1 Introduction

The aim of this paper is to extend some nowadays classical results about the energy conservation for the space-periodic 3D Euler equations (here \(\mathbb {T}^3:=\big ({\mathbb R}/2\pi \mathbb {Z}\big )^{3}\))

$$\begin{aligned} \begin{aligned} \partial _{t}v^E+(v^E\cdot \nabla )\,v^E+\nabla p^{E}&=0\qquad&\text { in } (0,T)\times \mathbb {T}^3,\\ {\text {div}}v^E&=0\qquad&\text { in }(0,T)\times \mathbb {T}^3,\\ v^E(0)&=v^E_0 \qquad&\text {in }\mathbb {T}^3, \end{aligned} \end{aligned}$$
(1.1)

to embrace a full space-time range of exponents. Recall that it is known since [10] that weak solutions of the Euler equations such that

$$\begin{aligned} v^E\in C_{w}(0,T;L^{2}(\mathbb {T}^3))\cap L^{3}(0,T;B^{\alpha }_{3,\infty }(\mathbb {T}^3)),\qquad \text {with}\quad \alpha >\frac{1}{3}, \end{aligned}$$
(1.2)

conserve the energy, where \(B^{\alpha }_{3,\infty }(\mathbb {T}^3)\) denotes a standard Besov space and the motivation for this result is the Onsager conjecture [19] following from Kolmogorov K41 theory. The Onsager conjecture (only recently solved also for the negative part, see Isett [17] and Buckmaster et al. [6]) suggested the threshold value \(\alpha =1/3\) for energy conservation. Here, we consider a combination of space-time conditions, identifying families of Besov spaces with the range (1/3, 1) for the exponent of regularity, balanced by a proper integrability exponent in time. Next, we consider also the limiting case \(\alpha =1\), and finally the connection of energy conservation with the vanishing viscosity limits. We recall that the first rigorous results about Onsager conjecture are probably those of Eyink [14, 15] in the Fourier setting and Constantin, E, and Titi [10] and we are mainly inspired by these references; for the vanishing viscosity limit we follow the same path as in Drivas and Eyink [12].

The original results we prove concern: 1) the extension of (1.2) to a full scale of exponents \(\frac{1}{3}<\alpha <1\), identifying the sharp conditions on the parameters, as previously done in the setting of Hölder continuous functions in [4]; 2) the extension to the case \(\alpha =1\), which means that we look for conditions on the gradient of \(v^E\) in standard Lebesgue spaces; 3) we identify hypotheses, uniform in the viscosity, on solutions to the Navier-Stokes equations (NSE), which allow us to pass to the limit as \(\nu \rightarrow 0\) and to construct weak solutions of the Euler equations satisfying the energy equality.

More precisely, concerning point 1) we extend the result of [10] to a wider range of exponents proving the following theorem:

Theorem 1

Let \(\frac{1}{3}<\alpha <1\), \(\alpha<\beta <1\), and \(v^E\) be a weak solution to the Euler equations (1.1) such that

$$\begin{aligned} v^E\in L^{1/\alpha }(0,T;B^\beta _{\frac{2}{1-\alpha },\infty }(\mathbb {T}^3)). \end{aligned}$$
(1.3)

Then, \(v^E\) conserves the energy.

Remark 1

In the sequel by conservation of energy, since we deal with functions \(v^{E}\) which are not strongly continuous with values in \(L^{2}(\mathbb {T}^3)\), we will always mean that \(\Vert v^{E}(t)\Vert _{2}=\Vert v^{E}(0)\Vert _{2}\) for almost all \(t\in [0,T]\).

Similar results have already been proved in the setting of Hölder continuous functions (see [3]) and in both cases, one can see that the limiting case \(\alpha \rightarrow 1^{-}\) leads formally to \(L^{1}(0,T;W^{1,\infty }(\mathbb {T}^3))\), which corresponds to the Beale-Kato-Majda criterion. It is also worth to mention the result in De Rosa, Inversi, and Stefani [11], which deals with energy conservation in more general spaces but includes also the case in which just the symmetric part of the gradient matrix belongs to \(L^1(0,T;L^\infty (\mathbb {T}^3))\). Nevertheless this improvement can be also extended in our case and is based on the observation that, for smooth enough vector fields u, v it holds

$$\begin{aligned} \int _{\mathbb {T}^3}(u\cdot \nabla )\, v\cdot u\,\text {d}x=\int _{\mathbb {T}^3}u\cdot Dv\cdot u\, \text {d}x,\qquad \text {with}\quad Dv=\frac{1}{2}(\nabla v+\nabla ^{T}v). \end{aligned}$$

Remark 2

Theorem 1, which identifies families of critical spaces, could be also modified to extend the results in Cheskidov et al. [9] involving the spaces \(B^{1/3}_{3,c_{{\mathbb N}}}\) and those of Fjordholm and Wiedemann [16] written in terms of iterated integrals. In our case, similar to Duchon and Robert [13] and Drivas and Eyink [12] we can prove the same results by using the spaces \(B^{\beta }_{\frac{2}{1-\alpha },c_{0}}\), which means to assume that there exists \(f\in L^{1/\alpha }(0,T)\) such that

$$\begin{aligned} \Vert v^{E}(\cdot +y)-v^{E}(\cdot )\Vert _{\frac{2}{1-\alpha }}\le f(t)|y|^{\beta }\sigma (|y|)\qquad a.e.\quad t\in (0,T), \end{aligned}$$

for some non-negative \(\sigma \) such that \(\sigma (s)\rightarrow 0\) as \(s\rightarrow 0^{+}\). A similar improvement is also discussed in [3] in the setting of classical Hölder spaces. Recent related results appeared also after submission of this paper in the technical report [20] and they generalize our result in the space \(L^p(0,T;B^{1/p}_{\frac{2p}{p-1},c_\mathbb {N}}(\mathbb {T}^3))\), for \(1<p\le 3\).

