Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

Abstract

We prove that any weak space-time $L^2$ vanishing viscosity limit of a sequence of strong solutions of Navier–Stokes equations in a bounded domain of ${\mathbb R}^2$ satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that $t-a.e.$ weak $L^2$ inviscid limits of solutions of 3D Navier–Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second-order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.

1 Introduction

Experimentally verified to a large degree, the two-thirds law and the law of finite energy dissipation are cornerstones of turbulence theory (Frisch 1995). The law of finite energy dissipation states the non-vanishing of the rate of dissipation of kinetic energy of turbulent fluctuations per unit mass, in the limit of zero viscosity. This can be expressed, if Navier–Stokes equations are used, as

$$\begin{aligned} \lim _{\nu \rightarrow 0}\nu \langle |\nabla u(x,t)|^2\rangle = \epsilon >0 \end{aligned}$$

(1.1)

where $\nu$ is the kinematic viscosity, u is the turbulent velocity fluctuation, $\nabla$ are spatial gradients, and $\langle \ldots \rangle$ is a relevant average.

The two-thirds law states that

$$\begin{aligned} \langle |u(x+y,t)-u(x,t)|^2\rangle \sim (\epsilon |y|)^{\frac{2}{3}} \end{aligned}$$

(1.2)

for |y| in the inertial range, that is, in a range of scales

$$\begin{aligned} \eta \le |y|\le L \end{aligned}$$

(1.3)

where L is a the integral scale of turbulence and $\eta$ is the Kolmogorov dissipation scale,

$$\begin{aligned} \eta = \nu ^{\frac{3}{4}}\epsilon ^{-\frac{1}{4}}. \end{aligned}$$

(1.4)

The expressions $s_p(y) = \langle |u(x+y,t)-u(x,t)|^p\rangle$ are called pth-order structure functions, and various turbulence theories argue about scaling properties of the type $s_p\sim |y|^{p\zeta _p}$ in the inertial range. Turbulence is parameterized by the Reynolds number

$$\begin{aligned} \frac{UL}{\nu } \end{aligned}$$

(1.5)

where U is a relevant velocity (for instance average r.m.s velocity). Often turbulence is generated at boundaries. Thin boundary layers carry significant changes of momentum or heat. Experimentally, in strong turbulence, these boundary layers detach and heat and momentum are transported to the bulk of the fluid. Much of the dissipation of kinetic energy takes place in the boundary layers.

An asymptotic description of the vanishing viscosity limit (the high Reynolds number limit, with U and L bounded) was proposed by Prandtl (1905). In it, boundary layers of size proportional to $\sqrt{\nu }$ are attached to boundaries. Outside them, the limit is given by the Euler equations. Inside them, a different equation is valid (the Prandtl equation) and there is matching between the two behaviors at the edges of the boundary layer. If such a description is valid, then zero viscosity limits of solutions of the Navier–Stokes equations inside the domain obey the Euler equations.

Much effort has been devoted to validate mathematically turbulence theories and the inviscid limit to the Euler equations. One of the most interesting connections between the two subjects has been made by Kato (1984). He proved the equivalence of four statements, for short time, in a regime in which the Euler equations are smooth and conserve energy. These are:

• 1. The strong convergence in $L^{\infty }(0,T; L^2(\Omega ))$

  $$\begin{aligned} \lim _{\nu \rightarrow 0}\sup _{t\in [0,T]}\left\| u^\mathrm{NS}(t)-u^\mathrm{E}(t)\right\| _{L^2(\Omega )} = 0. \end{aligned}$$

• 2. The weak convergence in $L^2(\Omega )$ for all fixed times of the velocity of the Navier–Stokes solution $u^\mathrm{NS}(t)$ to the velocity of the Euler solution, $u^\mathrm{E}(t)$.

• 3. The vanishing of the energy dissipation rate

  $$\begin{aligned} \lim _{\nu \rightarrow 0}\nu \int _0^T\left\| \nabla u^\mathrm{NS}(t)\right\| ^2_{L^2(\Omega )}\mathrm{d}t = 0. \end{aligned}$$

• 4. The vanishing of the energy dissipation rate in a very thin boundary layer of width proportional to $\nu$, $\Gamma _{\nu }$:

  $$\begin{aligned} \lim _{\nu \rightarrow 0}\nu \int _0^T\left\| \nabla u^\mathrm{NS}(t)\right\| ^2_{L^2(\Gamma _\nu )}\mathrm{d}t = 0. \end{aligned}$$

