带平流的耗散方程的随机变分原理
Stochastic variational principles for dissipative equations with advected quantities
摘要：
本文研究了位形空间为李群的带平流量的物质随机拉格朗日系统的对称约化。这样的变分原理产生确定性以及随机约束变分原理的耗散运动方程在空间表示。有限维情况下的一般理论。在无限维中，我们得到了偏微分方程和随机偏微分方程。当李群是，例如，一个同构群，一般的结果是不直接适用的，但设置和方法建议严格的证明，有效的无限维，导致类似的结果。我们应用这种技术的可压缩纳维-斯托克斯方程和磁流体力学的收费粘性可压缩流体。给出了一个随机Kelvin-Noether定理。我们得出，除其他外，经典的确定性耗散方程从纯粹的变分和随机的原则，没有任何上诉热力学。
英文摘要：
This paper presents symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group. Such variational principles yield deterministic as well as stochastic constrained variational principles for dissipative equations of motion in spatial representation. The general theory is presented for the finite dimensional situation. In infinite dimensions we obtain partial differential equations and stochastic partial differential equations. When the Lie group is, for example, a diffeomorphism group, the general result is not directly applicable but the setup and method suggest rigorous proofs valid in infinite dimensions which lead to similar results. We apply this technique to the compressible Navier-Stokes equation and to magnetohydrodynamics for charged viscous compressible fluids. A stochastic Kelvin-Noether theorem is presented. We derive, among others, the classical deterministic dissipative equations from purely variational and stochastic principles, without any appeal to thermodynamics.
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