Anyway, working directly with the velocity \(v^E\) in a Sobolev space, we obtain the following result:

Theorem 2

Let \(q>2\), \(r>\frac{5q}{5q-6}\), and let \(v^E\) be a weak solution to the Euler equations (1.1) such that

$$\begin{aligned} v^E\in L^{r}(0,T;W^{1,q}(\mathbb {T}^3)). \end{aligned}$$
(1.4)

Then, \(v^E\) conserves the energy.

The sharpness of this result comes by observing that we recover for the Euler equations the same conditions (at least in this range of exponents) which are known for Leray-Hopf weak solutions to the NSE, see the recent results in [1, 5].

Concerning point 3) we extend results from [4, 12] on the emergence of solutions to the Euler equations satisfying the energy equality as inviscid limits of Leray-Hopf weak solutions to the NSE (\(\nu >0\))

$$\begin{aligned} \begin{aligned} \partial _{t}v^\nu +(v^\nu \cdot \nabla )\,v^\nu -\nu \Delta v^\nu +\nabla p^{\nu }&=0\qquad&\text { in }(0,T)\times \mathbb {T}^3, \\ {\text {div}}v^\nu&=0\qquad&\text { in }(0,T)\times \mathbb {T}^3, \\ v^\nu (0 )&=v^\nu _0 \qquad&\text { in }\mathbb {T}^3. \end{aligned} \end{aligned}$$
(1.5)

This result generalizes to a wider range of exponents the result from [12] which deals with \(\alpha \sim 1/3\) and also the results in [4], which are in the setting of Hölder continuous functions, but with a more restrictive time-dependence in the case \(\alpha >1/2\). We have the following result:

Theorem 3

Let \(\frac{1}{3}<\alpha <1\), \(\alpha<\beta <1\), and let \(\{v^\nu \}_{\nu >0}\) be a family of weak solutions of the NSE (1.5) with the same initial datum \(v_{0}=v(0)\in H\cap B^\beta _{\frac{2}{1-\alpha },\infty }(\mathbb {T}^3)\). Let us also assume that there exists a constant \(C_{\alpha ,\beta }>0\), independent of \(\nu >0\), such that

$$\begin{aligned} \Vert v^\nu \Vert _{L^{1/\alpha }(0,T; B^\beta _{\frac{2}{1-\alpha },\infty })}\le C_{\alpha ,\beta },\qquad \forall \,\nu \in (0,1]. \end{aligned}$$
(1.6)

Then, in the limit \(\nu \rightarrow 0\) the family \(\{v^{\nu }\}\) strongly converges in \(L^2((0,T)\times \mathbb {T}^3)\) (up to a sub-sequence) to a weak solution \(v^E\) in [0, T] of the Euler equations satisfying the energy equality and explicit rates on the vanishing of the energy dissipation are proved.

Alternatively, the same result holds if for \(q>2\) and \(r>\frac{5q}{5q-6}\) we have \(v_{0}\in H\cap W^{1,q}(\mathbb {T}^3)\) and \(\Vert v^\nu \Vert _{L^{r}(0,T; W^{1,q}(\mathbb {T}^3))}\le C_{r,q}\), for all \(\nu \in (0,1]\).

In the above theorem one can also consider a family of initial data uniformly bounded \(H\cap B^\beta _{\frac{2}{1-\alpha },\infty }(\mathbb {T}^3)\) (or in \(H\cap W^{1,q}(\mathbb {T}^3)\) in the second case), in such a way that –by Sobolev type embeddings– one can infer that \(v_{0}^{\nu }\rightarrow v_{0}\) strongly in H, and the proof will remain the same. The problem of vanishing viscosity and construction of distributional (dissipative) solutions to the Euler equations has a long history and we mainly refer to Duchon and Robert [13] for similar results. We also wish to mention the Fourier-based approach recently developed by Chen and Glimm [7, 8] where spectral properties are used to deduce certain fractional regularity results, suitable to prove the inviscid limit, see also [11]. Our proof uses standard mollification and handling of the commutation terms. Even though the results use elementary techniques, they are new and rather sharp. Note also that Theorem 3 implies that “quasi-singularities” are required in Leray-Hopf weak solutions in order to account for anomalous energy dissipation, see the discussion and interpretation in [4, 12]. Moreover, due to results in [1, 5], weak solutions of the NSE with \(\nabla v^\nu \in L^{r}(0,T;L^{q}(\mathbb {T}^3))\), \(r>\frac{5q}{5q-6}\) conserve the energy and the same holds for the limiting solutions of the Euler equations.

Plan of the paper: In Sect. 2 we set up our notation by giving the definitions of the spaces and the solutions that we use throughout the paper. Moreover, we recall the basic properties of the mollification and the commutator formula that will be used extensively in the proofs of the theorems. In Sect. 3 we give the proofs of Theorem 1 and 2, investigating minimum regularity conditions for energy conservation, for weak solutions to the Euler equations. Finally, in Sect. 4, we give the proof of Theorem 3, dealing with the emergence of weak solutions of Euler in the limiting case \(\nu \rightarrow 0\).

