1 Introduction

In [2], we proposed the concept of dissipative weak (DW) solution to the compressible (isentropic) Euler system:

$$\begin{aligned} \begin{aligned}&\partial _t \varrho + \mathrm{div}_x\mathbf{m}= 0,\\&\partial _t \mathbf{m}+ \mathrm{div}_x\left( \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho } \right) + \nabla _xp(\varrho ) = 0,\ p(\varrho ) = a \varrho ^\gamma ,\ a> 0,\ \gamma > 1, \end{aligned} \end{aligned}$$
(1.1)

considered on a bounded domain \(\Omega \subset R^d\), \(d=1,2,3\), with impermeable boundary

$$\begin{aligned} \mathbf{m}\cdot \mathbf{n}|_{\partial \Omega } = 0, \end{aligned}$$
(1.2)

and the initial conditions

$$\begin{aligned} \varrho (0, \cdot ) = \varrho _0,\ \mathbf{m}(0, \cdot ) = \mathbf{m}_0. \end{aligned}$$
(1.3)

A dissipative solution is a trio \([\varrho , \mathbf{m}, E]\), where \(\varrho \), \(\mathbf{m}\) satisfy (in the sense of distributions) the augmented system

$$\begin{aligned} \begin{aligned}&\partial _t \varrho + \mathrm{div}_x\mathbf{m}= 0,\\&\partial _t \mathbf{m}+ \mathrm{div}_x\left( \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho } \right) + \nabla _xp(\varrho ) = - \mathrm{div}_x{\mathfrak {R}}, \end{aligned} \end{aligned}$$
(1.4)

with the “turbulent” total energy \(E = E(t)\)—a non-increasing function of t—satisfying

$$\begin{aligned} \begin{aligned} E(\tau \pm )&\le E_0 = \int _{\Omega } \left[ \frac{1}{2}\frac{|\mathbf{m}_0|^2}{\varrho _0} + P(\varrho _0) \right] \ \,\mathrm{d} {x},\\ E(\tau \pm )&\ge \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x}\quad \ \text{ for } \text{ any }\ \tau > 0, \\ P(\varrho )&\equiv \frac{a}{\gamma - 1} \varrho ^\gamma . \end{aligned} \end{aligned}$$
(1.5)

Note that the total energy is defined as a convex l.s.c. function of \([\varrho , \mathbf{m}] \in R^{d+1}\),

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

The quantity \({\mathfrak {R}}\) is a matrix-valued measure, specifically,

$$\begin{aligned} {\mathfrak {R}} \in L^\infty ([0, \infty ); {\mathcal {M}}^+(\overline{\Omega }; R^{d \times d}_{\mathrm{sym}})) \end{aligned}$$
(1.6)

called Reynolds defect. Here, the symbol \({\mathcal {M}}^+(\overline{\Omega }; R^{d \times d}_{\mathrm{sym}}))\) denotes the cone of positively semi-definite symmetric matrix-valued measures on \(\overline{\Omega }\), specifically,

$$\begin{aligned} \left\langle {\mathfrak {R}}:[\xi \otimes \xi ]; g \right\rangle \ge 0 \quad \ \text{ for } \text{ any }\ g \in C(\overline{\Omega }),\ g \ge 0, \xi \in R^d. \end{aligned}$$

The crucial property of (DW) solutions is the compatibility condition

$$\begin{aligned}&E(\tau +) - \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x} \nonumber \\&\quad \ge {\underline{d}} \int _{\overline{\Omega }} \mathrm{d}\ (\mathrm{trace}[{\mathfrak {R}}])(\tau ) \quad \ \text{ for } \text{ any }\ \tau \in [0, \infty ) \end{aligned}$$
(1.7)

for a certain constant \({\underline{d}} > 0\). A detailed definition is given in Sect. 2 below.

