arXiv subjects

Xuemin Deng

Publications and source records attributed to Xuemin Deng.

2 recordsLinked to original sources

arXiv2024

On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state

A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain $Ω=\mathbb{T}^2\times\mathbb{R}$ with $\mathbb{T}^2=[0, 1]^2$. More precisely, each velocity equation lacks its own directional dissipation, and the magnetic equation lacks vertical dissipation in the MHD system. The key point to obtain the stability result is that we decompose the solution $(u,b)$ into the zeroth horizontal mode and the non-zeroth modes and complete the desired bound with the strong Poincaré type inequalities in the treatment of several nonlinear terms. Then we focus on the large-time behavior of the solution, where the non-zeroth modes decay exponentially in $H^2$, and the solution converges to its zeroth horizontal mode.

math.AP↗

arXiv2023

Stability and inviscid limit of the 3D anisotropic MHD system near a background magnetic field with mixed fractional partial dissipation

A main result of this paper establishes the global stability of the three-dimensional MHD equations near a background magnetic field with mixed fractional partial dissipation with $α, β\in(\frac{1}{2}, 1]$. Namely, the velocity equations involve dissipation $(Λ_1^{2α} + Λ_2^{2α}+σΛ_3^{2α})u$ with the case $σ=1$ and $σ=0$. The magnetic equations without partial magnetic diffusion $Λ_i^{2β} b_i $ but with the diffusion $(-Δ)^βb$, where $Λ_i^{s} (s>0)$ with $i=1, 2, 3$ are the directional fractional operators. Then we focus on the vanishing vertical kinematic viscosity coefficient limit of the MHD system with the case $σ=1$ to the case $σ=0$. The convergent result is obtained in the sense of $H^1$-norm.

math.AP↗