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arXiv · 2510.12707

Nonlinear instability for the 3D MHD equations around the Taylor-Couette flow

Víctor Navarro-Fernández·David Villringer
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Abstract

We study the 3D magnetohydrodynamics (MHD) equations in an annular cylinder, perturbed around the explicit steady state given by the 3D Taylor-Couette velocity field and zero magnetic field. Combining a recent linear instability result for the magnetic field with the framework of Friedlander, Pavlovi\'c and Shvydkoy [Comm. Math. Phys. 264 (2006), no. 2, 335-347], we prove nonlinear instability of the solution around this steady state in $L^p$, for any $p>1$. In particular, our results are, to the best of our knowledge, the first rigorous instability results for 3D MHD without forcing, in which the instability is produced as a result of the (exponential) growth of the magnetic field. Furthermore, we offer a mathematical proof of the physically conjectured transfer of energy from the velocity field to the magnetic field in the MHD system.

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Víctor Navarro-Fernández, David Villringer. 2025-10-14. Nonlinear instability for the 3D MHD equations around the Taylor-Couette flow. https://arxiv.org/abs/2510.12707

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