On the vanishing viscosity limit of statistical solutions of the incompressible Navier–Stokes equations
Abstract.
We study statistical solutions of the incompressible Navier–Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures, which are probability measures accounting for spatial correlations, and moment equations is equivalent to statistical solutions in the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov’s self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Kármán–Howarth–Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures.
1. Introduction
The motion of an incompressible viscous fluid can be described by the Navier–Stokes equations
| (1.1) |
where is the fluid velocity and , the pressure, acting as a Lagrange multiplier to enforce the divergence constraint , and is the initial condition. Here, we take the spatial domain to be the -dimensional torus , and we denote the phase space by . The divergence is defined as and is the spatial gradient. The parameter denotes the viscosity and is proportional to the reciprocal of the Reynolds number. It is well-known that many flows of interest are characterized by high to very-high Reynolds numbers. Hence, one is interested in studying what happens when viscosity equal to zero. In this formal limit , one obtains the incompressible Euler equations, which are the prototypical models for an ideal fluid.
The question of whether the limit is a good approximation of (1.1) is of great practical relevance and has received considerable attention, both from a physical as well as mathematical point of view. Furthermore, it plays an essential role in computational fluid dynamics, as many numerical methods for the Euler equations, as well as large eddy simulations (LES) for Navier-Stokes equations, can be viewed as discretizations of (1.1) with of the order of the discretization parameter.
While in two spatial dimension, the convergence of a sequence of solutions of (1.1) to a solution of the Euler equations has been proved rigorously for many settings, see e.g., [14, 52, 13, 15], it turns out to be very challenging in 3D. Leray [47] proved in 1934 the existence of “Leray–Hopf solutions” of (1.1), which are weak solutions of (1.1), i.e., they satisfy (1.1) in the sense of distributions and in addition an energy inequality of the form
| (1.2) |
(Here and in the remainder, we suppress the dependence of on for convenience and write just .) Hence, if the initial data lies in , such solutions satisfy . (Here, and are the weakly divergence free functions in and , respectively.) However, as , the -bound on the gradient of , that stems from the energy inequality (1.2), no longer suffices for deriving sufficient compactness of the sequence in —which would be needed to pass to the limit in the nonlinear terms. It appears that, at least as far as global solutions are concerned, there is currently no means of gaining sufficient compactness through other conserved quantities or bootstrapping; in fact, it is unclear if global solutions of higher regularity than the one given by (1.2) exist in 3D. Closely related to this issue is the lack of stability estimates, i.e., well-posedness of Leray-Hopf solutions [22, 45]. The main obstruction to better regularity or stability estimates is caused by the nonlinear convective term . The role of the nonlinear term and possible instabilities in the Leray-Hopf solutions are often related to the issue of turbulence in fluid flows.
The theory of mathematical turbulence was initiated in the 1930s and 1940s by Taylor, Richardson, Kolmogorov and others, see [32] and references therein, and has since influenced fluid mechanics, as well as atmospheric sciences and plasma physics heavily. In his sequence of three papers [42, 41, 43], nowadays referred to as K41, Kolmogorov took a probabilistic approach to turbulence and formulated basic hypotheses about fluid flow at high Reynolds numbers and derived predictions based on these. Many of these have later been confirmed by experiments. The idea of studying equations (1.1) in a probabilistic setting has since been taken up again in many works, in different frameworks, by adding stochastic forcing terms to (1.1), see e.g. [26, 49, 10], or taking uncertain or measure-valued initial data, e.g. [17]. In the latter case, the solution of (1.1) may not be a function any more but instead a time-parametrized probability measure on the phase space. Global existence of such measure-valued solutions for incompressible flows has been shown in 3D, and even the passage to the limit can be made rigorous in this case [17]. However, measure-valued solutions are generally not unique, which can be shown by counterexample even in the case of Burgers’ equation [23]. Hence, measure-valued solutions are too broad a solution concept to resolve the problem of non-uniqueness, and more information or constraints need to be added.
To overcome this, in [24], it was suggested to take into account the (time) evolution of all possible multi-point spatial correlations. Instead of a single probability measure on the phase space , such a statistical solution is a family of probability measures on the phase space and products of the phase space , for , corresponding to the multi-point correlations. Hence, one can interpret the solution as a measure-valued solution augmented with information about higher order spatial correlations. From a practical point of view, this approach is very natural, as often only averaged quantities of interest of the fluid flow can be observed. Moreover, it is also in line with Kolmogorov’s turbulence theory, as this theory studies statistical properties of the fluid and makes predictions about these. The system of equations that arises for the higher order correlations is also known as the Friedman–Keller infinite chain of moment equations [39, 58] and finite closure relation for this infinite family of equations have been studied for small and large Reynolds numbers in [34, 36, 35, 37].