2 Notation

In the sequel we will use the Lebesgue \((L^{p}(\mathbb {T}^3),\Vert .\,\Vert _{p})\) and Sobolev \((W^{1,p}(\mathbb {T}^3),\Vert .\,\Vert _{1,p})\) spaces, with \(1\le p\le \infty \); for simplicity we denote by \((\,.\,,\,.\,)\) and \(\Vert .\,\Vert \) the \(L^{2}\) scalar product and norm, respectively, while the other norms are explicitly indicated. By H and V we denote the closure of smooth, periodic, and divergence-free vector fields in \(L^{2}(\mathbb {T}^3)\) or \(W^{1,2}(\mathbb {T}^3)\), respectively. Moreover, we will use the Besov spaces \(B^{\alpha }_{p,\infty }(\mathbb {T}^3)\), which are the same as Nikol’skiĭ spaces \({{N}}^{\alpha ,p}(\mathbb {T}^3)\). They are sub-spaces of \(L^{p}(\mathbb {T}^3)\) for which there exists \(c>0\), such that \(\Vert u(\cdot +h)-u(\cdot )\Vert _{p}\le c |h|^{\alpha }\), and the smallest constant is the semi-norm \([.\,]_{B^{\alpha }_{p,\infty }}\).

To properly set the problem we consider, we give the definitions of weak solutions:

Definition 1

(Weak solution to the Euler equations) Let \(v_0\in H\). A measurable function \(v^E:\,(0,T)\times \mathbb {T}^3\rightarrow {\mathbb R}^3\) is called a weak solution to the Euler equations (1.1) if \( v^E\in C_{w}(0,T;H)\), solves the equations in the following weak sense:

$$\begin{aligned}{} & {} \int _{\mathbb {T}^3}v(t)\cdot \phi (t)\,\text {d}x-\int _{\mathbb {T}^3}v_0\cdot \phi (0)\,\text {d}x\nonumber \\{} & {} \qquad -\int _0^{t}\int _{\mathbb {T}^3}\Big [v^E\cdot \partial _{t}\phi + (v^E\otimes v^E):\nabla \phi \Big ]\,\text {d}x\,d t=0, \end{aligned}$$
(2.1)

for all \(\phi \in C^{1}([0,T];C^{\infty }(\mathbb {T}^3))\) such that \({\text {div}}\phi =0\).

We also recall the definition of weak solutions to the Navier–Stokes equations.

Definition 2

(Space-periodic Leray-Hopf weak solution) Let \(v_0^{\nu }\in H\). A measurable function \(v^\nu :\,(0,T)\times \mathbb {T}^3\rightarrow {\mathbb R}^{3}\) is called a Leray-Hopf weak solution to the space-periodic NSE (1.5) if \( v^\nu \in L^\infty (0,T;H)\cap L^2(0,T;V)\) and the following hold true:

The function \(v^\nu \) solves the equations in the weak sense:

$$\begin{aligned} \int _0^{T}\int _{\mathbb {T}^3}\Big [v^\nu \cdot \partial _{t}\phi -\nu \nabla v^\nu :\nabla \phi + (v^\nu \otimes v^\nu ):\nabla \phi \Big ]\,\text {d}x\text {d}t= -\int _{\mathbb {T}^3}v^\nu _0\cdot \phi (0)\,\text {d}x,\nonumber \\ \end{aligned}$$
(2.2)

for all \(\phi \in {\mathcal {D}}_T=\Big \{\phi \in C^{\infty }_{0}([0,T[;C^{\infty }(\mathbb {T}^3)):\ {\text {div}}\phi =0\Big \}\);

The (global) energy inequality holds:

$$\begin{aligned} \frac{1}{2} \Vert v^\nu (t)\Vert ^2_{2}+\nu \int _0^t\Vert \nabla v^\nu (\tau )\Vert ^2_{2}\, d \tau \le \frac{1}{2}\Vert v^\nu _0\Vert ^2_{2},\qquad \forall \,t\in [0,T]; \end{aligned}$$
(2.3)

The initial datum is strongly attained: \(\lim _{t\rightarrow 0^+}\Vert v^\nu (t)-v^\nu _0\Vert =0\).

Definition 2 is the original definition of weak solutions to the NSE. Note that the strong \(L^{2}\)-continuity from the right at \(t=0\) is a part of the definition which could be derived from the energy inequality, after having redefined the solution a.e. in [0, T] to identify it with a \(L^{2}\)-weakly continuous function, with the so called “Hopf lemma.” Also with this assumption one can consider a weak formulation similar to that for the Euler equations with test functions with support in [0, T].

2.1 Mollification and Sobolev/Besov spaces

We use the classical tools of mollification to justify calculations and to this end we fix \(\rho \in C^{\infty }_{0}({\mathbb R}^{3})\) such that \(\rho (x)=\rho (|x|),\ \rho \ge 0,\ \text {supp }\rho \subset B(0,1)\subset {\mathbb R}^3,\ \int _{{\mathbb R}^{3}}\rho (x)\,\text {d}x=1\), and we define, for \(\epsilon \in (0,1]\), the Friedrichs family \(\rho _{\epsilon }(x):=\epsilon ^{-3}\rho (\epsilon ^{-1}x)\). Then, for any function \(f\in L^{1}_{\text {loc}}({\mathbb R}^{3})\) we define by the usual convolution

$$\begin{aligned} f_{\epsilon }(x):=\int _{{\mathbb R}^{3}}\rho _{\epsilon }(x-y)f(y)\,\text {d}y=\int _{{\mathbb R}^{3}} \rho _{\epsilon }(y)f(x-y)\,\text {d}y. \end{aligned}$$

If \(f\in L^{1}(\mathbb {T}^3)\), then \(f\in L^{1}_{\text {loc}}({\mathbb R}^{3})\), and it turns out that \(f_{\epsilon }\in C^{\infty }(\mathbb {T}^3)\) is \(2\pi \)-periodic along the \(x_{j}\)-direction, for \(j=1,2,3\). Moreover, if f is a divergence-free vector field, then \(f_{\epsilon }\) is a smooth divergence-free vector field. We report now the basic properties of the convolution operator we will use in the sequel, see for instance [2, 4, 10].

