1 Introduction

A largely used approach in the mathematical studies of hydrodynamic turbulence is to add certain stochastic perturbation to the model. For instance, Yakhot and Orszak [29] suggested that a stochastically perturbed Navier–Stokes system shall possess the same statistical properties as the deterministic Navier–Stokes system with general inhomogeneous boundary conditions. It is expected that turbulence is created at the boundary and therefore by boundary conditions, and, accordingly, the study of the associated boundary layer shall play an essential role. Due to the substantial theoretical difficulties, however, it turns out to be more easily amenable to rigorous analysis to replace boundary conditions by an external stochastic forcing term. This is one of the reasons why the field of stochastic PDEs related to motion of fluids gained a massive importance in the contemporary mathematics research.

Another approach how to observe turbulent behavior of fluids is increasing the Reynolds number, meaning following the so-called vanishing viscosity regime. On the formal level, the Navier–Stokes system then approaches the Euler system and the statistical behavior of the fluid flow shall be therefore reflected in the Euler equations in a certain sense. In the present paper, we investigate the question whether the turbulent randomness observed along the vanishing viscosity limit can be obtained from the Euler system directly by including a suitable stochastic perturbation.

As pointed out, the Euler system is supposed to capture the behavior of real (viscous) fluids in the vanishing viscosity or, more precisely, large Reynolds number regime. The viscosity solutions resulting from this process are in turn considered to be physically relevant. Note that solutions of the (compressible) Euler system are known to develop singularities in a finite time, whereas the initial-value problem is ill-posed in the class of weak solutions. Relevant examples in the incompressible case are discussed by Bressan and Murray [4], Buckmaster and Vicol [5], Elgindi and Jeong [13], and in the compressible case by Chiodaroli et al. [8], De Lellis and Székelyhidi [11], among others. The class of solutions obtained through the vanishing viscosity limit may represent a chance to restore well-posedness of the Euler system in some sense.

A rigorous identification of the vanishing viscosity limit is hampered by two principal difficulties:

  • the absence of sufficiently strong uniform bounds to perform the limit in nonlinear terms;

  • the viscous boundary layer created by the incompatibility of the no-slip boundary condition satisfied by the viscous fluid and the impermeability condition for the Euler system, see the survey by E [12].

The essential role of viscosity in a neighborhood of objects immersed in a fluid in motion is demonstrated by D’Alembert’s paradox, see Stewartson [28]. The interaction of the fluid with the physical boundary in the high Reynolds number regime gives rise to turbulent phenomena observed in real world experiments, see the monograph by Davidson [10] or the nice survey by Bonheure, Gazzola, and Sperone [1], among many others.

Leaving apart the rather complex problem of viscous boundary layer, one may still legitimately ask if the Euler system can be used to describe the vanishing viscosity limit out of the boundary. The appearance of wakes, fluid separation and similar phenomena observed in the turbulent regime suggests to use statistical methods to obtain a rigorous description. Apparently, the fluid is driven by the instabilities in the boundary area; whence one may anticipate the Euler system with complicated oscillatory boundary conditions to be the relevant model. The role of boundary conditions is often replaced by random forcing represented by a stochastic integral. The goal of this paper is to discuss to which extent such a scenario can be rigorously justified.

We consider a compressible, viscous Navier–Stokes fluid in an unbounded domain \(Q \subset R^d\), \(d=2,3\), exterior to a simply connected convex compact set B. We suppose the non-slip boundary conditions on \(\partial B\), and prescribe the far field values of the density \(\varrho \), and the velocity \(\mathbf{u}\) for \(|x| \rightarrow \infty \). The relevant system of equations reads:

$$\begin{aligned}&\partial _t \varrho + {\mathrm{div}}_x(\varrho \mathbf{u}) = 0,\end{aligned}$$
(1.1)
$$\begin{aligned}&\partial _t (\varrho \mathbf{u}) + {\mathrm{div}}_x(\varrho \mathbf{u}\otimes \mathbf{u}) + \nabla _xp(\varrho ) = {\mathrm{div}}_x{\mathbb {S}} (\nabla _x\mathbf{u}), \end{aligned}$$
(1.2)

with the boundary and far field conditions

$$\begin{aligned}&\mathbf{u}|_{\partial Q} = 0,\end{aligned}$$
(1.3)
$$\begin{aligned}&\mathbf{u}\rightarrow \mathbf{u}_\infty , \ \varrho \rightarrow \varrho _\infty \ \text{ as } \ |x| \rightarrow \infty . \end{aligned}$$
(1.4)

The viscous stress \({\mathbb {S}}\) is given by Newton’s rheological law

$$\begin{aligned} {\mathbb {S}} (\nabla _x\mathbf{u}) = \mu \left( \nabla _x\mathbf{u}+ \nabla _x^t \mathbf{u}- \frac{2}{d} {\mathrm{div}}_x\mathbf{u}{\mathbb {I}} \right) + \lambda {\mathrm{div}}_x\mathbf{u}{\mathbb {I}}, \ \mu > 0, \ \lambda \ge 0; \end{aligned}$$
(1.5)

the pressure p is related to the density through the isentropic equation of state

$$\begin{aligned} p(\varrho ) = a \varrho ^\gamma , \ a> 0, \ \gamma > 1. \end{aligned}$$
(1.6)

Our aim is to study the asymptotic behavior of solutions to the problem (1.1)–(1.4) in the vanishing viscosity regime \(\mu \searrow 0\), \(\lambda \searrow 0\). To fix ideas, suppose

$$\begin{aligned} \mu _n = \varepsilon _n \mu ,\ \lambda _n = \varepsilon _n \lambda ,\ \varepsilon _n \rightarrow 0 \ \text{ as }\ n \rightarrow \infty , \end{aligned}$$

and denote the corresponding solution \((\varrho _n, \mathbf{m}_n)\), \(\mathbf{m}_n= \varrho _n\mathbf{u}_n\). Extending the functions \(\mathbf{u}_\infty \), \(\varrho _\infty \) inside Q, \(\mathbf{u}_\infty |_{\partial Q} = 0\), we introduce the relative energy

$$\begin{aligned} E\left( \varrho , \mathbf{u}\ \Big | \varrho _\infty , \mathbf{u}_\infty \right) {=} \frac{1}{2} \varrho |\mathbf{u}{-} \mathbf{u}_\infty |^2 {+} P(\varrho ) - P'(\varrho _\infty ) (\varrho - \varrho _\infty ) - P(\varrho _\infty ),\ P(\varrho ) = \frac{a}{\gamma - 1} \varrho ^\gamma . \end{aligned}$$

Motivated by the numerical experiments of Elling [14, 15], we describe the vanishing viscosity limit in a statistical way. To this end, we introduce a suitable trajectory space \({\mathcal {T}} \) in which the solutions \((\varrho _n, \mathbf{m}_n)\) live and denote \({\mathfrak {P}}({\mathcal {T}})\) the set of Borel probability measures on \({\mathcal {T}}\). The Cesàro averages

$$\begin{aligned} {\mathcal {V}}_N = \frac{1}{N} \sum _{n=1}^N \delta _{(\varrho _n, \mathbf{m}_n)} \in {\mathfrak {P}} ({\mathcal {T}}),\ \delta _h \ \text{-Dirac } \text{ mass } \text{ centered } \text{ at } \ h, \end{aligned}$$

represent statistical solutions of the Navier–Stokes system in the vanishing viscosity regime. To perform the asymptotic limit \(N \rightarrow \infty \), we suppose the energy bound

$$\begin{aligned} \frac{1}{N} \sum _{n=1}^N \left[ \sup _{t \in (0,T) } \int _{{Q}} E\left( \varrho _n, \mathbf{u}_n\Big | \varrho _\infty , \mathbf{u}_\infty \right) \ \,{\mathrm{d}} {x} + \varepsilon _n \int _0^T \int _{{Q}} {\mathbb {S}} (\nabla _x\mathbf{u}_n) : \nabla _x\mathbf{u}_n \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \le \overline{{\mathcal {E}}}\nonumber \\ \end{aligned}$$
(1.7)

uniformly for \(N \rightarrow \infty \).

Our programme consists of several steps. First, we show that the family of measures \({\mathcal {V}}_N\) is tight; whence

$$\begin{aligned} {\mathcal {V}}_N \rightarrow {\mathcal {V}} \ \text{ narrowly } \text{ in } \ {\mathfrak {P}}({\mathcal {T}}), \end{aligned}$$

at least for a suitable subsequence. Then, we denote by \((r,\mathbf{w})\) the canonical process on \({\mathcal {T}}\) (see Sect. 3.1 for more details) and introduce the following notion of statistical equivalence.

Definition 1.1

(Statistical equivalence) Let \({\mathcal {P}}_{i}\), \(i=1,2\), be Borel probability measures on \({\mathcal {T}}\). We say that the two measures are statistically equivalent if the following holds:

  • Equality of expectations of density and momentum.

    $$\begin{aligned} \begin{aligned} {\mathbb {E}}_{{\mathcal {P}}_{1}}\left[ \int _0^T \int _{{Q}} r \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right]&= {\mathbb {E}}_{{{\mathcal {P}}_{2}}}\left[ \int _0^T \int _{{Q}} r \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] ,\\ {\mathbb {E}}_{{\mathcal {P}}_{1}}\left[ \int _0^T \int _{{Q}} \mathbf{w} \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right]&= {\mathbb {E}}_{{{\mathcal {P}}_{2}}}\left[ \int _0^T \int _{{Q}} \mathbf{w} \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] , \end{aligned} \end{aligned}$$
    (1.8)

    for any \(\varphi \in C^\infty _c((0,T) \times Q)\), \(\varvec{\varphi }\in C^\infty _c((0,T) \times Q; R^d)\).

  • Equality of expectations of kinetic, internal and angular energy.

    $$\begin{aligned} \begin{aligned} {\mathbb {E}}_{{\mathcal {P}}_{1}}\left[ \int _0^T \int _{{Q}} 1_{r> 0} \frac{|\mathbf{w} |^2}{r} \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right]&= {\mathbb {E}}_{{\mathcal {P}}_{2}}\left[ \int _0^T \int _{{Q}} 1_{r> 0} \frac{|\mathbf{w} |^2}{r} \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] , \\ {\mathbb {E}}_{{\mathcal {P}}_{1}}\left[ \int _0^T \int _{{Q}} P(r) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right]&= {\mathbb {E}}_{{{\mathcal {P}}_{2}}}\left[ \int _0^T \int _{{Q}} P(r) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] ,\\ {\mathbb {E}}_{{\mathcal {P}}_{1}}\left[ \int _0^T \int _{{Q}} 1_{r> 0} \frac{1}{r} ({\mathbb {J}}_{x_0} \cdot \mathbf{w} ) \cdot \mathbf{w} \ \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right]&= {\mathbb {E}}_{{{\mathcal {P}}_{2}}}\left[ \int _0^T \int _{{Q}} 1_{r > 0} \frac{1}{r} ({\mathbb {J}}_{x_0} \cdot \mathbf{w} ) \cdot \mathbf{w} \ \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] , \end{aligned}\nonumber \\ \end{aligned}$$
    (1.9)

    for any \(x_0 \in R^d\), and any \(\varphi \in C^\infty _c((0,T) \times Q)\), where

    $$\begin{aligned} {\mathbb {J}}_{x_0}(x) = |x - x_0|^2 {\mathbb {I}} - (x - x_0) \otimes (x - x_0). \end{aligned}$$

In (1.8) and (1.9), we tacitly assume that all the expectations are finite. In the same spirit, we say that two stochastic processes defined on possibly different probability spaces are statistically equivalent, if their probability laws are statistically equivalent.

Remark 1.2

The reader may have expected the statistical equivalence of two stochastic processes to be defined as their equivalence in law. We consider instead a much weaker notion, which postulates only equality of expectations of certain statistically and physically relevant quantities characterizing the fluid flow. In particular, expectations of other quantities or higher order moments do not need to coincide.

Next, we suppose that the limit \({\mathcal {V}}\) is statistically equivalent to the probability law induced by a process \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\), which is a solution of the stochastically driven Euler equation. Namely, the process \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) solves in the sense of distributions the system

$$\begin{aligned} \begin{aligned}&{\mathrm{d}}{{\tilde{\varrho }}}+ {\mathrm{div}}_x\widetilde{\mathbf{m}}\,{\mathrm{d}} t = 0, \\&{\mathrm{d}}\widetilde{\mathbf{m}}+ {\mathrm{div}}_x\left( \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}} \right) \,{\mathrm{d}} t + \nabla _xp({{\tilde{\varrho }}}) \,{\mathrm{d}} t = \mathbf{F} {\mathrm{d}}W, \end{aligned} \end{aligned}$$
(1.10)

where

$$\begin{aligned} \mathbf{F} {\mathrm{d}}W = \sum _{k \ge 1} \mathbf{F}_k {\mathrm{d}}W_k \end{aligned}$$

with a family of diffusion coefficients \((\mathbf{F}_k )_{k \ge 1}\) satisfying a suitable stochastic integrability assumption and a cylindrical Wiener process \((W_k)_{k \ge 1}\).

Remark 1.3

It becomes clear at this point that a stronger definition of statistical equivalence, such as equality in law, would be too restrictive and would immediately rule out the possibility of modeling the inviscid limit by the stochastic Euler system. Indeed it will be seen in the analysis below that the limit measure \({\mathcal {V}}\) is supported on momenta of bounded variation in time with values in some negative Sobolev space. But this is not the case for \(\widetilde{\mathbf{m}}\) satisfying (1.10) due to the limited regularity of the stochastic integral, unless \({\mathbf {F}}\equiv 0\). However, the statistical equivalence in the sense of Definition 1.1, specifically, equality of only several chosen moments, is a priori not excluded.

Similarly, note that strengthening the statistical bound (1.7) to

$$\begin{aligned} \sup _{n\in {\mathbb {N}}}\left[ \sup _{t \in (0,T) } \int _{{Q}} E\left( \varrho _n, \mathbf{u}_n\Big | \varrho _\infty , \mathbf{u}_\infty \right) \ \,{\mathrm{d}} {x} + \varepsilon _n \int _0^T \int _{{Q}} {\mathbb {S}} (\nabla _x\mathbf{u}_n) : \nabla _x\mathbf{u}_n \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \le \overline{{\mathcal {E}}} \end{aligned}$$

would imply an essentially uniform bound for the limit process, which is also not compatible with solutions to the stochastic Euler system (1.10). This motivates the more natural and essentially weaker statistically uniform bound in the sense of the average in (1.7).