The result is a stability result of the Euler path $S^\mathrm{E}(t)u_0$, conditioned on assumptions on the viscous dissipation at the boundary. There is a large literature concerned with related or similar conditional strong $L^2$ convergence results (a few examples are  Bardos and Titi 2007, 2013; Constantin et al. 2017, 2015; Kelliher 2007, 2008; Temam and Wang 1998; Wang 2001). Some strong $L^2$ unconditional convergence results for short time do exist. They are based on assumptions of real analytic data Sammartino and Caflisch (1998), or the vanishing of the Eulerian  initial vorticity in a neighborhood of the boundary Maekawa (2014). Symmetries can also lead to strong inviscid limits  (b); Kelliher (2009); Lopes Filho et al. (2008); Lopes Filho et al, (2008); Mazzucato and Taylor (2008). All these unconditional results are for short time, close to a smooth solution of Euler equation in laminar situations where energy dissipation rates vanish in the limit. The vast majority of the conditional results are also for short time, close to a smooth solution of Euler equation in laminar situations where energy dissipation rates vanish in the limit, and the conditions involve some uniform property of the Navier–Stokes solutions near the boundary such as bounds on derivatives (like the wall shear stress) or at least some uniform equicontinuity (Constantin et al. 2017). These short time results are not in contradiction with the Prandtl expansion accurately describing the asymptotic behavior. The Euler equations have very large classes of weak solutions, including non-dissipative ones (Bardos et al. 2014), but the inviscid limit can in some cases furnish a selection principle (Bardos et al. 2012).

What happens in the bulk for turbulent flows in domains with boundaries is a fundamental open problem. Is there a connection between the Euler equations and the inviscid limit when the limiting energy dissipation rate does not vanish?

Infinite time and the zero viscosity limit do not commute. This is obvious in the case of unforced Navier–Stokes equations in a smooth regime without boundaries, where the infinite time viscous limits are all zero, and the finite time inviscid limits are conservative smooth Euler solutions. This lack of interchangeability of limits is also true in the forced case. Consider, for example, a sequence of solutions of two-dimensional, spatially periodic solutions of the Navier–Stokes equations with Kolmogorov forcing f, i.e., forces which are eigenfunctions of the Stokes operator A:

$$\begin{aligned} Af = \lambda f. \end{aligned}$$

Unique, exact solutions of the Navier–Stokes equations are of the form $u(t) = y(t)f$ with the real valued function y(t) given by

$$\begin{aligned} y(t) = y_0e^{-\nu \lambda t} + \frac{1}{\nu \lambda }(1-e^{-\nu \lambda t}). \end{aligned}$$

Exact solutions of the Euler equations are of the same form, with

$$\begin{aligned} y(t) = y_0 + t. \end{aligned}$$

For any finite time the Navier–Stokes solutions converge to the Euler solution $S^\mathrm{NS}(t)u_0\rightarrow S^\mathrm{E}(t)u_0$ and the solutions are bounded as $\nu \rightarrow 0$, locally in time. By contrast, the infinite time limit at fixed viscosity is $u(t)\rightarrow u_f = \frac{1}{\nu \lambda }f$, and this sequence obviously diverges as $\nu \rightarrow 0$. Also, the initial data are forgotten in the infinite time limit. If the forcing has odd symmetry, the solutions obey Dirichlet boundary conditions as well.

Because of the lack of interchangeability of limits, it is important to distinguish between the short time zero viscosity limit, the arbitrary finite time limit, and the infinite time limit.

In this paper, we prove two results. They are for arbitrary finite time, and the conditions imposed are far away from boundaries. The results are of weak convergence on subsequences, allowing for non-unique, possibly dissipative Euler limit solutions.

For 2D flows at high Reynolds numbers, we prove that any $L^2(0,T; L^2(\Omega ))$ weak limit of a sequence of strong solutions of Navier–Stokes equations satisfies the Euler equations if interior local enstrophy bounds [see (2.6)] are uniform in viscosity. No assumptions need to be placed on the behavior of the Navier–Stokes solutions near the boundary. This is not a stability result of an Euler path, but rather a reflection interior good behavior of Navier–Stokes solutions uniform in viscosity. The limiting Euler solutions inherit interior enstrophy bounds, but the energy dissipation rate might be non-vanishing in the limit of zero viscosity.

For 3D we prove that if $S^\mathrm{NS}(t)u_0$ converge weakly in $L^2(\Omega )$ for almost all time to a function $u_{\infty }(t)$, and if a second-order structure function scaling from above is assumed locally uniformly (like in the two-thirds law, but with any positive exponent), then $u_{\infty }$ satisfies Euler equations. This is different than Kato’s condition 2 in that no assumptions are placed on $u_{\infty }$, and all time convergence is not required. In fact, the rate of dissipation of energy need not vanish in the limit, no Euler path is singled out, and the Euler solution may be wild.

We start by establishing the notation and make preliminary comments. Section 2 is devoted to 2D and Sect. 3 to 3D. A brief discussion concludes the paper.

