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

Víctor Navarro-Fernández, David Villringer

TL;DR

The paper establishes nonlinear instability in $L^p$ for the 3D MHD equations near the Taylor–Couette flow with zero magnetic field, by combining a linear magnetic-dynamo instability with a nonlinear instability framework. The authors decompose the dynamics into a Navier–Stokes part and a kinematic-dynamo part, prove analytic semigroup and resolvent bounds for the dynamo operator, and upgrade linear growth to nonlinear growth through a mild formulation and a bootstrap argument. The main result demonstrates exponential growth of the magnetic field and energy transfer from velocity to magnetic field in the absence of forcing, offering a rigorous mathematical confirmation of physically conjectured dynamo behavior. The work provides a rigorous bridge between linear dynamo mechanisms and nonlinear MHD instability in a geometrically relevant setting with perfectly conducting boundaries, highlighting the role of viscosity in enabling energy transfer to $B$.

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, Pavlović 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.

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

TL;DR

The paper establishes nonlinear instability in for the 3D MHD equations near the Taylor–Couette flow with zero magnetic field, by combining a linear magnetic-dynamo instability with a nonlinear instability framework. The authors decompose the dynamics into a Navier–Stokes part and a kinematic-dynamo part, prove analytic semigroup and resolvent bounds for the dynamo operator, and upgrade linear growth to nonlinear growth through a mild formulation and a bootstrap argument. The main result demonstrates exponential growth of the magnetic field and energy transfer from velocity to magnetic field in the absence of forcing, offering a rigorous mathematical confirmation of physically conjectured dynamo behavior. The work provides a rigorous bridge between linear dynamo mechanisms and nonlinear MHD instability in a geometrically relevant setting with perfectly conducting boundaries, highlighting the role of viscosity in enabling energy transfer to .

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, Pavlović and Shvydkoy [Comm. Math. Phys. 264 (2006), no. 2, 335-347], we prove nonlinear instability of the solution around this steady state in , for any . 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.
Paper Structure (6 sections, 6 theorems, 59 equations)

This paper contains 6 sections, 6 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Linear instability and the kinematic dynamo
  4. Setup and technical lemmas
  5. Proof of Lemma \ref{['lemma:abstract semigroup lemma']}
  6. Proof of nonlinear instability

Key Result

Theorem 1.1

Let $u_\mathrm{TC}$ denote the Taylor--Couette velocity field eq:TC. For any $s\geq 0$, $\nu>0$, and any $\varepsilon>0$ small enough, there exist $\chi,c>0$ so that the following holds true: For all $\delta>0$, there exists an initial configuration $(u_0,B_0)\in W^{s,p}$ with and a time $t_\star =c|\log(\delta)|$, so that the solution to the MHD equations eq:mhd satisfies Furthermore, if $\nu\g

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.1
  • Proposition 2.1: NFVillringer*Theorem 1.2, Lemma 3.5
  • Lemma 3.1
  • Lemma 3.2: Properties of the Hodge Laplacian (Agmon_Douglis_Nirenberg_1964Seeley_1971Seeley_1972)
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: Hairer, Proposition 4.45
  • proof : Proof of Lemma \ref{['lemma:abstract semigroup lemma']}
  • ...and 1 more