Let \((\varrho ,\mathbf{m})\) be a Skorokhod representation of the limit measure \({\mathcal {V}}\). Our main result asserts that validity of (1.10) necessarily implies:

  1. (1)

    \((\varrho , \mathbf{m})\) is a weak statistical solution of the deterministic Euler system, in particular, the genuinely stochastic model becomes irrelevant.

  2. (2)

    The sequence \((\varrho _n, \mathbf{m}_n)\) S-converges in the sense of [17] to a parametrized measure

    $$\begin{aligned}&\widetilde{{\mathcal {V}}} \in L^\infty _{\mathrm{weak-(*)}}([0,T] \times Q; {\mathfrak {P}}[R^{d+1}]),\\&\quad \int _0^T \int _{{Q}} \left\langle \widetilde{{\mathcal {V}}}_{t,x}; b \right\rangle \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t = {\mathbb {E}} \left[ \int _0^T \int _{{Q}} b(\varrho , \mathbf{m}) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\quad \text{ for } \text{ any }\ b \in C_c(R^{d + 1}), \ \text{ and } \text{ any } \ \varphi \in C_c((0,T) \times Q). \end{aligned}$$
  3. (3)

    If, in addition, the barycenter

    $$\begin{aligned} ({\overline{\varrho }}, {\overline{\mathbf{m}}}) = \int _{{\mathcal {T}}} (r, \mathbf{w}) {\mathrm{d}}{\mathcal {V}} \in {\mathcal {T}} \end{aligned}$$

    is a weak solution to the Euler system, then the limit is deterministic

    $$\begin{aligned} {\mathcal {V}} = \delta _{({\overline{\varrho }}, {\overline{\mathbf{m}}})}, \end{aligned}$$

    and the sequence \((\varrho _n, \mathbf{m}_n)\) statistically converges to \(({\overline{\varrho }}, {\overline{\mathbf{m}}})\), specifically,

    $$\begin{aligned} \frac{1}{N} \# \left\{ n \le N \Big | \Vert \varrho _n- {\overline{\varrho }} \Vert _{L^\gamma (K)} + \Vert \mathbf{m}_n- {\overline{\mathbf{m}}} \Vert _{L^{\frac{2 \gamma }{\gamma + 1}}(K; R^d)} > \varepsilon \right\} \rightarrow 0 \ \text{ as }\ N \rightarrow \infty \nonumber \\ \end{aligned}$$
    (1.11)

    for any \(\varepsilon > 0\), and any compact \(K \subset [0,T] \times Q\).

The result can be extended to the driving force

$$\begin{aligned} \sigma \cdot \nabla _x\mathbf{m}\circ {\mathrm{d}}W_1 + \mathbf{F} {\mathrm{d}}W_2 \end{aligned}$$

in the absence of the obstacle.

As a corollary, we obtain the dichotomy proved in [18] in the case \(Q = R^d\): If

$$\begin{aligned} \varrho _n \rightarrow \varrho \ \text{ weakly-(*) } \text{ in } \ L^\infty (0,T; L^\gamma _{\mathrm{loc}} (Q)), \ \mathbf{m}_n \rightarrow \mathbf{m}\ \text{ weakly-(*) } \text{ in } \ L^\infty (0,T; L^{\frac{2 \gamma }{\gamma + 1}}_{\mathrm{loc}} (Q)), \end{aligned}$$

then either

$$\begin{aligned} \varrho _n \rightarrow \varrho \ \text{ in } \ L^\gamma _{\mathrm{loc}}([0,T] \times Q),\ \mathbf{m}_n \rightarrow \mathbf{m}\ \text{ in } \ L^{\frac{2 \gamma }{\gamma + 1}}_{\mathrm{loc}}([0,T] \times Q; R^d), \end{aligned}$$

or \((\varrho , \mathbf{m})\) is not a solution of the Euler system (1.10) (with \({\mathbf {F}} = 0\).)

To summarize, the above results can be interpreted in the following way. If we adopt the statistical limit as our working hypothesis, then the limit fluid motion is never statistically equivalent to a stochastic Euler system. If, in addition, we accept the Kolmogorov hypothesis advocated by Chen and Glimm [7] for compressible turbulence (in the case \(Q = R^d\)), then the S-convergence is the right tool to identify the limit. Last, if the expected value (barycenter) of the limit statistical solution solves the Euler system, then the statistical limit is a single deterministic solution. It is worth noting that the results are independent of a specific choice of the initial data as well as the boundary conditions on \(\partial Q\). This fact gives rise to full flexibility to cover all the physically relevant situations. In addition, we do not postulate any form of energy balance, neither on the approximate nor on the limit level. In the light of the experimental evidence, cf. [1], our results strongly indicate that a stochastically driven Euler system is not a relevant model of compressible turbulence driven by a rigid obstacle. Indeed the graphic material collected in [1] is more reminiscent of the weak rather than strong convergence in the high Reynolds number regime, which, in view of the above results, excludes the Euler system to describe the statistical limit.

The paper is organized as follows. In Sect. 2, we collect the available results concerning the Navier–Stokes problem (1.1)–(1.6). In Sect. 3, we identify the measure \({\mathcal {V}}\)—the statistical limit of the sequence \((\varrho _n, \mathbf{m}_n)_{n \ge 1}\). In Sect. 4, we associate to the statistical limit \({\mathcal {V}}\) a defect measure—Reynolds stress tensor \({\mathfrak {R}}\)—characterizing possible oscillations and concentrations created in the limit process. It turns out that \({\mathfrak {R}}\) is a positive semi—definite tensor-valued finite measure on Q. Section 5 is the heart of the paper. We show that \({\mathfrak {R}}\) vanishes as long as the obstacle B is a convex set, and then we examine the statistical convergence in the vanishing viscosity limit. The paper is concluded by a short discussion in Sect. 6.

2 Navier–Stokes System in Exterior Domain

We recall the available results for the Navier–Stokes system (1.1)–(1.6). For the sake of simplicity, we suppose that \(\partial Q\) is regular and \(B = R^d \setminus Q\) is a simply connected compact set. In addition, we suppose that

$$\begin{aligned} \varrho _\infty > 0,\ \mathbf{u}_\infty \in R \ \text{ are } \text{ given } \text{ constant } \text{ fields. } \end{aligned}$$
(2.1)

Accordingly, we may extend \(\mathbf{u}_\infty \) inside Q in such a way that

$$\begin{aligned} \mathbf{u}_\infty \in C^\infty (Q; R^d), \ \mathbf{u}_\infty (x) = 0 \ \text{ for } \ |x| < L,\ \mathbf{u}_\infty (x) = \mathbf{u}_\infty \ \text{ for } \ |x| > 2 L \end{aligned}$$
(2.2)

for some \(L > 0\).

2.1 Weak Solutions

First we introduce finite energy weak solution to the problem (1.1)–(1.6). The total energy defined as

$$\begin{aligned} E\left( \varrho , \mathbf{m}\right) = \left\{ \begin{array}{l} \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \ \text{ if } \ \varrho > 0, \\ 0 \ \text{ if } \ \varrho = 0,\ \mathbf{m}= 0, \\ \infty \ \text{ otherwise } \end{array} \right. \ \mathbf{m}= \varrho \mathbf{u}\end{aligned}$$

is a convex l.s.c. function of \((\varrho , \mathbf{m}) \in R^{d + 1}\). In view of the non-zero far-field conditions, it is more convenient to consider the relative energy

$$\begin{aligned} E\left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{u}_\infty \right) = \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } - \mathbf{m}\cdot \mathbf{u}_\infty + \frac{1}{2} \varrho |\mathbf{u}_\infty |^2 + P(\varrho ) - P'(\varrho _\infty ) (\varrho - \varrho _\infty ) - P(\varrho _\infty ). \end{aligned}$$

As E is convex, the relative energy can be interpreted as the so-called Bregman distance between \((\varrho , \mathbf{m})\) and \((\varrho _\infty , \mathbf{m}_\infty )\), \(\mathbf{m}_\infty = \varrho _\infty \mathbf{u}_\infty \), specifically

$$\begin{aligned} E\left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{u}_\infty \right) = E(\varrho , \mathbf{m}) - \partial _{\varrho , \mathbf{m}} E(\varrho _\infty , \mathbf{m}_\infty ) \cdot (\varrho - \varrho _\infty , \mathbf{m}- \mathbf{m}_\infty ) - E(\varrho _\infty , \mathbf{m}_\infty ). \end{aligned}$$

Definition 2.1

(Weak solution to Navier–Stokes system) We say that \((\varrho , \mathbf{u})\) is finite energy weak solution to the Navier–Stokes system (1.1)–(1.6) in \((0,T) \times Q\) if:

  • Finite energy and dissipation rate

    $$\begin{aligned} \int _{{Q}} E\Big (\varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{u}_\infty \Big ) (\tau , \cdot ) \ \,{\mathrm{d}} {x} + \int _0^\tau \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}) : \nabla _x\mathbf{u} \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t < \infty \end{aligned}$$
    (2.3)

    holds for any \(0 < \tau \le T\);

  • Equation of continuity

    $$\begin{aligned} \left[ \int _{{Q}} \varrho \varphi \ \,{\mathrm{d}} {x} \right] _{t=\tau _1}^{t= \tau _2} = \int _{\tau _1}^{\tau _2} \int _{{Q}} \Big [ \varrho \partial _t \varphi + \varrho \mathbf{u}\cdot \nabla _x\varphi \Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \end{aligned}$$

    holds for any \(0< \tau _1< \tau _2 < T\) and \(\varphi \in C^1_c ((0,T) \times Q)\);

  • Momentum equation

    $$\begin{aligned} \begin{aligned} \left[ \int _{{Q}} \varrho \mathbf{u}\cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \right] _{ t = \tau _1 }^{t = \tau _2} = \int _{\tau _1}^{\tau _2} \int _{{Q}} \Big [ \varrho \mathbf{u}\cdot \partial _t \varvec{\varphi }+ (\varrho \mathbf{u}\otimes \mathbf{u}) : \nabla _x\varvec{\varphi }+ p(\varrho ) {\mathrm{div}}_x\varvec{\varphi }- {\mathbb {S}} (\nabla _x\mathbf{u}) : \nabla _x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \end{aligned} \end{aligned}$$

    holds for any \(0< \tau _1< \tau _2 < T\) and \(\varvec{\varphi }\in C^1_c ((0,T) \times Q; R^d)\).

Note that boundedness of the total energy and the dissipation rate stated in (2.3) yields the natural bound \(\varrho \ge 0\) and renders all integrals in the weak formulation finite. The definition is usually supplemented with the renormalized equation of continuity as well as the compatibility condition

$$\begin{aligned} (\mathbf{u}- \mathbf{u}_\infty ) \in L^2(0,T; D^{1,2}(Q; R^d) ) \end{aligned}$$
(2.4)

where \(D^{1,2}\) denotes the homogeneous Sobolev space. The energy bound (2.3) then follows from the associated energy inequality

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}}}{ \,{\mathrm{d}} t }&\int _{{Q}} E \left( \varrho , \mathbf{u}\Big | \varrho _\infty , \mathbf{u}_\infty \right) \ \,{\mathrm{d}} {x} + \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}) : \nabla _x\mathbf{u} \ \,{\mathrm{d}} {x} \\&\le -\int _{{Q}} \Big ( \varrho \mathbf{u}\otimes \mathbf{u}+ p(\varrho ) {\mathbb {I}} \Big ) : \nabla _x\mathbf{u}_\infty \ \,{\mathrm{d}} {x} + \frac{1}{2} \int _{{Q}} \varrho \mathbf{u}\cdot \nabla _x|\mathbf{u}_\infty |^2 \ \,{\mathrm{d}} {x} \\&\quad + \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}) : \nabla _x\mathbf{u}_\infty \ \,{\mathrm{d}} {x}. \end{aligned} \end{aligned}$$

We refer the reader to the monographs of Lions [23], Novotný and Straškraba [25, Section 7.12.6] or Kračmar, Nečasová, and Novotný [22] for the relevant existence results for the initial-value problem under various restrictions imposed on the adiabatic coefficient \(\gamma \).

Here we consider families of solutions that may fail to satisfy (2.4), meaning we do not really specify the boundary behavior of the velocity on the rigid body. Similarly, we also ignore the behavior of the initial state. The only piece of information necessary for our analysis is the uniform bound (2.3).

3 Statistical Limit

For a given sequence of \(\varepsilon _n \rightarrow 0\) we consider the vanishing viscosity limit

$$\begin{aligned} \mu _n = \varepsilon _n \mu , \ \lambda _n = \varepsilon _n \lambda , \ \mu > 0, \ \lambda \ge 0. \end{aligned}$$

We suppose that the Navier–Stokes system admits a related sequence of weak solutions \((\varrho _n, \mathbf{u}_n)_{n=1}^\infty \) in the sense of Definition 2.1. Our goal is to study the statistical properties of \((\varrho _n, \mathbf{u}_n)_{n=1}^\infty \). To this end, we associate to this sequence a family of measures \({\mathcal {V}}_N\) supported on the trajectory space

$$\begin{aligned} {\mathcal {T}}= & {} C_{\mathrm{weak}}([0,T]; L^\gamma _{\mathrm{loc}}(Q) \times L^{\frac{2 \gamma }{\gamma + 1}}_{\mathrm{loc}}(Q; R^d)),\nonumber \\ {\mathcal {V}}_N= & {} \frac{1}{N} \sum _{n=1}^N \delta _{(\varrho _n, \mathbf{m}_n)},\ \mathbf{m}_n = \varrho _n \mathbf{u}_n, \end{aligned}$$
(3.1)

where \(\delta \) denotes the Dirac mass. Note that any finite energy weak solution belongs to \({\mathcal {T}}\). Moreover, motivated by the energy bound (2.3), we assume

$$\begin{aligned} \frac{1}{N} \sum _{n=1}^N \left[ \sup _{0 \le \tau \le T}\int _{{Q}} E\left( \varrho _n, \mathbf{m}_n \Big | \varrho _\infty , \mathbf{u}_\infty \right) (\tau , \cdot ) \ \,{\mathrm{d}} {x} + \varepsilon _n \int _0^T \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}_n) : \nabla _x\mathbf{u}_n \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \le \overline{{\mathcal {E}}} \end{aligned}$$
(3.2)

uniformly for \(N \rightarrow \infty \). Note in particular that we do not assume a uniform bound of the form

$$\begin{aligned} \sup _{n}\left[ \sup _{0 \le \tau \le T}\int _{{Q}} E\left( \varrho _n, \mathbf{m}_n \Big | \varrho _\infty , \mathbf{u}_\infty \right) (\tau , \cdot ) \ \,{\mathrm{d}} {x} + \varepsilon _n \int _0^T \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}_n) : \nabla _x\mathbf{u}_n \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] <\infty . \end{aligned}$$

In fact, such an assumption would render parts of our analysis below trivial. We consider instead the weaker statistically uniform bound in the sense of the average in (3.2), which is more natural for the problem under consideration.

