Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers

We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.


Introduction
Navier-Stokes equation describes the velocity of incompressible viscous fluids. We shall consider this equation with periodic boundary conditions, namely, if u = u(t, x), x ∈ T, denotes this velocity at time t, with T being the flat torus, it reads ∂u ∂t + (u · ∇)u = ν∆u − ∇p, div(u) = 0, (1.1) where ν is a positive constant (the viscosity coefficient) and t ∈ [0, T ]. The function p = p(t, x) denotes the pressure and is also an unknown in the equation.
Lagrange's point of view consists in describing positions of particles: it concerns the flows driven by the velocity fields. Lagrangian trajectories for the Euler equation (the case where there is no viscosity term) have been identified as minimisers of the kinetic energy defined on the space of diffeomorphisms by V. Arnold in [3]. In other words they are geodesics for a L 2 metric on such space of curves. This geometric approach to the Euler equation was developed in the fundamental paper by D. Ebin and J. Marsden ( [8]) and gave rise to many subsequent works. It is well known that the pressure in an incompressible fluid acts like a Lagrange multiplier and one can, indeed, derive the Euler equation from a variational principle with such a multiplier (c.f. for example [6,10,13]).
Navier-Stokes equation, being a dissipative physical system, does not correspond to analogous deterministic variational principles. Nevertheless, by replacing the Lagrangian flows by stochastic ones, we may still derive this equation from a stochastic variational principle associated with the energy. Then the velocity field is identified with the drift of the Lagrangian diffusion process, indeed the (conditional) expectation of its time derivative. Inspired by [12] and [16], such a stochastic variational principle was proved in [5]. More recently it was generalised in the context of Lie groups in [1] and many other dissipative systems can be derived using the same ideas (c.f. also [4]). Moreover stochastic partial differential equations were also obtained by variational principles, corresponding to random perturbations of the action functionals, in [4].
In this paper we show that it is possible to derive the Navier-Stokes equation from a (stochastic) variational principle with a Lagrange multiplier expressed in terms of the pressure. Although we consider here a flat case, the principle can be extended to general manifolds following the construction in [2].
Stochastic Noether's theorem was introduced in [14], [15] in the context of stochastic processes associated with the heat equation. A conserved quantity corresponds there to a martingale. In the spirit of this theorem as well as of [7], we present a result about conserved quantities associated to our stochastic variational principle. The main difference with the the one of [7] is that we consider here the Lagrangian motion as a stochastic flow (with respect to its initial values x) and in the notion of symmetry we integrate with respect to the variable x.
It should be stressed, here, that in our derivation of the Navier-Stokes equation no random perturbation is added. What we advocate is an approach where the presence of the Laplacian in Navier-Stokes equation is interpreted as the underlying presence of diffusion processes, used afterwards for studying (1.1) in probabilistic terms. In the last section we show how to derive a variational approach to a randomly perturbed Navier-Stokes equation ((4.3)).

A stochastic variational principle
On a fixed probability space (Ω, P, P ) endowed with an increasing filtration P t that satisfies the standard assumptions, we consider ξ to be a semimartingale with values in T, where M t is the martingale part in the decomposition of ξ t and D t ξ its drift (for simplicity we do not write the probability parameter ω ∈ Ω in the formulae).
Recall the definition of generalised derivative, that we denote by D t : for F defined in when such (a.s.) limit exists, and where E t denotes the conditional expectation with respect to P t . This definition justifies in particular the notation used in (2.1), since the generalised derivative corresponds to the seminartingale's drift. If W t is a P t -adapted Wiener process, we denote by g t (·) diffusions on the torus T d of the form with x ∈ T, dW t the Itô differential. The drift function v is assumed to be regular enough so that g t (·) are diffeomorphisms (cf. [11]). Note that we do not require, a priori, the vector field u to be divergence free.
For the particular cases F (t, x) = x and F (t, x) = v(t, x), we have, respectively, and, using Itô's formula, for p ∈ H and where E denotes expectation (with respect to P ).
We have the following Proof. Using the notation δS(g, p) = d dε ε=0 S(g ǫ , p ǫ ), the variation of the first term in the action gives: The notation < ·, · > stands here for the L 2 (T d ) scalar product. By Itô's formula, where the last term denotes the Itô contraction, is the differential of a martingale (whose expectation vanishes); therefore .h(t, g t (x)))dtdx+E ((dD T g T (x).dh(t, g T (x))−(dD 0 g 0 (x).dh(t, g 0 (x))) Concerning the second part of the action functional, we have Since ϕ is arbitrary we conclude from (2.9) that critical points of the action are volumepreserving diffeomorphisms (det ∇g t (x) = 1) and therefore have divergence-free drifts. It follows immediately that (2.10) = 0 so we only have to compute (2.11). We have, +p(t, g t )h j (t, g t )∂ i (∇g t ) −1 ij , and we are in the periodic case, Notice that we already concluded that det ∇g t = 1. On the other hand, Indeed, derivating the equality det ∇g t = 1, we get Also, derivating equality Returning to the expression of δS 1 , we have We conclude that δS = 0 in the class of variations considered, is equivalent to the condition for every test function h, together with the incompressibility condition det ∇g t (x) = 1. Applying the expression of D t D t g t (x) = D t v(t, g t (x)) (formula (2.4)) and reversing the time in the drift function we obtain the Navier-Stokes equation.
Remark 1. Comparing with [5,1], the variations we have used here are defined by shifts, since we do not have to work a priori in the class of measure-preserving flows.

