The Dissipation Properties of Transport Noise

Abstract

The aim of this work is to present, in a compact way, the latest results about the dissipation properties of transport noise in fluid mechanics. Starting from the reasons why transport noise is natural in a passive scalar equation for the heat diffusion and transport, several results about enhanced dissipation due to the noise are presented. Rigorous statements are matched with numerical experiments in order to understand that the sufficient conditions stated are not yet optimal but give a first useful indication.

1 Introduction

In the last four years, a new understanding of heat diffusion in a turbulent fluid modeled by white noise has been developed. This model has the interesting feature of describing properly the dissipation properties of a turbulent fluid. The equation for the heat diffusion and transport, with a heat source q, is

$$\displaystyle \begin{aligned} \partial_{t}\theta+u\cdot\nabla\theta=\kappa\varDelta\theta+q {} \end{aligned} $$

(1)

where $\theta =\theta \left ( t,x\right )$ is the temperature, κ is the diffusion constant and $u=u\left ( t,x\right )$ is the velocity field of the fluid. The turbulent fluid is a priori described by a random field, Gaussian and white in time, with covariance structure given a priori (hence the temperature is a passive scalar). In this review we consider the following description for u:

$$\displaystyle \begin{aligned} u\left( t,x\right) =\sum_{k\in K}\sigma_{k}\left( x\right) \frac {dW_{t}^{k}}{dt} {} \end{aligned} $$

(2)

where σ _k are divergence free vector fields satisfying no slip-boundary conditions and $W_{t}^{k}$ are independent Brownian motions on a filtered probability space $\left ( \varOmega ,\mathcal {F},\left ( \mathcal {F}_{t}\right ) _{t\geq 0},\mathbb {P}\right )$; for simplicity, assume K is a finite set, but the case of a countable set can be studied without troubles at the price of additional summability assumptions. Some rigorous justification for describing the velocity of a turbulent fluid by Eq. (2) are available in Sect. 2.2. Here we want just give some ideas. Let us denote by u ^{ν, ε} the solution in a domain with boundary D of the SPDE

$$\displaystyle \begin{aligned} \begin{cases} \partial_t u^{\nu,\varepsilon}+\nabla p^{\nu,\varepsilon}& = \nu \varDelta u^{\nu,\varepsilon}-\frac{1}{\varepsilon} u^{\nu,\varepsilon}+\frac{1}{\varepsilon}\sum_{k\in K}\sigma_k \partial_t d W^k_t\\ \operatorname{div}\left(u^{\nu,\varepsilon}\right) &=0\\ u|{}_{\partial D}&=0, \end{cases} \end{aligned} $$

(3)

where the terms $-\frac {1}{\varepsilon } u^{\nu ,\varepsilon }+\frac {1}{\varepsilon }\sum _{k\in K}\sigma _k \partial _t d W^k_t$ describe the roughness of the boundary as stated in Sect. 2.2. Let, moreover, $W^{\nu ,\varepsilon }_t=\int _0^t u^{\nu ,\varepsilon }(s)\ ds$, then it can be proven than

$$\displaystyle \begin{aligned} \lim_{\varepsilon\rightarrow 0}\mathbb{E}\left[\operatorname{sup}_{t\in [0,T]}\lVert W^{\nu,\varepsilon}_t-\sum_{k\in K} \sigma_k W^{k}_t\rVert_{L^2(D)}^2\right]=0, \end{aligned}$$

see for example [6].

The correct interpretation of Eq. (1) when u has the form (2) is the Stratonovich equation

$$\displaystyle \begin{aligned} d\theta+\sum_{k\in K}\sigma_{k}\cdot\nabla\theta\circ dW_{t}^{k}=\left( \kappa\varDelta\theta+q\right) dt {} \end{aligned} $$

(4)

or equivalently the Itô equation with corrector $\mathcal {L}\theta$ given by the second order differential operator (7) below:

$$\displaystyle \begin{aligned} d\theta+\sum_{k\in K}\sigma_{k}\cdot\nabla\theta dW_{t}^{k}=\left( \kappa \varDelta\theta+\mathcal{L}\theta+q\right) dt. {} \end{aligned} $$

(5)

There are some motivations for the analysis of Eq. (4) based on the idea to extend to SPDE the remarkable principle of Wong-Zakai [20], see for example [2, 3, 14, 15, 18, 19, 16].

Assuming that the external source q and the initial temperature θ ₀ are deterministic, under suitable mild assumptions the deterministic function

$$\displaystyle \begin{aligned} \varTheta\left( t,x\right) =\mathbb{E}\left[ \theta\left( t,x\right) \right] \end{aligned}$$

is the solution of the deterministic parabolic equation

$$\displaystyle \begin{aligned} \partial_{t}\varTheta=\left( \kappa\varDelta+\mathcal{L}\right) \varTheta+q {} \end{aligned} $$

(6)

where $\mathbb {E}$ denotes the mathematical expectation on $\left ( \varOmega ,\mathcal {F},\mathbb {P}\right )$. The main results in the last years are quantitative estimates on the difference θ − Θ, some convergence properties of the solution of Eq. (5) to the stationary solution of Eq. (6) and the enhanced dissipative properties of the second order differential operator $\kappa \varDelta +\mathcal {L}$, see [7, 9, 10]. These kinds of results explained properly the dissipation properties of transport noise and are the core of this review article.