Relation (1.7) can be interpreted in the way that the energy defect dominates the Reynolds defect. As shown in [2], the (DW) solutions exist globally in time for any finite energy initial data. Moreover, they can be identified as limits of consistent approximations arising in numerical analysis (see [11, 12]) or as vanishing viscosity limits of solutions to the Navier–Stokes system (see [10]). Note that, despite the large number of ill-posedness results (see, e.g., Chiodaroli et al. [3,4,5,6]), the standard (admissible) weak solutions that correspond to the case \({\mathfrak {R}} = 0\) are not known to exist globally in time for arbitrary initial data. (DW) solutions share many important properties with the standard (admissible) weak solutions:

  • Compatibility. Any (DW) solution, for which \([\varrho , \mathbf{m}]\) are continuously differentiable functions, is a classical solution. In particular, \({\mathfrak {R}} = 0\).

  • Weak–strong uniqueness. A (DW) solution coincides with the strong solution starting from the same initial data as long as the latter exists.

Moreover, we have shown in [2] that the class of all (DW) solutions admits a semiflow selection. In particular, the selected solutions are minimal with respect to the relation “\(\prec \)”:

$$\begin{aligned} {[}\varrho _1, \mathbf{m}_1, E_1] \prec [\varrho _2, \mathbf{m}_2, E_2] \ \Leftrightarrow \ E_1(\tau \pm ) \le E_2 (\tau \pm ) \quad \ \text{ for } \text{ any }\ \tau > 0. \end{aligned}$$

The minimal solutions dissipate the maximal amount of the total energy, which is in agreement with the commonly accepted maximal dissipation principle, see, e.g., Dafermos [7,8,9].

In this note, we show another interesting property of minimal (DW) solutions, namely

$$\begin{aligned} E(\tau ) - \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x} \rightarrow 0 \ \text{ as }\ \tau \rightarrow \infty . \end{aligned}$$
(1.8)

In view of (1.6), (1.7), the Reynolds defect \({\mathfrak {R}}\) vanishes in the asymptotic limit for large times. This fact may be seen as another piece of evidence supporting physical relevance of (minimal) (DW) solutions.

The paper is organized as follows. In Sect. 2, we introduce the necessary preliminary material and state the main result. In Sect. 3, we prove (1.8).

2 Preliminaries and main result

We recall the concept of dissipative weak solution introduced in [2, Definition 2.1].

Definition 2.1

(Dissipative weak (DW) solution) Let \(\Omega \subset R^d\), \(d=1,2,3\) be a bounded domain. We say that \([\varrho , \mathbf{m}, E]\) is a dissipative weak (DW) solution of the Euler system (1.1)–(1.4) in \([0, \infty ) \times \Omega \) if the following holds:

  • \(\varrho \ge 0\), and

    $$\begin{aligned}&\varrho \in C_{\mathrm{weak,loc}}([0, \infty ); L^\gamma (\Omega )),\ \mathbf{m}\in C_{\mathrm{weak,loc}}([0, \infty ); L^{\frac{2 \gamma }{\gamma + 1}}(\Omega ; R^d)),\\&\quad \ E \in BV[0, \infty ),\ E \ge 0; \end{aligned}$$
  • $$\begin{aligned} \left[ \int _{\Omega } \varrho \varphi \ \,\mathrm{d} {x} \right] _{t = 0}^{t = \tau } = \int _0^\tau \int _{\Omega } \Big [ \varrho \partial _t \varphi + \mathbf{m}\cdot \nabla _x\varphi \Big ] \ \,\mathrm{d} {x} \,\mathrm{d} t \end{aligned}$$

    for any \(\tau \ge 0\), \(\varphi \in C^1_c([0, \infty ) \times \overline{\Omega })\);

  • $$\begin{aligned} \begin{aligned}&\left[ \int _{\Omega } \mathbf{m}\cdot \varvec{\varphi } \ \,\mathrm{d} {x} \right] _{t = 0}^{t = \tau } \\&\quad =\int _0^\tau \int _{\Omega } \left[ \mathbf{m}\cdot \partial _t \varvec{\varphi }+ \frac{\mathbf{m}\otimes \mathbf{m}}{\varrho } : \nabla _x\varvec{\varphi }+ p(\varrho ) \mathrm{div}_x\varvec{\varphi }\right] \ \,\mathrm{d} {x} \,\mathrm{d} t \\&\qquad + \int _0^\tau \left( \int _{\overline{\Omega }} \nabla _x\varvec{\varphi }: \mathrm{d}{\mathfrak {R}} (t) \right) \,\mathrm{d} t \end{aligned} \end{aligned}$$

    for any \(\tau \ge 0\), \(\varphi \in C^1_c([0, \infty ) \times \overline{\Omega }; R^d)\), \(\varvec{\varphi }\cdot \mathbf{n}|_{\partial \Omega } = 0\), where

