Given $\ \mathbf{u}\cdot \nabla c=\Delta c-a_{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.
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$\begingroup$ When $u$ is divergence free one can obtain good estimates on a solution $c$ even when there are essentially no regularity assumptions on $u$. Check my account because I asked a very similar question a while ago... $\endgroup$– CraigCommented Aug 1, 2013 at 2:21
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