STABILITY OF 3D MHD EQUATIONS WITH ANISOTROPIC DISSIPATION NEAR A HORIZONTAL BACKGROUND MAGNETIC FIELD

Yuning Wang; Limei Li

doi:10.11948/20250254

2027 Volume 17 Issue 1

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Yuning Wang, Limei Li. STABILITY OF 3D MHD EQUATIONS WITH ANISOTROPIC DISSIPATION NEAR A HORIZONTAL BACKGROUND MAGNETIC FIELD[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 39-53. doi: 10.11948/20250254

Citation:

Yuning Wang, Limei Li. STABILITY OF 3D MHD EQUATIONS WITH ANISOTROPIC DISSIPATION NEAR A HORIZONTAL BACKGROUND MAGNETIC FIELD[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 39-53. doi: 10.11948/20250254

STABILITY OF 3D MHD EQUATIONS WITH ANISOTROPIC DISSIPATION NEAR A HORIZONTAL BACKGROUND MAGNETIC FIELD

Yuning Wang^1,,
Limei Li^1, ,

1.
School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, Sichuan, China

Author Bio: Email: 20230801016@stu.sicnu.edu.cn(Y. Wang)
Corresponding author: Email: mathllm@163.com(L. Li)

Abstract

Abstract

Both physical experiments and mathematical theories indicate that the background magnetic field can stabilize the conducting fluid. Existing results (e.g., [7,20]) show that in the 3D case, either a background magnetic field in a single direction or a magnetic field that satisfies the Diophantine condition can provide a stabilizing mechanism for the fluid. In this paper, we examine a setting that lies between these two cases, specifically a horizontal background magnetic field $ B_0=(n_1, n_2, 0) $, $ n_1, n_2\ne 0 $. We investigate the 3D incompressible magnetohydrodynamics (MHD) equations with anisotropic dissipation: Fractional viscosity in the velocity field in the $ x_1 $-direction and fractional magnetic diffusion in the $ x_2 $-direction. Since dissipation acts only in one direction and is fractional, establishing global stability in $ H^3(\mathbb{T}^3) $ is nontrivial. We analyze the stabilizing effect of the horizontal magnetic field, combining both the velocity field and the weak dissipation mechanism of the magnetic field, and derive exponential decay in the horizontal directions, which is crucial for the stability proof.
- 3D MHD equations /
- anisotropic dissipation /
- horizontal background magnetic field /
- stability
MSC: 35B35, 35Q35, 35B40

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References

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