Stochastic Lagrangian perturbation of Lie transport and applications to fluids

Abstract

In this paper, we propose a novel stochastic Lagrangian formulation of dissipatively perturbed Lie transport, which is based on the statistical generalized Cauchy invariants equation. This formulation consists of, first, finding a convenient Lagrangian formulation of the Lie transport equation involving particle trajectories, for instance the backward generalized Cauchy invariants equation (a Lagrangian formulation for Lie-advected exact p-forms, which is the Hodge dual of a generalization of the Cauchy vorticity formula), and second, performing a stochastic perturbation of the velocity-driven particle trajectories by adding to them white noises. Finally, using Itô’s calculus, an ensemble average of the stochastically perturbated generalized Cauchy invariants equation allows us to obtain the Lie transport equation perturbed by a deterministic potentially dissipative term given by the sum of squares of some Lie derivative operators. A remarkable property of this equation is that it satisfies a statistical Kelvin–Helmholtz theorem of conservation of circulation and flux. These results are obtained on periodic Euclidean spaces as well as on smooth closed Riemannian manifolds. In particular, we recover and thus generalize the Constantin–Iyer results on the stochastic Lagrangian formulation of the incompressible Navier–Stokes equations to a larger class of (deterministic) dissipative PDEs. A first application of this stochastic Lagrangian formulation is the derivation of new Lagrangian formulations for non-ideal (dissipative) hydrodynamic and magnetohydrodynamic models on flat and curved spaces, and in particular we obtain stochastic-Lagrangian incompressible extended MHD equations. As a second application, we use this new stochastic Lagrangian formulation to study the local well-posedness, the non-resistive limit and the global existence of classical solutions for the non-ideal incompressible extended MHD on the flat torus. Global-in-time existence is proved for small magnetic Reynolds numbers, that is, either for small initial data or large resistivity.