An alternative point of view in this context, is to consider instead probability measures on a space of suitable initial conditions; in the case of (1.1) this would be . Equation (1.1) is then interpreted as a Liouville equation on an infinite dimensional function space and the solution is a mapping assigning to each time a probability measure on . This setting was first considered by Prodi [53] and later on extensively studied by Foiaş and collaborators [27, 28, 30, 29], see also [38]. A closely related notion of statistical solutions was studied by Vishik and Fursikov [58]. Foiaş and his collaborators proved existence of such solutions in 2D and 3D, uniqueness in 2D, and further properties related to turbulence [29]. The relations between the Foiaş and Prodi notion of statistical solutions and the Vishik-Fursikov version were explored in [31, 8, 9]. The latter work etends the notion of statistical solutions to other relevant PDEs in fluid mechanics.
Given this plurality of definitions of statistical solutions, it is natural to examine, if and under what conditions, these solution concepts are equivalent. The first goal of this paper is to prove that both these concepts of statistical solutions of the incompressible Navier-Stokes equations (1.1) are equivalent as long as a statistical version of the of the energy inequality (1.2) holds.
The second and main goal of this paper is to investigate the vanishing viscosity limit of the statistical solutions of incompressible Navier-Stokes equations. Under an weak scaling assumption on the Navier-Stokes statistical solutions, we will use compactness criteria, presented recently in [25], to prove that vanishing viscosity limits of the statistical solutions of Navier-Stokes equations are statistical solutions of the incompressible Euler equations.
Our weak scaling assumption is a significantly weaker version of the scaling hypothesis of Kolmogorov’s 1941 theory and allows for intermittent corrections. Our main technical tool is a statisical version of the well-known Kármán–Howarth–Monin relation [32, 16, 48], that relates the evolution of 2-point correlations to the longitudinal structure function , which is, roughly speaking, defined as
Here is a suitable average of the flow.
Thus, by characterizing this vanishing viscosity limit, we establish a rigorous relationship between the incompressible Navier-Stokes and Euler equations, while accommodating physically observed facts about turbulent flows in this description.
The remainder of this article is organized as follows: In Section 2, we introduce the concept of correlation measures and in Section 3 we show the equivalence of statistical solutions as introduced by Foiaş and Prodi with families of correlation measures satisfying the Friedman–Keller chain of moment equations. Then in Section 4, we consider the passage to the limit and conclude with an appendix with technical results.
2. Correlation measures
In this section, we recall the definition of correlation measures and some important properties of them from [24, 25]. We start by introducing the necessary notation.
2.1. Notation
For , , we denote the tensor products
If is a topological space then we let denote the Borel -algebra on , we let denote the set of signed Radon measures on , and we let denote the set of all probability measures on , i.e. all with (see e.g. [3, 7, 40]). For and a multiindex we write and , and we let be the vector of length consisting of the elements of for which is non-zero. For a vector we write . For a vector we write with the convention .
2.1.1. Carathéodory functions
If and are Euclidean spaces then a measurable function is called a Carathéodory function if is continuous for a.e. and is measurable for every (see e.g. [1, Section 4.10]). Given and a Carathéodory function we define the functional by
| (2.1) |
(It is not obvious that is continuous, or even well-defined; see [24].) We denote the set of Carathéodory functions depending on space and time by and its dual space by (see e.g. [5]).
In the following, we will focus on a specific type of Carathéodory functions. In particular, for we let denote the space of Carathéodory functions satisfying
| (2.2) |
for nonnegative functions , . We let denote the subspace of functions satisfying the local Lipschitz condition
| (2.3) |
for every , for some , for some nonnegative and and some . (Note that the term was not present in [25, Definition 2.2], but one can generalize the results of that paper to include such a term.)
We also denote for a parametrized probability measure and a Carathéodory function the pairing
(where is the usual duality pairing between Radon measures and continuous functions ).
2.2. Definitions
We are now in a position to define time-dependent correlation measures.
Definition 2.1.
A time-dependent correlation measure is a collection of functions such that
-
(i)
for a.e. , and the map is measurable for every and almost every . (In other words, is a Young measure from to .)
-
(ii)
integrability:
(2.4) -
(iii)
Diagonal continuity (DC):
(2.5) where
is called the modulus of continuity of .
We denote the set of all time-dependent correlation measures by .