We consider a bounded open domain $\Omega \subset {\mathbb R}^d$ $d=2,3$ with smooth boundary. We denote by $u:\Omega \times [0,T)\rightarrow {\mathbb R}^d$ a solution of the Navier–Stokes equation

$$\begin{aligned} \partial _t u -\nu \Delta u + u\cdot \nabla u + \nabla p = f \end{aligned}$$

(1.6)

in $\Omega$ with

$$\begin{aligned} \nabla \cdot u =0, \end{aligned}$$

(1.7)

boundary conditions

$$\begin{aligned} u_{\left| \right. \partial \Omega } =0, \end{aligned}$$

(1.8)

and initial data

$$\begin{aligned} u_{t=0} = u_0. \end{aligned}$$

(1.9)

The velocity $u = S^\mathrm{NS}(t)(u_0)$ obviously depends on $\nu >0$, space variable $x\in \Omega$, time variable $t\in [0,T)$, with T possibly infinite, body forces f, and initial data $u_0$.

We discuss weak limits in $L^2(0,T; L^2(\Omega ))$. The existence of weak limits of solutions of the Navier–Stokes equations is guaranteed by bounds

$$\begin{aligned} \int _0^T\int _{\Omega }|u(x,t)|^2\mathrm{d}x\mathrm{d}t \le E \end{aligned}$$

(1.10)

which are uniform for the ensemble of solutions. Conversely, if a weak limit exists for a sequence of functions, (1.10) is necessary. If a sequence $u_n$ converges weakly in $L^2(0,T; L^2(\Omega ))$ to a function u, it does not follow that $u_n(t)$ converges weakly to u(t) in $L^2(\Omega )$ for almost all t, not even on a subsequence. A subsequence of a weakly convergent sequence converges weakly to the same limit, and the subsequence might have some additional properties. In this paper, we use this fact to deduce additional information about the weak limits in two dimensions.

We say that function $u\in L^2(0,T;L^2(\Omega ))$ is a weak solution of the Euler equations if it is divergence-free and satisfies the Euler equations in the sense of distributions:

$$\begin{aligned} (u, \Phi _t) + ((u\otimes u):\nabla \Phi ) + (f,\Phi ) = 0 \end{aligned}$$

(1.11)

for any $\Phi \in C_0^{\infty }((0,T)\times \Omega )$ which is divergence-free. The notation M : N refers to the trace of the product of the two matrices. This is the distributional form of the incompressible Euler equations

$$\begin{aligned} \partial _t u + u\cdot \nabla u + \nabla p = f, \quad \nabla \cdot u = 0 \end{aligned}$$

forced by f. No boundary conditions nor initial data are part of the distributional formulation.

In order to verify that the limit u of a weakly convergent sequence $u_n \in L^2(0,T; L^2(\Omega ))$ of solutions of Navier–Stokes equations satisfies the Euler equations, it is enough to prove the convergence

$$\begin{aligned} N_{\Phi }(u_n)\rightarrow N_{\Phi }(u) \end{aligned}$$

(1.12)

for any fixed, divergence-free test function $\Phi \in C_0^{\infty }((0,T)\times \Omega )$ where

$$\begin{aligned} N_\Phi (u) = \int _0^T\int _{\Omega }(u\otimes u):\nabla \Phi \, \mathrm{d}x \mathrm{d}t. \end{aligned}$$

(1.13)

This is true of course only if we assume that the forces driving the Navier–Stokes equations converge weakly in $L^2$ to f. Then the linear terms (viscous term, time derivative term) and the forcing terms obviously converge.

In two dimensions, we use a vorticity formulation of the equations.

2 2D

We consider the vorticity $\omega = \partial _1u_2-\partial _2u_1 = \nabla ^{\perp }\cdot u$. For solutions of the 2D Navier–Stokes equations, the vorticity obeys

$$\begin{aligned} \partial _t \omega + u\cdot \nabla \omega -\nu \Delta \omega = g = \nabla ^{\perp }\cdot f. \end{aligned}$$

(2.1)

We recall that the velocity is obtained from a stream function

$$\begin{aligned} u = \nabla ^{\perp }\psi \end{aligned}$$

(2.2)

and that

$$\begin{aligned} \omega = \Delta \psi \end{aligned}$$

(2.3)

and therefore

$$\begin{aligned} \Delta u = \nabla ^{\perp }\omega \end{aligned}$$

(2.4)

holds. Note that $-\Delta u$ is not the Stokes operator applied to u. The identity

$$\begin{aligned} \int _{\Omega }|\omega (x,t)|^2\mathrm{d}x = \int _{\Omega }|\nabla u(x,t)|^2\mathrm{d}x \end{aligned}$$

(2.5)

is true in view of the boundary conditions (1.8). Indeed:

$$\begin{aligned} \int _{\Omega } \partial _j u_i(x,t) \partial _j u_i(x,t)\mathrm{d}x = -\int _{\Omega } u(x,t)\cdot \Delta u(x,t)\mathrm{d}x \\ =- \int _{\Omega }u(x,t)\cdot \nabla ^{\perp }\omega (x,t)\mathrm{d}x = \int _{\Omega }|\omega (x,t)|^2\mathrm{d}x. \end{aligned}$$

In the integrations by parts, we used only (1.8). For (2.4), the fact that $u(t)\in H^2(\Omega )$ for almost all time is true because u is a strong solution of NSE, but in fact the equality (2.5) is true for all divergence-free $u\in H_0^1(\Omega )$, by approximation.