In Sect. 2 we will present some motivations for the analysis of Eq. (4) as a good model for the heat diffusion in a turbulent fluid and we will introduce the main notations. In Sect. 3 we will present the main results, referring to [7, 9, 10] for some rigorous proofs. Lastly, in Sect. 4 we will present some cases where the coefficients σ _k introduce more dissipation in the model with respect to the theoretical predictions made by the rigorous sufficient conditions, exploiting real computations or numerical simulations following the ideas of [7, 10].

Remark 1

In this review we only considered the effects of the transport noise on passive scalars. Actually, some results can be stated also for the scalar vorticity of the fluid itself, in two space dimensions. We refer to [8, 11, 12] for further readings. The case of the influence on vector fields is much more difficult and still to be understood.

2 Well-Posedness and Motivations

2.1 Notations and Definitions

In this review we will denote by D a 2D domain with boundary, either a smooth bounded open set or an infinite 2D channel, namely $\mathbb {R}\times \left ( -1,1\right )$. We write the coordinates using the notation

$$\displaystyle \begin{aligned} x=\left( x_{1},z\right) \in D. \end{aligned}$$

Let Z be a separable Hilbert space, denote by $L^2(\mathcal {F}_{t_0},Z)$ the space of square integrable random variables with values in Z, measurable with respect to $\mathcal {F}_{t_0}$. Moreover, denote by $C_{\mathcal {F}}\left ( \left [ 0,T\right ] ;Z\right )$ the space of continuous adapted processes $\left ( X_{t}\right ) _{t\in \left [ 0,T\right ] }$ with values in Z such that

$$\displaystyle \begin{aligned} \mathbb{E}\left[ \sup_{t\in\left[ 0,T\right] }\left\Vert X_{t}\right\Vert {}_{Z}^{2}\right] <\infty \end{aligned}$$

and by $L_{\mathcal {F}}^{2}\left ( 0,T;Z\right )$ the space of progressively measurable processes $\left ( X_{t}\right ) _{t\in \left [ 0,T\right ] }$ with values in Z such that

$$\displaystyle \begin{aligned} \mathbb{E}\left[ \int_{0}^{T}\left\Vert X_{t}\right\Vert {}_{Z}^{2}dt\right] <\infty. \end{aligned}$$

Denote by $L^{2}\left ( D\right )$ and $W^{k,2}\left ( D\right )$ the usual Lebesgue and Sobolev spaces and by $W_{0}^{k,2}\left ( D\right )$ the closure in $W^{k,2}\left ( D\right )$ of smooth compact support functions. Set $H=L^{2}\left ( D\right )$, $V=W_{0}^{1,2}\left ( D\right )$, $D\left ( A\right ) =W^{2,2}\left ( D\right ) \cap V$. We denote by 〈⋅, ⋅〉 and $\lVert \cdot \rVert$ the inner product and the norm in H respectively.

Assume that K is a finite set and $\sigma _{k}\in \left (D\left ( A\right )\cap C_{b}^{\infty }\left ( D\right )\right )^2,\ \nabla \cdot \sigma _k=0$, k ∈ K (less is sufficient but we do not stress this level of generality). Define the matrix-valued function

$$\displaystyle \begin{aligned} Q\left( x,y\right) =\sum_{k\in K}\sigma_{k}\left( x\right) \otimes \sigma_{k}\left( y\right) . \end{aligned}$$

If we denote by $W\left ( t,x\right )$ the vector valued random field

$$\displaystyle \begin{aligned} W\left( t,x\right) =\sum_{k\in K}\sigma_{k}\left( x\right) W_{t}^{k} \end{aligned}$$

(the velocity field u given by (2) is the distributional time derivative of W) then we see that $Q\left ( x,y\right )$ is the space-covariance of $W\left ( 1,x\right )$:

$$\displaystyle \begin{aligned} Q\left( x,y\right) =\mathbb{E}\left[ W\left( 1,x\right) \otimes W\left( 1,y\right) \right] . \end{aligned}$$

The matrix-function $Q\left ( x,x\right )$ is elliptic:

$$\displaystyle \begin{aligned} \sum_{i,j=1}^{d}Q_{ij}\left( x,x\right) \xi_{i}\xi_{j}=\mathbb{E}\left[ \left\vert W\left( t,x\right) \cdot\xi\right\vert ^{2}\right] \geq0 \end{aligned}$$

for all $\xi =\left ( \xi _{1},\ldots ,\xi _{d}\right ) \in \mathbb {R}^{d}$. Associated to it define the bounded linear operator

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbb{Q}:L^2(D;\mathbb{R}^2)\rightarrow L^2(D;\mathbb{R}^2), \qquad \left(\mathbb{Q}v\right)\left( x\right)=\int_D Q(x,y)v(y)\ dy \end{array} \end{aligned} $$

and the quantities:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{q}(x):=\operatorname{min}_{\xi \neq 0}\frac{\xi ^TQ(x,x)\xi}{\lvert \xi\rvert^2},\\ \varepsilon_{Q}:=\lVert Q^{1/2}\rVert^2_{L^2(D;\mathbb{R}^2)\rightarrow L^2(D;\mathbb{R}^2)}. \end{array} \end{aligned} $$

