From Random Advection to Transport Noise and the LU Framework for the Navier–Stokes Equations

Introduction

In this work, we study the incompressible Navier–Stokes equations advected by a random velocity field which decorrelates rapidly in time, and we consider the diffusion limit when the correlation time goes to zero. This question is motivated by the modeling of unresolved scales in fluid dynamics, where stochastic transport models are now widely used. The idea is to split the Lagrangian velocity into a large-scale component, regular in time, and a rapidly fluctuating one. In the modeling under location uncertainty (LU), the fluctuating part is a velocity field which is white in time and correlated in space. Using this decomposition in the Reynolds transport theorem and in the conservation laws of an incompressible fluid, we obtain a stochastic Navier–Stokes equation with a transport noise, an effective diffusion, and a drift due to the spatial inhomogeneity of the unresolved velocity. We refer to Mémin [31] and Resseguier, Mémin and Chapron [36] for the derivation of LU equations in fluid dynamics, and to [7, 37] for applications, in particular to reduced flow models and to geophysical flows. A closely related approach is the stochastic advection by Lie transport (SALT) introduced by Holm [26] through a variational principle. In SALT, the noise acts on the momentum through a Lie derivative, which contains, besides the advection $(\sigma\cdot\nabla)u$, the stretching term $(\nabla\sigma)^{T}u$. In our setting, the momentum is transported by the fast velocity through pure advection, and the limit contains the advection noise $P((\sigma\cdot\nabla)u)$ of the LU model, with $P$ the Leray projection onto divergence-free fields.

A white noise in time is an idealization, since it corresponds to an infinite separation of time scales. In practice, the unresolved velocity has a small but positive correlation time. We believe that it is important to prove that the LU equations are limits of equations with more realistic noises, and this is the aim of this article.

Let $S \subset\mathbb{R}^{d}$, $d \in\{2,3\}$, be a smooth bounded domain, and let $m(t)$ be a centered, stationary, divergence-free Markov process with values in a space of smooth vector fields on $S$ vanishing on $\partial S$. The fast velocity is of the form $\varepsilon^{-1}m(t/\varepsilon^{2})$: its amplitude is of order $\varepsilon^{-1}$ and its correlation time of order $\varepsilon^{2}$. Along a Lagrangian trajectory, we write

The field $u_s$ is the Itô–Stokes drift. It comes from the spatial inhomogeneity of $m$, and it is subtracted so that the last bracket is asymptotically centered; its expression is given in § 3. In Eulerian variables, we obtain

where $m^{\varepsilon}(t)=m(t/\varepsilon^{2})$, $A$ is the Stokes operator with Dirichlet boundary conditions and $B$ is the usual projected nonlinear operator.

Our main result can be stated informally as follows. Precise assumptions are given in § 3, and the precise statement is Theorem 4.3.

Equation (3) is the Itô form of a Stratonovich transport equation. The amplitude $\varepsilon^{-1}$ and the correlation time $\varepsilon^{2}$ balance each other, so that a transport noise survives in the limit. Its covariance is the Green–Kubo covariance of $m$, and the spatial variations of this covariance produce the Itô–Stokes drift, which appears in the Itô correction as an advection by $-u_s$. Thus the drift and the diffusion of the LU model are not chosen independently: both come from the same unresolved velocity. Note that $m$ need not be Gaussian: at leading order, only its integrated covariance matters.

Note that both the large scales, represented by $u$, and the small scales, represented by $m$, converge: the integrated fast velocity $\varepsilon^{-1}\int_0^t m^{\varepsilon}(s)\,ds$ converges, jointly with $u^{\varepsilon}$, to $-Q^{1/2}W$. This justifies both the LU modeling and the representation of the small scales by a white noise.