Theorem 2.1

Let $\Omega \subset {\mathbb R}^2$ be a bounded open set with smooth boundary. Let $u_n$ be a sequence of solutions of Navier–Stokes equations with viscosities $\nu _n\rightarrow 0$. We assume that the solutions are driven by forces $f_n\in H^1(\Omega )$ that are uniformly bounded in $H^1(\Omega )$ and converge weakly in $H^1(\Omega )$ to f. We assume that the initial data $u_n(0)$ are divergence-free and are uniformly bounded in $H^1_0(\Omega )$. Let K be a compact, $K\subset \subset \Omega$. We assume that there exists a constant ${\mathcal {E}}_K$ which might depend on $H^1(\Omega )$ norms of initial data and f, on K and T, but is independent of viscosity, such that

$$\begin{aligned} \sup _{0\le t\le T}\int _K|\omega _n(x,t)|^2\mathrm{d}x\le {\mathcal E}_K \end{aligned}$$

(2.6)

where $\omega _n = \nabla ^{\perp }\cdot u_n$ are the vorticities.

Then any weak limit in $L^2(0,T; L^2(\Omega ))$ of the sequence $u_n$, $u_{\infty }$, is a weak solution of the Euler equations

$$\begin{aligned} \partial _t \omega _{\infty } + u_{\infty }\cdot \nabla \omega _{\infty } = g =\nabla ^{\perp }\cdot f \end{aligned}$$

(2.7)

with $\omega _{\infty } = \nabla ^{\perp }\cdot u_{\infty }$. The solution has bounded energy,

$$\begin{aligned} u_{\infty }\in L^{\infty }(0,T; L^2(\Omega )). \end{aligned}$$

(2.8)

Moreover, for any compact $K\subset \subset \Omega$ there exists a constant $\widetilde{\mathcal E}_K$ such that

$$\begin{aligned} \sup _{t\in [0,T]}\int _K |\omega _{\infty }(x,t)|^2\mathrm{d}x \le \widetilde{\mathcal E}_K \end{aligned}$$

(2.9)

holds.

Proof

The fact that $u_{\infty }$ has bounded energy is a simple consequence of the fact that under our conditions the sequence $u_n$ is bounded in time in $L^2(\Omega )$. Indeed, for time intervals I, $\chi _I(t) u_n$ converge weakly in $L^2(0,T; L^2(\Omega ))$ to $\chi _I(t)u_{\infty }$ where $\chi _I$ is the indicator function of I. Thus

$$\begin{aligned} \int _I\Vert u_{\infty }(t)\Vert ^2_{L^2(\Omega )}\mathrm{d}t \le \lim \inf _{n\rightarrow \infty }\int _I\Vert u_n(t)\Vert ^2_{L^2(\Omega )}\mathrm{d}t \le C|I|, \end{aligned}$$

and (2.8) follows. In order to prove that $u_{\infty }$ solves (2.7), we consider the nonlinear term, which is the only term whose behavior is in question. We take a compactly supported test function $\Phi \in C_0^{\infty }((0,T)\times \Omega )$ whose support is a compact $L\subset [t_1, T_1]\times K_1$ with $K_1\subset \subset \Omega$ compact, and $0<t_1<T_1<T.$ We consider a larger compact $K\subset \subset \Omega$ such that $K_1$ is included in the interior of K and a slightly larger time interval $[t_0, T_0]$ with $0<t_0<t_1$ and $T_1<T_0<T$. Let us also take a function $\chi _0\in C_0^{\infty }((t_0, T_0)\times K)$ which is identically 1 on a neighborhood of $[t_1, T_1]\times K_1$. We consider the sequence $\chi _0 u_n$. Because $u_n$ are uniformly bounded in $L^2(\Omega )$, it is clear in view of (2.6) that $\nabla ^{\perp }(\chi _0 u_n)$ is a bounded sequence in $L^{\infty }(0,T; L^2(\Omega ))$. Because $\nabla \cdot (\chi _0 u_n)$ is bounded in $L^{\infty }(0,T, L^2(\Omega ))$ as well, it follows that $\chi _0 u_n$ is bounded in $L^{\infty }(0,T; H_0^1(\Omega ))$. In order to use a Aubin–Lions lemma, and obtain some uniform control on time derivatives, it is best to take the curl of the equation, because the vorticity equation is local. The equations obeyed by the vorticities are

$$\begin{aligned} \partial _t \omega _n + u_n\cdot \nabla \omega _n -\nu \Delta \omega _n = g_n. \end{aligned}$$