Consider the divergence form elliptic operator $\mathcal {L}$ defined as

$$\displaystyle \begin{aligned} \left( \mathcal{L}\theta\right) \left( x\right) =\frac{1}{2}\sum _{i,j=1}^{d}\partial_{i}\left( Q_{ij}\left( x,x\right) \partial_{j} \theta\left( x\right) \right)\end{aligned} $$

(7)

for $\theta \in W^{2,2}\left ( D\right )$. Define the linear operator $A:D\left ( A\right ) \subset H\rightarrow H$ as

$$\displaystyle \begin{aligned} A\theta=\left( \kappa\varDelta+\mathcal{L}\right)\theta.\end{aligned} $$

It is the infinitesimal generator of an analytic semigroup of negative type, see [1, 4, 13, 17], that we denote by e ^tA, t ≥ 0. Moreover, if D is bounded, we denote by κλ the first eigenvalue of − κΔ and by $\kappa \lambda _{\mathcal {L}}$ the first eigenvalue of $-(\kappa \varDelta +\mathcal {L})$.

Definition 1

Given $\theta _{0}\in L^{2}(\mathcal {F}_{0},H)$ and q ∈ L ²(0, T;H), a stochastic process

$$\displaystyle \begin{aligned} \theta\in C_{\mathcal{F}}\left( \left[ 0,T\right] ;H\right) \cap L_{\mathcal{F}}^{2}\left( 0,T;V\right) \end{aligned}$$

is a mild solution of Eq. (5) if the following identity holds

$$\displaystyle \begin{aligned} \theta\left( t\right) =e^{tA}\theta_{0}+\int_{0}^{t}e^{\left( t-s\right) A}q\left( s\right) ds-\sum_{k\in K}\int_{0}^{t}e^{\left( t-s\right) A}\sigma_{k}\cdot\nabla\theta\left( s\right) dW_{s}^{k}\end{aligned} $$

for every $t\in \left [ 0,T\right ]$, $\mathbb {P}$-a.s.

Theorem 1

For every $\theta _{0}\in L^{2}(\mathcal {F} _0,H)$ and q ∈ L ²(0, T;H) there exists a unique θ mild solution of Eq. (5). Moreover θ depends continuously on θ ₀ and q.

Definition 2

Given $\theta _{0}\in L^{2}(\mathcal {F}_{0},H)$ and q ∈ L ²(0, T;H), we say that a stochastic process θ is a weak solution of Eq. (5) if

$$\displaystyle \begin{aligned} \theta \in C_{\mathcal{F}}([0,T];H)\cap L_{\mathcal{F}}^{2}(0,T;V) \end{aligned}$$

and for every ϕ ∈ D(A), we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \langle\theta(t),\phi\rangle=& \langle\theta_{0},\phi\rangle+\int_{0}^{t}\langle\theta(s),A\phi\rangle \ ds+\int_0^t \langle q(s),\phi\rangle \\ & +\sum_{k\in K}\int_{0}^{t} \langle \theta(s),\sigma_k\cdot\nabla\phi\rangle\ dW^{k}_{s} \end{array} \end{aligned} $$

for every $t \in [0,T],\ \mathbb {P}-a.s.$

Theorem 2

θ is a weak solution of problem (5) if and only if is a mild solution of problem (5). Moreover the Itô formula

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lVert\theta(t)\rVert^2-\lVert\theta(0)\rVert^2=& 2\int_0^t\langle \theta(s),q(s)\rangle\ ds+\sum_{k\in K} \int_0^t\lVert\sigma_k\cdot\nabla \theta(s)\rVert^2\ ds \\ & -2\int_0^t \langle(-A)^{\frac{1}{2}}\theta(s),(-A)^{\frac{1}{2}}\theta(s)\rangle\ ds \end{array} \end{aligned} $$

holds.

These results are classical and can be found in [5, 10] together with several generalizations.

2.2 Motivations

In this section we want to give some heuristics to accept Eq. (4) as a correct model for heat diffusion in a turbulent fluid. In the domain D we have a fluid with velocity u (pressure p, constant density = 1) and the heat θ. Both u and θ are equal to zero on ∂D:

$$\displaystyle \begin{aligned} \begin{array}{rcl} u|{}_{\partial D} & =0\\ \theta|{}_{\partial D} & =0. \end{array} \end{aligned} $$

The condition u|_∂D = 0 provokes several interesting technical questions. The equations are

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \begin{array}{rcl} \partial_{t}u+u\cdot\nabla u+\nabla p & =&f\\ \nabla \cdot u & =&0\\ \partial_{t}\theta+u\cdot\nabla\theta & =&\kappa\varDelta\theta+q \\ u|{}_{t=0} & =&u_{0}\\ \theta|{}_{t=0} & =&\theta_{0}. \end{array} \end{array} \end{aligned} $$

