Abstract
In this paper we extend the class of stochastic processes allowed to represent solutions of the Navier-Stokes equation on the two dimensional torus to certain non-Markovian processes which we call admissible. More precisely, using the variations of Ref. [3], we provide a criterion for the associated mean velocity field to solve this equation. Due to the fluctuations of the shift a new term of pressure appears which is of purely stochastic origin. We provide an alternative formulation of this least action principle by means of transformations of measure. Within this approach the action is a function of the law of the processes, while the variations are induced by some translations on the space of the divergence free vector fields. Due to the renormalization in the definition of the cylindrical Brownian motion, our action is only related to the relative entropy by an inequality. However we show that, if we cut the high frequency modes, this new approach provides a least action principle for the Navier-Stokes equation based on the relative entropy.
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1 Introduction
Let \((W_t)\) be a suitably renormalized Brownian motion on the space of vector fields on the two dimensional torus \(\mathbb {T}^2\) with a well chosen Sobolev regularity. In the case where \((u_t)\) is a deterministic vector field, it was shown that equations of the form
could model the Navier-Stokes flows (see for instance the review article [2] and references within). More precisely it was shown that \((u_t)\) solves the Navier-Stokes equation if and only if a certain associated action is stationary. Subsequently, models of the form
where considered in Ref. [1], together with a notion of generalized stochastic flows with fixed marginals. In these latter models, the shift \(\dot{v}_t(\omega )\) is allowed to be random: the drift changes from one realization of the noise to another which seems to fit accurately with the microscopic models of the Navier-Stokes equation one encounters in physics. In particular such processes are not necessarily Markovian.
In the case of (1.2) there is no reason why we should hope \(\dot{v}(\omega )\) to solve the Navier-Stokes equation for any \(\omega \) \(a.s.\), and we should focus on the mean velocity field
where \(\eta \) is the underlying probability on the canonical path space, and where \(T_x\mathbb {T}^2\) is the tangent space at \(x\).
We extend here the criterion of Ref. [2] from equations with the form (1.1) to equations of type (1.2) for a wide class of stochastic drifts. Namely we focus on drifts \(v\) associated with a probability \(\eta \) with finite entropy with respect to the law \(\mu \) of the renormalized Brownian motion on the corresponding path space. We exhibit a class of such drifts (they will be called admissible) whose mean velocity field solves the Navier-Stokes equation if and only if the associated action, which will be noted \(\mathcal {S}(\eta | \mu )\), is critical. We then prove that this notion naturally extends the variational principle of Ref. [2]. One of the aspects of this model is to allow that the fluctuations of the drift itself may contribute to the pressure. Then we provide an alternative formulation to the least action principle by means of transformation of measure. However in this case, due to the renormalization involved in the definition of the cylindrical Brownian motion, our action for a process with law \(\eta \) is only related to the corresponding relative entropy
by an inequality. Nevertheless, by introducing a cut-off, the action \(\mathcal {S}(\nu | \eta )\) becomes proportional to the relative entropy, and by cutting the high modes, we provide a least action principle to the Navier-Stokes equation by means of the relative entropy.
The structure of this paper is the following. In Sect. 2 we introduce the general framework as well as the main notations of the paper. In Sect. 3 we provide a characterization of solutions of the Navier-Stokes equation as critical flows of the action. In Sect. 4 this criterion is proved to extend those of Refs. [2, 3]. In Sect. 5 we introduce a cut-off in order to transform variations of the action in variations of the entropy. (Sect. 6).