(2.10)

We consider now another cutoff function $\chi$ which is still equal identically to 1 on a neighborhood of $[t_1,T_1]\times K_1$ but whose compact support is included in the region where $\chi _0$ is identically 1. We multiply by $\chi$ and consider the sequence $w_n = \chi \omega _n$. In view of (2.6) $w_n$ is bounded in $L^{\infty }(0,T; L^2(\Omega ))$. We use the equation (2.10) to examine $\partial _t w_n$.

The sequence $\chi \nu \Delta \omega _n$ is bounded in $L^{\infty }(0,T; H^{-2}(\Omega ))$, where $H^{-2}(\Omega )$ is the dual of $H_0^2(\Omega )$ because of (2.6). The terms $\partial _t \chi \omega _n$ and $\chi g_n$ are bounded in a better space, $L^{\infty }(0,T; L^2(\Omega ))$. It is well known that the term $u_n\cdot \nabla \omega _n$ is a second derivative. Indeed, dropping the subscript n for a moment in order to avoid confusion,

$$\begin{aligned} u\cdot \nabla \omega = \partial _1\partial _2\left( u_2^2-u_1^2\right) + \left( \partial _1^2-\partial _2^2\right) (u_1u_2) \end{aligned}$$

(2.11)

where now $u_1, u_2$ are components of the vector $u_n$. Therefore, because on the support of $\chi$ we have that $u_n = \chi _0 u_n$, it follows that the term $\chi u_n\cdot \nabla \omega _n$ is bounded in $L^{\infty }(0,T; H^{-2}(\Omega ))$. Indeed, using the continuous embedding $H_0^1(\Omega )\subset L^4(\Omega )$ we have that $\chi _0 u_n$ are uniformly bounded in $L^4(\Omega )$, and thus, after peeling off the two derivatives of $u_n\cdot \nabla \omega _n$ we are left with functions that are bounded uniformly in $L^{\infty }(0,T; L^2(\Omega ))$. By the Aubin–Lions lemma with spaces $L^2(\Omega )\subset \subset H^{-1}(\Omega )\subset H^{-2}(\Omega )$ (Lions 1969), we have that the sequence $w_n$ has a strongly convergent subsequence (relabelled $w_n$) in $L^2(0,T; H^{-1}(\Omega ))$. More precisely, we have that $\Lambda _D^{-1}w_n$ converges strongly in $L^2(0,T; L^2(\Omega ))$ to a function v, where $\Lambda _D = (-\Delta )^{\frac{1}{2}}$ with $-\Delta$ the Laplacian with homogeneous Dirichlet boundary conditions in $\Omega$. It is well known that $\Lambda _D:H_0^1(\Omega )\rightarrow L^2(\Omega )$ is an isometry. Taking a test function $\Psi (x,t)$, we have that

$$\begin{aligned} \int _0^T\int _{\Omega }\Lambda _D^{-1}w_n(x,t)\Psi (x,t) = -\int _0^T\int _{\Omega } u_n(x,t)\cdot \nabla ^{\perp } {\left( \chi \Lambda ^{-1}_D\Psi \right) }(x,t)\mathrm{d}x\mathrm{d}t. \end{aligned}$$

We pass to the limit in both sides, noting that $\nabla ^{\perp }{(\chi \Lambda ^{-1}_D\Psi )}(x,t)$ is an allowed test function because it belongs to $L^2(0,T; L^2(\Omega ))$, in view of the boundedness of the Riesz transforms $R_D = \nabla \Lambda _D^{-1}$ in $L^2(\Omega )$. It follows that

$$\begin{aligned} v= \Lambda _D^{-1}\left( \chi \nabla ^{\perp }\cdot u_{\infty }\right) = \Lambda _D^{-1}\chi \omega _{\infty } \end{aligned}$$