(8)

where f and q take care of interaction with external sources. In particular, physical boundaries are never completely smooth. Hence, the external source f want to model the effects of the roughness of the boundary and its influence to the velocity of the fluid. The instability of the flow at the boundary, originating vortices, is very strong, hence the frequency and intensity of creation of vortices at the boundary strongly suffers from the imprecision of the description of the true boundary. Replacing the true details of the boundary by a random mechanism of vorticity production would increase the realism of the model. Emergence of vortices near obstacles is commonly observed and we content ourselves with an ad hoc inclusion of this fact into the equations. Assume the velocity field at time t is u(t, x). Assume that, as a consequence of an instability near the boundary, a modification occurs and in a very short time we have a field u(t + Δt, x) which is not just equal to the smooth evolution of u(t, x). We may assume that at some time t we have a jump:

$$\displaystyle \begin{aligned} u(t+\varDelta t, x)= u(t,x)+\sigma(x)\end{aligned}$$

where σ(x) is presumably localized in space and corresponds to a vortex structure. After these preliminary comments we can accept to model the roughness via a friction term of intensity $-\frac {u}{\varepsilon }$ and a term of jump described by

$$\displaystyle \begin{aligned} W_N(t,x)=\sum_{k\in K}\frac{\sigma_k(x)}{N}\frac{N^{k,1}_{N^2t/\varepsilon^2}-N^{k,2}_{N^2t/\varepsilon^2}}{\sqrt{2}},\end{aligned}$$

where $N_t^{\cdot ,\cdot }$ are independent Poisson processes. More on this topic can be found in [6]. Applying a Donsker invariance principle to the stochastic process W _N(t, x), it converges in law to the gaussian process

$$\displaystyle \begin{aligned} W_t(x)=\frac{1}{\varepsilon}\sum_{k\in K}\sigma_k(x)W^k_t, \end{aligned}$$

where $W^k_t$ are independent Brownian motions. Parameterizing the solutions of system (8) by ε we arrive to the following stochastic coupled system

$$\displaystyle \begin{aligned} \begin{array}{rcl} \partial_{t}u^{\varepsilon}+u^{\varepsilon}\cdot\nabla u^{\varepsilon}+\nabla p^{\varepsilon} &=&-\frac{1}{\varepsilon}\left( u^{\varepsilon }-\partial_{t}W\right) \\ \nabla \cdot u^{\varepsilon} &=& 0\\ \partial_{t}\theta^{\varepsilon}+u^{\varepsilon}\cdot\nabla\theta^{\varepsilon} &=&\kappa\varDelta\theta^{\varepsilon}+q \\ u|{}_{t=0} &=&u_{0}\\ \theta|{}_{t=0} &=&\theta_{0}. \end{array} \end{aligned} $$

(9)

The last step for moving from system (9) to Eq. (4) is trying to understand the behavior of system (9) letting ε → 0 and it based on a result proved in [12] in the case of the 2D torus and under analysis in the case of general 2D domains with boundary. Thus just for the last sentence of this subsection we assume the $D=\mathbb {T}^2:=\mathbb {R}^2/(2\pi \mathbb {Z}^2).$

Theorem 3

Under previous assumptions on q and σ, if moreover:

• the coefficients σ _k are zero-mean and there exists l ≥ 1 such that

  σ _k ∈ W ^{l, ∞} ∀k ∈ K;

• $\forall x\in \mathbb {T}^2$ it holds $\sum _{k\in K}\left ((K* \sigma _k)\cdot \nabla \sigma _k\right )(x)=0;$

• $q\in L^1\left ([0,T];L^{\infty }\left (\mathbb {T}^2\right )\right );$

• $\theta _0\in L^{\infty }\left (\mathbb {T}^2\right ),$

then for every $f\in L^1(\mathbb {T}^2)$

$$\displaystyle \begin{aligned} \mathbb{E}\left[ \left\lvert \int_{\mathbb{T}^2} (\theta^{\varepsilon}_t-\theta_t)(x)f(x)\ dx \right\rvert\right]\rightarrow 0\qquad \mathit{\text{ as }} \varepsilon\rightarrow 0\end{aligned}$$

for every fixed t ∈ [0, T] and in L ^p([0, T]) for every finite p. Moreover, if $q\in L^1([0,T];Lip(\mathbb {T}^2))$ then the previous convergence holds uniformly for t ∈ [0, T] and $f\in Lip(\mathbb {T}^2)$ with $[f]_{Lip(\mathbb {T}^2)}\leq 1$ and $\lVert f\rVert _{L^{\infty }(\mathbb {T}^2)}\leq 1$.

3 Main Results

The results related to the analysis of these equations can be classified in three categories:

• 1. Convergence of the solution of Eq. (5) to some quantities related to Eq. (6).

• 2. Quantification of the dissipation of the function $\mathbb {E}\left [\Vert \theta (t)\rVert ^2\right ]$.

• 3. Enhanced dissipative properties of the second order differential operator $\kappa \varDelta +\mathcal {L}$.

Remark 2

Even if Q is a covariance operator, the third question is far to be trivial. In fact we assumed that σ _k|_∂D = 0. Thus the operator $\mathcal {L}$ degenerates at the boundary.