2 Preliminaries and Notations
2.1 A Basis of Vector Fields on the Two Dimensional Torus
Let \(M:=\mathbb {T}^2\) be the set of pairs \((\theta _1,\theta _2)\) of real numbers modulo \(2 \pi \), and denote \(m_{\mathbb {T}}=\frac{\lambda ^L\otimes \lambda ^L}{4\pi ^2}\) where \(\lambda ^L\) is the Lebesgue measure on \([0,2\pi ]\). Integration will often be noted \(dx\) instead of \(m_{\mathbb {T}}(dx)\). A basis of the tangent space \(T_xM\) at \(x=(\theta _1,\theta _2)\in M\) is given by \((\partial _{i} |_x):= (\frac{\partial }{\partial \theta _i} |_{x=(\theta _1,\theta _2)})\). We define a scalar product \(\langle ., .\rangle _{T_xM}\) on each \(T_xM\) by setting \(\langle \partial _i |_x,\partial _j |_x\rangle _{T_xM} =\delta ^{i,j}\) where \(\delta ^{i,j} =1\) if \(i=j\) and \(0\) if \(i\ne j\). When there is no ambiguity, we will sometimes note \(X.Y\) instead of \(\langle X,Y\rangle _{T_xM}\) for \(X,Y\in T_xM\). If \(\mathcal {X}(M)\) consists of sections of \(TM\), \(\mathcal {X}(M)=\{X:M\rightarrow T(M)\}\), and considering its \(L^2\) equivalence class, we set
which is a separable Hilbert space with the product
An Hilbertian basis of \(\mathcal {G}\) is given by a subset \((e_\alpha )_{\alpha =1}^\infty \), whose definition is the following. Let \(k:\alpha \in \mathbb {N} / \left\{ 0\right\} \rightarrow k(\alpha ):= (k_1(\alpha ), k_2(\alpha )) \in (\mathbb {Z} \times \mathbb {Z}) / \left\{ (0,0)\right\} \) be a bijection such that \(|k(\alpha )|:= \sqrt{k_1^2(\alpha )+ k_2(\alpha )^2} \uparrow \infty \); we set
where
and where, for \(k=(k_1,k_2)\in \mathbb {Z} \times \mathbb {Z}\) and \(x=(\theta _1,\theta _2) \in M\), \(k.x := k_1\theta _1 + k_2\theta _2\). Any \(X\in \mathcal {G}\) can be written
where \(X_j(x) =\sum _{\alpha } \langle X,e_\alpha \rangle _{\mathcal {G}} a^{\alpha ,j}(x) \). Let \(Y(x) := \sum _j Y_j(x) \partial _j|_x\) be another vector field: it is straightforward to check that we also have
We recall the following formulae
and
2.2 The Group of the Volume Preserving Homeomorphisms
Let \(G\) be the group of the homeomorphisms of \(M\) which leaves \(m_{\mathbb {T}}\) invariant
We note \(e\) the identity on \(G\) and \(\phi .\psi \) the group operation of \(\phi ,\psi \in G\) (given by the composition of the two maps). We recall [6] that the subset of \(G\) consisting of maps which are, together with their inverses, in the Sobolev class \(H^r\), for \(r>2\) is a Hilbert manifold and a topological group. It is not, strictly speaking, a Lie algebra since left translation is not smooth. \(T_e G\) is given by the set of the vector fields \(X:x\in M \rightarrow X_x\in T_xM\) such that \(\hbox {div} (X)=0\). Let \(X\in T_e G\), and let
be a smooth curve on \(G\) to which \(X\) is tangent. We recall that, by setting \(\widehat{c} : (t,x) \in \mathbb {R} \times M \rightarrow c_t(x) \in M \), the value of \(X\) at \(x\in M\) is given by
Furthermore \(X\) can be uniquely extended to a right invariant vector field \(\widehat{X}\) on \(G\) by setting
where \(\widehat{X}_\phi \) is given by
i.e. \(\widehat{X}_\phi \) is tangent to the curve \(c^\phi :t\in \mathbb {R} \rightarrow c_t.\phi \in G\). In particular for any smooth \(f\) on \(M\) and \(x\in M\) denote \(f^x \) the map \( \phi \in G \rightarrow f^x(\phi ) := f(\phi (x)) \in \mathbb {R} \). Then \(f^x\) is smooth on \(G\) and we have
In the sequel we will simply write \(X\) instead of \(\widehat{X}\) since it will be clear from the context whether we consider \(X\) as an element of the tangent space, or as a right-invariant vector field on \(G\). In order to kill the noise in the higher modes and to control the integrability of the derivatives, we introduce the following Sobolev spaces \((\mathcal {G}_\lambda )_{\lambda >1}\) and the associated abstract Wiener spaces \((W, H_\lambda , \mu _\lambda )\).