3.1 Statistical Convergence

Our goal is to show

$$\begin{aligned} {\mathcal {V}}_N \rightarrow {\mathcal {V}} \ \text{ narrowly } \text{ in } \ {\mathfrak {P}}[ {\mathcal {T}} ] \end{aligned}$$
(3.3)

at least for a suitable subsequence \(N_k \rightarrow \infty \) as \(k \rightarrow \infty \). According to Prokhorov theorem it is enough to show that \(({\mathcal {V}}_{N})_{N \ge 1}\) is tight. To this end, we denote by \((r,\mathbf{w})\) the canonical process on \({\mathcal {T}}\), that is,

$$\begin{aligned} (r,\mathbf{w}):{\mathcal {T}}\rightarrow {\mathcal {T}},\ (r,\mathbf{w})(\omega ,t)=\omega (t)\ \text{ for } \ \omega \in {\mathcal {T}},\, t\in [0,T]. \end{aligned}$$

We note that the velocity \(\mathbf{v}\) satisfying \(\mathbf{w}=r\mathbf{v}\) is well defined under each \({\mathcal {V}}_{N}\). For a probability measure \({\mathcal {Q}}\in {\mathfrak {P}}({\mathcal {T}})\), we denote by \({\mathbb {E}}_{{\mathcal {Q}}}\) the expected value in the probability space \(({\mathcal {T}},{\mathfrak {B}}[{\mathcal {T}}],{\mathcal {Q}})\). From Sect. 4 on, we also use the notation \({\mathbb {E}}\) without any subscript to denote the expectation on the standard probability space \((\Omega ,{\mathfrak {B}},{\mathcal {P}})\).

Lemma 3.1

Under the hypothesis (3.2), the family \(({\mathcal {V}}_{N})_{N \ge 1}\) is tight in \({\mathfrak {P}}[{\mathcal {T}}]\).

Proof

First observe, by virtue of Young’s inequality,

$$\begin{aligned} |\mathbf{w}|^{\frac{2 \gamma }{\gamma + 1}} + r^\gamma \lesssim E(r, \mathbf{w}). \end{aligned}$$
(3.4)

Consequently, it follows from (3.2) that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}_{{\mathcal {V}}_N} \left[ \sup _{0 \le \tau \le T} \int _K |\mathbf{w}|^{\frac{2 \gamma }{\gamma + 1}}\ \,{\mathrm{d}} {x}+ \sup _{0 \le \tau \le T} \int _K r^\gamma \ \,{\mathrm{d}} {x}\right] \\&\quad = \frac{1}{N}\sum _{n=1}^{N} \left[ \sup _{0 \le \tau \le T} \int _K |\mathbf{m}_{n}|^{\frac{2 \gamma }{\gamma + 1}}\ \,{\mathrm{d}} {x}+ \sup _{0 \le \tau \le T} \int _K \varrho _{n}^\gamma \ \,{\mathrm{d}} {x}\right] \le c (K, \overline{{\mathcal {E}}}) \end{aligned} \end{aligned}$$
(3.5)

for any compact \(K \subset \overline{Q}\).

To complete the proof, it is enough to show uniform bounds on the modulus of continuity of processes

$$\begin{aligned} t \in [0,T] \mapsto \int _{{Q}} r(t) \varphi \ \,{\mathrm{d}} {x},\ t \in [0,T] \mapsto \int _{{Q}} \mathbf{w}(t) \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x},\ \varphi \in C^1_c(Q),\ \varvec{\varphi }\in C^1_c (Q; R^d), \end{aligned}$$

under \({\mathcal {V}}_{N}\). As \(\varrho _n\), \(\mathbf{m}_n\) satisfy the equation of continuity, we deduce from (3.5)

$$\begin{aligned} {\mathbb {E}}_{{\mathcal {V}}_N} \left[ \sup _{0\le \tau _{1}<\tau _{2}\le T} \frac{\left| \int _{{Q}} \Big [ r(\tau _2, \cdot ) - r(\tau _1, \cdot ) \Big ] \varphi \ \,{\mathrm{d}} {x} \right| }{|\tau _{2}-\tau _{1}|}\right] \le c(\varphi , \overline{{\mathcal {E}}}). \end{aligned}$$
(3.6)

Similarly, the momentum equation yields

$$\begin{aligned} {\mathbb {E}}_{{\mathcal {V}}_N} \left[ \sup _{0\le \tau _{1}<\tau _{2}\le T} \frac{\left| \int _{{Q}} \Big [ \mathbf{w}(\tau _2, \cdot ) - \mathbf{w}(\tau _1, \cdot ) \Big ] \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \right| }{|\tau _{2}-\tau _{1}|^{\frac{1}{2}}}\right] \le c(\varvec{\varphi }, \overline{{\mathcal {E}}}). \end{aligned}$$
(3.7)

Here, we have used

$$\begin{aligned} \Vert \sqrt{\varepsilon _{n}} {\mathbb {S}}(\nabla _x\mathbf{u}_n) \Vert ^{2}_{L^2(Q; R^{d \times d})} \lesssim \varepsilon _{n}\max \{\mu ,\lambda \} \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}_n) : \nabla _x\mathbf{u}_n \ \,{\mathrm{d}} {x}. \end{aligned}$$
(3.8)

\(\square \)

As the weak topology is not metrizable, the space \({\mathcal {T}}\) is not a Polish space and Prokhorov theorem does not directly apply. However, \({\mathcal {T}}\) is a sub-Polish space in the sense of [3, Definition 2.1.3]. Therefore, the Jakubowski–Skorokhod theorem [21] implies (3.3), at least for a suitable subsequence. In particular,

$$\begin{aligned} \begin{aligned}&\frac{1}{N_{k}} \sum _{n=1}^{N_k} b \left( \int _{0}^{T}\int _{{Q}} \varrho _n \varphi _1 \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t , \dots , \int _{0}^{T}\int _{{Q}} \varrho _n \varphi _{m_{1}} \ \,{\mathrm{d}} {x}\,{\mathrm{d}} t , \right. \\&\quad \left. \int _{0}^{T}\int _{{Q}} \mathbf{m}_n \cdot \varvec{\varphi }_1 \ \,{\mathrm{d}} {x}\,{\mathrm{d}} t , \dots , \int _{0}^{T}\int _{{Q}} \mathbf{m}_n \cdot \varvec{\varphi }_{m_{2}} \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right) \\&\quad \rightarrow {\mathbb {E}}_{\mathcal {V}} \left[ b \left( \int _{{Q}} r \varphi _1 \ \,{\mathrm{d}} {x} , \dots , \int _{{Q}} r \varphi _{m_{1}} \ \,{\mathrm{d}} {x}, \int _{{Q}} \mathbf{w} \cdot \varvec{\varphi }_1 \ \,{\mathrm{d}} {x} , \dots , \int _{{Q}} \mathbf{w} \cdot \varvec{\varphi }_{m_{2}} \ \,{\mathrm{d}} {x} \right) \right] \ \text{ as }\ k \rightarrow \infty \end{aligned} \end{aligned}$$
(3.9)

for any \(\varphi _i \in C^1_c(Q)\), \(i=1,\dots , m_{1}\), \(\varvec{\varphi }_i \in C^1_c(Q; R^d)\), \(i=1,\dots , m_{2}\), \(m_{1},m_{2}\in {\mathbb {N}}\), and any \(b \in C_c \left( R^{m_{1}+m_{2}} \right) \). In addition, we may also define the barycenter of \({\mathcal {V}}\) on the trajectory space,

$$\begin{aligned}&({\overline{\varrho }}, {\overline{\mathbf{m}}}) = {\mathbb {E}}_{{\mathcal {V}}}\left[ (r, \mathbf{w}) \right] ,\\&\int _0^T \int _{{Q}} {\overline{\varrho }} \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t = {\mathbb {E}}_{{\mathcal {V}}} \left[ \int _0^T \int _{{Q}} r \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] = \lim _{k \rightarrow \infty } \frac{1}{N_k} \sum _{n=1}^{N_k} \left[ \int _0^T \int _{{Q}} \varrho _n\varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\int _0^T \int _{{Q}} {\overline{\mathbf{m}}} \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t = {\mathbb {E}}_{{\mathcal {V}}} \left[ \int _0^T \int _{{Q}} \mathbf{w} \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] = \lim _{k \rightarrow \infty } \frac{1}{N_k} \sum _{n=1}^{N_k} \left[ \int _0^T \int _{{Q}} \mathbf{m}_n\cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\quad \text{ for } \text{ any } \ \varphi \in C^1_c((0,T) \times Q), \ \varvec{\varphi }\in C^1_c ((0,T) \times Q; R^d). \end{aligned}$$

Summarizing we obtain:

Proposition 3.2

(Statistical vanishing viscosity limit) Let \((\varrho _n, \mathbf{u}_n)_{n=1}^\infty \) be a sequence of weak solutions to the Navier–Stokes system in the sense of Definition 2.1 with the viscosity coefficients

$$\begin{aligned} \mu _n = \varepsilon _n \mu , \ \lambda _n = \varepsilon _n \lambda , \ \mu > 0, \ \lambda \ge 0, \ \varepsilon _n \!\searrow 0. \end{aligned}$$

Suppose the Cesàro averages of the total (relative) energy are bounded as in (3.2).

Then there exists \(N_k \rightarrow \infty \) and a probability measure \({\mathcal {V}} \in {\mathfrak {P}}[{\mathcal {T}}]\) on the trajectory space \({\mathcal {T}}\) such that (3.9) holds.

4 Reynolds Defect

In order to identify the statistical limit \({\mathcal {V}}\) with a stochastic process, we apply Skorokhod representation theorem, or rather its generalized version by Jakubowski [21]. It is convenient to extend the class of variables \((\varrho _n, \mathbf{m}_n)\) to their nonlinear composition appearing in both the relative energy and the momentum equation. Specifically, we consider the quantities

$$\begin{aligned} 1_{\varrho _n > 0} \frac{\mathbf{m}_n\otimes \mathbf{m}_n}{\varrho _n} \in L^\infty _{\mathrm{weak-(*) }} (0,T; {\mathcal {M}}(Q; R^{d \times d}_{\mathrm{sym}} )) ,\ \text{ and } \ p(\varrho _n) \in L^\infty _{\mathrm{weak-(*)}} (0,T; {\mathcal {M}}(Q)), \end{aligned}$$
(4.1)

where the symbol \({\mathcal {M}}\) denotes the space of all Radon (not necessarily finite) measures on Q. In view of Riesz representation theorem (see e.g. Rudin [27, Chapter 2, Theorem 2.14]), we have

$$\begin{aligned} C_{c}(Q,R_{\mathrm{sym}}^{ d\times d})^* = {\mathcal {M}}(Q; R_{\mathrm{sym}}^{ d\times d}). \end{aligned}$$

In addition, as the space \(C_{c}(Q,R_{\mathrm{sym}}^{ d\times d})\) is separable, we conclude

$$\begin{aligned} L^\infty _{\mathrm{weak-(*)}} (0,T; {\mathcal {M}}(Q; R^{d \times d}_{\mathrm{sym}} )) = \big ( L^{1}(0,T; C_{c}(Q,R^{d\times d}_{\mathrm{sym}}))\big )^{*}, \end{aligned}$$

see e.g. Pedregal [26, Chapter 6, Theorem 6.14]. Consequently, the function spaces in (4.1) satisfy the assumptions of Jakubowski’s theorem [21]. We consider the extended measure

$$\begin{aligned} \begin{aligned} \overline{{\mathcal {V}}}_N&= \frac{1}{N} \sum _{n=1}^N \delta _{ \left[ (\varrho _n, \mathbf{m}_n), \ 1_{\varrho _n > 0} \frac{\mathbf{m}_n\otimes \mathbf{m}_n}{\varrho _n},\ p(\varrho _n) \right] }\\&\in {\mathfrak {P}} \left[ {\mathcal {T}} \times L^\infty _{\mathrm{weak-(*)}} (0,T; {\mathcal {M}}(Q; R^{d \times d}_{\mathrm{sym}} )) \times L^\infty _{\mathrm{weak-(*)}} (0,T; {\mathcal {M}}(Q)) \right] . \end{aligned} \end{aligned}$$

It follows from the energy bound (3.2) and Lemma 3.1 that the family \((\overline{{\mathcal {V}}}_N)_{N \ge 0}\) is tight. Now, we apply the version of Skorokhod representation theorem due to Jakubowski [21]. Hence there is a subsequence \(N_{k}\rightarrow \infty \) and a family of random variables

$$\begin{aligned} ({{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k)\in & {} {\mathcal {T}},\ 1_{{{\tilde{\varrho }}}_k > 0} \frac{\widetilde{\mathbf{m}}_k \otimes \widetilde{\mathbf{m}}_k}{{{\tilde{\varrho }}}_k} \in L^\infty _{\mathrm{weak - (*)}}(0,T; {\mathcal {M}}(Q; R^{d \times d}_{\mathrm{sym}})),\\ \ p({{\tilde{\varrho }}}_k)\in & {} L^\infty _{\mathrm{weak - (*)}}(0,T; {\mathcal {M}}(Q)) \end{aligned}$$

defined on the standard probability space \((\Omega , {\mathfrak {B}}, {\mathcal {P}})\) with the law \(\overline{{\mathcal {V}}}_{N_{k}}\) and such that

$$\begin{aligned} ({{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k)&\rightarrow (\varrho , \mathbf{m}) \ \text{ in } \ {\mathcal {T}} \ {\mathcal {P}}- \text{ a.s. }, \nonumber \\ 1_{{{\tilde{\varrho }}}_k> 0} \frac{\widetilde{\mathbf{m}}_k \otimes \widetilde{\mathbf{m}}_k}{{{\tilde{\varrho }}}_k}&\rightarrow \overline{ \left[ 1_{\varrho > 0} \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho } \right] } \ \text{ weakly-(*) } \text{ in }\ L^\infty _{\mathrm{weak - (*)}}(0,T; {\mathcal {M}}(Q; R^{d \times d}_{\mathrm{sym}})) \ {\mathcal {P}}- \text{ a.s. }, \nonumber \\ p({{\tilde{\varrho }}}_k)&\rightarrow \overline{p(\varrho )} \ \text{ weakly-(*) } \text{ in } \ L^\infty _{\mathrm{weak - (*)}}(0,T; {\mathcal {M}}(Q)) \ {\mathcal {P}}- \text{ a.s. } \end{aligned}$$
(4.2)