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

    is called Reynolds defect;

  • \(E: [0, \infty ) \rightarrow [0, \infty )\) is a non-decreasing function,

    $$\begin{aligned}&E(0-) \equiv \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}_0|^2}{\varrho _0} + P(\varrho _0) \right] \ \,\mathrm{d} {x},\nonumber \\&E(\tau +) - \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x} \nonumber \\&\quad \ge {\underline{d}} \int _{\overline{\Omega }} \mathrm{d}\ (\mathrm{trace} [{\mathfrak {R}}])(\tau ) \quad \ \text{ for } \text{ a } \text{ certain } \text{ constant }\ {\underline{d}} > 0 \end{aligned}$$
    (2.1)

    for any \(\tau \ge 0\).

As a matter of fact, the (DW) solutions introduced in [2] are defined as a barycenter of a Young measure \(\{ \nu _{t,x} \}_{t > 0, x \in \Omega }\), specifically

$$\begin{aligned} \varrho (t,x) = \left\langle \nu _{t,x}; {{\tilde{\varrho }}}\right\rangle ,\ \mathbf{m}(t,x) = \left\langle \nu _{t,x}; \tilde{\mathbf{m}}\right\rangle , \end{aligned}$$

with the associated total energy

$$\begin{aligned} E = \int _{\Omega } \left\langle \nu _{t,x}; \frac{1}{2} \frac{|\tilde{\mathbf{m}}|^2}{{{\tilde{\varrho }}}} + P({{\tilde{\varrho }}}) \right\rangle \ \,\mathrm{d} {x} + \int _{\overline{\Omega }} \mathrm{d}{\mathfrak {E}}, \end{aligned}$$

where \({\mathfrak {E}}\) is the so-called energy concentration defect. As observed in [10], the two definitions are equivalent.

Following [2], we introduce the relation \(\prec \) for two (DW) solutions \([\varrho _1, \mathbf{m}_1, E_1]\) and \([\varrho _2, \mathbf{m}_2, E_2]\) starting from the same initial data \([\varrho _0, \mathbf{m}_0]\),

$$\begin{aligned} {[}\varrho _1, \mathbf{m}_1, E_1] \prec [\varrho _2, \mathbf{m}_2, E_2]\ \Leftrightarrow E_1(\tau \pm ) \le E_2(\tau \pm ) \ \text{ for } \text{ all }\ \tau > 0. \end{aligned}$$

Finally, we introduce the admissible (DW) solution, cf. [2, Definition 2.3].

Definition 2.2

(Admissible (DW) solutions) A dissipative weak solution \([\varrho , \mathbf{m}, E]\) is called admissible if it is minimal with respect to the relation \(\prec \). Specifically, if \([{{\tilde{\varrho }}}, \tilde{\mathbf{m}}, {\widetilde{E}}]\) is another dissipative solution starting from the same initial data and such that

$$\begin{aligned} {[}{{\tilde{\varrho }}}, \tilde{\mathbf{m}}, {\widetilde{E}}] \prec [\varrho , \mathbf{m}, E], \end{aligned}$$

then

$$\begin{aligned} E(\tau \pm ) = {\widetilde{E}}(\tau \pm ) \quad \ \text{ for } \text{ any }\ \tau > 0. \end{aligned}$$

We are ready to state our main result.

Theorem 2.3

Let \(\Omega \subset R^d\) be a bounded Lipschitz domain. Let \([\varrho , \mathbf{m}, E]\) be an admissible (DW) solution of the isentropic Euler system in the sense of Definition 2.2.

