A Consistent Stochastic Large-Scale Representation of the Navier–Stokes Equations

Abstract

In this paper we analyze the theoretical properties of a stochastic representation of the incompressible Navier–Stokes equations defined in the framework of the modeling under location uncertainty (LU). This setup built from a stochastic version of the Reynolds transport theorem incorporates a so-called transport noise and involves several specific additional features such as a large scale diffusion term, akin to classical subgrid models, and a modified advection term arising from the spatial inhomogeneity of the small-scale velocity components. This formalism has been numerically evaluated in a series of studies with a particular interest on geophysical flows approximations and data assimilation. In this work we focus more specifically on its theoretical analysis. We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier–Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier–Stokes equation. similarly to the grid convergence property of well established large-eddies simulation strategies, this result allows us to give some guarantee on the interpretation of the LU Navier–Stokes equations as a consistent large-scale model of the deterministic Navier–Stokes equation.

A Mathematical Tools

In this part, we recall some results used to prove theorem 3.1.

1.1 A.1 Compact Embedding Result

The following results are proved in Flandoli and Gatarek [18] and are variations of the compactness theorems of Lions [28], Ch. I, Sect. 5, and Temam [45], Sect. 13.3.

Lemma A.1

Let $B_{0} \subset B \subset B_{1}$ be Banach spaces with compact embedding of $B_0$ in B. Let $p\in (1, \infty )$ and $\alpha \in (0,1)$ be given. Then the embedding

$$\begin{aligned} L^{p}([0,T]; B_0) \cap W^{\alpha , p}([0,T ]; B_{1}) \hookrightarrow L^{p}([0,T]; B) \quad \text {is compact.} \end{aligned}$$

Lemma A.2

Let $B_0 \subset B$ two Banach spaces with compact embedding. Let $\alpha \in (0,1)$ and $p>1$ be given such that $\alpha p >1$, then the injection

$$\begin{aligned} W^{\alpha , p}([0,T ]; B_0) \hookrightarrow C^{0}([0,T]; B) \quad \text {is compact.} \end{aligned}$$

1.1.1 Stochastic Inequalities

Let $(\Omega ,{\mathcal {F}}, ({\mathcal {F}}_t)_t, {\mathbb {P}})$ be a stochastic basis. Let X be a separable Hilbert space. Let U be a second separable Hilbert space, and let W be a cylindrical Wiener process with values in U, defined on the stochastic basis. For any progressively measurable process $\Phi \in L^{2}(\Omega \times [0, T];{\mathcal {L}}_{2}(U,X))$, we denote by $I(\Phi )$ the Ito integral defined for $t\in [0,T]$ as

$$\begin{aligned} I(\Phi )(t) = \int _{0}^{t} \Phi (s) \, \textrm{d} W_s. \end{aligned}$$

Lemma A.3

Let $p\ge 2$ and $\alpha < 1/2$ be given. Then, for any progressively measurable process $\Phi \in L^{p}(\Omega \times [0, T];{\mathcal {L}}_{2}(U,X))$, we have

$$\begin{aligned} I(\Phi ) \in L^{p}(\Omega ; W^{\alpha ,p}([0,T] \,; \, X)) \end{aligned}$$

and there exists a constant $C(p,\alpha ) > 0$ independent of $\Phi$ such that

$$\begin{aligned} {\mathbb {E}} \left[ \, \Vert I(\Phi ) \, \Vert ^{p}_{W^{\alpha ,p}([0,T] \,; \, X)} \right] \; \le \; C(p, \alpha ) \; {\mathbb {E}} \left[ \, \int _{0}^{T} \Vert \Phi \Vert ^{p}_{{\mathcal {L}}_{2}(U,X)} \, \textrm{d}t \; \right] . \end{aligned}$$

This result is proved in Flandoli and Gatarek [18]. Most notably for the analysis here, the Burkholder–Davis–Gundy inequality holds and is recalled in the next proposition.

Proposition A.1

(Burkholder–Davis–Gundy inequality) For all integer $r\ge 1$, we have

$$\begin{aligned} {\mathbb {E}} \left[ \, \sup _{t\in [0,T]} \left| \int _{0}^{t} \Phi (s) \, \textrm{d}W_s \, \right| ^{r} \; \right] \; \le \; C \, {\mathbb {E}} \left[ \,\left( \int _{0}^{T} \Vert \, \Phi (s)\, \Vert ^{2}_{{\mathcal {L}}_{2}(U,X)} \, \textrm{d}s \; \right) ^{r/2}\right] \end{aligned}$$

with a constant $C>0$ depending only on r.

In Sect. 4.4, the following lemma of Debussche et al. [16] is used to facilitate the passage to the limit in the Galerkin scheme.

