PreprintCC BY 4.0arXiv
Liouville Theorems for the Stationary Fractional Magnetohydrodynamic Equations
Abstract
We establish Liouville-type theorems for the three-dimensional stationary fractional magnetohydrodynamic (MHD) system with different orders of dissipation. Our first result shows that every smooth solution satisfying the natural homogeneous Sobolev energy conditions is trivial whenever 1≤α≤ 5/3 and 3≤α+2β≤5, without any additional integrability assumptions on the velocity or magnetic fields. This extends previous results to new parameter regimes, including cases with β<1. Beyond this region, we prove a complementary Liouville theorem under suitable integrability conditions only on the radial components of the velocity and magnetic fields. The proofs rely on the Caffarelli--Silvestre extension and localized energy estimates.
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BibTeX
@misc{ai2026liouville,
title = {{Liouville Theorems for the Stationary Fractional Magnetohydrodynamic Equations}},
author = {Albert Ai and Gaston Vergara-Hermosilla},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27494},
eprint = {2610.27494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27494},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Liouville Theorems for the Stationary Fractional Magnetohydrodynamic Equations
AU - Albert Ai
AU - Gaston Vergara-Hermosilla
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27494
UR - https://arxiv.org/abs/2610.27494
AB - We establish Liouville-type theorems for the three-dimensional stationary fractional magnetohydrodynamic (MHD) system with different orders of dissipation. Our first result shows that every smooth solution satisfying the natural homogeneous Sobolev energy conditions is trivial whenever 1≤α≤ 5/3 and 3≤α+2β≤5, without any additional integrability assumptions on the velocity or magnetic fields. This extends previous results to new parameter regimes, including cases with β<1. Beyond this region, we prove a complementary Liouville theorem under suitable integrability conditions only on the radial components of the velocity and magnetic fields. The proofs rely on the Caffarelli--Silvestre extension and localized energy estimates.
ER -