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On W2,1-Regularity for Rivière's System with a Local Hardy Drift
Abstract
In this paper, we study the endpoint regularity of weak solutions to the inhomogeneous Rivière system −Δu=Ω·∇ u+f in B1⊂ℝ2, with an antisymmetric potential Ω∈ L2 and a local Hardy drift term f∈ h1. We construct a continuous solution in W01,2, which fails to belong to W2,1 in every neighborhood of the origin, even when |Ω|L2 is arbitrarily small. Smooth truncations further show that the corresponding uniform interior W2,1 estimate fails even for smooth solutions and coefficients. Thus, the Llog L assumption ensuring local W2,1 regularity cannot in general be replaced by componentwise local Hardy data.
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BibTeX
@misc{kang2026w21regularity,
title = {{On $W^2,1$-Regularity for Rivière's System with a Local Hardy Drift}},
author = {Yu-Ting Kang and Wen-Juan Qi and Zhao-Min Sun},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01877},
eprint = {2611.01877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01877},
}
RIS
TY - UNPB
TI - On W^2,1-Regularity for Rivière's System with a Local Hardy Drift
AU - Yu-Ting Kang
AU - Wen-Juan Qi
AU - Zhao-Min Sun
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01877
UR - https://arxiv.org/abs/2611.01877
AB - In this paper, we study the endpoint regularity of weak solutions to the inhomogeneous Rivière system −Δu=Ω·∇ u+f in B_1⊂ℝ^2, with an antisymmetric potential Ω∈ L^2 and a local Hardy drift term f∈ 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 Llog L assumption ensuring local W^2,1 regularity cannot in general be replaced by componentwise local Hardy data.
ER -