Given $\ \mathbf{u}\cdot \nabla c=\Delta ca_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset %TCIMACRO{\U{211d} } %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ bounded, $div$$(\mathbf{u})=0$, $\mathbf{u\in L}^{2},$ and $\rho \in L^{2}$. $\ $and boundary Dirichlet conditions (or Neumann conditions!). What can I say about the existence of solutions and regularity results for this equation? (Gilbarg and Trudinger's book doesn't have an answer for this kind of problem because they ask for $\mathbf{u\in L}^{\infty }$ in the hypotheses). I would appreciate very much some precise references for this problem.
