Regularity Properties of Degenerate Diffusion Equations with Drifts
Inwon C. Kim, Yuming Paul Zhang · SIAM Journal on Mathematical Analysis · 2018
This paper considers a class of nonlinear degenerate drift-diffusion equations. We study well-posedness and regularity properties of the solutions with the goal of achieving uniform Hölder regularity in terms of the $L^p$-bound on the drift vector field. A formal scaling argument yields that the threshold for such estimates is $p=d$, while our estimates are for $p>d+\frac{4}{d+2}$. On the other hand, we are able to show by a series of examples that one needs $p>d$ for such estimates, even for divergence-free drift.