We will treat all the three problems above, sometimes specializing our general framework.

Theorem 4

Assume D is a bounded domain.

• 1. If $\theta _0\in L^2(\mathcal {F}_0,H),\ q\equiv 0$ . Then, ∀ϕ ∈ L ^∞(D),

  $$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbb{E}\left[\langle \phi,\theta(t)-\varTheta(t)\rangle^2\right]\leq \frac{\varepsilon_{Q}}{\kappa}\mathbb{E}\left[\lVert\theta_0\rVert^2\right]\lVert \phi\rVert_{L^{\infty}(D)}. \end{array} \end{aligned} $$

• 2. Moreover, if θ ₀ ≥ 0

  $$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbb{E}\left[\lVert \theta(t)\rVert^2 \right]\leq \left(\frac{\varepsilon_{Q}}{\kappa}+2\lvert D\rvert e^{-\kappa\lambda_{\mathcal{L}}t} \right)\mathbb{E}\left[\lVert \theta_0\rVert^2\right]. \end{array} \end{aligned} $$

Remark 3

A result similar to the first item can be proved also in the case of D infinite channel and q≢0 adapting the proof of Theorem 7 in [10] to such finite time case.

Thanks to previous theorem is evident that the dissipation properties of the solution of the stochastic Eq. (5) are influenced obviously by the first eigenvalue of the operator $\mathcal {L}$ but also by the operatorial norm of $\mathbb {Q}^{1/2}$. Thus, our next step will be state state some sufficient conditions in order to have ε _Q very small and $\kappa \lambda _{\mathcal {L}}\gg \kappa \lambda$.

For δ > 0 fixed, let us define

$$\displaystyle \begin{aligned} D_{\delta}:=\{ x\in D:\ dist(x,\partial D)>\delta\}. \end{aligned}$$

Then the following theorems hold.

Theorem 5

Assume that the family of coefficients $\left ( \sigma _{k}\left ( \cdot \right ) \right ) _{k\in K}$ has the following approximate orthogonality property: there exists a finite number $M\in \mathbb {N}$ and a partition K = K ₁ ∪… ∪ K _M such that

$$\displaystyle \begin{aligned} \left\langle \sigma_{k},\sigma_{k^{\prime}}\right\rangle =0 \mathit{\text{ for all }}k,k^{\prime}\in K_{i}\end{aligned}$$

for all i = 1, …, M. Then

$$\displaystyle \begin{aligned} \varepsilon_{Q}\leq M\sup_{k\in K}\left\Vert \sigma_{k}\right\Vert ^{2}. \end{aligned}$$

Theorem 6

Assuming that $\tilde {q}(x)\geq \sigma ^2\ \mathit {in}\ D_{\delta }$ , then for any κ > 0 fixed

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{(\sigma,\delta)\rightarrow (+\infty,0)}\kappa\lambda_{\mathcal{L}}=+\infty. \end{array} \end{aligned} $$

Theorem 7

There exists a constant C _D > 0 such that

$$\displaystyle \begin{aligned} \kappa \lambda_{\mathcal{L}}\geq C_D\operatorname{min}\left(\sigma^2,\frac{\kappa}{\delta}\right)\end{aligned}$$

for every Q such that

$$\displaystyle \begin{aligned} \tilde{q}(x)\geq \sigma^2\ \textit{in}\ D_{\delta}. \end{aligned}$$

When D is the unit ball, asymptotically as δ → 0, one can take C _D = 1 and

$$\displaystyle \begin{aligned} \kappa\lambda_{\mathcal{L}}\geq \frac{2\kappa}{\kappa+\delta\sigma^2}\sigma^2. \end{aligned}$$

From the last two theorems we understand that the dissipation properties enhance if ε _Q is very small and $\tilde {q}(x)$ is very large except for a small boundary layer around ∂D. Obviously ε _Q is related to the operatorial norm of $\mathbb {Q}^{1/2}$ and thus, loosely speaking, is related to the operatorial norm of $\mathbb {Q}$. Instead $\tilde {q}(x)$ is related to the trace of $\mathbb {Q}$, i.e.

$$\displaystyle \begin{aligned} Tr(\mathbb{Q})=\int_D Tr Q(x,x) \ dx.\end{aligned}$$

Consequently we want that the operatorial norm of $\mathbb {Q}$ is small and the trace of $\mathbb {Q}$ is arbitrarily large and, possibly, infinity. Hence the existence of such operators Q which increase the dissipativity properties of the equation is not surprising. The last issue related to this topic is the presentation of an operator Q which has a fluid dynamics interpretation and satisfies previous property. This definition for general domain D is a bit implicit. Thus in the last section we will present some more explicit computations.

Let us fix a parameter ℓ such that 0 < ℓ ≤ δ , consider a smooth probability density function $\varPsi :\mathbb {R}^2\rightarrow \mathbb {R}$ with compact support in $\overline {B(0,1)}$ and let us denote by K(x, y) the Biot-Savart kernel in D. We recall that a point vortex in x ₀ has vorticity $\delta _{x_0}$ and smoothing it by $\varPsi _{\ell }(x):=\frac {1}{{\ell }^2}\varPsi \left (\frac {x}{{\ell }}\right )$, then it has vorticity $\frac {1}{{\ell }^2}\varPsi \left (\frac {x-x_0}{{\ell }}\right )$.