4.1 Asymptotic Limit in the Equation of continuity

In view of (3.3),

$$\begin{aligned} {\mathcal {V}} = {\mathcal {L}}_{{\mathcal {T}}}[\varrho , \mathbf{m}], \end{aligned}$$
(4.3)

where \({\mathcal {V}}\) is the statistical limit identified in Proposition 3.2. Moreover, as the law of the new variables coincides with \(\overline{{\mathcal {V}}}_{N_{k}}\), the continuity equation is satisfied \({\mathcal {P}}-\)a.s.:

$$\begin{aligned} \left[ \int _{{Q}} {{\tilde{\varrho }}}_k \varphi \ \,{\mathrm{d}} {x} \right] _{t=\tau _1}^{t= \tau _2} = \int _{\tau _1}^{\tau _2} \int _{{Q}} \Big [ {{\tilde{\varrho }}}_k \partial _t \varphi + \widetilde{\mathbf{m}}_k \cdot \nabla _x\varphi \Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \end{aligned}$$
(4.4)

for any \(0< \tau _1< \tau _2 < T\) and \(\varphi \in C^1_c ((0,T) \times Q)\). Thus, letting \(k \rightarrow \infty \), we get

$$\begin{aligned} \left[ \int _{{Q}} \varrho \varphi \ \,{\mathrm{d}} {x} \right] _{t=\tau _1}^{t= \tau _2} = \int _{\tau _1}^{\tau _2} \int _{{Q}} \Big [ \varrho \partial _t \varphi + \mathbf{m}\cdot \nabla _x\varphi \Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \end{aligned}$$
(4.5)

for any \(0< \tau _1< \tau _2 < T\) and \(\varphi \in C^1_c ((0,T) \times Q)\) \({\mathcal {P}}-\)a.s.

4.2 Asymptotic Limit in the Momentum Equation

In accordance with (3.2) we have

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{0 \le \tau \le T} \int _{{Q}} E\left( {{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{u}_\infty \right) (\tau , \cdot ) \ \,{\mathrm{d}} {x} \right] \le \overline{{\mathcal {E}}}. \end{aligned}$$

Here and in the sequel, we denote by \({\mathbb {E}}\) (i.e. without any subscript) the expectation on the probability space \((\Omega ,{\mathfrak {B}},{\mathcal {P}})\). Consequently, it follows from (4.2) that

$$\begin{aligned} E\left( {{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{u}_\infty \right) \rightarrow \overline{E\left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{u}_\infty \right) }\ \text{ weakly-(*) } \text{ in }\ L^\infty _{\mathrm{weak - (*)}}(0,T; {\mathcal {M}} (Q) ), \end{aligned}$$
(4.6)

or, more specifically,

$$\begin{aligned} \int _0^T \psi \int _{{Q}} E\left( {{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{u}_\infty \right) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \rightarrow \int _0^T \psi \int _Q \varphi \ {\mathrm{d}}\overline{E\left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{u}_\infty \right) } \,{\mathrm{d}} t \end{aligned}$$

for any \(\psi \in L^1(0,T), \ \varphi \in C_c(Q)\) \({\mathcal {P}}-\)a.s. Moreover, passing to expectations we get, by Fatou’s lemma,

$$\begin{aligned} {\mathbb {E}} \left[ \int _0^T \psi \int _Q \varphi \ {\mathrm{d}}\overline{E\left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{u}_\infty \right) } \,{\mathrm{d}} t \right]\le & {} \liminf _{k \rightarrow \infty } {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} E\left( {{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{u}_\infty \right) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \nonumber \\\le & {} \overline{{\mathcal {E}}} \Vert \psi \Vert _{L^1(0,T)} \Vert \varphi \Vert _{C(Q)}, \ \psi , \varphi \ge 0. \end{aligned}$$
(4.7)

Consequently, unlike its components in (4.2), the limit relative energy is a finite measure on \((0,T) \times Q\).

Finally, since \(({{\tilde{\varrho }}}_{k},\widetilde{\mathbf{m}}_{k})\) has the law \({\mathcal {V}}_{N_{k}}\), it satisfies the corresponding momentum equation

$$\begin{aligned} \begin{aligned} \left[ \int _{Q}\widetilde{\mathbf{m}}_{k}\cdot \varvec{\varphi }\ \,{\mathrm{d}} {x}\right] _{t=\tau _{1}}^{t=\tau _{2}}&=\int _{\tau _{1}}^{\tau _{2}} \int _{{Q}} \left[ \widetilde{\mathbf{m}}_k \cdot \partial _t \varvec{\varphi }+ 1_{{{\tilde{\varrho }}}_k > 0} \frac{ \widetilde{\mathbf{m}}_k \otimes \widetilde{\mathbf{m}}_k }{{{\tilde{\varrho }}}_k} : \nabla _x\varvec{\varphi }+ p({{\tilde{\varrho }}}_k) {\mathrm{div}}_x\varvec{\varphi }\right] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad -\sum _{n=1}^{N_{k}}1_{({{\tilde{\varrho }}}_{k},\widetilde{\mathbf{m}}_{k})=(\varrho _{n},\mathbf{m}_{n})}\varepsilon _{n}\int _{\tau _{1}}^{\tau _{2}} \int _{{Q}} \Big [{\mathbb {S}}(\nabla _x\mathbf{u}_{n}):\nabla _x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \end{aligned} \end{aligned}$$

for any \(0< \tau _1< \tau _2 < T\) and \(\varvec{\varphi }\in C^1_c ((0,T) \times Q; R^d)\). We shall prove that the right hand side vanishes as \(k\rightarrow \infty \). In view of (3.8) and (3.2), we obtain

$$\begin{aligned} \frac{1}{N_{k}}\sum _{n=1}^{N_{k}}\left[ \int _{0}^{T}\Vert \sqrt{\varepsilon _{n}}{\mathbb {S}}(\nabla _x\mathbf{u}_{n})\Vert _{L^{2}(Q;R^{d\times d})}^{2}\ \,{\mathrm{d}} t \right] \lesssim \overline{{\mathcal {E}}}, \end{aligned}$$

hence

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\left[ \left| \sum _{n=1}^{N_{k}}1_{({{\tilde{\varrho }}}_{k},\widetilde{\mathbf{m}}_{k})=(\varrho _{n},\mathbf{m}_{n})}\varepsilon _{n}\int _{0}^{T} \int _{{Q}} \Big [{\mathbb {S}}(\nabla _x\mathbf{u}_{n}):\nabla _x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right| \right] \\&\quad ={\mathbb {E}}_{{\mathcal {V}}_{N_{k}}}\left[ \left| \sum _{n=1}^{N_{k}}1_{(r,\mathbf{w})=(\varrho _{n},\mathbf{m}_{n})}\varepsilon _{n}\int _{0}^{T} \int _{{Q}} \Big [{\mathbb {S}}(\nabla _x\mathbf{u}_{n}):\nabla _x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right| \right] \\&\quad =\frac{1}{N_{k}}\sum _{i=1}^{N_{k}}\left[ \left| \sum _{n=1}^{N_{k}}1_{(\varrho _{i},\mathbf{m}_{i})=(\varrho _{n},\mathbf{m}_{n})}\varepsilon _{n}\int _{0}^{T} \int _{{Q}} \Big [{\mathbb {S}}(\nabla _x\mathbf{u}_{n}):\nabla _x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right| \right] \\&\quad =\frac{1}{N_{k}}\sum _{i=1}^{N_{k}}\left[ \left| \varepsilon _{i}\int _{0}^{T} \int _{{Q}} \Big [{\mathbb {S}}(\nabla _x\mathbf{u}_{i}):\nabla _x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right| \right] \\&\quad \lesssim \Vert \varvec{\varphi }\Vert _{C^{1}((0,T)\times Q)}\left( \frac{1}{N_{k}}\sum _{i=1}^{N_{k}}\varepsilon _{i}\right) ^{\frac{1}{2}}\left( \frac{1}{N_{k}}\sum _{i=1}^{N_{k}} \left[ \int _{0}^{T} \Vert \sqrt{\varepsilon _{i}}{\mathbb {S}}(\nabla _x\mathbf{u}_{i})\Vert _{L^{2}(Q;R^{d\times d})}^{2} \,{\mathrm{d}} t \right] \right) ^{\frac{1}{2}}\\&\quad \lesssim \sqrt{\overline{{\mathcal {E}}}}\Vert \varvec{\varphi }\Vert _{C^{1}((0,T)\times Q)}\left( \frac{1}{N_{k}}\sum _{i=1}^{N_{k}}\varepsilon _{i}\right) ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

Since the above right hand side vanishes as \(k\rightarrow \infty \), we conclude

$$\begin{aligned}&\int _0^T \left( \int _{{Q}} \mathbf{m}\cdot \partial _t \varvec{\varphi } \ \,{\mathrm{d}} {x} + \int _Q \nabla _x\varvec{\varphi }: {\mathrm{d}}\ \overline{ \left[ 1_{\varrho > 0} \frac{ \mathbf{m}\otimes \mathbf{m}}{\varrho } + p(\varrho ) {\mathbb {I}} \right] } \right) \,{\mathrm{d}} t \nonumber \\&\quad = 0 \ \text{ for } \text{ any } \ \varvec{\varphi }\in C^1_c((0,T) \times Q) \end{aligned}$$
(4.8)

\({\mathcal {P}}-\)a.s.

4.3 Defect Measure in the Momentum Equation

Following [18] we rewrite equation (4.8) in the form

$$\begin{aligned} \begin{aligned} \int _0^T&\int _{{Q}} \left[ \mathbf{m}\cdot \partial _t \varvec{\varphi }+ 1_{\varrho> 0} \frac{ \mathbf{m}\otimes \mathbf{m}}{\varrho }: \nabla _x\varvec{\varphi }+ p(\varrho ) {\mathrm{div}}_x\varvec{\varphi }\right] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&= - \int _0^T \int _Q \nabla _x\varvec{\varphi }: \left( {\mathrm{d}}\overline{ \left[ 1_{\varrho> 0} \frac{ \mathbf{m}\otimes \mathbf{m}}{\varrho } + p(\varrho ) {\mathbb {I}} \right] } - \left[ 1_{\varrho > 0} \frac{ \mathbf{m}\otimes \mathbf{m}}{\varrho } + p(\varrho ) {\mathbb {I}} \right] \,{\mathrm{d}} {x}\right) \,{\mathrm{d}} t . \end{aligned} \end{aligned}$$

The quantity

$$\begin{aligned} {\mathfrak {R}} = \overline{ \left[ 1_{\varrho> 0} \frac{ \mathbf{m}\otimes \mathbf{m}}{\varrho } + p(\varrho ) {\mathbb {I}} \right] } - \left[ 1_{\varrho > 0} \frac{ \mathbf{m}\otimes \mathbf{m}}{\varrho } + p(\varrho ) {\mathbb {I}} \right] \in \ L^\infty _{\mathrm{weak-(*)}} (0,T; {\mathcal {M}}(Q; R^{d \times d}_{\mathrm{sym}})) \end{aligned}$$

is called Reynolds defect. A simple but crucial observation is that the tensor-valued measure \({\mathfrak {R}}\) is positively semi-definite in the sense that

$$\begin{aligned} \int _0^T \psi \int _{{Q}} {\mathfrak {R}} : (\xi \otimes \xi ) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t\ge & {} q 0 \ \text{ for } \text{ any }\ \xi \in R^d \ \text{ and } \text{ any }\ \psi \in L^1(0,T), \ \varphi \\\in & {} C_c(Q), \ \psi , \varphi \ge 0 \ {\mathcal {P}}- \text{ a.s. } \end{aligned}$$

Indeed this follows from convexity of the function

$$\begin{aligned} (\varrho , \mathbf{m}) \mapsto \frac{|\mathbf{m}\cdot \xi |^2}{\varrho } + p(\varrho )|\xi |^2 \ \text{ for } \text{ any }\ \xi \in R^d, \end{aligned}$$

cf. [18] for details.