In [25] (and see [24] for a time-independent version), the following equivalence between time-dependent correlation measures and parametrized probability measures on was proved:
Theorem 2.2.
For every time-dependent correlation measure there is a unique (up to subsets of of Lebesgue measure ) map such that
-
(i)
the map
(2.6) is measurable for all ,
-
(ii)
is -bounded:
(2.7) -
(iii)
is dual to : the identity
(2.8) holds for a.e. , every and all .
Conversely, for every satisfying (i) and (ii), there is a unique correlation measure satisfying (iii).
We also have the following “Compactness” Theorem for time-dependent correlation measures [25, Theorem 2.21]
Theorem 2.3.
Let for be a sequence of correlation measures such that
| (2.9) | |||
| (2.10) |
for some and all . Then there exists a subsequence and some such that
-
(i)
as , that is, for every and every
-
(ii)
for a.e.
-
(iii)
for every and
-
(iv)
for , let and be nonnegative, and let . Then
(2.11) -
(v)
Assume moreover that is compact, and that have uniformly bounded support, in the sense that
(2.12) with being dual to , then the following observables converge strongly:
(2.13) for every .
3. Statistical solutions
The goal of this section is to show that the statistical solutions of Navier–Stokes as introduced by Foiaş and Prodi [27, 28, 29, 30, 53] are equivalent to families of correlation measures as introduced in [24] that satisfy the Friedman–Keller system of moment equations. For the sake of simplicity we will assume that the support of the initial measure lies in a bounded set , that is,
| (3.1) |
3.1. The Leray projector
We recall first that the Helmholtz–Leray projector, or simply Leray projector, is the linear map that projects a vector field to its divergence free component, that is with and . One can show that is orthogonal in to any function ,
For functions in the tensor product space we let denote the Leray projector in the th component, i.e., where , and with .
3.2. Definitions
We will start by recalling the different definitions of statistical solutions introduced in [29, 31, 24].
Definition 3.1 (Definition 3.2 in [24]).
Let . The Friedman–Keller system of moment equations, defined for time-dependent correlation measures , is the hierarchy of equations
| (3.2) |
for all , for all with for all , along with the divergence constraint
| (3.3) |
where , , for all , , with for all .
If solves the Friedman–Keller system of moments equations and in addition satisfies the energy inequality
| (3.4) | ||||
for all and , such that is a nonnegative, nondecreasing polynomial for for sufficiently large related to the support of the correlation measure (see (2.12)), then we call a Friedman–Keller statistical solution of the Navier–Stokes (when ) or Euler (when ) equations.
Remark 3.2.
Remark 3.3.
By a standard argument for weak solutions to continuity equations, the map is weakly continuous for every ; see e.g. [2, Remark 2.2].
Remark 3.4 (Formulation of (3.2) with non-divergence free test functions).
Denoting (cf. Section 3.1), we can replace the divergence-free test function in (3.2) by for an arbitrary . Using the fact that the Leray projection is self-adjoint and that, by (3.3),
we observe that
Therefore,
Similarly,
and
the last one being true due to the fact that the Laplacian and the Leray projection commute on the torus. Hence, the weak formulation (3.2) can be rewritten as
| (3.5) |
where . The terms
| (3.6) |
correspond to the pressure in the deterministic setting.
To define statistical solutions in the sense of Foiaş and Prodi, we need to introduce some notation. We denote by the space of divergence free -vector fields and by the space of divergence free functions in (these can be obtained as the closures of in and , respectively, with suitable integral conditions:
for periodic boundary conditions. We denote the -inner product by
and for the -inner product by
Define the Stokes operator by
where is the Leray projector, and the skew-symmetric trilinear form by
| (3.7) |
We can then write the Navier–Stokes equations in the functional formulation: Let , , find with such that
| (3.8) |
and in a suitable sense. This corresponds to the weak formulation
| (3.9) |
If we denote
| (3.10) |
the functional formulation becomes
| (3.11) |
We need the following class of test functions:
Notation 3.5.
[29] Let denote the class of cylindrical test functions consisting of the real-valued functionals that depend on a finite number of components of , that is,
where and . Let denote the subset of such functions which satisfy . We denote by the differential of in , which can be expressed as
where is the derivative of with respect to its th component.
We can now define statistical solutions in the sense of Foiaş and Prodi. We will use the definition as it stated in their newer work [31, Def. 3.2]:
Definition 3.6 (Foiaş–Prodi [29, 27, 30, 31]).
