I'm considering a partial differential equation of the form

$$\nabla^2 u + \mathbf{a}\cdot\nabla u = 0$$

with Dirichlet boundary conditions, where $\mathbf{a}$ is a (smooth, nonconstant) vector field. Can anybody give me a hint how to find sufficient conditions on $\mathbf{a}$ so that the solution is unique? Thank you.