I am interested in the following PDE.
Suppose $u_m$ is a smooth solution of a elliptic equation of the form

$$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on $\partial B_1$. $B_1 $ is the open unit ball centered at the origin in $ R^N$.
$a_m(x)$ is a smooth divergence free drift whose suppose is contained in $B_\frac{1}{m}$ and whose $L^N $ norm is constant (but maybe not small). Here $f_m$ is bounded in, say $L^\infty$. My question is can one obtain uniform Holder bounds on $u_m$?

I know these problems are very well studied lately but I am having trouble googling for the correct answer. thanks for any remarks Craig



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