Lemma A.4

Consider a sequence of stochastic bases $S_n = (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t}^{n})_{t\in [0,T]},{\mathbb {P}},W^n)$ with $W^n$ a cylindrical Wiener process (over U) with respect to ${\mathcal {F}}_{t}^{n}$. Assume that $(\psi ^n)_{n\ge 1}$ is a collection of X-valued ${\mathcal {F}}_{t}^{n}$ predictable processes such that $\psi ^{n} \in L^{2}\left( [0,T],{\mathcal {L}}_2(U,X)\right)$ a.s. Finally consider $S = (\Omega ,{\mathcal {F}}, {\mathbb {P}}, W)$ with W a cylindrical Wiener process (over U) and $\psi \in L^{2}\left( [0,T],{\mathcal {L}}_2(U,X)\right)$, which is $F_t$ predictable. If

$$\begin{aligned} W^{n} \underset{n\rightarrow +\infty }{\longrightarrow }\ W \qquad \text {in probability in } \; C^{0}\left( [0,T] ; U_0 \,\right) \\ \psi ^{n} \underset{n\rightarrow +\infty }{\longrightarrow }\ \psi \qquad \text {in probability in } \; L^{2}\left( [0,T] , {\mathcal {L}}_{2}(U , X)\,\right) \end{aligned}$$

then

$$\begin{aligned} \displaystyle \int _{0}^{t} \psi ^n \, \textrm{d}W^{n}_s \underset{n\rightarrow +\infty }{\longrightarrow }\ \int _{0}^{t} \psi \, \textrm{d}W_s \qquad \text {in probability in} \; L^{2} \left( [0,T]; X \right) \end{aligned}$$

1.2 A.2 Skohorod and Prokhorov Theorems

We recall here two classical theorems whose proofs can be found in Da Prato and Zabczyk [14].

Theorem A.1

(Prokhorov) A set $\Lambda$ of probability measures on $(E,{\mathcal {B}})$ is relatively compact if and only if it is tight.

Theorem A.2

(Skorohod) Assuming there is a sequence $\{\mu _n\}_{n\ge 1}$ converging weakly to a measure $\mu$, then there exists a probability space $(\overline{\Omega },\overline{{\mathcal {F}}}, \overline{{\mathbb {P}}})$ and a sequence of X-valued random variables $(\overline{Y}_n)_{n\ge 0}$ (relative to this space) such that $\overline{Y}_n$ converges almost surely to the random variable $\overline{Y}$ and such that the laws of $\overline{Y}_n$ and $\overline{Y}$ are $\mu _n$ and $\mu$, respectively, i.e. $\mu _n(E) = {\mathbb {P}}(\overline{Y}_n \in E)$, $\mu (E) = {\mathbb {P}}(\overline{Y} \in E)$, for all $E \in {\mathcal {B}}(X)$.

1.3 A.3 Deterministic Navier Stokes Equation

Let $v_0\in H$. The standard deterministic Navier–Stokes equations for incompressible fluids reads

$$\begin{aligned} \left\{ \begin{array}{l} d_t v(t) \; + \; \dfrac{1}{\, R_e\, } \Delta v(t) \, \textrm{d}t \; + \; \bigl (v(t) \varvec{\cdot }\nabla \bigr ) v(t)\, \textrm{d}t \; = \; -\dfrac{1}{\rho } \; \nabla p {\textrm{d}}t, \\ \nabla \varvec{\cdot }v = 0, \\ v(0)=v_0. \end{array}\right. \end{aligned}$$

(89)

This equation can be written as the following pressure-free abstract problem using the Leray projection and the operator A and B defined Sect. 4.1:

$$\begin{aligned} \left\{ \begin{array}{l} d_t v(t) \; + \; Av(t) \, \textrm{d}t \; + B v(t)\, \textrm{d}t \; = \; 0, \\ v(0)=v_0. \end{array}\right. \end{aligned}$$

(90)

We say that v is a weak solution of (90) if $v\in L^{2}([0,T], V) \cap {L^\infty }([0,T], H)$ and verifies for all $z\in V$ and $t\in [0,T]$ the following equality

$$\begin{aligned} \bigl ( v(t) \, , \, z\bigr )_{H} - \bigl ( v_0 \, , \, z\bigr )_{H} + \int _{0}^{t} \bigl ( A\,v(r)\,, \, z\bigr )_{H} \textrm{d}r \; + \; \int _{0}^{t} \left( B\,v(r)\, , \, z \right) _{H} \textrm{d}r \; = 0 . \end{aligned}$$

For $d=2$ or 3, system (90) admits a weak solution. This solution is unique and belongs to C([0, T], H) when $d=2$.