A family of probability measures on is a Foiaş–Prodi statistical solution of the Navier–Stokes equations on with initial data if
-
(a)
The function
(3.12) is measurable on for every ;
-
(b)
satisfies the weak formulation
(3.13) for all and all cylindrical test functions , where is given in (3.10).
-
(c)
satisfies the strengthened mean energy inequality: For any nonnegative, nondecreasing with bounded derivative and , the inequality
(3.14) holds.
-
(d)
The function
(3.15) is continuous at from the right, for any function nonnegative, nondecreasing with bounded derivative.
Remark 3.7.
Note that, as a consequence of the energy inequality (3.14) for , the function
belongs to and the function
| (3.16) |
belongs to . Notice also that (3.13) implies that
for a cylindrical test function, is continuous since
is locally integrable. Combining this fact with condition (c), conditionn (d) follows directly.
3.3. Equivalence between the solution concepts
Next, we show that the Friedman–Keller statistical solutions in Definition 3.1 and the Foiaş–Prodi statistical solutions in Definition 3.6 are in fact the same.
Theorem 3.8 (Foiaş–Temam statistical solutions satisfy the Friedman–Keller system).
Conversely, we have:
Theorem 3.9 (Friedman–Keller solutions are Foiaş–Prodi statistical solutions).
Let be a Friedman–Keller statistical solution of Navier–Stokes (cf. Definition 3.1) with bounded support, i.e.,
| (3.17) |
for some , every , and almost every . Then corresponds to a probability measure on a bounded set of which is a Foiaş–Prodi statistical solution of the Navier–Stokes equations (cf. Definition 3.6).
The proofs of these two results are given in Appendix A.
Foiaş et al. have shown existence of Foiaş–Prodi statistical solutions for the (forced) Navier–Stokes equations, see e.g. [27, 28, 29]. Using these equivalence theorems, this implies existence of statistical solutions via correlation measures as in Definition 3.1.
Remark 3.10.
The equivalence theorems 3.8 and 3.9 are restricted to probability measures with bounded support. It should be possible to extend these results to probability measures having sufficiently fast decay near infinity; however, the proofs would become significantly more technical. We have therefore decided to restrict ourselves to probability measures with bounded support.
4. Vanishing viscosity limit of statistical solutions of Navier–Stokes
The goal of this section is to pass to the inviscid limit under the assumption of weak statistical scaling (c.f. Section 4.3, Assumption 1). We will first prove a rigorous result on the longitudinal third order structure function
| (4.1) |
and then relate it to the similarly defined second order structure function using the weak scaling assumption. Together with weak statistical anisotropy, this yields diagonal continuity of the correlation measures that is needed to apply the compactness theorem 2.3 and pass to the limit. The proof of the scaling estimate for the third order structure function (4.1) in Lemma 4.2 and 4.3 largely follows the proof of a similar result for martingale solutions of stochastic Navier–Stokes equations in [6]. To simplify notation, we will omit writing the dependence of and on in the following sections.
4.1. Kármán–Howarth–Monin relation
The key to deriving an estimate on the behavior of the third order structure function (4.1) is the so-called Kármán–Howarth–Monin (KHM) relation [16] that describes the evolution of the second correlation marginal. Similar relations have been derived before for various settings (stochastic, forced, etc.), see [32, 50, 16, 48, 6, 21, 19]. For statistical solutions we derive:
Proposition 4.1.
Let be a Friedman–Keller statistical solution of the Navier–Stokes equations. Then the second correlation marginal satisfies the Kármán–Howarth–Monin relation for correlation measures:
| (4.2) |
for any , where is any smooth, compactly supported, isotropic rank 2 tensor – that is, any of the form
| (4.3) |
for .
Proof.
We consider equation (3.5) for with the test function (for simplicity replacing and by and and writing in component form ),
| (4.4) |
Since solves and solves , we have and and so (up to additive constants). Using this and changing the integration variables to and , we obtain
| (4.5) |
where . The cubic terms can be rewritten using the following simple fact (which can also be found in Frisch [32, equation (6.13)], and in a similar, weak form in [6]):
| (4.6) |
The proof of this is postponed to the end of this proof. Using this, we can rewrite (4.5) as
| (4.7) |
Since
we can rewrite
Moreover, we have for symmetric, smooth and compactly supported rank 2 tensors of the form
| (4.8) |
with ,
| (4.9) |
whose proof is postponed to the end of this proof. Using this, (4.7) becomes
| (4.10) |
Let be a sequence of smooth, uniformly bounded functions with the property that for every as . If we now use a test function
| (4.11) |
where is of the form (4.3), then we can use the weak continuity in time of the moments to obtain for any , as ,
| (4.12) |
Proof of (4.6).