Moreover, because $\Vert \Lambda _D^{-1}w_n(t)\Vert _{L^2(\Omega )}^2$ converge strongly in $L^1(0,T)$, there is a subsequence, relabelled by n such that $\Lambda _D^{-1}w_n(t)$ converges strongly in $L^2(\Omega )$ for almost all $t\in [0,T]$ to v(t). Testing with a test function $\phi$ we have that, on the one hand

$$\begin{aligned} \left| \int _{\Omega }\phi w_n(t)\mathrm{d}x\right| =\left| \int _{\Omega }\chi \omega _n(t)\phi \mathrm{d}x\right| \le {\mathcal E}_K^{\frac{1}{2}}\Vert \phi \Vert _{L^2(\Omega )}, \end{aligned}$$

and on the other

$$\begin{aligned} \int _{\Omega }(\Lambda _D \phi )v(t)\mathrm{d}x = \lim _{n\rightarrow \infty }\int _{\Omega }\Lambda _D\phi \left( \Lambda _D^{-1}w_n(t)\right) \mathrm{d}x \end{aligned}$$

holds for almost all t. Thus

$$\begin{aligned} \int _{\Omega }\left| \chi \nabla ^{\perp }\cdot u_{\infty }(x,t)\right| ^2\mathrm{d}x \le {\mathcal E}_K \end{aligned}$$

holds a.e. in time, that is

$$\begin{aligned} \Vert \chi \omega _{\infty }\Vert _{L^{\infty }(0,T; L^2(\Omega ))}\le {\mathcal E}_K^{\frac{1}{2}}. \end{aligned}$$

(2.12)

We now pass to another subsequence, relabelled again by n such that $\chi _0 u_n$ converges weakly in $L^2(0,T; H_0^1(\Omega ))$. It is obvious that the limit is $\chi _0 u_{\infty }$. We consider the nonlinear term

$$\begin{aligned} \int _0^T\int _\Omega (u_n\cdot \nabla \Phi )\omega _n\mathrm{d}x\mathrm{d}t = N_{\Phi }(n) \end{aligned}$$

Because on the support of $\Phi$, we have $u_n = \chi _0 u_n$ and $\omega _n = w_n$, this integral is the duality pairing between a weakly convergent sequence in $L^2(0,T; H_0^1(\Omega ))$, namely $\chi _0u_n\cdot \nabla \Phi$ and a strongly convergent sequence in $L^2(0,T; H^{-1}(\Omega ))$, namely $w_n$:

$$\begin{aligned} N_{\Phi }(n) = \int _0^T\int _{\Omega } \Lambda _D(\chi _0u_n\cdot \nabla \Phi )\Lambda _D^{-1}w_n\mathrm{d}x \mathrm{d}t. \end{aligned}$$

Therefore $N_\Phi (n)$ convergences as the scalar product between weakly convergent and strongly convergent $L^2$ functions, and

$$\begin{aligned} \lim _{n\rightarrow \infty } N_{\Phi }(n)= & {} \int _0^T\int _{\Omega }(\chi _0u_{\infty }\cdot \nabla \Phi )\chi \omega _{\infty }\mathrm{d}x\mathrm{d}t \nonumber \\= & {} \int _0^T\int _{\Omega }(u_{\infty }\cdot \nabla \Phi )\omega _{\infty }\mathrm{d}x\mathrm{d}t \end{aligned}$$

(2.13)

$\square$

Remark 2.2

We note that from the proof it follows that $\chi \omega _n(t)$ converge weakly in $L^2$ to $\chi \omega _{\infty }(t)$ on a subsequence, for almost all t. No convergence is implied for $u_n(t)$ in $L^2(\Omega )$: the global behavior may depend on viscosity.

3 3D

We consider families of solutions of Navier Stokes equations in a bounded domain $\Omega \subset {\mathbb R}^3$.

Theorem 3.1

Let $u_n$ be a sequence of weak solutions of the Navier–Stokes equations

$$\begin{aligned} \partial _t u_n + u_n\cdot \nabla u_n -\nu _n\Delta u_n + \nabla p_n = f_n \end{aligned}$$

(3.1)

with $\nabla \cdot u_n =0$, $f_n$ bounded in $L^2(0,T; L^2(\Omega ))$, converging weakly to f, $u_n(0)$ divergence-free and bounded in $L^2(\Omega )$ and $\nu _n\rightarrow 0$. We assume that for any $K\subset \subset \Omega$ there exists a constant $E_K$, a constant $\epsilon >0$ and a constant $\zeta _2>0$ such that

$$\begin{aligned} \sup _{n}\int _0^T\int _K |u_n(x+y,t)-u_n(x,t)|^2\mathrm{d}x\mathrm{d}t \le E_K |y|^{2\zeta _2} \end{aligned}$$

(3.2)

holds for $|y|< \mathrm{dist}(K,\partial \Omega )$ in the inertial range

$$\begin{aligned} |y|\ge \epsilon ^{-\frac{1}{4}}\nu _n^{\frac{3}{4}} = \eta (n) \end{aligned}$$

(3.3)

Assume that $u_n(t)$ converge weakly in $L^2(\Omega )$ to $u_{\infty }(t)$ for almost all $t\in (0,T)$. Then $u_{\infty }$ is a weak solution of the Euler equations.

Remark 3.2

The domain need not be bounded. Local uniform energy bounds are enough. In exterior domains local uniform (in viscosity) energy bounds can be obtained for suitable weak solutions.

