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arXiv · 2502.16300

On the fractional regularity for an elliptic nonlinear singular drift equation

Oscar Jarrin
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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.

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Oscar Jarrin. 2025-02-22. On the fractional regularity for an elliptic nonlinear singular drift equation. https://arxiv.org/abs/2502.16300

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