We expand the right hand side:
The first term on the right hand side is zero since is compactly supported (after changing the integration variable from to in one of the terms). The second term on the right hand side vanishes using the divergence constraint (3.3). Using that and are symmetric, the last two terms are identical and so
which proves the claim. ∎
Proof of (4.9).
We consider the second expression, assume is of the form
| (4.13) |
where , are compactly supported in the torus. Then using that (the first term is treated in a similar way)
We note that since have compact support, we can write in polar coordinates
and so
where we used the divergence theorem in the second to last identity and the divergence constraint (3.3) for the last identity. ∎
4.2. Scaling of third order structure functions
Next, we use the KHM-relation (4.2) to derive a scaling relation for the averaged third order structure function in terms of the measure ,
| (4.14) |
which will be more convenient to work with for this purpose. We have:
Lemma 4.2.
Let be a Foiaş–Prodi statistical solution of the Navier–Stokes equations (cf. Definition 3.6). Then
| (4.15) |
where is the initial energy,
| (4.16) |
Proof.
We take a test function of the form in the KHM-relation (4.2) with having compact support in . A little bit of algebra yields (denoting )
and so (4.2) for this particular test function reads
| (4.17) |
In terms of the statistical solution this is
| (4.18) |
The last term can also be written as
| (4.19) |
Changing to spherical coordinates and using the definition of , (4.14), we obtain
We denote
Since is supported on functions in , is a continuous function. Moreover, notice that due to the a priori bounds following from the energy inequality (3.14), both and are uniformly bounded and continuous in and (for the continuity in of the first quantity, one needs weak time continuity of the moments which follows the fact that they satisfy the equations (4.2) where all the terms are integrable). We obtain
which is an ODE in the sense of distributions for , and because the right hand side is uniformly bounded and continuous, we can consider it in the strong sense (note that boundary terms when integrating the term by parts vanish):
or
The energy inequality (3.14) and the Cauchy–Schwarz inequality imply that and are both bounded by (defined in (4.16)), uniformly in . Hence,
| (4.20) |
(See also [6, Proposition 1.9] for a related result.) ∎
Using this lemma, we can derive a scaling relation for the averaged longitudinal structure function , where
| (4.21) |
Lemma 4.3.
Let be a Foiaş–Prodi statistical solution of the Navier–Stokes equations (cf. Definition 3.6). Then
| (4.22) |
where is some constant independent of and is the initial energy,
| (4.23) |
Proof.
Again, we start with the KHM relation (4.2). This time we use the test function where is an even function. We have
Therefore, (4.2) becomes
| (4.24) |
Again, in terms of , this means
| (4.25) |
Similar to the computation in (4.19), we have
so equation (4.25) becomes (after switching to polar coordinates)
| (4.26) |
Denote
Writing , (4.26) becomes
Again, we note that due to the estimates from the energy inequality (3.14), , and are continuous and bounded quantities in and . And so we can consider this ODE in the sense of distributions as an ODE in the strong sense,
or
| (4.27) |
By the energy bound and Cauchy–Schwarz inequality, and are uniformly bounded in for all . Moreover from Lemma 4.2, we have that is uniformly bounded in by . Hence,
| (4.28) |
for some independent of . ∎
4.3. Scaling assumption
In order to pass to the limit , we need to an additional assumption about the behavior of structure functions. Specifically, we need
Assumption 1 (Weak statistical scaling).
For any , let be a Foiaş–Prodi statistical solution of the incompressible Navier–Stokes equations. We assume that for , the second and third order longitudinal structure functions (4.21) are related by
| (4.31) |
where is a constant independent of and .
Remark 4.5 (Weak statistical scaling).
Assumption 1 is inspired by the following stronger scaling assumption often encountered in turbulence theory: For any with , the -th and -th order longitudinal structure functions (4.21) are related by
| (4.32) |
where is a constant independent of and for where . In Kolmogorov’s 1941 (“K41”) theory [42, 41, 43], . However, this cannot be confirmed with physical experiments [4, 54]. Various physicists therefore suggested intermittency corrections to account for the deviation from Kolmogorov’s original theory, among others, Kolmogorov himself in 1962 [44] in his refined theory of turbulence, Frisch et al. the -model [33], as well as Novikov and Stewart [51]. Assumption (4.32) can also accommodate the frequently used model by She and Leveque [55] who suggested
| (4.33) |
Remark 4.6.