Lemma 4

Let \(\rho \) be as above. If \(u\in L^{q}(\mathbb {T}^3)\), then \(\exists \, C>0\) (depending only on \(\rho \)) such that

$$\begin{aligned} \Vert u_{\epsilon }\Vert _{r}\le \frac{C}{\epsilon ^{3(\frac{1}{q}-\frac{1}{r})}} \Vert u\Vert _{q}\quad \text {for all }r\ge q; \end{aligned}$$
(2.4)

If \(u\in B^{\beta }_{q,\infty }(\mathbb {T}^3)\), then

$$\begin{aligned}&\Vert u(\cdot +y )-u(\cdot )\Vert _{q}\le [u]_{B^{\beta }_{q,\infty }}|y |^{\beta }, \end{aligned}$$
(2.5)
$$\begin{aligned}&\Vert u-u_{\epsilon }\Vert _{q}\le [u]_{B^{\beta }_{q,\infty }}\, \epsilon ^{\beta }, \end{aligned}$$
(2.6)
$$\begin{aligned}&\Vert \nabla u_{\epsilon }\Vert _{q}\le C [u]_{B^{\beta }_{q,\infty }}\,\epsilon ^{\beta -1}, \end{aligned}$$
(2.7)

while if \(u\in W^{1,q}(\mathbb {T}^3)\), then

$$\begin{aligned}&\Vert u(\cdot +y )-u(\cdot )\Vert _{q}\le \Vert \nabla u\Vert _{q}|y |, \end{aligned}$$
(2.8)
$$\begin{aligned}&\Vert u-u_{\epsilon }\Vert _{q}\le \Vert \nabla u\Vert _{q}\, \epsilon , \end{aligned}$$
(2.9)
$$\begin{aligned}&\Vert \nabla u_{\epsilon }\Vert _{q}\le C \Vert u\Vert _{q}\,\epsilon ^{-1}. \end{aligned}$$
(2.10)

In the sequel, the following well-known commutator formula derived in [10] and known as the “Constantin-E-Titi commutator” will be crucial:

$$\begin{aligned} (u\otimes u)_{\epsilon }=u_{\epsilon }\otimes u_{\epsilon } +r_{\epsilon }(u,u)-(u-u_{\epsilon })\otimes (u-u_{\epsilon }), \end{aligned}$$
(2.11)

with

$$\begin{aligned} r_{\epsilon }(u,u):=\int _{\mathbb {T}^3}\rho _{\epsilon }(y)(\delta _{y}u(x) \otimes \delta _{y}u(x))\,\text {d}y,\qquad \text {for}\quad \delta _{y}u(x):=u(x-y)-u(x). \end{aligned}$$

3 On the conservation of energy for ideal fluids

We prove Theorem 1 and, for \(\alpha \in (\frac{1}{3},1)\), we investigate the minimum Besov regularity that is needed so that weak solutions of the Euler equations conserve their kinetic energy.

Proof of Theorem 1

We test the Euler equations against \(\varphi =(v^E_\epsilon )_\epsilon \); the argument would require a further smoothing with respect to time, which is standard (cf. [10]) and inessential to the proof. We obtain

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Vert v^E_\epsilon (t)\Vert _{2}^2&=\frac{1}{2}\Vert v^E_\epsilon (0)\Vert _{2}^2-\int _0^t\int _{\mathbb {T}^3} r_\epsilon (v^E,v^E):\nabla v^E_\epsilon \,\text {d}x\text {d}s\\&+ \int _0^t\int _{\mathbb {T}^3}(v^E-v^E_\epsilon )\otimes (v^E-v^E_\epsilon ):\nabla v^E_\epsilon \,\text {d}x\text {d}s, \end{aligned} \end{aligned}$$
(3.1)

since \( \int _0^t\int _{\mathbb {T}^3} v^E_\epsilon \otimes v^E_\epsilon :\nabla v^E_\epsilon \,\text {d}x\text {d}s=0\), due to \(v^E_{\epsilon }\) being smooth and divergence-free. We now estimate the last two terms from the right-hand side, using the properties of Besov spaces.

Indeed, for \(0<\eta <2\) and \(q>1\), such that \(q-(1+\eta )>0\) we write:

$$\begin{aligned} \begin{aligned} I_1&:=\left| \int _0^t\int _{\mathbb {T}^3}(v^E-v^E_\epsilon )\otimes (v^E-v^E_\epsilon ):\nabla v^E_\epsilon \,\text {d}x\text {d}s\right| \\&\le \int _0^T\int _{\mathbb {T}^3}|v^E-v^E_\epsilon |^\eta |v^E-v^E_\epsilon |^{2-\eta }|\nabla v^E_\epsilon |\,\text {d}x\text {d}s\\&\le \int _0^T\Vert v^E-v^E_\epsilon \Vert _{q}^\eta \, \Vert v^E-v^E_\epsilon \Vert _{\frac{(2-\eta )q}{q-(1+\eta )}}^{2-\eta }\Vert \nabla v^E_\epsilon \Vert _{q}\,\text {d}s, \end{aligned} \end{aligned}$$

and in the second line we used Hölder’s inequality. Since a weak solution \(v^E\) is in \(L^\infty (0,T;H)\), if \(\frac{(2-\eta )q}{q-(1+\eta )}=2\), we can use the energy bound to infer

$$\begin{aligned} \Vert v^E(t)-v^E_\epsilon (t)\Vert _{2}\le & {} \Vert v^E(t)\Vert _{2}+\Vert v^E_\epsilon (t)\Vert _{2}\le 2\Vert v^E(t)\Vert _{2}\\\le & {} 2\,\text {esssup}_{t\in (0,T)}\Vert v^E(t)\Vert _{2}\le C, \end{aligned}$$

and thus by (2.6)-(2.7)

$$\begin{aligned} \begin{aligned} I_1&\le C\int _0^T\Vert v^E-v^E_\epsilon \Vert _{q}^\eta \,\Vert \nabla v^E_\epsilon \Vert _{q}\,\text {d}s\le C\epsilon ^{\beta \eta +\beta -1}\int _0^T[v^E(t)]_{B_{q,\infty }^\beta }^{\eta +1}\text {d}s, \end{aligned} \end{aligned}$$

where \(C>0\) does not depend on \(\epsilon >0\).

