二维电阻磁流体动力学方程在周期域上具有恒定磁场的周期剪切流的长波不稳定性
Long-wave instability of periodic shear flows with constant magnetic field for the 2D resistive MHD equations
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中文总结 AI 辅助
研究二维电阻磁流体动力学方程在特定周期域上围绕周期剪切流与恒定磁场的长波线性稳定性和不稳定性，通过建立条件、结合范式变换与渐近展开，得到相空间的不稳定和稳定子空间及解的指数增长或衰减特性。
中文摘要 AI 辅助
我们研究二维粘性、电阻磁流体动力学（MHD）方程在周期域${\mathbb T}_\alpha\times {\mathbb T}$上涡度 - 电流形式下围绕周期剪切流$(U(y),0)$与恒定背景磁场${\bf b}=({\rm b}_1,{\rm b}_2)$的长波线性稳定性和不稳定性。这是将科伦坡等人关于纳维 - 斯托克斯方程的近期论文扩展到MHD情形。我们建立了关于剪切流剖面$U(y)$的明确条件，涉及粘度$\nu$、电阻率$\eta$和背景磁场分量${\bf b}$，以在$\alpha\ll 1$区域获得线性长波稳定性和不稳定性。证明结合了非微扰范式变换以及关于参数$\alpha$的从零未受扰特征值分叉的特征值的尖锐渐近展开。作为动力学结果，我们得到相空间分裂为不稳定和稳定子空间，其上解在索伯列夫范数下指数增长或衰减。
英文摘要
We investigate the long-wave linear stability and instability of the two-dimensional viscous, resistive Magnetohydrodynamic (MHD) equations, in vorticity-current formulation, on the periodic domain ${\mathbb T}_α\times {\mathbb T} = \Big( {\mathbb R}/(\frac{2 π}α {\mathbb Z}) \times {\mathbb R}/(2 π{\mathbb Z}) \Big)$, around a periodic shear flow $(U(y),0)$ coupled with a constant background magnetic field ${\bf b}=({\rm b}_1,{\rm b}_2)$. It is a non-trivial extension of a recent paper for the Navier-Stokes equations by Colombo, Dolce, Montalto & Ventura to the MHD setting in the spirit of the classical works of Kolmogorov, Meshalkin, Sinai and Yudovich. We establish explicit conditions on the shear flow profile $U(y)$ involving the viscosity $ν$, the resistivity $η$ and the components of the background magnetic field ${\bf b}$ to obtain linear long-wave stability and instability in the regime $α\ll 1$. The proof combines a non-perturbative normal form transformation decoupling the zero Fourier mode from the non-zero modes with sharp asymptotic expansions of the eigenvalues bifurcating from the zero unperturbed eigenvalue with respect to the parameter $α$. As a dynamical consequence, we obtain a splitting of the phase space into unstable and stable subspaces, on which solutions grow or decay exponentially in Sobolev norm.