Our final goal in this section is to show that all components of \({\mathfrak {R}}\) are finite measures on Q. To see this, we compute its trace

$$\begin{aligned} 0 \le \mathrm{trace}[{\mathfrak {R}}] = \overline{ \frac{|\mathbf{m}|^2}{{{\tilde{\varrho }}}} + d p(\varrho ) } - \left[ \frac{|\mathbf{m}|^2}{\varrho } + d p(\varrho ) \right] , \end{aligned}$$

and compare it with the defect of relative energy. More precisely, we first observe

$$\begin{aligned} \begin{aligned}&\overline{ \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + P(\varrho ) } - \left[ \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + P(\varrho ) \right] \\&\quad \le \max \left\{ \frac{1}{2}, \frac{1}{(\gamma - 1)d} \right\} \left( \overline{ \frac{|\mathbf{m}|^2}{\varrho } + d p(\varrho ) } - \left[ \frac{|\mathbf{m}|^2}{\varrho } + d p(\varrho ) \right] \right) \\&\quad \le \max \left\{ \frac{1}{2}, \frac{1}{(\gamma - 1)d} \right\} \max \left\{ 2 , (\gamma - 1)d \right\} \left( \overline{ \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + P(\varrho ) } - \left[ \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + P(\varrho ) \right] \right) . \end{aligned} \end{aligned}$$
(4.9)

Next, we write

$$\begin{aligned} \begin{aligned}&\int _0^T \psi \int _Q \varphi \left( {\mathrm{d}}\overline{ \left[ \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + P(\varrho ) \right] } - \left[ \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + P(\varrho ) \right] \,{\mathrm{d}} {x}\right) \,{\mathrm{d}} t \\&\quad = \lim _{k \rightarrow \infty } \int _0^T \psi \left[ \varphi \left( \left[ \frac{1}{2} \frac{|\widetilde{\mathbf{m}}_k|^2 }{{{\tilde{\varrho }}}_k} + P({{\tilde{\varrho }}}_k) \right] - \left[ \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + P(\varrho ) \right] \right) \right] \,{\mathrm{d}} t \\&\quad = \lim _{k \rightarrow \infty } \int _0^T \psi \left[ \varphi \left( \frac{1}{2} \frac{|\widetilde{\mathbf{m}}_k|^2 }{{{\tilde{\varrho }}}_k} + 2 \widetilde{\mathbf{m}}_k \cdot \mathbf{u}_\infty - \frac{1}{2} {{{\tilde{\varrho }}}_{k}} |\mathbf{u}_\infty |^2\right. \right. \\&\left. \left. \qquad + P({{\tilde{\varrho }}}_k) - P'(\varrho _\infty )({{\tilde{\varrho }}}_k - \varrho _\infty ) - P(\varrho _\infty ) \right) \!\!\right] \,{\mathrm{d}} t \\&\qquad - \int _0^T \psi \int _{{Q}} \varphi \left( \frac{1}{2} \frac{|\mathbf{m}|^2 }{\varrho } + 2 \mathbf{m}\cdot \mathbf{u}_\infty \right. \\&\left. \qquad - \frac{1}{2} {\varrho } |\mathbf{u}_\infty |^2 + P(\varrho ) - P'(\varrho _\infty )(\varrho - \varrho _\infty ) - P(\varrho _\infty ) \right) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad = \lim _{k \rightarrow \infty } \int _0^T \psi \int _{{Q}} \varphi E \left( {{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{m}_{\infty } \right) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t - \int _0^T \psi \int _{{Q}} \varphi E \left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{m}_{\infty } \right) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad = \int _0^T \psi \int _Q \varphi \ {\mathrm{d}}\overline{ E \left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{m}_{\infty } \right) } \,{\mathrm{d}} t - \int _0^T \psi \int _{{Q}} \varphi E \left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{m}_{\infty } \right) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \end{aligned}\nonumber \\ \end{aligned}$$
(4.10)

for any \(\psi \in L^1(0,T)\), \(\varphi \in C_c(Q)\) \({\mathcal {P}}-\)a.s., where we have used

$$\begin{aligned}&\int _0^T \psi \int _{{Q}} \varphi {{\tilde{\varrho }}}_k \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \rightarrow \int _0^T \psi \int _{{Q}} \varphi \varrho \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t ,\ \int _0^T \psi \int _{{Q}} \varphi \widetilde{\mathbf{m}}_{k} \cdot \mathbf{u}_\infty \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad \rightarrow \int _0^T \psi \int _{{Q}} \varphi \mathbf{m}\cdot \mathbf{u}_\infty \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \end{aligned}$$

\({\mathcal {P}}\)-a.s.

Thus, combining (4.7) with (4.9) we obtain the desired conclusion

$$\begin{aligned} {\mathbb {E}} \left[ \int _0^T \psi \left( \int _Q \varphi \ {\mathrm{d}}\, \mathrm{trace}[{\mathfrak {R}}] \right) \,{\mathrm{d}} t \right] \le c \overline{{\mathcal {E}}} \Vert \psi \Vert _{L^1(0,T)} \Vert \varphi \Vert _{C(Q)}, \ \psi \in L^1(0,T),\ \varphi \in C_c(Q). \end{aligned}$$
(4.11)

5 Limit Problem

In the preceding two sections, we have identified the limit problem in the vanishing viscosity regime as a stochastic process \((\varrho , \mathbf{m})\) on the probability space \((\Omega , {\mathfrak {B}}, {\mathcal {P}})\), with paths in the trajectory space \({\mathcal {T}}\), satisfying \({\mathcal {P}}-\)a.s.

$$\begin{aligned}&\int _0^T \int _{{Q}} \Big [ \varrho \partial _t \varphi + \mathbf{m}\cdot \nabla _x\varphi \Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t = 0 \ \text{ for } \text{ any }\ \varphi \in C^1_c((0,T) \times Q),\end{aligned}$$
(5.1)
$$\begin{aligned}&\int _0^T \int _{{Q}} \Big [ \mathbf{m}\cdot \partial _t \varvec{\varphi }+ 1_{\varrho > 0} \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho }: \nabla _x\varvec{\varphi }+ p(\varrho ) {\mathrm{div}}_x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t = - \int _0^T \int _Q \nabla _x\varvec{\varphi }: \, {\mathrm{d}}{\mathfrak {R}} \,{\mathrm{d}} t \end{aligned}$$
(5.2)

for any \(\varvec{\varphi }\in C^1_c((0,T) \times Q; R^d)\), where \({\mathfrak {R}}\) is the Reynolds stress,

$$\begin{aligned} {\mathfrak {R}}\in & {} L^\infty _{\mathrm{weak-(*)}}(0,T; {\mathcal {M}}^+(Q; R^{d \times d}_{\mathrm{sym}})),\nonumber \\&{\mathbb {E}} \left[ \int _0^T \psi \int _Q \varphi \ {\mathrm{d}}\ \mathrm{trace}[{\mathfrak {R}}] \,{\mathrm{d}} t \right] \le c \overline{{\mathcal {E}}} \Vert \psi \Vert _{L^1(0,T)} \Vert \varphi \Vert _{C(Q)}. \end{aligned}$$
(5.3)

In other words, the limit satisfies pathwise the compressible Euler system in the generalized sense introduced in [2] with the difference that no energy inequality is postulated. Moreover, since the limit is a stochastic process, it can be regarded as a statistical dissipative solution in the spirit of [16].

5.1 Stochastic Euler System

As the next step, we investigate the question, whether the randomness accumulated along the statistical vanishing viscosity limit can be directly modeled by a stochastic perturbation in the limiting Euler system. We suppose that the limit process \((\varrho , \mathbf{m})\) is statistically equivalent, in the sense specified in Definition 1.1, to a weak martingale solution \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) of the Euler system driven by the noise of Itô’s type:

$$\begin{aligned} {\mathrm{d}}{{\tilde{\varrho }}}+ {\mathrm{div}}_x\widetilde{\mathbf{m}}\,{\mathrm{d}} t = 0,\ {\mathrm{d}}\widetilde{\mathbf{m}}+ {\mathrm{div}}_x\left( \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}} \right) \,{\mathrm{d}} t + \nabla _xp({{\tilde{\varrho }}}) \,{\mathrm{d}} t = {\mathbf {F}} {\mathrm{d}}W. \end{aligned}$$
(5.4)

Here, \(W=(W_{k})_{k\ge 1}\) is a cylindrical Wiener process and the diffusion coefficient \({\mathbf {F}}=({\mathbf {F}}_{k})_{k\ge 1}\) is stochastically integrable, that is, progressively measurable and satisfies

$$\begin{aligned} {\mathbb {E}} \left[ \int _{0}^{T}\sum _{k\ge 1}\Vert {\mathbf {F}}_{k}\Vert _{W^{-\ell ,2}(Q;R^{d})}^{2}\,{\mathrm{d}} t \right] <\infty \end{aligned}$$
(5.5)

where \(W^{-\ell ,2}(Q;R^{d})\) is a possibly negative Sobolev space. A priori, the coefficient \({\mathbf {F}}\) may depend on the solution \(({{\tilde{\varrho }}},\widetilde{\mathbf{m}})\) provided the above stochastic integrability condition is satisfied. In this setting, the stochastic integral in (5.4) is a martingale, in particular, it has a zero expectation.

Since we only compare certain statistical properties of the two processes \((\varrho ,\mathbf{m})\) and \(({{\tilde{\varrho }}},\widetilde{\mathbf{m}})\), they can be possibly defined on different probability spaces. Nevertheless, without loss of generality we assume for notational simplicity that they are both defined on the probability space \((\Omega ,{\mathfrak {B}},{\mathcal {P}})\) with expectation \({\mathbb {E}}\).

It holds \({\mathcal {P}}-\)a.s.

$$\begin{aligned}&\int _0^T \int _{{Q}} \Big [ \widetilde{\mathbf{m}}\cdot \varvec{\varphi }\partial _t \psi + \psi 1_{{{\tilde{\varrho }}}> 0} \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}}: \nabla _x\varvec{\varphi }+ \psi p({{\tilde{\varrho }}}) {\mathrm{div}}_x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \nonumber \\&\quad = -\int _0^T \psi \sum _{k \ge 1} \left( \int _{{Q}} \mathbf{F}_k \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \right) {\mathrm{d}}W_k \end{aligned}$$
(5.6)

for any (deterministic) \(\psi \in C^1_c(0,T)\), \(\varvec{\varphi }\in C^1_c(Q; R^d)\). Thus passing to expectations, we obtain

$$\begin{aligned} {\mathbb {E}} \left[ \int _0^T \int _{{Q}} \Big [ \widetilde{\mathbf{m}}\cdot \varvec{\varphi }\partial _t \psi + \psi 1_{{{\tilde{\varrho }}}> 0} \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}}: \nabla _x\varvec{\varphi }+ \psi p({{\tilde{\varrho }}}) {\mathrm{div}}_x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] = 0. \end{aligned}$$
(5.7)

Similarly, we consider expectation of the limit equation (5.2) obtaining

$$\begin{aligned}&{\mathbb {E}} \left[ \int _0^T \int _{{Q}} \Big [ \mathbf{m}\cdot \varvec{\varphi }\partial _t\psi + \psi 1_{\varrho > 0} \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho }: \nabla _x\varvec{\varphi }+ \psi p(\varrho ) {\mathrm{div}}_x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \nonumber \\&\quad = - {\mathbb {E}} \left[ \int _0^T \psi \int _Q \nabla _x\varvec{\varphi }: \, {\mathrm{d}}{\mathfrak {R}} \,{\mathrm{d}} t \right] . \end{aligned}$$
(5.8)

Comparing (5.7), (5.8) and using the fact that the random processes are statistically equivalent and that \(p(\varrho ) = (\gamma - 1) P(\varrho )\), \(p({{\tilde{\varrho }}}) = (\gamma -1 ) P({{\tilde{\varrho }}})\), we may infer that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \left[ \int _0^T \psi \left( \int _Q \nabla _x\varvec{\varphi }: {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \\&\quad = {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} \left( 1_{{{\tilde{\varrho }}}> 0} \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}} - 1_{\varrho > 0} \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho } \right) : \nabla _x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \end{aligned} \end{aligned}$$
(5.9)

for any \(\psi \in C^1_c(0,T)\), \(\varvec{\varphi }\in C^1_c (Q)\).

Our goal is to show that (5.9), together with the fact that \((\varrho , \mathbf{m})\), \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) are statistically equivalent, imply \({\mathfrak {R}} = 0\) \({\mathcal {P}}-\)a.s. as soon as Q is an exterior domain to a convex body B.

5.1.1 Domains Exterior to a Convex Body

The following result may be of independent interest.

Proposition 5.1

Let \(Q = R^d \setminus B\) where B is a bounded set and let \(B_R\) be a (closed) ball in \(R^d\) of radius R containing B. Suppose

$$\begin{aligned} {\mathfrak {R}} \in L^\infty _{\mathrm{weak-(*)}} (0,T; {\mathcal {M}}^+(Q, R^{d \times d}_{\mathrm{sym}})) \end{aligned}$$

satisfies (5.9), where \((\varrho , \mathbf{m})\), \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) are statistically equivalent in the sense of Definition 1.1.

Then \({\mathfrak {R}}|_{R^d \setminus B_R} = 0\) \({\mathcal {P}}-\)a.s.

Proof

Without loss of generality, we may suppose that the ball is centered at the origin, \(B_R = \{ |x| \le R \}\). Keep in mind that this amounts to replacing x by \(x - x_0\) in the following computations. Consider a smooth convex function

$$\begin{aligned} F(Z)= & {} 0 \ \text{ for } \ 0 \le Z \le R^2,\ 0< F'(Z) \le \overline{F} \ \text{ for } \ R^2< Z < R^2 + 1,\ F'(Z)\\= & {} \overline{F} \ \text{ if } \ Z \ge R^2 + 1, \end{aligned}$$

together with a cut-off function

$$\begin{aligned} \chi \in C^\infty _c[0, \infty ), \ \chi (Z) = 1 \ \text{ for } \ Z \le 1,\ \chi (Z) = 0 \ \text{ for } \ Z \ge 2. \end{aligned}$$
(5.10)

Now, we take

$$\begin{aligned} \varvec{\varphi }_L (x) = \chi \left( \frac{|x|}{L} \right) \nabla _xF(|x|^2) ,\ \varvec{\varphi }\in C^1_c(Q),\ L\ge 1. \end{aligned}$$

as a test function in (5.9). The integral on the left-hand side of (5.9) reads

$$\begin{aligned}&{\mathbb {E}} \left[ \int _0^T \psi \left( \int _Q \nabla _x\varvec{\varphi }_L : {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \\&\quad = {\mathbb {E}} \left[ \int _0^T \psi \left( \int _{R^d \setminus B_R} \chi \left( \frac{|x|}{L} \right) \nabla _{x}^2 F(|x|^2) : {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \\&\qquad + \frac{2}{L} {\mathbb {E}} \left[ \int _0^T \psi \left( \int _{L \le |x| \le 2L} \chi ' \left( \frac{|x|}{L} \right) F'(|x|^2)|x| \left( \frac{x}{|x|} \otimes \frac{x}{|x|} \right) : {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] , \end{aligned}$$

where, in view of (5.3), \({\mathrm{d}}{\mathfrak {R}}\) is a finite measure, and consequently,

$$\begin{aligned} \begin{aligned}&\frac{2}{L} {\mathbb {E}} \left[ \int _0^T \psi \left( \int _{L \le |x| \le 2L} \chi ' \left( \frac{|x|}{L} \right) F'(|x|^2)|x| \left( \frac{x}{|x|} \otimes \frac{x}{|x|} \right) : {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \\&\quad \le 4 {\mathbb {E}} \left[ \int _0^T \psi \left( \int _{L \le |x| \le 2L} \chi ' \left( \frac{|x|}{L} \right) F'(|x|^2) \left( \frac{x}{|x|} \otimes \frac{x}{|x|} \right) : {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \rightarrow 0 \ \text{ as }\ L \rightarrow \infty . \end{aligned} \end{aligned}$$

Computing

$$\begin{aligned} \nabla _{x}^2 F(|x|^2) = 2 \nabla _{x} \left( F'(|x|^2) x \right) = 4 F''(|x|^2) (x \otimes x) + 2 F'(|x|^2) {\mathbb {I}} \end{aligned}$$

and using convexity of F together with the positive semi-definitness of \({\mathfrak {R}}\), we obtain

$$\begin{aligned} \begin{aligned}&\lim _{L \rightarrow \infty } {\mathbb {E}} \left[ \int _0^T \psi \left( \int _Q \nabla _x\varvec{\varphi }_L : {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \\&\quad = \lim _{L \rightarrow \infty } {\mathbb {E}} \left[ \int _0^T \psi \left( \int _{R^d \setminus B_R} \chi \left( \frac{|x|}{L} \right) \nabla _{x}^2 F(|x|^2) : {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \\&\quad \ge 2 {\mathbb {E}} \left[ \int _0^T \psi \int _{R^d \setminus B_R} F'(|x|^2 ) {\mathrm{d}}\ \mathrm{trace}[{\mathfrak {R}}]\,{\mathrm{d}} t \right] \end{aligned} \end{aligned}$$
(5.11)

for any \(\psi \in C^1_c(0,T)\), \(\psi \ge 0\).