Then,

$$\begin{aligned} \lim _{\tau \rightarrow \infty } E(\tau ) = \lim _{\tau \rightarrow \infty } \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x}, \end{aligned}$$
(2.2)

in particular,

$$\begin{aligned} \mathrm{ess} \sup _{t > \tau } \left\| {\mathfrak {R}}(t) \right\| _{{\mathcal {M}}(\overline{\Omega }; R^{d \times d}_{\mathrm{sym}})} \rightarrow 0 \ \text{ as }\ \tau \rightarrow \infty . \end{aligned}$$

The rest of the paper is devoted to the proof of Theorem 2.3.

3 Asymptotic behavior: proof of Theorem 2.3

The analysis leans on the following two results proved in [2].

Proposition 3.1

([2, Proposition 3.2])

Let \(T \ge 0\) and the initial data \(\varrho _T\), \(\mathbf{m}_T\),

$$\begin{aligned} \varrho _T \ge 0, \ E_T = \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}_T|^2}{\varrho _T} + P(\varrho _T) \right] \ \,\mathrm{d} {x} < \infty , \end{aligned}$$

be given.

Then, the Euler system admits a global in time dissipative solution \([\varrho , \mathbf{m}, E]\) in \([T; \infty )\) in the sense of Definition 2.1, specifically,

$$\begin{aligned}&\varrho \in C_{\mathrm{weak, loc}}([T,\infty ); L^\gamma (\Omega )),\ \mathbf{m}\in C_{\mathrm{weak, loc}}([T, \infty ); L^{\frac{2 \gamma }{\gamma + 1}}(\Omega ; R^d)), \nonumber \\&\quad E \in BV[T, \infty ) \ \text{ non-increasing }, \end{aligned}$$

such that

$$\begin{aligned} \begin{aligned}&0 \le E(\tau \pm ) \le E_T ,\\&E(\tau +) - \int _{\Omega } \frac{1}{2} \frac{|\mathbf{m}(\tau , \cdot ) |^2}{\varrho (\tau , \cdot )} P(\varrho (\tau , \cdot )) \ \,\mathrm{d} {x}&\ge \min \left\{ \frac{1}{2}, \frac{1}{\gamma - 1} \right\} \int _{\overline{\Omega }} \mathrm{d}\ (\mathrm{trace}[{\mathfrak {R}}]) \end{aligned} \end{aligned}$$

for all \(\tau > T\).

Proposition 3.2

([2, Theorem 2.5]) Given the initial data \(\varrho _0\), \(\mathbf{m}_0\),

$$\begin{aligned} \varrho _0 \ge 0, \ E_0 = \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}_0|^2}{\varrho _0} + P(\varrho _0) \right] \ \,\mathrm{d} {x} < \infty , \end{aligned}$$

the Euler system admits a global in time admissible (DW) solution

$$\begin{aligned} \varrho \in C_{\mathrm{weak, loc}}([0,\infty ); L^\gamma ({\mathbb {T}}^d)),\ \mathbf{m}\in C_{\mathrm{weak, loc}}([0, \infty ); L^{\frac{2 \gamma }{\gamma + 1}}({\mathbb {T}}^d; R^d)),\ E \in BV[0, \infty ) \end{aligned}$$

in the sense of Definition 2.2.

We are ready to prove Theorem 2.3. Let \([\varrho , \mathbf{m}, E]\) be an admissible (DW) solution of the Euler system in \([0, \infty ) \times \Omega \), the existence of which is guaranteed by Proposition 3.2. As E is a non-increasing function, it admits a limit

$$\begin{aligned} E_\infty = \lim _{\tau \rightarrow \infty } E(\tau ) \ge 0. \end{aligned}$$

Moreover, in view of (2.1),

$$\begin{aligned} E_\infty \ge \limsup _{\tau \rightarrow \infty } \int _{\Omega } \left[ \frac{1}{2} \frac{ |\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x}. \end{aligned}$$
(3.1)

Next, we claim the following result.

