On the fractional regularity for an elliptic nonlinear singular drift equation

Abstract

We consider an elliptic equation with the fractional Laplacian operator $(-\hat{I}”{)}^{\frac{\hat{I}}{\pm }2}$ in the dissipative term, a singular integral operator $\mathbf{A}(\cdot )$ in the nonlinear term, and an external source $f$. The key example is the stationary (time-independent) counterpart of the surface quasi-geostrophic equation. Under suitable assumptions on $f$ and natural assumptions on $\mathbf{A}(\cdot )$ in the setting of Sobolev spaces, our main result examines how the fractional power $\hat{I}\pm$ propagates and optimally improves the regularity of weak $L^p$-solutions to this equation.

19 pages