2.3 Sobolev Vector Fields
To any positive real number \(\lambda > 1\) we associate a sequence \((\lambda _\alpha )_{\alpha \in \mathbb {N}}\) defined by
where \([.]\) is the floor function and where \(K(\lambda )\) is chosen so that
Such a \(K(\lambda )\) exists from standard results on Riemann series since \(\lambda >1\), and we have \(K(\lambda ) \uparrow \infty \) as \(\lambda \downarrow 1\). For \(\lambda >1\), let \(S_\lambda \) be the positive, definite, trace class operator defined by
and let
which is an Hilbert space for the scalar product \(\langle .,.\rangle _{\mathcal {G}_\lambda }\) characterized by
A natural Hilbertian basis of \(\mathcal {G}_\lambda \) is given by \((H_\alpha ^\lambda )_{\alpha =1}^\infty \) where
We set
so that
and
Since \(\sqrt{S}_\lambda \) is Hilbert-Schmidt, it is well known that \(|.|_{\mathcal {G}}\) is a measurable semi-norm on the Hilbert space \(\mathcal {G}_\lambda \) (see [9]). In particular \((\mathcal {G}_\lambda , \mathcal {G})\) is an abstract Wiener space [9, 12], which allows to regard the cylindrical Brownian motion below as a Brownian sheet (note that we could have defined a Wiener measure directly on the Wiener space \((\mathcal {G}_\lambda , \mathcal {G})\), but we won’t use this in the sequel since we are interested in the path space).
2.4 Associated Wiener Spaces
The space
is an Hilbert space whose product will be noted \(\langle .,.\rangle _\lambda \). On the other hand the space
is a separable Banach space for the uniform convergence norm. We denote by \(i_\lambda \) the injection of \(H_\lambda \) in \(W\). Since for \(\lambda >1\) \(|.|_\mathcal {G}\) is a measurable semi-norm on \(\mathcal {G}_\lambda \), it is a classical result on Wiener spaces that \((i_\lambda , W, H_\lambda )\) is also an abstract Wiener space. If \(\mu _\lambda \) is the standard Wiener measure on \(W\) for the A.W.S. \((W, H_\lambda , i_\lambda )\), we recall that under this probability the coordinate process \(t\rightarrow W_t(\omega ) = \omega (t) \in \mathcal {G}\) is an abstract Brownian motion with respect to its own filtration \( \left( \mathcal {F}_t \right) \) (see for instance [10, 12]). From the Itô Nisio theorem, we have \(\mu _\lambda -\)a.s.
with \(W^\alpha _t := \widehat{\delta }H_\alpha (W_t) \), and where \(\left\{ \widehat{\delta }(X), X \in \mathcal {G}_\lambda \right\} \) is the isonormal Gaussian process on \(\mathcal {G}_\lambda \). We recall that under \(\mu _\lambda \), \(\left\{ \widehat{\delta }(X)(W_s), X\in \mathcal {G}_ \lambda ,s\in [0,1] \right\} \) is a Gaussian process with covariance
so that \((W^\alpha _.)\) is a family of real valued independent Brownian motions under \(\mu _\lambda \). Under \(\mu _{\lambda }\), the coordinate process \(t\rightarrow W_t\) is called the cylindrical Brownian motion. The difference with respect to the case where the state space is finite dimensional is that it is a renormalized sum of independent Brownian motions, the renormalization appearing in (2.3). For a measure \(\eta \ll \mu _\lambda \) and a \(u\in L^0_a(\eta ,H_\lambda )\), the stochastic integral \(\delta ^W u := \int _0^1 \dot{u}_s dW_s\) is well defined as an abstract stochastic integral [10, 12]. Let \(\eta \) be a probability which is absolutely continuous with respect to \(\mu _\lambda \). Then there is a unique \(v\in L^0_a(\eta , H_\lambda )\) such that \(\eta -a.s.\)
Moreover \(W^\eta := I_W - v\) is a \(\left( \mathcal {F}_t\right) \)-Brownian motion on \((W,\mathcal {F},\eta )\). We call \(v\) the velocity field associated to \(\eta \). The famous formula of [7] (which in fact holds in a more general framework: [10, 12]) reads
where
is the relative entropy of \(\eta \) with respect to \(\mu _\lambda \). Note that since \(\mathcal {G}_\lambda \subset \mathcal {G} \subset T_eG\) it makes sense to consider \((Xf)(\phi )\) for \(\phi \in G\), for \(f \) smooth on \(G\) and for \(X \in \mathcal {G}_\lambda \) or \(X\in \mathcal {G}\).