Now let us consider a random variable X ₀ distributed uniformly on D _2δ, a real random variable Γ ₀ such that

$$\displaystyle \begin{aligned} \mathbb{E}\left[ \varGamma_{0}\right] =0,\qquad \varepsilon_{0}^{2}:=\mathbb{E} \left[ \varGamma_{0}^{2}\right] <\infty \end{aligned}$$

and set

$$\displaystyle \begin{aligned} u\left( x\right) & =\varGamma_{0}\int_{D}K\left( x,y\right) \theta_{\ell }(y-X_{0})dy \\ & =:\varGamma_{0}K_{\ell}(x,X_{0}). \end{aligned} $$

If we consider in Eq. (5) the Brownian motion $W\left ( t,x\right )$, with covariance operator

$$\displaystyle \begin{aligned} Q_{\ell}\left( x,y\right) =\varepsilon_{0}^{2}\mathbb{E}\left[ K_{\ell} (x,X_{0})\otimes K_{\ell}(y,X_{0})\right] \end{aligned}$$

one has

$$\displaystyle \begin{aligned} v^{T}Q_{\ell}\left( x,x\right) v=\varepsilon_{0}^{2}\mathbb{E}\left[ \left\vert K_{\ell}(x,X_{0})\cdot v\right\vert ^{2}\right] \qquad \text{for } v\in\mathbb{R}^{2} \end{aligned}$$

$$\displaystyle \begin{aligned} \left\langle \mathbb{Q}_{\ell} w,w\right\rangle =\varepsilon_{0}^{2}\mathbb{E}\left[ \left( \int_{D}w\left( x\right) \cdot K_{\ell}(x,X_{0})dx\right) ^{2}\right] \qquad \text{for } w\in L^2(D;\mathbb{R}^2) . \end{aligned}$$

Inside the previous identities there is the key to have $v^{T}Q\left ( x,x\right ) v$ large and $\left \langle \mathbb {Q}w,w\right \rangle$ small. Moreover, the law of u on the space of divergence free square integrable vector fields with null normal trace, heuristically, is a Poisson Point Process generating smoothed point vortices (and the associated velocity field) in random positions of D. Thus this kind of noise is reasonable for model what we expect from the heuristic analysis described in Sect. 2.

Theorem 8

• There exists a constant C > 0 such that

  $$\displaystyle \begin{aligned} \left\langle \mathbb{Q}_{\ell}v,v\right\rangle \leq C\varepsilon_{0} ^{2}\left\Vert v\right\Vert {}_{H}^{2} \end{aligned}$$

  for every v ∈ H and ℓ > 0.

• For every x ∈ D, let $q_{\ell }\left ( x\right ) \geq 0$ be the largest number such that

  $$\displaystyle \begin{aligned} v^{T}Q_{\ell}\left( x,x\right) v\geq q_{\ell}\left( x\right) \left\vert v\right\vert ^{2} \end{aligned}$$

  for all $v\in \mathbb {R}^{2}$ . Then

  $$\displaystyle \begin{aligned} \lim_{\ell\rightarrow\infty}\inf_{x\in D_{2\delta}}q_{\ell}\left( x\right) =+\infty.\end{aligned} $$

In the last result of this section the presence of the external source q is crucial. Moreover we assume that q is independent of time and introduce the stationary solution of Eq. (6)

$$\displaystyle \begin{aligned} \varTheta_{st}:=-A^{-1}q. \end{aligned}$$

In fact we want to study the convergence of the solution of the stochastic Eq. (5) to Θ _st.

Set

$$\displaystyle \begin{aligned} C_{\infty}\left( \theta_{0},q\right) :=\sup_{t\geq0}\mathbb{E}\left[\Vert \theta\left( t\right) \rVert _{\infty}^{2}\right]. \end{aligned}$$

Theorem 9

If $\theta _0\in L^2(\mathcal {F}_0,D(A))$ and q ∈ D(A), then

• $C_{\infty }\left ( \theta _{0},q\right ) <\infty$.

• For every ϕ ∈ H

  

In order to be of interest for applications, this theorem requires two conditions:

• (1) that ε _Q is small.

• (2) that Θ _st is significantly affected by the noise.

Obviously if κλ _L ≫ κλ then Θ _st is significantly affected by the noise. Thus we reconduct ourselves to the previous framework already treated. In Sect. 4 we will show a concrete example where this phenomenon appears.

4 Explicit Computations

Theorem 8 is not completely suitable for numerical simulations because the definition of K(x, y) is not explicitly available for every domain smooth and bounded. In this section we will present an explicit construction with a fluid dynamics interpretation, again based on vortex structures, which satisfies both ε _Q arbitrarily small and $\tilde {q}$ arbitrarily large outside a boundary layer. Moreover we will show numerically that, even relaxing the conditions in this construction, the noise influences the behavior of the stationary solution.