We will combine Assumption 1 with the following lemma, which is Lemma 1 by Drivas [18], translated to the setting of statistical solutions. The proof is given in Appendix C:
Lemma 4.7 (Weak anisotropy).
Let be a statistical solution of the Navier–Stokes equation. Then satisfies
| (4.34) |
4.4. Passage to the limit
Now we are in a position to prove our main result. We will keep track of the superscript again in order to distinguish between the approximating sequence and the limiting measure for .
Theorem 4.8.
Let be a sequence of (either Foiaş–Temam or Friedman–Keller) statistical solutions to the Navier–Stokes equations with initial data with bounded support (cf. (3.1)). Assume that all satisfy Assumption 1. Then, as , converges (along a subsequence) to a correlation measure on with bounded support (cf. (3.17))
| (4.37) |
for some , any and that satisfies the “inviscid Friedman–Keller system”:
| (4.38) |
for all , for all with , a.e. for all and (corresponding to the divergence constraint)
| (4.39) |
where , , for all , , and .
Proof.
From the condition on the initially bounded support (3.1) and the energy inequality (3.4), we obtain that the sequence satisfies (2.9) for uniformly in . The reasoning of Subsection 4.1, 4.2 and 4.3 resulting in (4.36) imply that is uniformly diagonal continuous as in (2.10). Hence, using Theorem 2.3, we obtain, up to subsequence, the existence of a limiting correlation measure . So it remains to check whether satisfies the equations (4.38) and (4.39). We note that the functions
| (4.40) |
for , , are all functions in . Hence, we can pass to the limit in all the terms in the Friedman–Keller system (3.2). The term that is multiplied by vanishes because it is a uniformly bounded in quantity that is multiplied by . For the divergence constraint (4.39), we note that the function
lies in for any , and , and with . Passing in and using that satisfy the divergence constraint (3.3), we can conclude, by the arbitrariness of , that (4.39) holds for a.e. . ∎
Remark 4.9.
By the equivalence theorem 2.2, we know that the limiting correlation measure corresponds to a parametrized measure that satisfies
| (4.41) |
for all cylindrical test functions that satisfy , and the energy inequality
| (4.42) |
The proof of this fact follows along the lines of the proof of Theorem 3.9 while ignoring the terms involving and not attempting to recover as it may be unbounded.
5. Discussion
It is well-known that many incompressible fluid flows of interest are characterized by very-high Reynolds number. Hence, a precise characterization of the vanishing viscosity () limit of the Navier-Stokes equations (1.1) is of great interest. Formally, one would expect that the vanishing viscosity limit of Navier-Stokes equations is related to the incompressible Euler equations. However as mentioned in the introduction, rigorous results in this direction are only available in two space dimensions, even in the case of periodic boundary conditions. The key aim of this article was to investigate the vanishing viscosity limit of the Navier-Stokes equations, including in three space dimensions.
It is well known that fluid flows at high Reynolds numbers are characterized by turbulence, loosely speaking, marked by the presence of energy containing eddies at smaller and smaller scales. This phenomenon is clearly linked to the lack of compactness in the Leray-Hopf Navier-Stokes solutions as well as their possible instabilities/non-uniqueness.
Hence, one needs to make further assumptions on the Leray-Hopf solutions that can yield additional information and facilitate passage to the limit. One avenue for making such assumptions, which are realistic and possibly observed in experiments, comes from physical theories of turbulence. In particular, Kolmogorov’s well-known K41 theory is based on several verifiable assumptions on the underlying fluid flow and results in a precise characterization of quantities such as structure functions and energy spectra.
In [12], Chen and Glimm relate the K41 energy spectra to compactness results on the Leray-Hopf solutions, in appropriate Sobolev and Hölder spaces. Consequently, under the assumption of the K41 energy spectrum, the authors prove that the underlying Leray-Hopf solutions converge to weak solutions of the incompressible Euler equations as . However, Kolmogorov’s derivation of the decay of energy spectra is based on a probabilistic charectization of the underlying fluid flow. In particular, assumptions such as (statistical) homogeneity, isotropy and scaling, which form the foundation of Kolmogorov’s theory, are too stringent if imposed at the deterministic level, as done in [12]. Moreover, it is now well-established that the strong scaling assumptions of Kolmogorov might not hold in real fluid flows and intermittent corrections are necessary. Hence, the applicability of the assumptions and results of [12] can be questioned from this perspective.
