On the fractional regularity for an elliptic nonlinear singular drift equation

TL;DR

This paper investigates how fractional Laplacian operators influence the regularity of solutions to a nonlinear elliptic equation with singular drift, with applications to geophysical fluid dynamics.

Contribution

It provides new insights into the regularity propagation for solutions of elliptic equations involving fractional Laplacians and singular nonlinear terms.

Findings

Fractional power $oldsymbol{rac{oldsymbol{ ext{alpha}}}{2}}$ enhances solution regularity.

Optimal regularity results for weak solutions in Sobolev spaces.

Application to stationary surface quasi-geostrophic equations.

Abstract

We consider an elliptic equation with the fractional Laplacian operator $(-\Delta)^{\frac{\alpha}{2}}$ in the dissipative term, a singular integral operator ${\bf A}(\cdot)$ 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 ${\bf A}(\cdot)$ in the setting of Sobolev spaces, our main result examines how the fractional power $\alpha$ propagates and optimally improves the regularity of weak $L^p$-solutions to this equation.