if $D(u)=0$. Therefore we take $D(u)$ to be nonzero. I do not see why this is the case. If $D(u)=0$, the PDE would be $ \mathbf{v}\cdot\nabla u = 0$. How do I know that this has no solution?