Finally, in accordance with (1.9), the integral on the right-hand side of (5.9) vanishes. Indeed we easily compute

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} \left( 1_{{{\tilde{\varrho }}}> 0} \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}} - 1_{\varrho> 0} \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho } : \nabla _x\varvec{\varphi }_L \right) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\quad = {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 4 {F''(|x|^2)} \left( 1_{{{\tilde{\varrho }}}> 0} \frac{ |\widetilde{\mathbf{m}}\cdot x|^2 }{{{\tilde{\varrho }}}} - 1_{\varrho> 0} \frac{|\mathbf{m}\cdot x|^2 }{\varrho } \right) \chi \left( \frac{|x|}{L} \right) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\qquad + {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 2 {F'(|x|^2)} \left( 1_{{{\tilde{\varrho }}}> 0} \frac{ |\widetilde{\mathbf{m}}|^2 }{{{\tilde{\varrho }}}} - 1_{\varrho> 0} \frac{|\mathbf{m}|^2 }{\varrho } \right) \chi \left( \frac{|x|}{L} \right) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\qquad + \frac{2}{L} {\mathbb {E}} \left[ \int _0^T \psi \int _{L \le |x| \le 2L} {F'(|x|^2)} \frac{1}{|x|^2} \left( 1_{{{\tilde{\varrho }}}> 0} \frac{ |\widetilde{\mathbf{m}}\cdot x|^2 }{{{\tilde{\varrho }}}} - 1_{\varrho > 0} \frac{|\mathbf{m}\cdot x|^2 }{\varrho } \right) \chi ' \left( \frac{|x|}{L} \right) \,{\mathrm{d}} {x}\,{\mathrm{d}} t \right] = 0. \end{aligned} \end{aligned}$$

Indeed using statistical equivalence of the kinetic and angular energies (1.9) we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 1_{\varrho> 0} \frac{{|\mathbf{m}\cdot x|^2}}{\varrho } \varphi (x) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\quad = - {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 1_{\varrho> 0} ({\mathbb {J}}_0 \cdot \mathbf{m}) \cdot \mathbf{m}\varphi (x) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] + {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 1_{\varrho> 0} \frac{|\mathbf{m}|^2 }{\varrho } \varphi (x) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\quad = - {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 1_{{{\tilde{\varrho }}}> 0} ({\mathbb {J}}_0 \cdot \widetilde{\mathbf{m}}) \cdot \widetilde{\mathbf{m}}\varphi (x) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] + {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 1_{{{\tilde{\varrho }}}> 0} \frac{|\widetilde{\mathbf{m}}|^2 }{{{\tilde{\varrho }}}} \varphi (x) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\quad = {\mathbb {E}} \left[ \int _0^T \psi \int _{{Q}} 1_{{{\tilde{\varrho }}}> 0} \frac{{|\widetilde{\mathbf{m}}\cdot x|^2}}{\varrho } \varphi (x) \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \end{aligned} \end{aligned}$$

for any \(\varphi \in C_c(Q)\).

Going back to (5.11) we may infer that

$$\begin{aligned} {\mathbb {E}} \left[ \int _0^T \psi \int _{R^d \setminus B_R} F'(|x|^2 ) {\mathrm{d}}\ \mathrm{trace}[{\mathfrak {R}}]\,{\mathrm{d}} t \right] = 0 \end{aligned}$$

for any \(\psi \in C^1_c(0,T)\), \(\psi \ge 0\), which yields the desired conclusion as \(F'(|x|^2) > 0\) for \(|x| > R\). \(\square \)

Corollary 5.2

Let \(Q = R^d \setminus B\) where B is a compact convex set. Suppose

$$\begin{aligned} {\mathfrak {R}} \in L^\infty _{\mathrm{weak-(*)}} (0,T; {\mathcal {M}}^+(Q, R^{d \times d}_{\mathrm{sym}})) \end{aligned}$$

satisfies (5.9), where \((\varrho , \mathbf{m})\), \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) are statistically equivalent in the sense of Definition 1.1.

Then \({\mathfrak {R}} = 0\) \({\mathcal {P}}-\)a.s.

Proof

As B is convex, any \(x \in Q\) possesses an open neighborhood U(x) such that

$$\begin{aligned} U(x) \subset Q \setminus B_R \end{aligned}$$

for some ball \(B_R\) containing B. By Proposition 5.1,

$$\begin{aligned} {\mathfrak {R}}|_{U(x)} = 0 \end{aligned}$$

\({\mathcal {P}}-\)a.s. \(\square \)

5.2 Convergence

Going back to relation (4.10), we also obtain

$$\begin{aligned} \overline{ E \left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{m}_\infty \right) } = E \left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{m}_\infty \right) \ {\mathcal {P}}- \text{ a.s. } \end{aligned}$$
(5.12)

As shown in [18], relation (5.12) implies local strong convergence for the Skorokhod representation, more specifically,

$$\begin{aligned} \left\| {{\tilde{\varrho }}}_k - \varrho \right\| _{L^\gamma ((0,T) \times K)}^\gamma + \left\| 1_{{{\tilde{\varrho }}}_k> 0} \frac{\widetilde{\mathbf{m}}_k}{\sqrt{{{\tilde{\varrho }}}_k}} - 1_{\varrho > 0} \frac{\mathbf{m}}{\sqrt{\varrho }} \right\| _{L^2((0,T) \times K; R^d)}^2 \rightarrow 0 \end{aligned}$$
(5.13)

for any compact \( K \subset Q\) \({\mathcal {P}}\)–a.s.

Finally, we translate the convergence result (5.13) in terms of the original sequence \((\varrho _n, \mathbf{m}_n)_{n=1}^\infty \). Consider \(b \in C_c(R^{d + 1})\). It follows from (5.13) that

$$\begin{aligned}&\int _0^T \int _{{Q}} \frac{1}{N_k} \sum _{k=1}^{N_k} b(\varrho _n, \mathbf{m}_n) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad = {\mathbb {E}} \left[ \int _0^T \int _{{Q}} b({{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \rightarrow {\mathbb {E}} \left[ \int _0^T \int _{{Q}} b(\varrho , \mathbf{m}) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \end{aligned}$$

for any \(\varphi \in C^\infty _c((0,T) \times Q)\). This can be interpreted as

$$\begin{aligned}&\frac{1}{N_k} \sum _{k=1}^{N_k} b(\varrho _n, \mathbf{m}_n) \rightarrow {\mathbb {E}}\left[ b(\varrho , \mathbf{m}) \right] \\&\quad = {\mathbb {E}}_{{\mathcal {V}}}\left[ b(r, \mathbf{w}) \right] \ \text{ weakly-(*) } \text{ in } \ L^\infty ((0,T) \times Q) \ \text{ for } \ b \in C_c(R^{d + 1}). \end{aligned}$$

Moreover, identifying

$$\begin{aligned} \frac{1}{N_k} \sum _{k=1}^{N_k} b(\varrho _n, \mathbf{m}_n) = {\mathbb {E}}\left[ b({{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k) \right] , \end{aligned}$$

we compute

$$\begin{aligned} \left\| {\mathbb {E}}\left[ b({{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k) \right] - {\mathbb {E}}\left[ b(\varrho , \mathbf{m}) \right] \right\| _{L^1((0,T) \times K)} \le {\mathbb {E}} \left[ \left\| b({{\tilde{\varrho }}}_k, \widetilde{\mathbf{m}}_k) - b(\varrho , \mathbf{m}) \right\| _{L^1((0,T) \times K)} \right] \rightarrow 0 \end{aligned}$$

for any compact \(K \subset Q\). In particular, modulo a subsequence \((N_k)_{k \ge 1}\), the sequence \((\varrho _n, \mathbf{m}_n)_{n=1}^\infty \) is S-convergent in the sense of [17] as a consequence of Theorem 2.4 in [17].

We have proved the following result.

Theorem 5.3

(Strong statistical limit) Suppose that \(Q = R^d \setminus B\), where B is a convex compact set. Let \((\varrho _n, \mathbf{m}_n)_{n=1}^\infty \) be a sequence of weak solutions to the Navier–Stokes system in the sense of Definition 2.1 with the viscosity coefficients

$$\begin{aligned} \mu _n = \varepsilon _n \mu , \ \lambda _n = \varepsilon _n \lambda , \ \mu > 0, \ \lambda \ge 0, \ \varepsilon _n \!\searrow 0, \end{aligned}$$

and satisfying the total energy bound (3.2). Let \((\varrho , \mathbf{m})\) be the Skorokhod representation of the limit \({\mathcal {V}}\) identified in Proposition 3.2. Suppose that \((\varrho , \mathbf{m})\) is statistically equivalent to a solution \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) of the stochastic Euler system (5.4), driven by a stochastic forcing \({\mathbf {F}} {\mathrm{d}}W\) of Itô’s type satisfying (5.5).

Then

  • \((\varrho , \mathbf{m})\) is a (weak) statistical solution of the deterministic compressible Euler system.

  • The sequence \((\varrho _n, \mathbf{m}_n)_{n=1}^\infty \), modulo a subsequence \((N_k)_{k \ge 1}\), is S-convergent in the sense of [17]. Specifically, there is a sequence \(N_k \rightarrow \infty \) such that

    $$\begin{aligned} \frac{1}{N_k} \sum _{k=1}^{N_k} b(\varrho _n, \mathbf{m}_n) \rightarrow {\mathbb {E}}_{{\mathcal {V}}} \left[ b(r, \mathbf{w}) \right] \ \text{(strongly) } \text{ in }\ L^1((0,T) \times K) \ \text{ as }\ k \rightarrow \infty \nonumber \\ \end{aligned}$$
    (5.14)

    for any compact \(K \subset Q\) and any \(b\in C_c(R^{d + 1})\).

5.3 Drift force of Stratonovich Type

Revisiting the proof of Theorem 5.3, we may observe that the only used property of the Itô integral is its vanishing expectation. In other words, the same result remains valid if we replace the stochastic integral by an arbitrary random variable with zero expectation. A natural question is therefore whether our result applies to other random perturbations with generally non-zero expected value. Our goal is to extend Theorem 5.3 to a larger class of driving forces including, in particular, a Stratonovich type drift term, which is also widely used in the literature. In particular, a physical justification of a noise of this form in the context of fluid dynamics can be found in [24]. In [20], it was even proved that a transport noise of Stratonovich type provides regularization of the incompressible Navier–Stokes system in vorticity form.

For technical reasons, we restrict ourselves to the space dimension \(d=2\). Unfortunately, we are able to show the result only in the absence of the obstacle, meaning \(Q = R^2\).

We suppose that the limit process \((\varrho , \mathbf{m})\) is statistically equivalent to a solution \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) of the problem

$$\begin{aligned} {\mathrm{d}}{{\tilde{\varrho }}}+ {\mathrm{div}}_x\widetilde{\mathbf{m}}\,{\mathrm{d}} t = 0,\ {\mathrm{d}}\widetilde{\mathbf{m}}+ {\mathrm{div}}_x\left( \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}} \right) \,{\mathrm{d}} t + \nabla _xp({{\tilde{\varrho }}}) \,{\mathrm{d}} t = (\sigma \cdot \nabla _x) \widetilde{\mathbf{m}}\circ {\mathrm{d}}W_1 + {\mathbf {F}} \ {\mathrm{d}}W_2, \end{aligned}$$
(5.15)

where \(\sigma \in R^2\) is a constant vector. In particular, the weak formulation of the momentum equation reads

$$\begin{aligned} \begin{aligned}&\int _0^T \int _{R^2} \Big [ \widetilde{\mathbf{m}}\cdot \varvec{\varphi }\partial _t \psi + \psi 1_{{{\tilde{\varrho }}}> 0}\frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}}: \nabla _x\varvec{\varphi }+ \psi p({{\tilde{\varrho }}}) {\mathrm{div}}_x\varvec{\varphi }\Big ] \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad = -\int _0^T \psi \sum _{k \ge 1} \left( \int _{R^2} \mathbf{F}_k \cdot \varvec{\varphi } \ \,{\mathrm{d}} {x} \right) {\mathrm{d}}W_{2,k} + \int _0^T \psi \left( \int _{R^2} \widetilde{\mathbf{m}}\cdot \nabla _x(\varvec{\varphi }\otimes \sigma ) \ \,{\mathrm{d}} {x} \right) \circ {\mathrm{d}}W_1. \end{aligned} \end{aligned}$$
(5.16)

for any \(\psi \in C^1_c(0,T)\), \(\varvec{\varphi }\in C^1_c(R^2; R^2)\).