3 Navier-Stokes Flows with Stochastic Drifts
Henceforth and until the end of Sect. 5 we assume that the renormalization sequence is fixed for a \(\lambda \ge 2\), and we drop the indices \(\lambda \) of the notations except for \(\mathcal {G}_\lambda \).
3.1 Constraints on the Kinematics: Regular and Admissible Flows
Definition 1
A probability \(\eta \) which is absolutely continuous with respect to \(\mu \) with finite entropy (\(\mathcal {H}(\eta | \mu ) <\infty \)) is called a regular flow if \( u \in C^1([0,1]\times M)\) and \(dt\)- \(a.s.\) \(\partial _t u \in \mathcal {G} \), where \(u(t,x):=E_\eta \left[ \dot{v}_t(x)\right] \), and where \(v:=\int _0^. \dot{v}_s ds \) is the velocity field of \(\eta \) (see (2.5)). We call \(u\) the mean velocity field of \(\eta \). Moreover we say that a regular flow is admissible if there is a \(C^1 ([0,1]\times M)\) mapping \(p^\star : [0,1]\times M \rightarrow \mathbb {R}\) such that
i.e. for \(i,j\in \mathbb {N} \cap [1,d]\)
where \((\dot{v}^j_t(x))\) denotes the \(j\)th (random) component of \((\dot{v}_t^j)\) at \(x\) i.e. it is such that \(\dot{v}_t(x)= \sum _j \dot{v}_t^j(x) \partial _j |_x\), and where \(u_t^j(x) := E_\eta [\dot{v}_t^j(x)]\).
3.2 Constraints on the Dynamics: Critical Flows
Definition 2
Let \(\eta \) be a regular flow whose velocity field is denoted by \(v^\eta \) (see (2.5)). For any \(k\in C^1([0,1];\mathcal {G})\) we set
The probability \(\eta \) is said to be critical if and only if for any \(k\in C_{0}^1([0,1],\mathcal {G})\)
where
The dynamic of the mean velocity field of a critical flow is given by the following theorem
Theorem 1
Let \(\eta \) be a regular flow with a velocity field \(v\) and a mean velocity field \(u\in \mathcal {G}_\lambda \). Then \(\eta \) is critical (Definition 2) if and only if there is a function \(\widehat{p}(t,x)\) such that
In other words, let
Then \(u\) solves, in the weak \(L^2\) sense, the following equation :
Proof
For any \(k\in {C_0}^1([0,1];\mathcal {G})\) we have \(k(0,.)=k(1,.)=0\), so that an integration by parts yields
from which we obtain (3.8). Since
we obtain (3.10) from (3.8). \(\square \)
3.3 Navier-Stokes Flows
Definition 3
A regular flow \(\eta \) (see Definition 1) is a Navier-Stokes flow if its mean velocity field \(u\) solves the Navier-Stokes equation, i.e. if and only if there is a function \(p:[0,1]\times M \rightarrow \mathbb {R}\) which is such that \(u\) solves, in the weak \(L^2\) sense, the Navier-Stokes equation
we have:
Corollary 1
An admissible flow is a Navier-Stokes flow if and only if it is critical.
Proof
Let \(\eta \) be an admissible flow. We recall that by definition there exists a mapping \(p^\star \) such that
where \(v:=\int _0^.\dot{v}_s ds\) is the velocity field of \(\eta \) (see (2.5)). We also recall that
The idea is to apply Theorem 1 and to set
We have
Indeed (repeated indices are summed over) we have
Assumption (3.12) then yields \(\beta ^i(t,x) = \partial _i p^\star \) i.e.
\(\square \)
Remark 1
Note that by this proof, for critical flows, \(p^\star \) appears as a part of the pressure which is originated from the stochastic model. Specifically it expresses the fluctuations of the drift itself. Indeed by (3.13) and (3.9) for an admissible flow \(\eta \) we have
where \(p^\star \) is the function associated to the admissible flow \(\eta \) by formula (3.7).
4 Interpretation of Critical Flows by Means of the Stochastic Exponential
In this section we prove that the quantities \(L_k\mathcal {S}(\eta | \mu )\) defined in Definition 2 can still be interpreted in terms of certain variations along deterministic paths which extend those of Ref. [3].
