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Weak solutions of Moffatt's magnetic relaxation equations

Jin Tan

TL;DR

This work proves global-in-time existence and uniqueness of weak solutions to Moffatt's magnetic relaxation equations for γ > γ_c with rough initial data, and establishes global well-posedness at the borderline γ_c under additional initial-data conditions, along with energy equality and regularity propagation. The authors develop a Lions-type compactness framework for linear active vector equations, a refined Beale–Kato–Majda-type criterion for γ>d/2, and a Besov/Littlewood–Paley analysis to manage rough data and nonlocal operators. Uniqueness is achieved via a Lagrangian reformulation that yields contraction estimates in appropriate Besov spaces, with separate treatments for integer and non-integer γ. The results illuminate the relaxation dynamics of magnetic fields while preserving topology and connecting to Euler equilibria, providing explicit bounds and a deeper understanding of the role of the regularization parameter.

Abstract

We investigate the global-in-time existence and uniqueness of weak solutions for a family of equations introduced by Moffatt to model magnetic relaxation. These equations are topology-preserving and admit all stationary solutions to the classical incompressible Euler equations as steady states. In the work of Beekie, Friedlander and Vicol, global regularity results have been established for initial magnetic field B0 $\in$ Hs(Td)(s > d/2+1) when the regularization parameter $γ$ in the equations satisfies $γ$ > $γ$c := d/2+1. Global regularity for $γ$ $\in$ [0, $γ$c] is left as an open problem, as well as the existence of weak solutions with rough initial data for any $γ$ $\ge$ 0. In this paper, we show that for any solenoidal magnetic field B0 $\in$ L2(Td) there exists a unique global weak solution when $γ$ > $γ$c. Moreover, the solution can propagate higher-order Sobolev regularity. These results hold true for the borderline case $γ$ = $γ$c only if B0 $\in$ L2+(Td).

Weak solutions of Moffatt's magnetic relaxation equations

TL;DR

This work proves global-in-time existence and uniqueness of weak solutions to Moffatt's magnetic relaxation equations for γ > γ_c with rough initial data, and establishes global well-posedness at the borderline γ_c under additional initial-data conditions, along with energy equality and regularity propagation. The authors develop a Lions-type compactness framework for linear active vector equations, a refined Beale–Kato–Majda-type criterion for γ>d/2, and a Besov/Littlewood–Paley analysis to manage rough data and nonlocal operators. Uniqueness is achieved via a Lagrangian reformulation that yields contraction estimates in appropriate Besov spaces, with separate treatments for integer and non-integer γ. The results illuminate the relaxation dynamics of magnetic fields while preserving topology and connecting to Euler equilibria, providing explicit bounds and a deeper understanding of the role of the regularization parameter.

Abstract

We investigate the global-in-time existence and uniqueness of weak solutions for a family of equations introduced by Moffatt to model magnetic relaxation. These equations are topology-preserving and admit all stationary solutions to the classical incompressible Euler equations as steady states. In the work of Beekie, Friedlander and Vicol, global regularity results have been established for initial magnetic field B0 Hs(Td)(s > d/2+1) when the regularization parameter in the equations satisfies > c := d/2+1. Global regularity for [0, c] is left as an open problem, as well as the existence of weak solutions with rough initial data for any 0. In this paper, we show that for any solenoidal magnetic field B0 L2(Td) there exists a unique global weak solution when > c. Moreover, the solution can propagate higher-order Sobolev regularity. These results hold true for the borderline case = c only if B0 L2+(Td).
Paper Structure (14 sections, 20 theorems, 190 equations)

This paper contains 14 sections, 20 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. A few remarks on the main results
  4. Global regularity for the borderline case TEXT
  5. A refined regularity criteria for $\gamma>d/2$
  6. The global regularity
  7. Proof of existence in Theorem \ref{['Th2-1']} and \ref{['Th2']}
  8. Proof of existence
  9. Proof of Lemma \ref{['Lemma3.1']}
  10. Proof of uniqueness
  11. The Lagrangian coordinates
  12. The case TEXT
  13. The case TEXT
  14. Function spaces and useful inequalities

Key Result

Theorem 1.2

Let $\gamma>\gamma_c.$ Given any initial data $B_0\in L^2({\mathbb T}^d)$ and $\hbox{\rm div}\, B_0=0,$ then the Cauchy problem MRE--eq-indata admits a unique global-in-time weak solution $(B, u)\in (L^\infty({\mathbb R}_+; L^2)\cap \mathcal{C}_{\rm loc}({\mathbb R}_+; L^2))\times L^\infty({\math

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6: BFV
  • Corollary 1.7
  • proof
  • Proposition 2.1
  • proof
  • ...and 21 more