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Stability and Sharp Decay Estimates for 3D MHD Equations with Only Vertical Dissipation Near a Background Magnetic Field

Suhua Lai, Jiahong Wu, Jianwen Zhang, Xiaokui Zhao

Abstract

This paper is concerned with the stability and large-time behavior of 3D incompressible MHD equations with only vertical dissipation near a background magnetic field. By making full use of the dissipation generated by the background magnetic field, we first establish the global stability of the solutions in $H^3$-norm. Then, the optimal decay rates of the solutions are obtained, which are consistent with the 2D classical heat equation. Moreover, some enhanced decay rates of $(u_1,b_1)$ are also achieved. In other words, the decay estimates of the second or third component of velocity/magnetic field coincide with those of 2D heat kernel, while the first component behaves like the 3D heat kernel. This is mainly due to the divergence-free condition and the anisotropic structure. The results obtained improve the previous ones due to Lin-Wu-Zhu [24,25].

Stability and Sharp Decay Estimates for 3D MHD Equations with Only Vertical Dissipation Near a Background Magnetic Field

Abstract

This paper is concerned with the stability and large-time behavior of 3D incompressible MHD equations with only vertical dissipation near a background magnetic field. By making full use of the dissipation generated by the background magnetic field, we first establish the global stability of the solutions in -norm. Then, the optimal decay rates of the solutions are obtained, which are consistent with the 2D classical heat equation. Moreover, some enhanced decay rates of are also achieved. In other words, the decay estimates of the second or third component of velocity/magnetic field coincide with those of 2D heat kernel, while the first component behaves like the 3D heat kernel. This is mainly due to the divergence-free condition and the anisotropic structure. The results obtained improve the previous ones due to Lin-Wu-Zhu [24,25].
Paper Structure (17 sections, 15 theorems, 440 equations)

This paper contains 17 sections, 15 theorems, 440 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some elementary inequalities
  4. Integral representation and the kernels
  5. Mathematical tools for the decay estimates
  6. Global Stability and Proof of Theorem \ref{['thm1.1']}
  7. A Priori Estimates (I)
  8. A Priori Estimates (II)
  9. Proof of Theorem \ref{['thm1.1']}
  10. Decay Rates and Proof of Theorem \ref{['thm1.2']}
  11. Estimates of nonlinear terms.
  12. Decay rates of $(u,b)$.
  13. Enhanced decay rates of $(u_1,b_1)$.
  14. Improved Decay Rates and Proof of Theorem \ref{['thm1.3']}
  15. Estimates of nonlinear terms.
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that $(u_0,b_0)\in H^3$ with ${\rm \nabla\cdot}\,u_0={\rm \nabla\cdot}\,b_0=0$. There exists an absolutely positive constant $\varepsilon>0$, depending only on $\mu$ and $\eta$, such that if then the problem PMHD has a unique global solution $(u, b)$ on $\mathbb{R}^3\times[0,\infty)$, satisfying

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 20 more