4.1 The Stochastic Exponential
Let \(C_G= C_e\left( [0,1],G \right) \) be the space of continuous paths starting from \(e\) and with values in \(G\). The coordinate function \((t,\gamma )\in [0,1]\times C_G \rightarrow \gamma _t(\omega )\) generates a filtration \(\left( \mathcal {F}^G_t\right) \) and we denote \(\mathcal {F}^G:= \mathcal {F}^G_1\).
Proposition 1
The equation
has a continuous strong solution on the space \(\left( W, \mathcal {F}_.^{W}, \mu \right) \) with the canonical Brownian \(t \rightarrow W_t\in \mathcal {G}\). We note \(g\) this solution. By this we mean that for \(\mu {-}a.s.\) \(g \in C_G\) and, for any smooth \(f\) on \(G\),
where \(\circ \) denotes the Stratonovich integral.
Proof
See [11]. \(\square \)
Girsanov theorem on \((W, H, \mu )\) implies the following:
Proposition 2
Let \(\eta \) be a probability which is absolutely continuous with respect to \(\mu \) whose velocity field is noted \(v\), and set \(\widetilde{W}:= I_W-v\). Then \(( g , \widetilde{W})\) is a solution of
on \((W ,\mathcal {F}_., \eta )\).
Proof
We have
Since \( \widetilde{W}_\star \eta = \mu \), \(\widetilde{W}_.^\alpha := \widehat{\delta }(H_\alpha ) (\widetilde{W}_.)\) are independent Brownian motions on \((W, H, \eta )\), by Itô’s formula we have, \(\eta -a.s.\),
i.e.
\(\square \)
Proposition 3
Let \(\eta \) be a probability absolutely continuous with respect to \(\mu \), \(v:=\int _0^. \dot{v}_s ds\) the associated velocity field, \(\widetilde{W} = I_W- v\) and \(\widetilde{W}^\alpha _.= \widehat{\delta }(H_\alpha )(\widetilde{W}_.)\). For any smooth function \(f\) on \([0,1]\times M\) we have \(\eta {-}a.s.\)
and \(\eta {-}a.s.\)
Proof
Let \(x\in M\), \(f\in C^\infty (M)\). The main part of the proof will be to prove that
To see this recall that \(f^x: \phi \in G \rightarrow f(\phi (x))\in \mathbb {R}\). We have
so that by iterating (4.20) we obtain
On the other hand
Indeed by using the fact that for any \(\alpha \) the vector field \(H^\alpha \) is divergence free together with (2.4) we obtain
Finally by putting together (4.21) and (4.22) we get (4.19) which yields
On the other hand by the Girsanov theorem, \((\widetilde{W}_t)\) is a \((\mathcal {F}_t)\)-Brownian motion on \((W, \eta )\) so that (4.18) follows from (4.17). \(\square \)
4.2 Perturbations of the Energy Along Deterministic Paths
For \(k\in C^0([0,1], \mathcal {G}_\lambda )\), \(k:=\int _0^. \dot{k}_s ds\), we define \(e(k)\) to be the solution of the ordinary differential equation on \(G\)
i.e. for any smooth \(F:G\rightarrow \mathbb R\),
Note that \(e_.(0_{H})=e\) i.e. the exponential of the function which is constant and equal to \(0_H\) is constant and equal to \(e\). We denote by \((e^i_t(k))\) the ith component of \((e_t(k))\) in the canonical chart.