Lemma 3.3

Let \(T > 0\) be arbitrary and denote

$$\begin{aligned} E_T = \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (T, \cdot ) \ \,\mathrm{d} {x}. \end{aligned}$$

Then,

$$\begin{aligned} E_\infty \le E_T. \end{aligned}$$

Proof

Supposing the contrary, meaning

$$\begin{aligned} E_T > E_\infty , \end{aligned}$$
(3.2)

we may use Proposition 3.1 to construct a solution \({{\tilde{\varrho }}}\), \(\tilde{\mathbf{m}}\) defined on the interval \([T, \infty )\), with the initial data

$$\begin{aligned} {{\tilde{\varrho }}}(T, \cdot ) = \varrho (T, \cdot ),\ \tilde{\mathbf{m}}(T, \cdot ) = \mathbf{m}(T, \cdot ), \end{aligned}$$

with the non-decreasing total energy \({\widetilde{E}}\) such that

$$\begin{aligned} {\widetilde{E}}(\tau \pm ) \le E_T \ \text{ for } \text{ all }\ \tau \in (T,\infty ). \end{aligned}$$

Finally, set

$$\begin{aligned} {\widehat{\varrho }} = \left\{ \begin{array}{l} \varrho \quad \ \text{ for }\ t \in [0,T),\\ \\ {{\tilde{\varrho }}}\quad \ \text{ for }\ t \in [T, \infty ), \end{array} \right. \quad {\widehat{\mathbf{m}}} = \left\{ \begin{array}{l} \mathbf{m}\ \text{ for }\ t \in [0,T),\\ \\ \tilde{\mathbf{m}}\ \text{ for }\ t \in [T, \infty ), \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \widehat{E}(t) = \left\{ \begin{array}{l} E \ \quad \text{ for }\ t \in [0,T),\ \\ E(T-) (\ge E_T \ge ) \widetilde{E}(T+) ,\ t = T, \\ \widetilde{E} \quad \ \text{ for }\ t \in (T, \infty ). \end{array} \right. \end{aligned}$$

Obviously, \([{\widehat{\varrho }}, {\widehat{\mathbf{m}}}]\) with the energy \({\widehat{E}}\) is a dissipative solutions (cf. [2, Proposition 5.1 - continuation property]), and, in view of (3.2),

$$\begin{aligned} {[}{\widehat{\varrho }}, {\widehat{\mathbf{m}}}, {\widehat{E}}] \prec [\varrho , \mathbf{m}, E] \ \text{ and }\ \lim _{\tau \rightarrow \infty } {\widehat{E}}(\tau ) \le E_T < E_\infty \end{aligned}$$

in contrast with maximality of \([\varrho , \mathbf{m}, E]\). \(\square \)

In view of Lemma 3.3, any maximal (DW) solution satisfies

$$\begin{aligned} E_\infty = \lim _{\tau \rightarrow \infty } E(\tau ) \le \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (T, \cdot ) \ \,\mathrm{d} {x} \end{aligned}$$

for any \(T > 0\), in particular,

$$\begin{aligned} E_\infty \le \liminf _{\tau \rightarrow \infty } \int _{\Omega } \left[ \frac{1}{2} \frac{ |\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x}, \end{aligned}$$

which, together with (3.1), yields (2.2). We have proved Theorem 2.3.

4 Conclusion

We have shown that the “turbulent” energy E and the “intrinsic” energy

$$\begin{aligned} \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] \ \,\mathrm{d} {x} \end{aligned}$$

of any admissible (DW) solution \([\varrho ,\mathbf{m}]\) of the compressible Euler system coincide in the asymptotic limit as \(\tau \rightarrow \infty \), in particular, the limit

$$\begin{aligned} \int _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m}|^2}{\varrho } + P(\varrho ) \right] (\tau , \cdot ) \ \,\mathrm{d} {x} \rightarrow E_\infty \quad \ \text{ as }\ \tau \rightarrow \infty \end{aligned}$$

exists. Accordingly, the Reynolds defect measure \({\mathfrak {R}}\) in the momentum equation (1.4) vanishes for \(\tau \rightarrow \infty \), and the (DW) solutions behave asymptotically as the standard weak solutions. As turbulent phenomena are usually attributed to the properties of the system in the long run, this may be seen as a positive argument concerning physical relevance of the (DW) solutions. We expect similar properties to hold for the (DW) solutions of the complete Euler system introduced in [1].