4.1 Explicit Construction

We will construct a noise of the form $\varGamma \sum _{k\in K} u_{k}\left ( x\right ) dW_{t}^{k}$ with

$$\displaystyle \begin{aligned} u_{k}\left( x\right) =w_{r}\left( x -x_{k}\right) ,\qquad w_{r}\left( {x}\right) =r^{-1}{w}\left( \frac{{x}}{r}\right) \end{aligned}$$

for suitable r and w. Thus, the covariance of this noise is

$$\displaystyle \begin{aligned} Q\left( {x},{y}\right) =\varGamma^{2}\sum_{{z}\in \varLambda_{N}}{w}_{r}\left( {x}-{z}\right) \otimes {w}_{r}\left( {y}-{z}\right) . \end{aligned}$$

We need to choose x _k, called the “centers” of the vortex blobs, and a suitable vector field w. The vector field w must satisfy several conditions:

• 1. w is smooth and ∇⋅ w = 0;

• 2. w has compact support contained in $\overline {B(0,1)}$;

• 3. w is close to $\frac {1}{2\pi }\frac {x^{\perp }}{|x|{ }^2}$ near x = 0.

The first two properties are useful in order to have that the u _j’s model the velocity of an incompressible fluid at rest. The third one is close to our idea of vortex structures.

Now we choose the centers. For a fixed δ > 0, we choose a positive integer N such that $\frac {1}{N}\leq \delta$. Then we consider the set Λ _N of all points of D _δ having coordinates of the form $\big (\frac {k}{N},\frac {h}{N}\big )$ with $k,h\in \mathbb {Z}$. Thanks to this choice we have

$$\displaystyle \begin{aligned} \min_{{z}_{1}\neq{z}_{2}\in\varLambda_{N}}\left\vert {z} _{1}-{z}_{2}\right\vert =\frac{1}{N},\qquad \min_{{z}\in \varLambda_{N}}d\left( {z},\partial D\right) \geq\delta. \end{aligned}$$

We choose another positive integer M and we decompose the set Λ _N as the disjoint union of the sets

$$\displaystyle \begin{aligned} \varLambda_{N}=\bigcup_{(k_{0},h_{0})\in\{0,1,\ldots,M-1\}^{2}}\varLambda_{N}^{\left( M,k_{0},h_{0}\right) } \end{aligned}$$

where $\big (\frac {k}{N},\frac {h}{N}\big ) \in \varLambda _{N}^{\left ( M,k_{0},h_{0}\right ) }$ if k = Mn + k ₀, h = Mm + h ₀, with $n,m\in \mathbb {Z}$. In this way, we have

$$\displaystyle \begin{aligned} \min_{{z}_{1}\neq{z}_{2}\in\varLambda_{N}^{\left( M,k_{0} ,h_{0}\right) }}\left\vert {z}_{1}-{z}_{2}\right\vert =\frac {M}{N} \end{aligned}$$

for each $\left ( k_{0},h_{0}\right ) \in \left \{ 0,1,\ldots ,M-1\right \} ^{2}$. We have introduced M and the sets $\varLambda _{N}^{\left ( M,k_{0},h_{0}\right ) }$ in order to have that each couple of u _j and u _k in the same class have disjoint supports for r small enough and this is sufficient for our estimates, because it implies that the vector fields are “almost” orthogonal in the sense of Theorem 5. In order to have the supports disjoint for elements of $\varLambda _{N}^{\left ( M,k_{0},h_{0}\right ) }$ and the action of the noise covers the full set D _2δ we ask $r\leq \frac {M}{2N}$. Now we can focus on the vector field w. In order of being divergence free we set w = ∇^⊥ ψ. Thus, we look for a smooth function ψ on $\mathbb {R}^{2}$, compactly supported in $\overline {B\left ( 0,1\right ) }$, close to $\frac {1}{2\pi }\log \left \vert {x}\right \vert$ near x = 0. A possible construction is the following one:

$$\displaystyle \begin{aligned} \psi(x)=\int_{\mathbb{R}^2}\psi_0(x-y)f_{\varepsilon}(y)\ dy\end{aligned}$$

where f _ε is a mollifier with support in B(0, ε) and ψ ₀ is a $C^{\infty }(\mathbb {R}^2 \setminus \{0\})$ radial function such that

$$\displaystyle \begin{aligned} \psi_0(x)=\frac{\operatorname{log}|x|}{2\pi}\ \ \ \text{for}\ \ |x|\leq \frac{1}{3}\ \ \text{and}\ \ \psi_0(x)=0 \ \ \ \text{for}\ \ |x|>\frac{2}{3}.\end{aligned} $$

Moreover, it can be proved that w defined above satisfies

$$\displaystyle \begin{aligned} \left\Vert {w}\right\Vert^2 \leq C\log\frac{1}{\varepsilon}, \qquad \lVert w_r\rVert^2=\lVert w\rVert^2.\end{aligned} $$