On conserved quantities
In this section we present a Noether-type result where only transformations in space of the Lagrangian function are considered. A more general study of symmetries for equations obtained by stochastic variational principles will be considered in a forthcoming work.
Let us consider transformations of the following form: with η smooth, η(0, ·) = η(T, ·) = 0. We say that the Lagrangian used in the definition of the action functional (2.5), is invariant under the transformation associated with η if there exits a function G : [0, T ] × T → R such that, P -a.e., Theorem. If L is invariant under the transformation associated with η then, denoting L t = ∂ ∂t + (v ·∇)+ ν∆ where v(t, ·) = −u(T − t, ·) and u is the solution of the Navier-Stokes equation considered above, and, by the arguments in the proof of last section's Theorem, We know that (det ∇g t (x) − 1) = 0 on the critical points of the action functional, therefore the first term in the r.h.s. of last equality vanishes. The second one also vanishes after integration in x, as we consider periodic boundary conditions. We are therefore left with the equality, valid for g t critical of the action functional, (D t g t (x).D t η(t, g t (x))) + (∇p(t, g t (x)).η(t, g t (x))) dx = D t G(t, g t (x))dx On the other hand, recalling (2.8), we have D t (D t g t (x).η(t, g t (x))) = (D t D t g t (x).η(t, g t (x))) + (D t g t (x).D t η(t, g t (x))) (3.5) From the two last equalities and the fact that D t D t g t (·) = −∇p(t, g t (·)), we deduce that We have D t g t (x) = v(t, g t (x)) and (dD t g t (x).dη(t, g t (x)))dt = (∇v.∇η)(t, g t (x))dt. By the incompressibility condition, the flow g t (·) keeps the measure dx invariant (a.s.) and the result follows from the expression of the operator D t .
We remark the main difference with the finite-dimensional Noether's theorem of [14], [15], where there is no integration with respect to the initial values and martingales are obtained (since martingales are precisely characterised by the condition that their drift vanishes).

A stochastic Navier-Stokes equation
In this section we show that it is also possible to derive random perturbations of the Navier-Stokes equation by a stochastic variational principle.
Let ξ be a semimartingale with values in T of the form (2.1). We consider the random action functional Variations of ξ and p are taken as in Section 1, namely g t (·) → g ǫ t (·) = g t (·) + ǫh(t, g t (·)) p(t, ·) → p ǫ (t, ·) = p(t, ·) + ǫϕ(t, g t (·)) except that here we allow h and ϕ to be random. We want to characterise critical points ofS of the form dg t (x) = √ 2νdW t + v(t, g t (x))dt, g 0 (x) = x now considering the vector field v to be random. We proceed as in the theorem of section 1. The computations are analogous and we have to add, in the the variations of S, those of the second and third new terms of this functional. This gives, T 0 [(h(t, g t ). √ 2νdW t ) + (D t g t .(∇h(t, g t ).dW t )) − (h(t, g t ). √ 2νdW t )]dx that reduces to T 0 (v(t, g t (x)).(∇h(t, g t (x)).dW t )dx = (∇v(t, x).h(t, x)).dW t ) equality which holds P -almost surely. We therefore conclude that a diffusion process of the form dg t (x) = √ 2νdW t + v(t, g t (x))dt, g 0 (x) = x is critical for the action functionalS iff its (random) drift v(t, ·) = −u(T − t, ·) satisfies the following Navier-Stokes stochastic partial differential equation: du + (u.∇)u = √ 2ν∇u.dW t + ν∆u − ∇p, div u(t, ·) = 0 (4.3) with x ∈ T, t ∈ [0, T ]. This stochastic equation can be also regarded as a (Stratonovich) perturbation of Euler. Denoting by •dW the Stratonovich differential, it can be written as du + (u.∇)u = √ 2ν∇u • dW t − ∇p, div u(t, ·) = 0. (4.4)