Remark 3.3

Obviously, the scaling assumption (3.2) does not imply regularity, because it is limited to y bounded away from zero. Also, the exact Kolmogorov form of $\eta (n)$ is not needed. All that is used is that $\eta (n)$ converges to zero as $n\rightarrow \infty$. The power law behavior $|y|^{2\zeta _2}$ is not needed either, any uniform modulus of continuity can be used instead. Finally, there is no connection between the rate of vanishing of $\eta (n)$ as $n\rightarrow \infty$ and the assumed uniform modulus of continuity.

Remark 3.4

The result and the proof apply to 2D flows, mutandis mutatis.

Proof

We consider a nonnegative smooth function j(z) supported in the annulus $1<|z|<2$ and with integral equal to 1, $\int _{{\mathbb R}^3} j(z)\mathrm{d}z =1$. We assume also that $j(-z)=j(z)$. We fix a compact $K\subset \subset \Omega$ and denote, for a function u, for $x\in K$, and $2r<\mathrm{dist} (K, \partial \Omega )$,

$$\begin{aligned} u_r(x) = \int _{1\le |z|\le 2}u(x-rz)j(z)\mathrm{d}z. \end{aligned}$$

(3.4)

We note the identity (see Constantin 1994)

$$\begin{aligned} (uv)_r(x)- u_r(x)v_r(x) = \rho _r(u,v)(x) \end{aligned}$$

(3.5)

with

$$\begin{aligned} \rho _r(u,v)(x)= & {} \int _{1\le |z|\le 2} j(z)(u(x-rz)-u(x))(v(x-rz)-v(x))\mathrm{d}z \nonumber \\&- (u(x)-u_r(x))(v(x)-v_r(x)). \end{aligned}$$

(3.6)

Let us take a divergence-free test function $\Phi (x,t)\in (C_0^{\infty }((0,T)\times \Omega ))^3$ and investigate the behavior of the nonlinear term

$$\begin{aligned} N_\Phi (n) = \int _0^T\int _{\Omega }(u_n\otimes u_n):\nabla \Phi \, \mathrm{d}x \mathrm{d}t. \end{aligned}$$

(3.7)

We take a compact K such that the support of $\Phi$ is included in $[t_0, T_0]\times K$, where $0<t_0<T_0<T$. Let us start by noting that

$$\begin{aligned} \left| \int _0^T\int _{\Omega }(u_n\otimes u_n) : (\nabla \Phi -(\nabla \Phi )_r)\,\mathrm{d}x\mathrm{d}t\right| \le C_{\Phi }r\Vert u_n\Vert ^2_{L^2(0,T; L^2(K))} \end{aligned}$$

and thus

$$\begin{aligned} \left| \int _0^T\int _{\Omega }(u_n\otimes u_n) : (\nabla \Phi -(\nabla \Phi )_r)\,\mathrm{d}x\mathrm{d}t\right| \le rC_{\Phi }E \end{aligned}$$

(3.8)

with E a uniform bound on the local time average of energy,

$$\begin{aligned} \Vert u_n\Vert ^2_{L^2(0,T; L^2(K))}\le E. \end{aligned}$$

(3.9)

Then, we note, using $j(z)= j(-z)$ that

$$\begin{aligned} \int _0^T\int _{\Omega }u_n\otimes u_n : (\nabla \Phi )_r \,\mathrm{d}x\mathrm{d}t = \int _0^T\int _{\Omega }(u_n\otimes u_n)_r : \nabla \Phi \,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.10)

Now we use the identity (3.5)

$$\begin{aligned}&\int _0^T\int _{\Omega }(u_n\otimes u_n)_r: (\nabla \Phi ) \,\mathrm{d}x\mathrm{d}t =\nonumber \\&\int _0^T\int _{\Omega }(u_n)_r\otimes (u_n)_r : \nabla \Phi \,\mathrm{d}x\mathrm{d}t + \int _0^T\int _{\Omega }\rho _r(u_n, u_n) : (\nabla \Phi ) \,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.11)

We take n large enough so that $\eta (n)\le r$. We estimate the second term using the assumption (3.2):

$$\begin{aligned} \left| \int _0^T\int _{\Omega }\rho _r(u_n, u_n) : (\nabla \Phi ) \,\mathrm{d}x\mathrm{d}t\right| \le C_{\Phi }E_K r^{2\zeta _2} \end{aligned}$$

Therefore we have proved so far that

$$\begin{aligned} \left| N_{\Phi }(n)- \int _0^T\int _{\Omega }(u_n)_r\otimes (u_n)_r : \nabla \Phi \,\mathrm{d}x\mathrm{d}t \right| \le C_\Phi \left( E_K r^{2\zeta _2} + E r\right) \end{aligned}$$