Nevertheless, the connection with Kolmogorov’s theories of turbulence and their variants forms the basis of our work. We start with the realization that a probabilistic description of the solutions of Navier-Stokes equations is necessary to relate physical theories of turbulence to rigorous mathematical statements. To this end, we focus on statistical solutions of Navier-Stokes equations. Two possible frameworks of such statistical solutions are available, namely the Foiaş-Prodi statistical solutions (see Definition 3.6) and the Friedman-Keller statistical solutions (see Definition 3.1), which is based on the concept of correlation measures of [24]. We prove that both these solution concepts are equivalent as long as a statistical version of the energy inequality holds. This also allows us to prove the existence of Friedmann-Keller statistical solutions of the incompressible Navier-Stokes equations.
Then, we investigated the vanishing viscosity limit of the statistical solutions of the incompressible Navier-Stokes equations. To this end, we derived a suitable statistical version of the well-known Kármán-Howarth-Monin relation and used it to prove precise rates for the asymptotic decay of a averaged third-order structure function in Lemma 4.2. However, these estimates do not suffice to pass to the limit. To this end, we assumed a weak statistical scaling of the Navier-Stokes statistical solutions (see Assumption 1). This assumption is a weaker version of Kolmogorov’s scaling assumptions in his K41 theory. Moreover, it is consistent with and incorporates different variants of scaling that are proposed in the physics literature to explain intermittent corrections to Kolmogorov’s theory. Under this assumption, we proved a weak anisotropy result and invoked compactness results of [25] to rigorously prove that the statistical solutions of the Navier-Stokes equations converge, in a suitable sense, to a statistical solution of the incompressible Euler equations. Thus, we were able to characterize the vanishing viscosity limit of the Navier-Stokes equations in a relevant regime.
At this juncture, it is essential to point that that no assumption, other than weak statistical scaling, is made in our results and all other estimates are derived rigorously. This should be contrasted with the results of [12] where the authors directly assume a decay of the energy spectrum for the weak solutions of the Navier-Stokes equations. It is currently unclear if one can relax the weak statistical scaling assumption or even if it holds for all incompressible fluid flows. Experimental evidence strongly supports that this assumption is verified in practice, see e.g., [4, 55, 54].
To the best of our knowledge, the only rigorous study of the vanishing viscosity limit of the (Foiaş-Prodi) statistical solutions was carried out by Chae in [11] where he proved that these statistical solutions converge to a measure-valued solution of the incompressible Euler equations. In contrast, we prove convergence to statistical solutions of the incompressible Euler equations and recall that statistical solutions are much more informative than measure-valued solutions as they also incorporate knowledge of all multi-point correlations.
Finally, our characterization of the vanishing viscosity limit can be viewed in connection to recent results in [46] where the authors proved convergence of numerical spectral viscosity approximations to the statistical solutions of the Euler equations under very similar weak scaling assumptions.
Appendix A Equivalence of different definitions of statistical solutions for the incompressible Navier–Stokes equations
This appendix is devoted to the proof of Theorems 3.8 and 3.9. For convenience, we restate the result:
Theorem A.1 (Foiaş–Prodi statistical solutions are Friedman–Keller solutions).
Let be a Foiaş–Prodi statistical solution such that the initial condition has bounded support, , for some large enough. Then is a Friedmann–Keller statistical solution (cf. Definition 3.1).
Proof.
It is shown in [27, Theorem 2, Section 3] that Foiaş–Prodi statistical solutions with initial measure having bounded support in have bounded support for all times, i.e., . Therefore, we can assume that has uniformly bounded support. This implies in particular that have bounded moments:
| (A.1) |
Moreover, by [27, Lemma 5, Section 3], we have that statistical solutions of Navier–Stokes satisfy
| (A.2) |
for any test function that is Fréchet differentiable on with near and and for all and and some constants (equation () and condition (3.8) in [27]). Therefore, we can choose test functions
for a polynomial on and and , in (A.2), so that we get
Note that we can integrate by parts in the terms involving and , , so that all the derivatives are on the test functions , and :
Now take , so that the last identity becomes (denote )
which is, denoting and , equivalent to
Since we assume that the support of is bounded (and therefore also the support of for almost all ), we can use a density argument to conclude that the above identity holds for all , that is, all with for all . Observe next that all the terms in the above identity have the form required to apply Theorem 2.2 (recall that has bounded support). Therefore, by the same theorem, there exists a unique correlation measure corresponding to satisfying
which is (3.2). By previous arguments, each measure has bounded support (and in particular bounded moments), so we can take any nonnegative, nondecreasing polynomial , of degree , for in the strengthened energy inequality (3.14), so that we get
| (A.3) |
Thanks to the bounded support of , all the moment terms have the form required to apply Theorem 2.2, moreover, with Lemma B.3, we can rewrite the gradient term such that we obtain for a.e. , the energy inequality (3.4). Indeed, we have with Theorem 2.2
Similarly, with Lemma B.3, (via Lemma B.1)
and a similar computation for the term involving yields (3.4). The bounded support of implies, using the equivalence theorem [24, Theorem 2.7],
for any and thus the boundedness of all terms in (3.4). It remains to show that each satisfies (3.3). We let , , , with , and compute using the equivalence theorem 2.2,
since is supported on divergence free functions. This concludes the proof. ∎
Let us show the reverse direction now, that is, that any correlation measure that solves the Friedman–Keller system and satisfies in addition an energy inequality, is a statistical solution of the Navier–Stokes equations in the sense of Foiaş–Prodi.