Proposition 4
If \(\eta \) is a probability of finite entropy with respect to \(\mu \), for any \(k\in C_0^1([0,1], \mathcal {G}_\lambda )\) we have
where \(L_k\mathcal {S}(\eta | \mu )\) has been defined in Definition 2 and where \(D^\eta e_t(\epsilon k).g_t(x)\) is defined a.e. by
Proof
By (4.18) of Proposition 3 we first obtain
On the other hand let \(x\in M\) and denote by \(f\) a smooth function on \(M\). Considering \(F:=f^x\) in (4.23) we have
Since \(e_.(0_{H})(x)=e(x)=x\), we get :
so that
By (4.26) and (4.27) we obtain
For convenience of notations we denote by \(A\) the right hand term of (4.24). By first differentiating the product, then by applying (4.26) at \(\epsilon =0\), then by applying (4.28), and finally by using that \(g_t\) preserves the measure we obtain
which proves (4.24). \(\square \)
5 Variations of the Energy Along Translations
Let \(\eta \) be a probability which is absolutely continuous with respect to \(\mu \) (as mentioned in the beginning of Sect. 3 we work with a fixed \(\lambda \ge 2\)) and with velocity field \(v^\eta \). The stochastic action of \(\eta \) is defined by
The motivation for this definition is that, by taking \(\epsilon =0\) in (4.26) and using the fact that \(g_t\) preserves the measure, we also have
with the notations of Proposition 4. By (2.6), \(\mathcal {G}_\lambda \subset \mathcal {G}\) implies that whenever the entropy is finite we have
as well. More accurately, by a classical result on abstract Wiener spaces together with (2.6), there exists a \(c>0\) such that for any \(\eta \ll \mu \)
In this section we introduce another kind of variations for the functional \(\mathcal {S}(\eta | \mu )\), namely we study its variations along translations, These variations are generally different from those introduced above; however, when restricted to admissible flows, they are the same. We also investigate similar variations for the relative entropy. Proposition 5 computes the values of the variations of these quantities along deterministic translations.
Proposition 5
Let \(\eta \) be a probability absolutely continuous with respect to \(\mu \) with velocity field \(v^\eta \) and mean velocity \(u_s (x):= E_\eta [\dot{v}^\eta _s (x)]\). If \(\mathcal {S}(\eta | \mu ) <\infty \) we have,
and if \(\mathcal {H}(\eta | \mu ) <\infty \) we have
where \(\tau _h\eta \) is the image measure of \(\eta \) by the mapping \(\tau _h\) defined by
Proof
A straightforward application of the Cameron-Martin theorem shows that for any \(h:=\int _0^. \dot{h}_sds \in H\), the velocity field field \(v^{\tau _h\eta }\) of \(\tau _h\eta \) is given by
Hence by (5.29) we have
which yields (5.30). Similarly (5.31) follows by (2.6) and (5.32). \(\square \)
Let
and let \(\Pi \) be the Helmoltz projection on divergence free vector fields. We set
so that it makes sense to say that any \(h\in \mathcal {K}_{0}^\eta \) is associated to a \(k\in C^n_0([0,1],\mathcal {G}_{\lambda +2})\). For \(n\) sufficiently large we have \(\mathcal {K}^\eta \subset H\).
Proposition 6
Let \(\eta \) be a smooth flow whose mean velocity field is given by \(u\). Then \(u\) solves the Navier-Stokes equation if and only if for any \(h\in \mathcal {K}_0^\eta \)
Proof
By Proposition 5, and by definition of \(\Pi \), for any \(h\) (which is associated to \(k\)) we have
and, since \(k(0,.)=k(1,.)=0\), the result directly follows from an integrating by parts. \(\square \)
We now relate these variations to the ones of Sect. 4. Namely we prove that, for admissible flows, these variations of measure by quasi-invariant transformations yield exactly the same variations as the exponential variations of Sect. 4.
Proposition 7
Let \(\eta \) be an admissible flow. Then, for any \(h\in \mathcal {K}_{0}^\eta \) (see (5.34)) associated with a \(k\in C^n_{0}([0,1],\mathcal {G}_{\lambda +2})\) (see (5.33)) we have
Proof
Let \(u\) be the mean velocity field of \(\eta \). Since \(\eta \) is admissible we have, by (3.14)
Hence, using (5.36),
which is exactly the definition of \( L_k\mathcal {S}(\eta |\mu )\) (Definition 2). \(\square \)
6 Generalized Flows with a Cut-off
In Sect. 5 we have seen that in the infinite dimensional case, the relative entropy was generally not proportional to the action \(\mathcal {S}(\cdot | \mu )\). The reason is that the renormalization procedure gives a different weight to the different modes: hard modes have a weaker weight in the energy than in the relative entropy. However if instead of renormalizing we introduce a cutoff, and rescale the noise accordingly, \(\mathcal {S}(\cdot | \mu )\) becomes proportional to the relative entropy \(\mathcal {H}(\cdot | \mu )\). Within this framework, we investigate the existence of generalized flows with a given marginal.
