On the fractional regularity for an elliptic nonlinear singular drift equation
Oscar Jarrin
TL;DR
This work analyzes a fractional-elliptic nonlinear drift equation $(-\Delta)^{\frac{\alpha}{2}}u + \mathbf{A}(u)\cdot \nabla u = f$ with $\operatorname{div}\mathbf{A}(u)=0$, exploring how the dissipative fractional term propagates regularity of weak $L^p$-solutions under natural assumptions on $f$ and the singular integral operator $\mathbf{A}$. The authors develop a Lorentz-space fixed-point framework, representing the nonlocal linear part via a kernel $K_\alpha$ and proving contraction in suitable Lorentz spaces for $1<\alpha<\frac{n}{2}+1$, yielding existence of weak solutions and enabling a sharp bootstrap to obtain $u\in \dot{W}^{s+\alpha,r}$ when $f\in \dot{W}^{s,r}$. The main result shows that, for $\alpha>1$, the fractional dissipation produces an optimal gain in regularity, with corollaries giving Hölder continuity and, in the homogeneous case, smoothness; the paper also discusses limitations for $0<\alpha\leq1$ and validates these ideas on a toy model with weaker nonlinearity. Collectively, these results advance understanding of regularity propagation in nonlinear fractional elliptic equations and connect to stationary SQG-type dynamics.
Abstract
We consider an elliptic equation with the fractional Laplacian operator $(-Δ)^{\fracα{2}}$ 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 $α$ propagates and optimally improves the regularity of weak $L^p$-solutions to this equation.