Using the Itô-Stratonovich correction formula we can write the Stratonovich integral in (5.16) as a martingale plus the correction term

$$\begin{aligned} -\frac{1}{2}\int _0^T \psi \int _{R^2} \widetilde{\mathbf{m}}\cdot [ (\sigma \otimes \sigma ) : \nabla _x^2] \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t , \end{aligned}$$

provided

$$\begin{aligned} {\mathbb {E}} \left[ \int _{0}^{T}\left( \psi \int _{R^2}\widetilde{\mathbf{m}}\cdot \nabla _{x}(\varvec{\varphi }\otimes \sigma )\ \,{\mathrm{d}} {x}\right) ^{2}\,{\mathrm{d}} t \right] <\infty . \end{aligned}$$
(5.17)

We take this stochastic integrability condition as an additional assumption. Obviously, the correction term can be written in the form

$$\begin{aligned} -\frac{1}{2}\int _0^T \psi \int _{R^2} \widetilde{\mathbf{m}}\cdot [ (\sigma \otimes \sigma ) : \nabla _x^2] \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t = -\frac{1}{2}\int _0^T \psi \int _{R^2} (\widetilde{\mathbf{m}}- \mathbf{m}_\infty ) \cdot [ (\sigma \otimes \sigma ) : \nabla _x^2] \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t , \end{aligned}$$

with \(\mathbf{m}_\infty = \varrho _\infty \mathbf{u}_\infty \).

Similarly to Sect. 5.1, passing to expectations in (5.16) we obtain

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \left[ \int _0^T \psi \left( \int _{R^2} \nabla _x\varvec{\varphi }: {\mathrm{d}}{\mathfrak {R}} \right) \,{\mathrm{d}} t \right] \\&\quad = {\mathbb {E}} \left[ \int _0^T \psi { \int _{R^2} (\widetilde{\mathbf{m}}- \mathbf{m}_\infty ) \cdot [ (\sigma \otimes \sigma ) : \nabla _x^2] \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t } \right] \\&\qquad + {\mathbb {E}} \left[ \int _0^T \psi \int _{R^2} \left( 1_{{{\tilde{\varrho }}}> 0} \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}} - 1_{\varrho > 0} \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho } \right) : \nabla _x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \right] \\&\qquad \text{ for } \text{ any } \ \psi \in C^1_c(0,T), \ { \varvec{\varphi }\in C^1_c (R^2; R^2)}. \end{aligned} \end{aligned}$$
(5.18)

The following steps are inspired by Chae [6]. Similarly to the proof of Proposition 5.1, we consider the test function

$$\begin{aligned} \varvec{\varphi }(x) = \chi \left( \frac{|x|}{L} \right) x. \end{aligned}$$
(5.19)

where \(\chi \) is the cut-off function introduced in (5.10). Accordingly,

$$\begin{aligned} \nabla _x\varvec{\varphi }= & {} \frac{1}{L} \chi ' \left( \frac{|x|}{L} \right) \frac{x \otimes x}{|x|} + \chi \left( \frac{|x|}{L} \right) {\mathbb {I}},\\ \nabla ^2_x \varvec{\varphi }= & {} \frac{1}{L^2} \chi '' \left( \frac{|x|}{L} \right) \frac{x \otimes x}{|x|^2} x - \frac{1}{L}\chi ' \left( \frac{|x|}{L} \right) \frac{x \otimes x}{|x|^2} \frac{x}{|x|}\\&+ \frac{1}{L} \chi ' \left( \frac{|x|}{L} \right) {\mathbb {I}} \frac{x}{|x|} + \frac{1}{L} \chi ' \left( \frac{|x|}{L} \right) \frac{1}{|x|} \nabla _x(x \otimes x). \end{aligned}$$

Exactly as in the proof of Proposition 5.1 we deduce

$$\begin{aligned} {\mathbb {E}} \left[ \int _0^T \psi \left( \int _{R^2} \ {\mathrm{d}}\ \mathrm{trace} [{\mathfrak {R}}] \right) \,{\mathrm{d}} t \right] \le \lim _{L \rightarrow \infty } \frac{1}{L} {\mathbb {E}} \left[ \int _0^T \int _{L \le |x| \le 2L} {|\widetilde{\mathbf{m}}- \mathbf{m}_\infty |} \,{\mathrm{d}} {x}\,{\mathrm{d}} t \right] \end{aligned}$$

Consequently, we obtain the same conclusion as in Proposition 5.1 if we show

$$\begin{aligned} \lim _{L \rightarrow \infty } \frac{1}{L} {\mathbb {E}} \left[ \int _0^T \int _{L \le |x| \le 2L} {|\widetilde{\mathbf{m}}- \mathbf{m}_\infty |} \,{\mathrm{d}} {x}\,{\mathrm{d}} t \right] = 0. \end{aligned}$$
(5.20)

We check easily by direct manipulation that

$$\begin{aligned} \begin{aligned} 1_{ \left\{ \frac{1}{2} \varrho _\infty \le {{\tilde{\varrho }}}\le 2 \varrho _\infty \right\} } { |\widetilde{\mathbf{m}}- \mathbf{m}_\infty |^2}&\lesssim E\left( {{\tilde{\varrho }}}, \widetilde{\mathbf{m}}\Big | \varrho _\infty , \mathbf{u}_\infty \right) ,\\ 1_{ \left\{ {{\tilde{\varrho }}}< \frac{1}{2} \varrho _\infty \ \mathrm {or}\ {{\tilde{\varrho }}}> 2 \varrho _\infty \right\} } {|\widetilde{\mathbf{m}}- \mathbf{m}_\infty |^{\frac{2 \gamma }{\gamma + 1}} }&\lesssim E\left( {{\tilde{\varrho }}}, \widetilde{\mathbf{m}}\Big | \varrho _\infty , \mathbf{u}_\infty \right) . \end{aligned} \end{aligned}$$

Denoting

$$\begin{aligned} \widetilde{\mathbf{m}}_1 = 1_{ \left\{ \frac{1}{2} \varrho _\infty \le {{\tilde{\varrho }}}\le 2 \varrho _\infty \right\} } { (\widetilde{\mathbf{m}}- \mathbf{m}_\infty ) } , \ \widetilde{\mathbf{m}}_2 = 1_{ \left\{ {{\tilde{\varrho }}}< \frac{1}{2} \varrho _\infty \ \mathrm {or}\ {{\tilde{\varrho }}}> 2 \varrho _\infty \right\} } { (\widetilde{\mathbf{m}}- \mathbf{m}_\infty ) }, \end{aligned}$$

we get, by Hölder’s inequality,

$$\begin{aligned}&\frac{1}{L} {\mathbb {E}} \left[ \int _0^T \int _{L \le |x| \le 2L} |\widetilde{\mathbf{m}}_1 | \,{\mathrm{d}} {x}\,{\mathrm{d}} t \right] \nonumber \\&\quad \lesssim L^{\frac{d - 2}{2}} {\mathbb {E}} \left[ \int _0^T \left( \int _{L \le |x| \le 2L} E\left( {{\tilde{\varrho }}}, \widetilde{\mathbf{m}}\Big | \varrho _\infty , \mathbf{u}_\infty \right) \,{\mathrm{d}} {x}\right) ^{\frac{1}{2}} \,{\mathrm{d}} t \right] , \end{aligned}$$
(5.21)

and

$$\begin{aligned}&\frac{1}{L} {\mathbb {E}} \left[ \int _0^T \int _{L \le |x| \le 2L} |\widetilde{\mathbf{m}}_2 | \,{\mathrm{d}} {x}\,{\mathrm{d}} t \right] \nonumber \\&\quad \lesssim L^{d \frac{\gamma - 1}{2 \gamma } - 1} {\mathbb {E}} \left[ \int _0^T \left( \int _{L \le |x| \le 2L} E\left( {{\tilde{\varrho }}}, \widetilde{\mathbf{m}}\Big | \varrho _\infty , \mathbf{u}_\infty \right) \,{\mathrm{d}} {x}\right) ^{\frac{\gamma + 1}{2\gamma }} \,{\mathrm{d}} t \right] . \end{aligned}$$
(5.22)

However, as \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) is statistically equivalent to \((\varrho , \mathbf{m})\) we have

$$\begin{aligned} {\mathbb {E}} \left[ \int _0^T \int _{L \le |x| \le 2L} E\left( {{\tilde{\varrho }}}, \widetilde{\mathbf{m}}\Big | \varrho _\infty , \mathbf{u}_\infty \right) \,{\mathrm{d}} {x}\,{\mathrm{d}} t \right] = {\mathbb {E}} \left[ \int _0^T \int _{L \le |x| \le 2L} E\left( \varrho , \mathbf{m}\Big | \varrho _\infty , \mathbf{u}_\infty \right) \,{\mathrm{d}} {x}\,{\mathrm{d}} t \right] \end{aligned}$$

In particular, if \(d = 2\), both integrals (5.21), (5.22) vanish in the asymptotic limit \(L \rightarrow \infty \).

We have proved the following result.

Theorem 5.4

Let \(Q = R^2\). Suppose that \((\varrho _n, \mathbf{m}_n)_{n=1}^\infty \) is a sequence of weak solutions to the Navier–Stokes system in the sense of Definition 2.1 with the viscosity coefficients

$$\begin{aligned} \mu _n = \varepsilon _n \mu , \ \lambda _n = \varepsilon _n \lambda , \ \mu > 0, \ \lambda \ge 0, \ \varepsilon _n \!\searrow 0, \end{aligned}$$

and satisfying the total energy bound (3.2). Let \((\varrho , \mathbf{m})\) be the Skorokhod representation of the limit \({\mathcal {V}}\) identified in Proposition 3.2. Suppose that \((\varrho , \mathbf{m})\) is statistically equivalent to a weak solution \(({{\tilde{\varrho }}}, \widetilde{\mathbf{m}})\) of the stochastic Euler system (5.15) driven by a stochastic forcing

$$\begin{aligned} (\sigma \cdot \nabla _x) \widetilde{\mathbf{m}}\, \circ {\mathrm{d}}W_1 + {\mathbf {F}} \,{\mathrm{d}}W_2 \end{aligned}$$

and satisfying (5.17) for any \(\psi \in C^1_c(0,T),\)\(\varvec{\varphi }\in C^1_c (R^2; R^2)\).

Then

  • \((\varrho , \mathbf{m})\) is a (weak) statistical solution of the deterministic compressible Euler system.

  • The sequence \((\varrho _n, \mathbf{m}_n)_{n=1}^\infty \), modulo a subsequence \((N_k)_{k \ge 1}\), is S-convergent in the sense of [17]. Specifically, there is a sequence \(N_k \rightarrow \infty \) such that

    $$\begin{aligned} \frac{1}{N_k} \sum _{k=1}^{N_k} b(\varrho _n, \mathbf{m}_n) \rightarrow {\mathbb {E}}_{{\mathcal {V}}}\left[ b(r, \mathbf{w}) \right] \ \text{(strongly) } \text{ in }\ L^1((0,T) \times K) \ \text{ as }\ k \rightarrow \infty \end{aligned}$$

    for any compact \(K \subset R^2\) and any \(b\in C_c(R^{3})\).

Similarly to the Itô case, we may extend the validity of Theorem 5.4 to random perturbations of the form \( M +N \), where M has zero expectation and N satisfies a suitable assumption so that the corresponding term vanishes in the required sense as it was the case for the Itô-Stratonovich correction through (5.20).

Remark 5.5

A short inspection of formula (5.18) and the specific form of the test function (5.19) reveals that the hypothesis concerning statistical equivalence of the angular energies can be relaxed in (1.9).

Remark 5.6

It is interesting to note that the same technique can be used to eliminate system with “effective” viscosity (cf. Davidson [10,  Chapter 4]),

$$\begin{aligned} \partial _t {{\tilde{\varrho }}}+ {\mathrm{div}}_x\widetilde{\mathbf{m}}= 0,\ \partial _t \widetilde{\mathbf{m}}+ {\mathrm{div}}_x\left( \frac{\widetilde{\mathbf{m}}\otimes \widetilde{\mathbf{m}}}{{{\tilde{\varrho }}}} \right) + \nabla _xp({{\tilde{\varrho }}}) \,{\mathrm{d}} t = \mu _{\mathrm{eff}} \Delta _x{{{\tilde{\mathbf{u}}}}},\ \end{aligned}$$

as the turbulent limit provided the velocity field \({{{\tilde{\mathbf{u}}}}}\) enjoys certain integrability properties.

5.3.1 Three Dimensional Setting

We observe that the proof of Theorem 5.4 goes through also in three spatial dimensions provided certain stronger assumptions are postulated. We assume that the far field density vanishes \(\varrho _{\infty }=0\), and that the adiabatic constant belongs to the physically relevant range \(\gamma \in (1,3]\). In that case, the term \({\widetilde{\mathbf{m}}}_{1}\) does not appear and the bound for \({\widetilde{\mathbf{m}}}_{2}\) indeed vanishes as \(L\rightarrow \infty \), which yields the claim of Theorem 5.4.

We can also allow for more complicated spatial domains provided suitable boundary conditions for both the approximate and the limit systems are considered. As the spatial domain we may take for instance a domain exterior to a cone,

$$\begin{aligned} {Q = R^3 \setminus Q},\ Q = \left\{ (x_1, x_2,x_{3}) \ \Big | x_{3}\ge 0,\ x^{2}_{3}\le \lambda ^{2}(x_{1}^{2}+x_{2}^{2}) \right\} ,\ \lambda >0. \end{aligned}$$

We suppose that the solutions of the Navier–Stokes system satisfy the complete slip condition,

$$\begin{aligned} \mathbf{u}\cdot \mathbf{n} |_{\partial Q} = 0, \ ({\mathbb {S}} \cdot \mathbf{n}) \times \mathbf{n}|_{\partial Q} = 0. \end{aligned}$$

Accordingly, the limit Euler system is endowed with the standard impermeability condition

$$\begin{aligned} \mathbf{m}\cdot \mathbf{n}|_{\partial Q} = 0. \end{aligned}$$

Under these circumstances, the function \(\varvec{\varphi }\) introduced in (5.19) is still an admissible test function for both the Navier–Stokes and the Euler system, and the arguments of Sect. 5.3 can be used as soon as \(\varrho _\infty = 0\), \(\mathbf{u}_\infty = 0\). Clearly, such a choice of boundary conditions eliminates the turbulent boundary layer. The same can be done also in two spatial dimensions where the assumption \(\varrho _{\infty }=0\) is not necessary.

5.4 Deterministic Limit

As our final goal, we identify the asymptotic limit in the situation of Theorem 5.3 and under the additional assumption that the barycenter

$$\begin{aligned} ({\overline{\varrho }}, {\overline{\mathbf{m}}}) = {\mathbb {E}}_{{\mathcal {V}}}\left[ (r, \mathbf{w}) \right] \end{aligned}$$

is a weak solution of the Euler system.

