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Global existence and stability of viscous Alfvén waves in the large-box limit for MHD systems

Li Xu, Jiahui Zhang

Abstract

This paper rigorously analyzes how the {\it large box limit} fundamentally alters the global existence theory and dynamics behavior of the incompressible magnetohydrodynamics (MHD) system with small viscosity/resistivity $(0<μ\ll 1)$ on periodic domains $Q_L=[-L,L]^3$, in presence of a strong background magnetic field. While the existence of global solutions (viscous Alfvén waves) on the whole space $\R^3$ was previously established in \cite{He-Xu-Yu}, such results cannot be expected for general finite periodic domains. We demonstrate that global solutions do exist on the torus $Q_L=[-L,L]^3$ precisely when the domain exceeds a size $L_μ>e^{\f1μ}$, providing the first quantitative characterization of the transition to infinite-domain-like behavior.

Global existence and stability of viscous Alfvén waves in the large-box limit for MHD systems

Abstract

This paper rigorously analyzes how the {\it large box limit} fundamentally alters the global existence theory and dynamics behavior of the incompressible magnetohydrodynamics (MHD) system with small viscosity/resistivity on periodic domains , in presence of a strong background magnetic field. While the existence of global solutions (viscous Alfvén waves) on the whole space was previously established in \cite{He-Xu-Yu}, such results cannot be expected for general finite periodic domains. We demonstrate that global solutions do exist on the torus precisely when the domain exceeds a size , providing the first quantitative characterization of the transition to infinite-domain-like behavior.

Paper Structure

This paper contains 37 sections, 29 theorems, 653 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and literature review
  3. Difficulties, key observations and strategies
  4. The characteristic geometries
  5. Main results
  6. Local well-posedness.
  7. Global well-posedness and dynamics
  8. Main A Priori Estimates and Proof of Theorem \ref{['thm:a1']}
  9. Ansatz for the method of continuity
  10. Preliminary estimates
  11. Analysis of the weights
  12. Technical lemmas involving characteristic hypersurfaces.
  13. Estimates of the weights involving boundaries.
  14. Div-curl Lemma
  15. Energy estimates for viscous linear equations
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $B_0 = (0,0,1)$, $N^* \in \mathbb{Z}_{\geq 7}$, $0<\mu\ll1$, $R\gg 1$ and $L\geq e^{\frac{1}{\mu}}$. There exists a universal constant $\varepsilon_0\in (0,1)$, independent of both the viscosity coefficient $\mu$ and the domain scale $L$, such that if the initial divergence-free vector fields $( then eq:MHD-initial admits a unique solution $(z_{+}(t,x),z_{-}(t,x))$ on $[0,\log L]\times Q_L$ sa

Theorems & Definitions (60)

  • Theorem 1.1: Local existence of the solution
  • Remark 1.1
  • Theorem 1.2: Global existence and dynamics
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 2.3: weighted Sobolev inequality
  • proof
  • ...and 50 more