Next, we estimate the remainder term in the commutator as follows:

$$\begin{aligned} \begin{aligned} \left| r_\epsilon (v^E,v^E)\right|&= \left| \int _{\mathbb {T}^3}\rho _\epsilon (y)(v^E(x-y)-v^E(x))\otimes (v^E(x-y)-v^E(x))\,\text {d}y\right| \\&\overset{y=\epsilon z}{=}\ \left| \int _{\mathbb {T}^3}\rho (z)(v^E(x-\epsilon z))-v^E(x))\otimes (v^E(x-\epsilon z))-v^E(x))\,\text {d}z\right| \\&\le \int _{\mathbb {T}^3}|v^E(x-\epsilon z)-v^E(x)|^2\,\text {d}z. \end{aligned} \end{aligned}$$

Then, as above, we can write for \(0<\eta <2\) such that \(\frac{(2-\eta )q}{q-(1+\eta )}=2\):

$$\begin{aligned} \begin{aligned} I_2&:= \left| \int _0^t\int _{\mathbb {T}^3} r_\epsilon (v^E,v^E):\nabla v^E_\epsilon \,\text {d}x\text {d}s\right| \\&\le \int _0^T\int _{\mathbb {T}^3}\int _{B(0,1)}|v^E(x-\epsilon z)-v^E(x)|^2|\nabla v^E_\epsilon |\,\text {d}z\text {d}x\text {d}s\\&= \int _0^T\int _{\mathbb {T}^3}\int _{B(0,1)}|v^E(x-\epsilon z)-v^E(x)|^\eta |v^E(x-\epsilon z)-v^E(x)|^{2-\eta }|\nabla v^E_\epsilon |\,\text {d}z\text {d}x\text {d}s\\&\le C\int _0^T\int _{\mathbb {T}^3}\Vert \nabla v^E_\epsilon \Vert _{q}\,\Vert v^E(\cdot -\epsilon z)-v^E(\cdot )\Vert _{q}^\eta \,\Vert v^E(\cdot -\epsilon z)-v^E(\cdot )\Vert _{\frac{(2-\eta )q}{q-(1+\eta )}}^{2-\eta }\,\text {d}x\text {d}s. \end{aligned} \end{aligned}$$

and using (2.5)-(2.7) we arrive at

$$\begin{aligned} \begin{aligned} I_2&\le C\epsilon ^{\beta \eta +\beta -1}\int _0^T\int _{\mathbb {T}^3}|z|^{\eta \beta }[v^E(t)]_{B_{q,\infty }^\beta }^{\eta +1}\,\text {d}z\text {d}s\le C\epsilon ^{\beta \eta +\beta -1}\int _0^T\Vert v^E(t)\Vert _{B_{q,\infty }^\beta }^{\eta +1}\text {d}s, \end{aligned} \end{aligned}$$

with \(C>0\) independent of \(\epsilon \).

Hence, for fixed \(\beta \in (\frac{1}{3},1)\), we want to find \((q,\eta )\in (1,+\infty )\times (0,2)\) such that \(\eta +1\) (the exponent of the Besov semi-norm) is the smallest possible, subject to the following set of constraints:

$$\begin{aligned} \left\{ \begin{array}{cc} \beta \eta +\beta -1>0 \\ \eta <q-1 \\ \frac{(2-\eta )q}{q-(1+\eta )}=2 \end{array}\right. . \end{aligned}$$

Hence, we set \(\eta =\frac{1-\alpha }{\alpha }\) and \(q=\frac{2}{1-\alpha }\) and the third constraint is satisfied. The second constraint is satisfied for \(\alpha >1/3\) (leading to \(q>3\)); next, if \(\beta >\alpha \), then \(\beta \eta +\beta -1=\frac{\beta -\alpha }{\alpha }>0\) and consequently

$$\begin{aligned} 0\le I_1+I_2\le C\epsilon ^\frac{\beta -\alpha }{\alpha }\int _0^T[v^E]^{1/\alpha }_{B_{\frac{2}{1-\alpha },\infty }^\beta } \text {d}t\overset{\epsilon \rightarrow 0}{\longrightarrow }0, \end{aligned}$$

and letting \(\epsilon \rightarrow 0\) in (3.1) gives

$$\begin{aligned} \frac{1}{2}\Vert v^E(t)\Vert _{L^2}^2 =\frac{1}{2}\Vert v^E(0)\Vert _{L^2}^2, \end{aligned}$$
(3.2)

and this is valid for almost all \(t\in [0,T]\). Therefore, for \(\alpha \in (\frac{1}{3},1)\), the “critical” space for energy conservation is \(L^{1/\alpha }(0,T;B^\alpha _{\frac{2}{1-\alpha },\infty })\). \(\square \)

We now prove the second theorem, corresponding to conditions on the gradient of \(v^E\), which would, formally, be the same with \(\alpha =1\), but in fact the result here is much stronger, since the bound on the gradient allows us to make sense of the convective term in a more precise manner.