Thanks to these relations we can obtain, easily, an estimate of ε _Q

$$\displaystyle \begin{aligned} & \int\int{v}\left( {x}\right) ^{T}Q\left( {x} ,{y}\right) {v}\left( {y}\right) d{x} d{y}=\varGamma^{2}\sum_{{z}\in\varLambda_{N}}\left( \int {w}_{r}\left( {x}-{z}\right) \cdot{v}\left( {x}\right) d{x}\right) ^{2}\\ & =\left\Vert {w}\right\Vert \varGamma^{2}\sum_{\left( k_{0},h_{0}\right) \in\left\{ 0,1,\ldots,M-1\right\} ^{2}}\sum_{{z} \in\varLambda_{N}^{\left( M,k_{0},h_{0}\right) }}\left( \int\frac {{w}_{r}\left( {x}-{z}\right) }{\left\Vert {w}\right\Vert {}_{L^{2}}}\cdot{v}\left( {x}\right) d{x}\right) ^{2}\\ & \leq M^{2}\left\Vert {w}\right\Vert ^{2}\varGamma^{2}\left\Vert {v}\right\Vert ^{2}.\end{aligned} $$

Thus, taking $\varepsilon =\frac {1}{N}$ we get

$$\displaystyle \begin{aligned} \varepsilon_{Q}\leq M^{2}\varGamma^{2}C\log N\end{aligned} $$

which is small if, given N, Γ is small enough.

For what concern the analysis of a lower bound for $\tilde {q}(x)$ in D _2δ, the computations are a bit more involving and we refer to [7] for a complete discussion which is out of our scope. We just claim that if

$$\displaystyle \begin{aligned} \begin{array}{rcl} r\geq \frac{12}{N},\qquad M>24,\qquad N \text{ is large enough} \end{array} \end{aligned} $$

then

$$\displaystyle \begin{aligned} \tilde{q}(x)\geq \frac{\varGamma^2 N}{16 \pi} \qquad \text{in }D_{2\delta}.\end{aligned}$$

4.2 Numerical Simulation

Summing up the results of previous subsection, we have seen that if

$$\displaystyle \begin{aligned} \begin{array}{rcl} r\leq\frac{M}{2N}, \qquad r\leq \delta, \qquad r\geq \frac{12}{N},\qquad \varepsilon=\frac{1}{N}\leq \frac{1}{6},\qquad N \text{ is large enough} \end{array} \end{aligned} $$

we have

$$\displaystyle \begin{aligned} \varepsilon_{Q}\leq M^{2}\varGamma^{2}C\log N, \qquad \tilde{q}(x)\geq \frac{\varGamma^2 N}{16 \pi} \qquad \text{in }D_{2\delta}. \end{aligned}$$

These conditions are strong from the numerical point of view: the cardinality of K must be very large and a finite but not small M is required. Certain supports have to overlap so that the noise acts everywhere. However in [10] it has been shown, numerically, that these conditions are overabundant and much less is required to see the influence of the noise on the solution, namely that Θ _st differs significantly from the parabolic profile even for relatively modest sets K and for M = 1. In this subsection we are working in an infinite 2D channel, suspend the requirement that q, Θ have to decay at infinity, although not strictly covered by the theory described in Sect. 2.1. We assume that the function $q\left ( x\right )$ is equal to a constant q.

For numerical reasons we consider the problem in the bounded domain

$$\displaystyle \begin{aligned} \tilde{D}=(\operatorname{tan}(-1.54),\operatorname{tan}(1.54))\times(-0.1,0.1).\end{aligned}$$

In order to have that the σ _k’s model a fluid at rest, we can take

$$\displaystyle \begin{aligned} r\leq max_{k\in K}d(\partial \tilde{D},x_k)\ \textit{and} \ \ \varepsilon<\frac{1}{6}.\end{aligned}$$

These are the real constraints on the parameters of our numerical simulation. The other parameters Γ, K, {x _k}_{k ∈ K} can be chosen more arbitrarily in order to have satisfactory results.

Differently from [10], here the vortex structures have not been chosen on a grid equally spaced in both directions. In particular the points thicken in the x ₁ direction. We have chosen 2 points in the z direction between − 0.05 and 0.05 and for what concern the x ₁ direction we have chosen 2 points between 0 and 0.2, 4 points between 0.2 and 0.4 and 8 points between 0.4 and 0.6. In order to improve the smoothness of the solution, avoiding a shock in the number of vortices, we prefer to consider some few vortices for x ₁ > 0.6. They only slightly affect the behavior of our solution in the critical region of interest x ₁ < 0.5. Thus we consider 4 points between 0.6 and 0.8 and 2 points between 0.8 and 1. Obviously we avoid repetition of the vortices. In conclusion we have 34 vortices. Moreover, we take r = 0.05, ε = 0.1 and Γ = 0.03. The other parameters of the problem are κ = 0.05 and q ≡ 1. In this way the quantity M and N are not well defined and the impact of the operator $\mathcal {L}$ is related to a small portion of the domain $\tilde {D}$, however we can completely appreciate how it changes the profile of the solution.

Figures 1 and 2 illustrate the modification of the profile, from the standard parabolic one of free diffusion in a steady medium, to the case of turbulent decay. Even if we use just a really reduced number of vortices we can observe a significant decay modification of the profile due to turbulence where vortices thicken.

Fig. 1

Solution in the critical region

Fig. 2

Profiles at different values of x ₁