6.1 General Framework for a Cut-off at Scale \(n\)
We recall that \((e_\alpha )\) denotes the Hilbertian basis of \(\mathcal {G}\) of Sect. 2. By induction we define \((I_l)_{l=1}^\infty \) by \(I_1=1\) and
For \(N\in \mathbb {N}, N >1\) we set
We define \(\mathcal {G}^n = Vect( e_1,\ldots ,e_n) \subset \mathcal {G}\) and recall that we work under the hypothesis
The cut-off has been chosen so that \(\exists S(N)\) such that
where \(\mathcal {S}(N) \uparrow \infty \). We note
and \(\langle .,.\rangle _{H_n}\) the associated scalar product. We set \(W_n:= C([0,1], \mathcal {G}^n)\) endowed with the norm of uniform convergence, and \(\mu _n\) the Wiener measure on \((W_n,H_n)\) with a parameter
\(t\rightarrow W_t\) is the coordinate process. Define \(g^n\) to be the solution of
on the Wiener space \((W_n,H_n,\mu _n)\) i.e., satisfying, for every smooth \(f\),
where \(W^\alpha := \langle W_t,e_\alpha \rangle _{G_n}\). We are now working with the Wiener measure with parameter \(\sigma (N)\). Still by the Girsanov theorem, for any \(\eta \ll \mu _n\) there is a unique \(v\in L^0(\eta , H_n)\) such that
and \(\widetilde{W}:= I_W -\sigma (N) v \) is a Brownian motion with parameter \(\sigma (N)\) under \(\eta \). We call \(v\) the velocity field of \(\eta \). Furthermore, Föllmer’s formula (c.f. [8]) then reads
Hence \((g, \widetilde{W})\) is a solution to
on the probability space \((W_n,\eta )\) for the filtration generated by the coordinate process \(t\rightarrow W_t\), i.e., for every smooth \(f\),
Within this framework, by an admissible flow we mean a probability \(\eta \) of finite entropy with respect to \(\eta \) satisfying the same conditions as in Definition 1 with \(\mu _n\) (resp. \(\mathcal {G}_n\)) instead of \(\mu \) (resp. of \(\mathcal {G}\)).
6.2 Variations of the Action
We now define the action for the cutoff \(n\in \mathbb {N}\) by
Therefore
Similarly to Proposition 7 we note
where \(\pi _n\) is the orthogonal projection \(\pi _n :\mathcal {G} \rightarrow \mathcal {G}_n\) and we say that a \(h\in \mathcal {K}^\eta _0(n)\) is associated to a \(k \in C^1_{0}([0,1],{\mathcal {G}^n})\).
Proposition 8
For any smooth flow \(\eta \)
solves the Navier-Stokes equation if and only if for any \(h\in \mathcal {K}^\eta _0(n)\) we have
for any \(h\) associated with a \(k\in C^1_{0}([0,1],{\mathcal {G}^n})\). Moreover whenever \(\eta \) is an admissible flow, and \(h\in \mathcal {K}^\eta _0(n)\) is associated to \(k\in C^1_{0}([0,1],{\mathcal {G}^n})\) we have
where the notations are those of Sect. 4.
Proof
The first part of the proof is the same as in Proposition 6. We now prove the second part of the claim which is similar to Proposition 7. As in the first subsection we have
Therefore by setting
and using the fact \(g_t\) preserves the measure we get
If \(\eta \) is assumed to be admissible, then similarly to the proof of Proposition 7 we obtain
\(\square \)
Concerning existence of Lagrangian Navier-Stokes flows with a cut-off they have been shown to exist in Ref. [4] for deterministic \(L^2\) drifts. Examples of random solutions of Navier-Stokes equations were constructed in Ref. [5] but we did not prove existence of the corresponding flows.
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Acknowledgments
We thank the anonymous referee for a careful reading of the manuscript and valuable remarks. We acknowledge support from the FCT project PTDC/MAT/120354/2010.
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Cruzeiro, A.B., Lassalle, R. (2014). On the Stochastic Least Action Principle for the Navier-Stokes Equation. In: Crisan, D., Hambly, B., Zariphopoulou, T. (eds) Stochastic Analysis and Applications 2014. Springer Proceedings in Mathematics & Statistics, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-11292-3_6
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