Theorem A.2 (Friedman–Keller solutions are Foiaş–Prodi solutions).
Let be a Friedman–Keller statistical solution of Navier–Stokes (cf. Definition 3.1) with bounded support, i.e.,
| (A.4) |
for some , every , and almost every . Then corresponds to a probability measure on a bounded set of which is a Foiaş–Prodi statistical solution of the Navier–Stokes equations (cf. Definition 3.6).
Proof.
From the equivalence theorem 2.2, we obtain that corresponds to a family of measures with bounded support, that is for almost every . Property (a) of Definition 3.6 – the fact that (3.12) is measurable for all – follows from a monotone class argument, which we include here. By property (i) of Theorem 2.2, the function
is measurable for every . Let be the collection of sets
By the monotone convergence theorem, is a monotone class, that is, it is closed under (countable) unions of increasing sequences of sets and intersections of decreasing sequences of sets. By the same argument as in the proof of [24, Proposition 2.12], contains the collection of cylinder sets , that is, all sets of the form for some , a Borel set and . Since is an algebra which generates (cf. e.g. [24, Appendix]), it follows from the monotone class lemma that . Approximating an arbitrary by simple functions now gives the desired conclusion.
We claim that is supported on . This follows from Lemma B.5: Since has bounded support on we may take in that lemma, which is continuous and bounded on any compact subset of and satisfies . Then by the lemma, for any ,
Hence, by Chebychev’s inequality,
Since was arbitrary and is separable, this implies that is supported on -functions that are weakly divergence free, which is exactly the space (see e.g. [57, Section 1, Chapter 1]).
Next we claim that satisfies condition (c) in Definition 3.6. As is assumed to satisfy (3.4), we can apply [24, Theorem 2.7] combined with Lemma B.4 to each of the terms and obtain that satisfies (A.3) for any nonnegative and nondecreasing polynomial on . This implies in particular that is supported on , for a.e. , and (3.16). Now any differentiable nondecreasing function on a bounded interval can be approximated by nondecreasing polynomials [56], from which (3.14) follows after passing to the limit in a suitable polynomial approximation. Specifically, for a given , let be a sequence of nonnegative, nondecreasing polynomials () satisfying
Then, for , by the compact support property of on ,
and
where the last inequality follows from (3.16).
It remains to show that the correlation measures satisfy the evolution equation (3.13) for all . From this also part (d) of the definition will follow (see Remark 3.7). Let be an arbitrary function in with and let . Since is continuously differentiable and bounded on , we can approximate it arbitrarily well by polynomials thanks to the Weierstrass approximation theorem. Let
for some large enough, be approximations of that satisfy
Let be an arbitrary compactly supported test function. Since by the equivalence theorem, for any , , ,
and if , then
where the last equality follows because is divergence free because is supported on -functions by the energy inequality (3.14). Furthermore, we have
again, the last equality following because is divergence free and is supported on -functions for almost every . We obtain, by combining the terms to the Friedman–Keller system, that if
then satisfies the equation
| (A.5) |
for every test function . Next, note that
| (A.6) |
because has support on . Moreover,
| (A.7) |
where we used Young’s inequality for the last inequality. Thanks to (A.6) and (A.7), we can pass to the limit in (A.5) and obtain
| (A.8) |
for every . It follows that the distributional derivative of is
and since the right-hand side lies in we see that is absolutely continuous. Consequently, we can (through a standard approximation procedure) insert in (A.8) and conclude that (3.13) is true.
∎
Appendix B Auxiliary lemmata
Lemma B.1 (Representation of the gradient).
Let be a measure on satisfying
| (B.1) |
Define for the finite difference gradient by
where is the th unit vector. Then we have for any (or in the case that is the torus also for )
| (B.2) |