Stochastic Variational Principles for Dissipative Equations with Advected Quantities

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.