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Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations

Youyi Zhao

Abstract

We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by $m$, is large compared to the perturbations. It was proved by Jiang--Jiang that the highest-order derivatives of the velocity increase with $m$, and the convergence rate of the nonlinear system towards a linearized problem is of $m^{-1/2}$ in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy estimate. This strategy prevents the appearance of terms that grow with $m$, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to $m$ does not appear. Additionally, we use the vorticity estimate to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or $m$ approaches infinity. Notably, our analysis reveals that the convergence rate in $m$ is faster compared to the finding of Jiang--Jiang. Finally, a key contribution of our work is the identification of an integrable time-decay of the lower dissipation, which can replace the time-decay of lower energy in closing the highest-order energy estimate. This finding significantly relaxes the regularity requirements for the initial perturbations.

Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations

Abstract

We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by , is large compared to the perturbations. It was proved by Jiang--Jiang that the highest-order derivatives of the velocity increase with , and the convergence rate of the nonlinear system towards a linearized problem is of in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy estimate. This strategy prevents the appearance of terms that grow with , and thus the increasing behavior of the highest-order derivatives of the velocity with respect to does not appear. Additionally, we use the vorticity estimate to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or approaches infinity. Notably, our analysis reveals that the convergence rate in is faster compared to the finding of Jiang--Jiang. Finally, a key contribution of our work is the identification of an integrable time-decay of the lower dissipation, which can replace the time-decay of lower energy in closing the highest-order energy estimate. This finding significantly relaxes the regularity requirements for the initial perturbations.
Paper Structure (14 sections, 13 theorems, 127 equations)

This paper contains 14 sections, 13 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Reformulation in Lagrangian coordinates
  4. Main result
  5. Notations
  6. Global-in-time existence of large perturbation solutions
  7. Vanishing phenomena of the nonlinear interactions
  8. Proof of Theorem \ref{['thm:2022n']}
  9. Well-known estimates and inequalities
  10. Preliminary estimates
  11. Energy estimate
  12. Highest-order derivative energy evolution
  13. Stability estimate
  14. Proof of Theorem \ref{['thm:2022nn']}.

Key Result

Theorem 2.1

Let $\bar{M}:=\varpi\omega$ and the unit vector $\omega\in\mathbb{R}^3$ satisfy the Diophantine condition 202207041032. There exist positive constants $c_1\geqslant4$, $c_2>0$ and a sufficiently small constant $c_3\in(0,1]$ such that, for any initial data $(\eta^0, u^0)\in (H^{13}_{*}\cap\underline{ the MHD problem 202109221247nn admits a unique solution $(\eta, u, q)\in (H^{13}_{*}\cap\underline{

Theorems & Definitions (13)

  • Theorem 2.1
  • Proposition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 3 more