Consider the averages

$$\begin{aligned} \frac{1}{N_k} \sum _{n=1}^{N_k} \varrho _n = \overline{\varrho }_k,\ \frac{1}{N_k} \sum _{n=1}^{N_k} \mathbf{m}_n = \overline{\mathbf{m}}_k. \end{aligned}$$

Due to convexity of the relative energy, we have

$$\begin{aligned} \int _{{Q}} E \left( \overline{\varrho }_k, \overline{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{u}_\infty \right) \ \,{\mathrm{d}} {x} \le \frac{1}{N_k} \sum _{n=1}^{N_k} \int _{{Q}} E \left( \varrho _n, \mathbf{m}_n \Big | \varrho _\infty , \mathbf{u}_\infty \right) \ \,{\mathrm{d}} {x}, \end{aligned}$$

and the uniform bound (3.2) yields

$$\begin{aligned} \sup _{0 \le t \le \tau } \int _{{Q}} E \left( \overline{\varrho }_k, \overline{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{u}_\infty \right) \ \,{\mathrm{d}} {x} \le \overline{{\mathcal {E}}}. \end{aligned}$$
(5.23)

Thus it follows from (5.14) that

$$\begin{aligned}&\overline{\varrho }_k \rightarrow {\overline{\varrho }} \ \text{ in }\ L^p_{\mathrm{loc}}([0,T] \times Q),\ 1 \le p< \gamma ,\\&\overline{\mathbf{m}}_k \rightarrow {\overline{\mathbf{m}}} \ \text{ in } \ L^q_{\mathrm{loc}}([0,T] \times Q; R^d),\ 1 \le q < \frac{2 \gamma }{\gamma + 1}. \end{aligned}$$

As \((\varrho _n, \mathbf{m}_n)\) satisfy the momentum equation, we get

$$\begin{aligned} \begin{aligned} \int _0^T&\int _{{Q}} \overline{\mathbf{m}}_k \cdot \partial _t \varvec{\varphi }+ 1_{\overline{\varrho }_k> 0} \frac{\overline{\mathbf{m}}_k \otimes \overline{\mathbf{m}}_k }{\overline{\varrho }_k} : \nabla _x\varvec{\varphi }+ p(\overline{\varrho }_k) {\mathrm{div}}_x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&= \frac{1}{N_k} \sum _{n=1}^{N_k} \varepsilon _n \int _0^T \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}_n) : \nabla _x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\qquad - \int _0^T \int _{{Q}} \left( \frac{1}{N_k} \sum _{n=1}^{N_k}\left[ 1_{\varrho _n> 0} \frac{ \mathbf{m}_n \otimes \mathbf{m}_n }{\varrho _n} + p(\varrho _n) {\mathbb {I}} \right] \right. \\&\left. \qquad - \left[ 1_{\overline{\varrho }_k > 0} \frac{\overline{\mathbf{m}}_k \otimes \overline{\mathbf{m}}_k }{\overline{\varrho }_k} + p(\overline{\varrho }_k) {\mathbb {I}} \right] \right) : \nabla _x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t , \end{aligned} \end{aligned}$$

or, as \(({\overline{\varrho }}, {\overline{\mathbf{m}}})\) is a weak solution of the Euler system,

$$\begin{aligned} \begin{aligned} \int _0^T&\int _{{Q}} \left( \frac{1}{N_k} \sum _{n=1}^{N_k}\left[ 1_{\varrho _n> 0} \frac{ \mathbf{m}_n \otimes \mathbf{m}_n }{\varrho _n} + p(\varrho _n) {\mathbb {I}} \right] - \left[ 1_{\overline{\varrho }_k> 0} \frac{\overline{\mathbf{m}}_k \otimes \overline{\mathbf{m}}_k }{\overline{\varrho }_k} + p(\overline{\varrho }_k) {\mathbb {I}} \right] \right) : \nabla _x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad = \int _0^T \int _{{Q}} \left( \left[ 1_{\overline{\varrho }_k> 0} \frac{\overline{\mathbf{m}}_k \otimes \overline{\mathbf{m}}_k }{\overline{\varrho }_k} + p(\overline{\varrho }_k) {\mathbb {I}} \right] - \left[ 1_{{{\overline{\varrho }}} > 0} \frac{{{\overline{\mathbf{m}}}} \otimes {{\overline{\mathbf{m}}}} }{{{\overline{\varrho }}}} + p({\overline{\varrho }}) {\mathbb {I}} \right] \right) : \nabla _x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\qquad + \frac{1}{N_k} \sum _{n=1}^{N_k} \varepsilon _n \int _0^T \int _{{Q}} {\mathbb {S}}(\nabla _x\mathbf{u}_n) : \nabla _x\varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\qquad + \int _0^T \int _{{Q}} ({\overline{\mathbf{m}}} - \overline{\mathbf{m}}_k) \cdot \partial _t \varvec{\varphi } \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \rightarrow 0 \ \text{ as }\ k \rightarrow \infty \end{aligned} \end{aligned}$$

for any \(\varvec{\varphi }\in C^1_c((0,T) \times Q)\).

Next, observe that the tensor

$$\begin{aligned} {\mathfrak {R}}^1_k = \left( \frac{1}{N_k} \sum _{n=1}^{N_k}\left[ 1_{\varrho _n> 0} \frac{ \mathbf{m}_n \otimes \mathbf{m}_n }{\varrho _n} + p(\varrho _n) {\mathbb {I}} \right] - \left[ 1_{\overline{\varrho }_k > 0} \frac{\overline{\mathbf{m}}_k \otimes \overline{\mathbf{m}}_k }{\overline{\varrho }_k} + p(\overline{\varrho }_k) {\mathbb {I}} \right] \right) \end{aligned}$$

is positively semi-definite as

$$\begin{aligned} \begin{aligned}&\left( \frac{1}{N_k} \sum _{n=1}^{N_k}\left[ 1_{\varrho _n> 0} \frac{ \mathbf{m}_n \otimes \mathbf{m}_n }{\varrho _n} + p(\varrho _n) {\mathbb {I}} \right] - \left[ 1_{\overline{\varrho }_k> 0} \frac{\overline{\mathbf{m}}_k \otimes \overline{\mathbf{m}}_k }{\overline{\varrho }_k} + p(\overline{\varrho }_k) {\mathbb {I}} \right] \right) : (\xi \otimes \xi )\\&\quad = \left( \frac{1}{N_k} \sum _{n=1}^{N_k}\left[ 1_{\varrho _n> 0} \frac{ |\mathbf{m}_n \cdot \xi |^2 }{\varrho _n} + p(\varrho _n) |\xi |^2 \right] - \left[ 1_{\overline{\varrho }_k > 0} \frac{|\overline{\mathbf{m}}_k \cdot \xi |^2 }{\overline{\varrho }_k} + p(\overline{\varrho }_k) |\xi |^2 \right] \right) \ge 0 \end{aligned} \end{aligned}$$

in view of convexity of the function

$$\begin{aligned} (\varrho , \mathbf{m}) \mapsto \frac{|\mathbf{m}\cdot \xi |^2}{\varrho } + p(\varrho )|\xi |^2. \end{aligned}$$

Note that boundedness of the energy implies

$$\begin{aligned} \frac{|\mathbf{m}_n|^2}{\varrho _n} = 1_{\varrho _n> 0} \frac{|\mathbf{m}_n|^2}{\varrho _n},\ \frac{|\overline{\mathbf{m}}_k|^2}{\overline{\varrho }_k} = 1_{\overline{\varrho }_k > 0} \frac{| \overline{\mathbf{m}}_k|^2}{\overline{\varrho }_k} \ \text{ a.a. } \end{aligned}$$

We show that

$$\begin{aligned} \mathrm{\mathrm{ess}} \sup _{0 \le \tau \le T} \int _{{Q}} \mathrm{trace} [{\mathfrak {R}}^1_k ] \ \,{\mathrm{d}} {x} \le c(\overline{{\mathcal {E}}}). \end{aligned}$$

As we have observed above, this is equivalent to a similar statement for the energy, namely

$$\begin{aligned} \mathrm{\mathrm{ess}} \sup _{0 \le \tau \le T} \int _{{Q}} \left( \frac{1}{N_k} \sum _{k=1}^{N_k} E(\varrho _n, \mathbf{m}_n) - E(\overline{\varrho }_k, \overline{\mathbf{m}}_k ) \right) \ \,{\mathrm{d}} {x}\le c(\overline{{\mathcal {E}}}). \end{aligned}$$

However, a simple computation yields

$$\begin{aligned} \frac{1}{N_k} \sum _{k=1}^{N_k} E(\varrho _n, \mathbf{m}_n) - E(\overline{\varrho }_k, \overline{\mathbf{m}}_k ) = \frac{1}{N_k} \sum _{k=1}^{N_k} E \left( \varrho _n, \mathbf{m}_n \Big | \varrho _\infty , \mathbf{m}_\infty \right) - E\left( \overline{\varrho }_k, \overline{\mathbf{m}}_k \Big | \varrho _\infty , \mathbf{m}_\infty \right) , \end{aligned}$$

and the desired conclusion follows from the energy bound (3.2).

Repeating the arguments of Sect. 5.1.1 we conclude

$$\begin{aligned} \begin{aligned}&\lim _{k \rightarrow \infty } \int _0^T \int _{{Q}} \frac{1}{N_k} \sum _{k=1}^{N_k} E(\varrho _n, \mathbf{m}_n) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t = \int _0^T \int _{{Q}} E({\overline{\varrho }}, {\overline{\mathbf{m}}}) \varphi \ \,{\mathrm{d}} {x} \,{\mathrm{d}} t \\&\quad \text{ for } \text{ any }\ \varphi \in C_c((0,T) \times R^d),\ \varphi \ge 0. \end{aligned} \end{aligned}$$
(5.24)

As shown in [17, Section 5], relation (5.24) implies

$$\begin{aligned} \lim _{k \rightarrow \infty } \frac{1}{N_k} \sum _{k=1}^{N_k} b(\varrho _n, \mathbf{m}_n) \rightarrow b({\overline{\varrho }}, {\overline{\mathbf{m}}}) \end{aligned}$$

for any \(b \in C_c(R^{d+1})\). This yields the final conclusion

$$\begin{aligned} \frac{1}{N_k} \sum _{n=1}^{N_k} \Big ( \Vert \varrho _n - {\overline{\varrho }} \Vert _{L^1((0,T) \times K)} + \Vert \mathbf{m}_n - {\overline{\mathbf{m}}} \Vert _{L^1((0,T) \times K; R^d)} \Big ) \rightarrow 0 \ \text{ as } \ k \rightarrow \infty . \end{aligned}$$

Consequently, \({\mathcal {V}} = \delta _{({\overline{\varrho }},{\overline{\mathbf{m}}})}\) and \((\varrho _n, \mathbf{m}_n)_{n=1}^\infty \) statistically converges to \(({\overline{\varrho }},{\overline{\mathbf{m}}})\).

Theorem 5.7

In addition to the hypotheses of Theorem 5.3, suppose that

$$\begin{aligned} ({\overline{\varrho }},{\overline{\mathbf{m}}}) = {\mathbb {E}}_{{\mathcal {V}}}\left[ (r, \mathbf{w}) \right] \end{aligned}$$

is a weak solution to the Euler system.

Then

$$\begin{aligned} {\mathcal {V}} = \delta _{({\overline{\varrho }},{\overline{\mathbf{m}}})} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{N_k} \sum _{n=1}^{N_k} \Big ( \Vert \varrho _n - {\overline{\varrho }} \Vert _{L^1((0,T) \times K)} + \Vert \mathbf{m}_n - {\overline{\mathbf{m}}} \Vert _{L^1((0,T) \times K; R^d)} \Big ) \rightarrow 0 \ \text{ as }\ k \rightarrow \infty . \end{aligned}$$
(5.25)

As shown by Connor [9], relation (5.25) yields statistical convergence, modulo the subsequence \((N_k)_{k \ge 1}\), of the sequence \((\varrho _n, \mathbf{m}_n)\): For any \(\varepsilon > 0\),

$$\begin{aligned} \frac{\# \left\{ n \le N_k \ \Big | \Big ( \Vert \varrho _n - {\overline{\varrho }} \Vert _{L^1((0,T) \times K)} + \Vert \mathbf{m}_n - {\overline{\mathbf{m}}} \Vert _{L^1((0,T) \times K; R^d)} \Big ) > \varepsilon \right\} }{N_k} \rightarrow 0 \ \text{ as }\ k \rightarrow \infty . \end{aligned}$$

6 Concluding Remarks

We have studied the vanishing viscosity limit for the compressible Navier–Stokes fluid flow around a convex obstacle. We have shown that the statistical limit cannot be a solution of the associated Euler system driven by stochastic forcing of Itô’s type. As a consequence, there are two basic scenarios to describe the statistical limit:

  • Oscillatory limit. The limit is in the weak sense and can be described in terms of a Young measure. In that case, the weak limit is not a weak solution of the Euler system. Statistically, however, the asymptotic limit is a singleton (Dirac measure). Note that this scenario is compatible with the hypothesis that the limit is independent of the choice of \(\varepsilon _n \searrow 0\).

  • Statistical limit. The limit is a statistical solution of the Euler system. If this is the case, the sequence of the Navier–Stokes system S-converges in the sense of [17]. This is apparently in agreement with the numerical experiments performed by Fjordholm et al. [19]. At the theoretical level, this alternative is also in agreement with the Kolmogorov hypothesis concerning turbulent flow that predicts compactness in the strong Lebesgue sense, see Chen and Glimm [7]. We point out that this scenario is not compatible with the hypothesis that the limit is independent of \(\varepsilon _n \searrow 0\) unless it is a monoatomic measure in which case the convergence must be strong.

The phenomena of oscillatory and statistical limits are in a way complementary. The former asserts weak (oscillatory) convergence to a single limit, the latter means strong convergence for any suitable subsequence but non-existence of a single limit. In reality, they can be, of course, mixed up. In both cases, the generating sequence of solutions of the Navier–Stokes system is S-convergent in the sense of [17].

The results depend essentially on the properties of the compressible Euler system, and also on the geometry of the physical space—an unbounded domain exterior to a convex set in \(R^d\). They might be extended to the case of the full Euler system, however, the zero viscosity limit here is more delicate as the existence theory for the Navier–Stokes–Fourier system is available only for a particular class of constitutive equations.