(3.12)

holds for n large enough, depending on r. $\square$

We note that if $u_n(t)$ converges weakly in $L^2(\Omega )$ to $u_{\infty }(t)$, then $(u_n(t))_r(x)$ converges pointwise in K to $(u_{\infty }(t))_r(x)$ at fixed r, just because it is the scalar product

$$\begin{aligned} (u_n(t))_r(x) = \int _{x-rA}u_n(y,t)j\left( \frac{x-y}{r}\right) r^{-3}\mathrm{d}y \end{aligned}$$

where we denoted by A the annulus $1\le |z|\le 2$. For $x\in K$ we observe that $x-rA\subset \subset \Omega$. We also have

$$\begin{aligned} |(u_n(t))_r(x)|\le Cr^{-\frac{3}{2}}\Vert u_n(t)\Vert _{L^2(\Omega )}\le Er^{-\frac{3}{2}} \end{aligned}$$

which allows us to use the dominated convergence theorem and pass to the limit. By the triangle inequality, we obtain

$$\begin{aligned} \left| N_{\Phi }(n)- \int _0^T\int _{\Omega }(u_{\infty })_r\otimes (u_{\infty })_r : \nabla \Phi \,\mathrm{d}x\mathrm{d}t \right| \le C_\Phi \left( E_K r^{2\zeta _2} + E r\right) \end{aligned}$$

(3.13)

for n sufficiently large, depending on r, and r small enough. Now we use the identity (3.5) in reverse, and the fact that translation is strongly continuous in $L^2$ to deduce that $(u_{\infty })_r$ converges to $u_\infty$ strongly in $L^2(0,T; L^2(K))$. Thus, given $\delta >0$ we can choose $r=r(\delta )$ small enough so that

$$\begin{aligned} \int _0^T\int _K|u_{\infty }-(u_{\infty })_r|^2\mathrm{d}x\mathrm{d}t \le \frac{\delta }{2} \end{aligned}$$

and, using (3.13) and making sure that r is small enough that

$$\begin{aligned} C_\Phi \left( E_K r^{2\zeta _2} + E r\right) \le \frac{\delta }{2} \end{aligned}$$

holds as well, we obtain

$$\begin{aligned} \left| N_{\Phi }(n)- \int _0^T\int _{\Omega }(u_{\infty })\otimes (u_{\infty }) : \nabla \Phi \,\mathrm{d}x\mathrm{d}t \right| \le \delta \end{aligned}$$

(3.14)

for n large enough. We have thus

$$\begin{aligned} \lim _{n\rightarrow \infty } N_{\Phi }(n) = \int _0^T\int _{\Omega }(u_{\infty })\otimes (u_{\infty }) : \nabla \Phi \,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

(3.15)

and this concludes the proof.

Remark 3.5

It is possible to remove the assumption of almost all time $L^2(\Omega )$ convergence, and replace it with the weak convergence in $L^2(0,T; L^2(\Omega ))$, at the price of demanding space-time second-order structure function scaling.

$$\begin{aligned} \int _0^T\int _K |u_n(x+y,t+s)-u_n(x,t)|^2\mathrm{d}x\mathrm{d}t \le E_K\left( |y|^{2\zeta _2} + |s|^\beta \right) \end{aligned}$$

(3.16)

for $\eta (n)\le |y|< \mathrm{dist}(K; \partial \Omega )$, $t+s\in [0,T]$, $|s|\ge \tau (n)$, $\tau (n)\rightarrow 0$, and $\beta >0$. The proof is the same; we translate in space-time. If $\tau (n)=0$ the requirement is strong, it implies the sequence bounded in $C^{\beta }(0,T; L^2(\Omega ))$, and in particular the $L^2(\Omega )$ convergence on each time slice.

Remark 3.6

By Fatou’s lemma in time and our assumptions, it follows that the limit solution of Euler equations satisfies the local bounds

$$\begin{aligned} \int _0^T\int _K |u_{\infty }(x+y,t)-u_{\infty }(x,t)|^2\mathrm{d}x\mathrm{d}t \le E_K |y|^{2\zeta _2} \end{aligned}$$

(3.17)

for $|y|<\mathrm{dist}(K, \partial \Omega )$ and any compact $K\subset \subset \Omega$.

4 Discussion

The vanishing of the dissipation rate follows from weak convergence in $L^2(\Omega )$ for all times only if the Euler equation is conservative. We proved results of emergence of weak, possibly dissipative solutions of Euler equations in 3D if the ensemble of Navier–Stokes solutions obeys a local-in-space but uniform in the ensemble second-order structure function scaling from above. In two dimensions, we proved the emergence of weak solutions form arbitrary families of strong solutions of Navier–Stokes equations with uniform interior (local) enstrophy bounds. There might be dissipative solutions among them, although an example is not available at this time.