Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids (original) (raw)

Lagrangian chaos and scalar advection in stochastic fluid mechanics

Journal of the European Mathematical Society, 2022

We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics subject to white-in-time, H sin space stochastic forcing in a periodic box. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive. Our main results are for the Navier-Stokes equations on T 2 and the hyper-viscous regularized Navier-Stokes equations on T 3 (at arbitrary Reynolds number and hyper-viscosity parameters), subject to forcing which is non-degenerate at high frequencies. As an application, we study statistically stationary solutions to the passive scalar advection-diffusion equation driven by these velocities and subjected to random sources. The chaotic Lagrangian dynamics are used to prove a version of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies that the scalar satisfies Yaglom's law of scalar turbulence-the analogue of the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems, namely the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.

Circulation and Energy Theorem Preserving Stochastic Fluids

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2019

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called thestochastic Euler–Poincaréandstochastic Navier–Stokes–Poincaréequations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems...

Stochastic variational principles for dissipative equations with advected quantities

2015

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...

Generalized stochastic flows and applications to incompressible viscous fluids

Bulletin des Sciences Mathématiques, 2013

We introduce a notion of generalized stochastic flows on manifolds, that extends to the viscous case the one defined by Brenier for perfect fluids. Their kinetic energy extends the classical kinetic energy to Brownian flows, defined as the L 2 norm of their drift. We prove that there exists a generalized flow which realizes the infimum of the kinetic energy among all generalized flows with prescribed initial and final configuration. We also construct generalized flows with prescribed drift and kinetic energy smaller than the L 2 norm of the drift.

Variational principles for stochastic fluid dynamics

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalizati...

Generalized stochastic Lagrangian paths for the Navier-Stokes equation

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2018

In the note added in proof of the seminal paper [Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebin and Marsden introduced the so-called correct Laplacian for the Navier-Stokes equation on a compact Riemannian manifold. In spirit of Brenier's generalized flows for the Euler equation, we introduce a class of semimartingales on a compact Riemannian manifold. We prove that these semimartingales are critical points to the corresponding kinetic energy if and only if its drift term solves weakly the Navier-Stokes equation defined with Ebin-Marsden's Laplacian. We also show that for the case of torus, classical solutions of the Navier-Stokes equation realize the minimum of the kinetic energy in a suitable class.

Stochastic Navier-Stokes Equations

Acta Applicandae Mathematicae, 1995

The purpose of this article is to survey some results related to the theory of stochastic Navier-Stokes equations (SNSE). The interest of SNSE arises from modelling turbulence. We begin to show how SNSE can be introduced intuitively from the random motion of particles. We then review briefly the deterministic theory and present the main core of existence theory for NSE. We also discuss uniqueness issues. We end up by showing how the splitting-up method provides a useful constructive approach to existence, and by presenting some extensions, like weakening assumptions or considering the special case of small initial data.

Stochastic variational derivations of the Navier-Stokes equation

Letters in Mathematical Physics, 1981

Stochastic calculus of variations in terms of sample paths is shown to be equivalent to ordinary calculus of variations in terms of local drift fields when applied to Markov processes. Tiffs equivalence enables us to derive the Navier-Stokes equation directly through a variational principle. 6 f ~(x)([]-KZ)~(x) d4x = 0, (2) a/~(x)(i~-k)qz(x) d4x = O, (3) where K and k are both mass parameters, ~ indicates the relativistic adjoint, [] = 3u3u , ~ = 3u7" and 3 ,u's are Dirac matrices. Recently we developed stochastic calculus of variations in terms of sample paths which may be understood as an extension of ordinary calculus of variations to stochastic processes [1-5].