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

2024/03/12 by Suhua Lai, Jiahong Wu, Lai, Suhua +5
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2403.07293

openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 H³-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 (u₁,b₁) 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].

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