论文标题
liouville- type定理用于稳定MHD和Hall-MHD方程$ \ r^2 \ times \ t $
Liouville--type Theorems for Steady MHD and Hall--MHD Equations in $\R^2 \times \T$
论文作者
论文摘要
在本文中，我们研究了具有周期性边界条件的平板中的三维固定不可压缩MHD和Hall-MHD系统的Liouville定理。我们表明，在假设的假设下，$（u^θ，b^θ）$或$（u^r，b^r）$是轴对称的，或$（ru^r，rb^r）$是有界的，对MHD或Hall-MHD的任何平滑界面解决方案与本地Dirichlet一起成长为dirichlet，随着任意功能的增长，必须稳定。这大大改善了\ cite [theorem 1.2] {pan2021liouville}的结果，其中$ \ mathbf {u} $的dirichlet积分被认为是有限的。由\ cite [bang-gui-wang-xie，2022，{\ it arxiv：2205.13259}] {bang20222222222222222liouvilletype}，我们的证明依赖于建立与我们的问题相关的圣徒的估计，并在当前的论文中及时显示了该工具的结果。 \ cite {bang2022liouvilletype} to MHD和Hall-MHD方程。为了实现这一目标，需要更复杂的估计来正确处理涉及$ \ mathbf {b} $的术语。
In this paper, we study the Liouville--type theorems for three--dimensional stationary incompressible MHD and Hall--MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that $(u^θ,b^θ)$ or $(u^r,b^r)$ is axisymmetric, or $(ru^r,rb^r)$ is bounded, any smooth bounded solution to the MHD or Hall--MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This hugely improves the result of \cite[Theorem 1.2]{pan2021Liouville}, where the Dirichlet integral of $\mathbf{u}$ is assumed to be finite. Motivated by \cite[Bang--Gui--Wang--Xie, 2022, {\it arXiv:2205.13259}]{bang2022Liouvilletype}, our proof relies on establishing Saint--Venant's estimates associated with our problem, and the result in the current paper extends that for stationary Navier--Stokes equations shown by \cite{bang2022Liouvilletype} to MHD and Hall--MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving $\mathbf{b}$ properly.