Our basic tool is the perturbed test function method. It was introduced by Papanicolaou, Stroock and Varadhan [34] in finite dimension, and this work is part of a program on diffusion approximation for infinite dimensional systems. Let us mention for instance [18] for stochastic kinetic limits, [17] for kinetic-fluid systems with mixing Markov forcing, [5] for singular dispersive limits, or [6] for Schrödinger equations with a random potential. The terms of size $\varepsilon^{-1}$ in (2) prevent any estimate uniform in $\varepsilon$ from the energy identity, and the generator of the pair $(u^\varepsilon,m^\varepsilon)$ contains terms of order $\varepsilon^{-2}$ and $\varepsilon^{-1}$. Both problems are solved by adding correctors to the test functions; these are obtained from Poisson equations for the generator $M$ of the fast process. We use this idea twice: first on the energy, to get bounds uniform in $\varepsilon$, and then on cylindrical observables, to prove tightness and identify the limit. We do not assume that $m$ is bounded. As in [6], the correctors are then not uniformly bounded, and we use a stopping time to control the large values of the fast field. Also, a second corrector is already needed for the energy, in order to replace the order-one terms which are quadratic in the fast variable by their averages.

The main difficulty is due to the boundary. The correctors which compensate the transport term are linear in $u$, with coefficients such as $P((n\cdot\nabla)h)$. These fields are divergence free, but the Leray projector does not preserve the Dirichlet condition, so that they do not belong to $D(A^{1/2})$. The perturbed generator then contains terms like $(Au,P((n\cdot\nabla)h))$, which cannot be controlled by the energy, see § 6.1. We overcome this problem by using modified correctors, obtained by inverting $M-\varepsilon^2 A$ instead of $M$. The Stokes operator then acts on the corrector, which satisfies the boundary condition, and the bad terms cancel in Itô’s formula (Corollary 6.3). The counterpart is that the modified correctors have a boundary layer of thickness $\varepsilon$, the distance over which viscosity acts during one correlation time $\varepsilon^2$. Their $V$-norm is of order $\varepsilon^{-1+2\theta}$ for every $\theta<1/4$, roughly $\varepsilon^{-1/2}$. This is compensated by the powers of $\varepsilon$ in front of the remainders, so that the boundary layer does not affect the limit, but we need a stronger moment condition on $m$ ($4p<q$ in Assumption 3.1). On the torus, or whenever the correctors can be taken in $D(A^{1/2})$, the remainders are of order $\varepsilon$. The modified correctors may be compared with the frequency cut-off used in [14].

In [15], we have constructed martingale solutions of the LU Navier–Stokes equation in dimensions two and three, proved pathwise uniqueness in dimension two, and studied the vanishing-noise limit. The limit equation obtained here is this LU equation, up to the additional drift $-\frac{1}{2}\sum_j B(\sigma_j,A\sigma_j)$. This term is the Itô–Stratonovich correction of the interaction between the transport noise and the additive noise $AQ^{1/2}\,\mathrm{d}W$, which comes from the viscous force acting on the fast component of the velocity; it is not taken into account in the LU model (see Remark 2.3). It vanishes for spatially homogeneous noise on the torus (see § 12), so that the difference with [15] is an effect of the inhomogeneity of the noise. We do not know which modeling is the most relevant. We give a self-contained proof of uniqueness in dimension two (Proposition 11.1).

Stochastic Navier–Stokes equations with transport noise go back to Brzeźniak, Capiński and Flandoli [10], whose framework covers two-dimensional equations with noise depending on first-order derivatives of the velocity. Mikulevičius and Rozovskii derived stochastic fluid equations from stochastic particle trajectories [32] and proved existence and uniqueness for a large class of such equations [33]. The LU framework has the same transport structure, and makes the covariance-induced diffusion and the Itô–Stokes drift explicit. Transport noise has also been obtained from scale separation in two-dimensional fluid models by Flandoli and Pappalettera [22, 23], and Debussche and Pappalettera [16] have obtained transport noise and an Itô–Stokes drift for coupled slow–fast systems, including Navier–Stokes, surface quasi-geostrophic and primitive equations. A rough-path approach to a related two-scale system is given in [13]. More recently, randomly advected Navier–Stokes equations with both a fast time scale $\varepsilon^2$ and a vanishing correlation length $\delta$ have been studied in [14]. When $\varepsilon=o(\delta)$, the limit is deterministic, with an enhanced viscosity in which the Leray projector plays a role, and Gaussian fluctuations appear in dimension two. Our setting is complementary since the spatial covariance is fixed and the noise survives at leading order. Navier–Stokes equations with transport noise have also been studied directly, see for instance [4, 21, 35].