Proof of Theorem 2

In the case of a condition on the gradient the approach is similar as before and we just need to control the commutator terms, after testing the equations by \((v^E_\epsilon )_\epsilon \) (we make explicit computations only for this one, since the remainder can be handled as we have done before). We get in fact

$$\begin{aligned} I_1:&= \left| \int _0^t\int _{\mathbb {T}^3}(v^E-v^E_\epsilon )\otimes (v^E-v^E_\epsilon ):\nabla v^E_\epsilon \,\text {d}x\text {d}s\right| \\&\le \int _0^T\int _{\mathbb {T}^3}|v^E-v^E_\epsilon |^2|\nabla v^E_\epsilon |\,\text {d}x\text {d}s\\&\le \int _0^T\Vert v^E-v^E_\epsilon \Vert ^2_{2p}\,\Vert \nabla v^E_\epsilon \Vert _{p'}\,\text {d}x\text {d}s\\&\le \int _0^T\Vert v^E-v^E_\epsilon \Vert _{2}^{2\theta }\,\Vert v^E-v^E_\epsilon \Vert _{q}^{2(1-\theta )}\Vert \nabla v^E_\epsilon \Vert _{p'}\,\text {d}x\text {d}s, \end{aligned}$$

where in the second step we used Hölder’s inequality with conjugate exponents p and \(p'\) (to be determined), and in the third one convex interpolation such that \(\frac{1}{2p}=\frac{\theta }{2}+\frac{1-\theta }{q}\), with \(2p<q\). Now, using the fact that \(v^E\in L^\infty (0,T;L^2(\mathbb {T}^3))\) and inequality (2.4) for the gradient of \(v^E\)

$$\begin{aligned} \Vert \nabla v^E_\epsilon \Vert _{p'}\le C\epsilon ^{-3\left( \frac{1}{q}-\frac{1}{p'}\right) }\Vert \nabla v^E\Vert _{q}, \qquad \text {for }p'>q, \end{aligned}$$

we obtain:

$$\begin{aligned} I_1&\le C\epsilon ^{-3\left( \frac{1}{q}-\frac{1}{p'}\right) }\int _0^T\Vert v^E-v^E_\epsilon \Vert _{q}^{\frac{2q(p-1)}{p(q-2)}}\Vert \nabla v^E\Vert _{q}\,\text {d}t\end{aligned}$$

and (2.9)-(2.10) yield:

$$\begin{aligned} I_1\le C\epsilon ^{\frac{2q(p-1)}{p(q-2)}-3\left( \frac{1}{q}-\frac{1}{p'}\right) }\int _0^T\Vert \nabla v^E(t)\Vert _{q}^{\frac{2q(p-1)}{p(q-2)}+1}\,\text {d}t. \end{aligned}$$

Now, we want to choose p such that the exponent of \(\epsilon \) is strictly positive corresponding to

$$\begin{aligned} p>\frac{(5q-6)q}{5q^2-9q+6}, \end{aligned}$$

and notice that the expression on the denominator is strictly positive. Moreover, the constraints \(2p<q\) and \(p'>q\) imply that \(p<\min \left\{ \frac{q}{2},\frac{q}{q-1}\right\} \) and thus the range of allowed p is

$$\begin{aligned} \frac{(5q-6)q}{5q^2-9q+6}<p<\min \left\{ \frac{q}{2},\frac{q}{q-1}\right\} , \end{aligned}$$

which is non-empty since \( \frac{(5q-6)q}{5q^2-9q+6}\) is always strictly smaller than \(\min \left\{ \frac{q}{2},\frac{q}{q-1}\right\} \) for \(q>2\).

The second term on the right-hand side of (3.1) is handled the same way and yields the same range for p. Then, for the infimum value of p that makes the exponent of \(\epsilon \) strictly positive, we get that the exponent r of the \(L^q\) norm of the gradient becomes \(r=\frac{5q}{5q-6}\) and thus the “critical” space for energy conservation is \(\nabla v^E\in L^{\frac{5q}{5q-6}}(0,T;L^q(\mathbb {T}^3))\). \(\square \)

4 Inviscid limit from Navier–Stokes to Euler

In this section we prove Theorem 3, dealing with the inviscid (singular) limit \(\nu \rightarrow 0\) and identifying sufficient conditions to construct weak solutions of the Euler equations conserving the kinetic energy.

Proof of Theorem 3

We give the proof only in the Besov case, since the other one is pretty similar. In the weak formulation (2.2) of the NSE we set \(\varphi =(v_\epsilon ^\nu )_\epsilon \). Note that since \(v^\nu _0\in L^2(\Omega )\), we deduce, being \(v^\nu \) a Leray-Hopf solution, that

$$\begin{aligned} \Vert v^\nu \Vert _{L^\infty (0,T;L^2(\mathbb {T}^3))}+ \Vert \sqrt{\nu }\nabla v^\nu \Vert _{L^2((0,T)\times \mathbb {T}^3)}\le C. \end{aligned}$$

