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arXiv · 2611.01877

On $W^{2,1}$-Regularity for Rivière's System with a Local Hardy Drift

Yu-Ting Kang·Wen-Juan Qi·Zhao-Min Sun
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Abstract

In this paper, we study the endpoint regularity of weak solutions to the inhomogeneous Rivière system $$ -Δu=Ω\cdot\nabla u+f \qquad\text{in }B_1\subset\mathbb R^2, $$ with an antisymmetric potential $Ω\in L^2$ and a local Hardy drift term $f\in h^1$. We construct a continuous solution in $W_0^{1,2}$, which fails to belong to $W^{2,1}$ in every neighborhood of the origin, even when $\|Ω\|_{L^2}$ is arbitrarily small. Smooth truncations further show that the corresponding uniform interior $W^{2,1}$ estimate fails even for smooth solutions and coefficients. Thus, the $L\log L$ assumption ensuring local $W^{2,1}$ regularity cannot in general be replaced by componentwise local Hardy data.

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Yu-Ting Kang, Wen-Juan Qi, Zhao-Min Sun. 2026-11-01. On $W^{2,1}$-Regularity for Rivière's System with a Local Hardy Drift. https://arxiv.org/abs/2611.01877

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