It is also known that transport noise has regularizing and dissipative effects. Flandoli and Luo [20] have shown that high-mode transport noise improves the control of the vorticity in three-dimensional Navier–Stokes equations and gives well-posedness with high probability (see also [19]). Agresti and Veraar [4] have developed a theory in critical spaces, with stochastic Serrin-type blow-up criteria. Agresti has shown that transport noise can induce anomalous enstrophy dissipation in two dimensions, even for small noise intensity [2], and that suitable transport noise gives global smooth solutions with high probability for hyperviscous Navier–Stokes equations with dissipation exponent below the Lions threshold $5/4$ [1, 30], and also for the three-dimensional Navier–Stokes equations [3]. Regularization is not our purpose, but these results show that the effective diffusion created by transport noise is important for the dynamics. In our setting, it is not postulated but obtained in the limit.

Compared with coupled slow–fast systems, the unresolved velocity is here an abstract stationary Markov field, and not the solution of a second fluid equation. We only assume polynomial moments and a polynomially weighted mixing of its mean and of its two-time correlations, and the limiting covariance is written directly in terms of its Green–Kubo tensor. The covariance symmetry (22) could be relaxed, at the price of an additional drift in the limit. We also think that similar results hold for other geophysical models for which LU versions are available.

The article is organized as follows. In § 2, we derive the model and introduce the functional setting. The assumptions on the fast process and the limiting operators are given in § 3, and the main result is stated in § 4. § 5 deals with fixed $\varepsilon$. The modified correctors are introduced in § 6. In § 7, we explain why we work with Galerkin approximations and prove the uniform estimates. In § 8 to § 11, we construct the correctors for cylindrical observables, identify the limit, prove tightness and conclude. Finally, the spatially homogeneous case is discussed in § 12.

Throughout the article, $C$ denotes a constant whose value may change from one line to the other and which, unless explicitly stated, does not depend on $\varepsilon$ nor on the Galerkin parameter $N$.

Modeling and functional setting

Lagrangian description with finite-correlation random advection

The starting point is the Lagrangian description underlying modeling under location uncertainty. In the LU framework the trajectory of a fluid particle is decomposed into a smooth-in-time large-scale velocity and a rapidly fluctuating velocity which, in the idealized model, is white in time,

The white-in-time component represents an infinite separation of temporal scales. At finite scale separation, however, the unresolved velocity retains a non-zero correlation time. We therefore replace the white-noise velocity by a centered stationary random field $m$ with a short correlation time. Under the diffusion scaling the Lagrangian trajectory is described by

The process $m$ is divergence free and sufficiently regular in space; its precise assumptions are given in § 3. The correlation time is of order $\varepsilon^{2}$ while the amplitude is of order $\varepsilon^{-1}$.

Although $m(t,x)$ is centered for each fixed $x$, the random field evaluated along the trajectory, $m(t/\varepsilon^2,X^\varepsilon(t))$, is not centered. Since the small scales should average to zero, a correction has to be added. The coupling between the spatial dependence of $m$ and the displacement generated by $m$ itself produces, at order one, the drift which becomes the Itô–Stokes drift of the limiting LU model. We reproduce the formal argument leading to this correction; its purpose is to identify the physical structure of the model.

Formal emergence of the Itô–Stokes drift

For clarity, suppose that $m$ has a dependence range in time bounded by one, i.e. $(m(r))_{r\le t-1}$ and $(m(r))_{r\ge t}$ are independent for every $t$, and that the initial position is independent of $m$. Since $X^\varepsilon(t-\varepsilon^2)$ depends on $m$ only through $(m(r))_{r\le t/\varepsilon^2-1}$, we have $\mathbb{E}[m(t/\varepsilon^2,X^\varepsilon(t-\varepsilon^2))]=0$ and

In the increment of $X^\varepsilon$, the contribution of the large-scale velocity is of order $\varepsilon$ and vanishes formally. The leading contribution is generated by the fast field itself. Since $X^\varepsilon$ has a time derivative of order $\varepsilon^{-1}$ and $t-r$ is of order $\varepsilon^2$, we may freeze the spatial argument, $m(r/\varepsilon^2,X^\varepsilon(r))=m(r/\varepsilon^2,X^\varepsilon(t))+O(\varepsilon)$. After the change of variable $r=t-\varepsilon^2s$,