随机受迫MHD的渐近分析
Asymptotic Analysis for Randomly Forced MHD
摘要：
本文考虑三维磁流体力学（MHD）方程在空间简并随机强迫作用下作为地球流体核中磁转湍流的模型。我们研究了Rossby数$ε$和磁性Rossby数$δ$的多参数奇异极限，并建立了：（i）极限随机驱动的有源标量方程（与$ε=δ=0$）具有唯一的遍历不变测度，和（ii）任何合适的序列的统计不变状态的完整的MHD系统弱收敛，作为$ε,δ\rightarrow 0$，极限方程的唯一不变测度。后一种收敛结果不需要任何关于$\varepsilon, δ$衰减的相对速率的条件。我们的极限方程的分析依赖于最近发展起来的无限维随机动力系统的亚椭圆理论。我们进行了详细的研究，非线性和随机项之间的相互作用，以证明一个Hörmander括号条件是满足的，这产生了一个收缩性能的极限方程在一个合适的Wasserstein度量。这种收缩性质减少了多参数极限下不变状态的收敛性，使其解在有限时刻收敛。但鉴于小参数系统与极限方程之间存在相空间不匹配，且由于问题的多参数性质，需要进一步分析以建立奇异极限。特别是，我们开发的方法，以解除压缩的极限方程的扩展相空间，包括速度和磁场。此外，在有限时间的解决方案的收敛性，我们利用概率修改的Grönwall不等式，依赖于一个微妙的停时参数。
英文摘要：
We consider the three-dimensional magnetohydrodynamics (MHD) equations in the presence of a spatially degenerate stochastic forcing as a model for magnetostrophic turbulence in the Earth's fluid core. We examine the multi-parameter singular limit of vanishing Rossby number $ε$ and magnetic Reynold's number $δ$, and establish that: (i) the limiting stochastically driven active scalar equation (with $ε=δ=0$) possesses a unique ergodic invariant measure, and (ii) any suitable sequence of statistically invariant states of the full MHD system converge weakly, as $ε,δ\rightarrow 0$, to the unique invariant measure of the limit equation. This latter convergence result does not require any conditions on the relative rates at which $\varepsilon, δ$ decay. Our analysis of the limit equation relies on a recently developed theory of hypo-ellipticity for infinite-dimensional stochastic dynamical systems. We carry out a detailed study of the interactions between the nonlinear and stochastic terms to demonstrate that a Hörmander bracket condition is satisfied, which yields a contraction property for the limit equation in a suitable Wasserstein metric. This contraction property reduces the convergence of invariant states in the multi-parameter limit to the convergence of solutions at finite times. However, in view of the phase space mismatch between the small parameter system and the limit equation, and due to the multi-parameter nature of the problem, further analysis is required to establish the singular limit. In particular, we develop methods to lift the contraction for the limit equation to the extended phase space, including the velocity and magnetic fields. Moreover, for the convergence of solutions at finite times we make use of a probabilistic modification of the Grönwall inequality, relying on a delicate stopping time argument.
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