Moreover, since \(v^\nu \in L^{1/\alpha }(0,T;B^\beta _{2/(1-\alpha ),\infty })\), it has a derivative in the sense of distributions in the space \(L^{1/2\alpha }(0,T;B_{1/(1-\alpha ),\infty }^{\beta -2})\), given by \(\frac{\text {d} v^\nu }{\text {d}t}=-\mathbb {P}{\text {div}}(v^\nu \otimes v^\nu )+\nu \Delta v^\nu \), where \(\mathbb {P}\) is the Leray projector. Indeed, by comparison

$$\begin{aligned} \int _0^T \big \langle \partial _tv^\nu ,\phi \big \rangle \,\text {d}t=\int _0^T \big \langle -\mathbb {P}{\text {div}}(v^\nu \otimes v^\nu )+\nu \Delta u^{\nu },\phi \big \rangle \,\text {d}t,\qquad \forall \,\phi \in C^\infty (\mathbb {T}^3), \end{aligned}$$

where \(\langle \cdot ,\cdot \rangle \) is the duality pairing between elements of \({{D}}'(\mathbb {T}^3)\) and \({{D}}(\mathbb {T}^3)=C^\infty (\mathbb {T}^3)\). Choosing \(\phi (x,t)=\psi (t)\varphi (x)\), with \(\psi \in C_{0}^\infty (0,T)\) and \(\varphi \in C^\infty (\mathbb {T}^3)\) we obtain

$$\begin{aligned} \int _0^T \big \langle \partial _tv^\nu ,\psi \,\varphi \big \rangle \,\text {d}t=\int _0^T\psi (t)\big \langle \left[ -\mathbb {P}{\text {div}}(v^\nu \otimes v^\nu )+\nu \Delta u^{\nu }\right] ,\varphi \big \rangle \,\text {d}t, \end{aligned}$$

and note that

$$\begin{aligned} \begin{aligned} \Vert \mathbb {P}{\text {div}}(v^\nu \otimes v^\nu )\Vert _{L^{1/2\alpha }(0,T;B_{1/(1-\alpha ),\infty }^{\beta -2})}&\le C\Vert v^\nu \otimes v^\nu \Vert _{L^{1/2\alpha }(0,T;B_{1/(1-\alpha ),\infty }^{\beta -1})} \\&{\le C\Vert v^\nu \Vert ^2_{L^{1/\alpha }(0,T;B_{2/(1-\alpha ),\infty }^{\beta -1})} } \\&\le C\Vert v^\nu \Vert ^2_{L^{1/\alpha }(0,T;B_{2/(1-\alpha ),\infty }^{\beta })}, \end{aligned} \end{aligned}$$

where the first estimate is proved in [12] and also in Kozono, Ogawa and Taniuchi [18], by properties of singular integral operators.

Moreover, since \(\mathbb {T}^3\) is bounded and \(T<+\infty \), the embeddings

$$\begin{aligned} L^{1/\alpha }(0,T;B_{2/(1-\alpha ),\infty }^{\beta -2})\hookrightarrow L^{1/2\alpha }(0,T;B_{2/(1-\alpha ),\infty }^{\beta -2})\hookrightarrow L^{1/2\alpha }(0,T;B_{1/(1-\alpha ),\infty }^{\beta -2}), \end{aligned}$$

are continuous, thus

$$\begin{aligned} \begin{aligned} \Vert \Delta v^\nu \Vert _{L^{1/2\alpha }(0,T;B_{1/(1-\alpha ),\infty }^{\beta -2})}&\le C\Vert \Delta v^\nu \Vert _{L^{1/2\alpha }(0,T;B_{2/(1-\alpha ),\infty }^{\beta -2})} \\&\le C\Vert \Delta v^\nu \Vert _{L^{1/\alpha }(0,T;B_{2/(1-\alpha ),\infty }^{\beta -2})} \\&\le C\Vert v^\nu \Vert _{L^{1/\alpha }(0,T;B_{2/(1-\alpha ),\infty }^{\beta })}, \end{aligned} \end{aligned}$$

and we conclude that \(\partial _tv^\nu \in L^{1/2\alpha }(0,T;B_{1/(1-\alpha ),\infty }^{\beta -2})\).

From the uniform bounds and standard pre-compactness arguments we get (up to a sub-sequence)

$$\begin{aligned} \begin{aligned} \sqrt{\nu }\nabla v^\nu&\rightharpoonup 0 ~\text {weakly in}~ L^2(0,T;H) \\ \partial _t v^\nu&\rightharpoonup \partial _t v ~\text {weakly in}~ L^{1/2\alpha }(0,T;B_{1/(1-\alpha ),\infty }^{\beta -2}), \end{aligned} \end{aligned}$$

Moreover, from the bound in (1.6), the compact embedding of \(B^\beta _{\frac{2}{1-\alpha },\infty }(\mathbb {T}^3)\) in \(L^{\frac{2}{1-\alpha }}(\mathbb {T}^3)\) (which follows by Riesz-Frechet-Kolmogorov theorem), and the bound proved for the time derivative, we get by the Aubin-Lions’ lemma that there exists a sub-sequence (not relabeled) such that:

$$\begin{aligned} \begin{aligned} v^\nu&\rightarrow v \quad \text {strongly in}~ L^q(0,T;H) \qquad \forall \, q\in (1,\infty ), \end{aligned} \end{aligned}$$

which is enough to pass to the limit as \(\nu \rightarrow 0\) in (2.2), proving that v is a solution of the Euler equations. \(\square \)

Remark 3

We also observe that since \(\{v^{\nu }\}\) is bounded uniformly in \(L^{1/\alpha }(0,T; B^\beta _{\frac{2}{1-\alpha },\infty })\), then by lower semicontinuity also \(v\in L^{1/\alpha }(0,T;B^{\beta }_{\frac{2}{1-\alpha },\infty })\). This would be enough to prove that the limit solution of the Euler equations conserves the energy. Nevertheless, the proof links in a quantitative way the subgrid scale of the regularization parameter with the viscosity to provide explicit rates in the convergence.

As a final step, we now show that the dissipation in the energy equation goes to zero as \(\nu \rightarrow 0\), yielding an energy equation for the limiting solution v, as well. Indeed, for \(\frac{1}{3}<\alpha \le \frac{1}{2}\) we have

$$\begin{aligned} \begin{aligned} \nu \int _0^t\Vert \nabla v_{\epsilon }^\nu \Vert _{2}^2\,\text {d}s&\le C\nu \int _0^t\Vert \nabla v_{\epsilon }^\nu \Vert _{\frac{2}{1-\alpha }}^2\,\text {d}s\le C\nu \epsilon ^{2(\beta -1)}\int _0^T\Vert v^{\nu }_{\epsilon }\Vert _{B^\beta _{\frac{2}{1-\alpha },\infty }}^2\,\text {d}s\\&\le C\nu \epsilon ^{2(\beta -1)}\Vert v^\nu \Vert _{L^\frac{1}{\alpha }(0,T;B^\beta _{\frac{2}{1-\alpha },\infty })}^2, \end{aligned} \end{aligned}$$

where \(C>0\) does not depend on \(\epsilon \) or \(\nu \) and we need to choose \(\epsilon ^{2(1-\beta )}\) going to zero slower than \(\nu \). This is the extension of the result in [4] for the Hölder case.

In the case \(\frac{1}{2}<\alpha <1\) we prove here a slightly better result, since

$$\begin{aligned} \begin{aligned} \nu \int _0^t\Vert \nabla v_{\epsilon }^\nu \Vert _{2}^2\,\text {d}s&= \nu \int _0^t\Vert \nabla v_{\epsilon }^\nu \Vert _{2}^{2-\frac{1}{\alpha }}\Vert \nabla v_{\epsilon }^\nu \Vert _{2}^{\frac{1}{\alpha }}\,\text {d}s\le C \nu \int _0^T\Vert \nabla v_{\epsilon }^\nu \Vert _{2}^{2-\frac{1}{\alpha }}\Vert \nabla v_{\epsilon }^\nu \Vert _{\frac{2}{1-\alpha }}^{\frac{1}{\alpha }}\,\text {d}s\\&\le C \nu \int _0^T\left( \frac{1}{\epsilon }\Vert v^{\nu }_{\epsilon }\Vert _{2}\right) ^{2-\frac{1}{\alpha }}\left( \epsilon ^{\beta -1} \Vert v^{\nu }_{\epsilon }\Vert _{B^\beta _{\frac{2}{1-\alpha },\infty }}\right) ^{\frac{1}{\alpha }}\,\text {d}s\\&\le C\nu \epsilon ^{\beta /\alpha -2}\Vert v^\nu \Vert _{L^\frac{1}{\alpha }\Big (0,T;B^\beta _{\frac{2}{1-\alpha },\infty }\Big )}^{1/\alpha } \\&\le C\nu \epsilon ^{-1}\Vert v^\nu \Vert _{L^\frac{1}{\alpha }\Big (0,T;B^\beta _{\frac{2}{1-\alpha },\infty }\Big )}^{1/\alpha }, \end{aligned} \end{aligned}$$

since \(\beta >\alpha \), with \(C>0\) independent of \(0<\epsilon <1\) and \(\nu >0\). One needs to choose \(\epsilon \) going to zero slower than \(\nu \).

So, in the case \(\frac{1}{3}<\alpha \le \frac{1}{2}\) we have \( \nu \int _0^t\Vert \nabla v_{\epsilon }^\nu \Vert _{2}^2\,\text {d}s=O(\nu \epsilon ^{2(\beta -1)})\), and thus

$$\begin{aligned} \frac{1}{2}\Vert v^\nu _{\epsilon }(t)\Vert _{2}^2-\frac{1}{2}\Vert v_{\epsilon }(0)\Vert _{2}^2 =O(\epsilon ^{\frac{\beta -\alpha }{\alpha }}) +O(\nu \epsilon ^{2(\beta -1)}); \end{aligned}$$

on the other hand in the case \(\frac{1}{2}<\alpha <1\) we have \(\nu \int _0^t\Vert \nabla v_{\epsilon }^\nu \Vert _{2}^2\,\text {d}s=O(\nu \epsilon ^{-1})\), and thus

$$\begin{aligned} \frac{1}{2}\Vert v^\nu _{\epsilon }(t)\Vert _{2}^2-\frac{1}{2}\Vert v_{\epsilon }(0)\Vert _{2}^2=O (\epsilon ^{\frac{\beta -\alpha }{\alpha }}) +O(\nu \epsilon ^{-1}). \end{aligned}$$

Since \(\epsilon >0\) is arbitrary, we can optimize the upper bound, the same way it was performed in [12], by balancing the contribution of the nonlinear flux with the one of the dissipation. Choosing \(\epsilon \sim \nu ^{\alpha /(\alpha +\beta -2\alpha \beta )}\) in the first case and \(\epsilon \sim \nu ^{\alpha /\beta }\) in the second one, yields the upper bounds

$$\begin{aligned} \frac{1}{2}\Vert v^\nu _{\epsilon }(t)\Vert _{2}^2-\frac{1}{2}\Vert v_{\epsilon }(0)\Vert _{2}^2 =O({\nu ^\frac{\beta -\alpha }{\alpha -2\alpha \beta +\beta } }), \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2}\Vert v^\nu _{\epsilon }(t)\Vert _{2}^2-\frac{1}{2}\Vert v_{\epsilon }(0)\Vert _{2}^2 =O({\nu ^\frac{\beta -\alpha }{\beta }}), \end{aligned}$$

respectively, hence showing that as \(\epsilon ,\nu \rightarrow 0\), linked with the above rates, the kinetic energy is conserved for almost all \(t\in (0,T)\). Note that from the strong convergence of \(v^{\nu }\rightarrow v\) in \(L^q(0,T;H)\), also follows, for almost all \(t\in (0,T)\), the strong convergence \(v^{\nu }(t)\rightarrow v(t)\) in \(L^{2}(\mathbb {T}^3)\), which is enough to conclude by the properties of the convolution.