Semiflow selection for the compressible Navier–Stokes system

Abstract

Although the existence of dissipative weak solutions for the compressible Navier–Stokes system has already been established for any finite energy initial data, uniqueness is still an open problem. The idea is then to select a solution satisfying the semigroup property, an important feature of systems with uniqueness. More precisely, we are going to prove the existence of a semiflow selection in terms of the three state variables: the density, the momentum, and the energy. Finally, we will show that it is possible to introduce a new selection defined only in terms of the initial density and momentum; however, the price to pay is that the semigroup property will hold almost everywhere in time.

1 Introduction

Consider the compressible Navier–Stokes system

$$\begin{aligned}&\partial _t \varrho + {{\,\mathrm{div}\,}}_x (\varrho \mathbf{u }) =0, \end{aligned}$$

(1)

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

(2)

where $\varrho =\varrho (t,x)$ denotes the density, $\mathbf{u }=\mathbf{u }(t,x)$ the velocity, $p=p(\varrho )$ the pressure, and ${\mathbb {S}}={\mathbb {S}}(\nabla _x \mathbf{u })$ the viscous stress. We will consider the system on the set $(t,x)\in (0,\infty )\times \Omega$, where $\Omega \subset {\mathbb {R}}^N$, $N=2,3$ is a bounded domain with $\partial \Omega$ of class $C^{2+\nu }$ for a certain $\nu >0$. As our goal is to handle a potentially ill-posed problem, we have deliberately omitted the case $N=1$, for which the problem is known to be well posed, see Kazhikhov [8].

We impose the no-slip boundary condition for the velocity

$$\begin{aligned} \mathbf{u }|_{\partial \Omega }=0 \quad \text{ for } \text{ all } t\in [0,\infty ), \end{aligned}$$

(3)

and we prescribe the initial conditions

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

(4)

Finally, we assume a barotropic pressure $p\in C[0,\infty ) \cap C^1(0,\infty )$ such that $p(0)=0$ and

$$\begin{aligned} {\left\{ \begin{array}{ll} p'(\varrho ) \ge a_1 \varrho ^{\gamma -1} -b &{}\text{ for } \text{ all } \varrho >0 \\ p(\varrho ) \le a_2 \varrho ^{\gamma } +b &{}\text{ for } \text{ all } \varrho \ge 0 \end{array}\right. } \end{aligned}$$

(5)

for certain constants $a_1>0$, $a_2$ and b, with $\gamma > \frac{N}{2}$ the adiabatic exponent, and the viscous stress tensor to be a linear function of the velocity gradient, more specifically to satisfy the Newton’s rheological law

$$\begin{aligned} {\mathbb {S}}(\nabla _x \mathbf{u })= \mu \left( \nabla _x \mathbf{u }+\nabla _x^T \mathbf{u }- \frac{2}{N}({{\,\mathrm{div}\,}}_x \mathbf{u }){\mathbb {I}}\right) + \lambda ({{\,\mathrm{div}\,}}_x \mathbf{u }){\mathbb {I}}, \end{aligned}$$

(6)

with $\mu >0$ and $\lambda \ge 0$. We would like to point out that (5) allows the pressure to be a general non-monotone function of the density, besides the standard case $p(\varrho )=a\varrho ^{\gamma }$ with $a>0$ and $\gamma >\frac{N}{2}$. Still, as we shall see below, the problem admits global-in-time weak solutions and retains other fundamental properties of the system, notably the weak–strong uniqueness, see [5].

We will consider dissipative weak solutions, i.e., solutions satisfying Equations (1) and (2) in a distributional sense along with the energy inequality, see Sect. 1.1. Although the existence of global in time solutions has already been established for any finite energy initial data, see, e.g., [10] and [6], uniqueness is still an open task. Then, a natural question is whether it is possible or not to select a solution satisfying at least the semiflow property, an important feature of systems with uniqueness: letting the system run from time 0 to time s and then restarting and letting it run from time s to time t gives the same outcome as letting it run directly from time 0 to time t.

The result presented in this manuscript can be seen as the deterministic version of the stochastic paper done by Breit, Feireisl and Hofmanová [2]. The construction of the semigroup arises from the theory of Markov selection in order to study the well-posedness of certain systems; it was first developed by Krylov [9] and later adapted by Flandoli and Romito [7], Cardona and Kapitanski [4] in the context of the incompressible Navier–Stokes system.

Breit, Feireisl, and Hofmanová [3] used the deterministic version motivated by [4] to show the existence of the semiflow selection for dissipative measure-valued solutions of the isentropic Euler system. Following the same strategy, we will establish the existence of a semiflow selection for the compressible Navier–Stokes system (1)–(6). Specifically, introducing the momentum $\mathbf{m }=\varrho \mathbf{u }$, we show the existence of a measurable mapping

$$\begin{aligned} V: [t, \varrho _0, \mathbf{m }_0] \mapsto [\varrho (t), \mathbf{m }(t)], \quad t\ge 0, \end{aligned}$$

satisfying the semigroup property:

$$\begin{aligned} V[t_1+t_2, \varrho _0, \mathbf{m }_0] = V \left[ t_2, V[t_1, \varrho _0, \mathbf{m }_0] \right] \quad \text{ for } \text{ a.e. } t_1,t_2 \ge 0, \end{aligned}$$

(7)

where $[\varrho ,\mathbf{m }= \varrho \mathbf{u }]$ represents a dissipative weak solution to (1)–(6). At this stage, we would like to point out the main essential difference between the present paper and [3]. The semigroup constructed for the Euler system in [3] contains the total energy as one of the state variables. This may be seen as a kind of drawback as the energy should be determined in terms of the basic state variables $[\varrho , \mathbf{m }]$. This is, however, a delicate issue for the Euler flow as the energy contains also the defect due to possible concentrations and/or oscillations. Such a problem does not occur for the Navier-Stokes system, where the energy is indeed a function of $[\varrho , \mathbf{u }]$ at least for a.a. $t \in [0, \infty )$, cf. (7).

The paper is organized as follows: The remaining part of this section contains the definitions of a dissipative weak solution and admissibility. In Sect. 2, we fix the topologies on the space of the initial data and the trajectory space, and we introduce the concept of a semiflow selection in terms of the three state variables: the density $\varrho _0$, the momentum $\mathbf{m }_0$, and the energy $E_0$. In Sect. 3, we analyze the properties (compactness, non-emptiness, the shift invariance, and continuation properties) of the solution set for a given initial data, while Sect. 4 is devoted to the proof of the existence of a semiflow selection. Finally, in Sect. 5, we study a new selection defined only in terms of the initial density $\varrho _0$ and the momentum $\mathbf{m }_0$.

1.1 Dissipative weak solution

Following [6], we can give the definition of a dissipative solution to the compressible Navier–Stokes system.

Definition 1.1

The pair of functions $\varrho$, $\mathbf{u }$ is called dissipative weak solution of the Navier–Stokes system (1)–(6) with the total energy E and initial data

$$\begin{aligned}{}[\varrho _0, (\varrho \mathbf{u })_0, E_0] \in L^{\gamma }(\Omega ) \times L^{\frac{2\gamma }{\gamma +1}}(\Omega ; {\mathbb {R}}^N) \times [0,\infty ) \end{aligned}$$

if the following holds:

• (i) regularity class:

  $$\begin{aligned}&{[\varrho , \varrho \mathbf{u }, E]} \in C_\mathrm{weak,loc}([0,\infty ); L^{\gamma }(\Omega )) \times C_\mathrm{weak,loc}([0,\infty ); L^{\frac{2\gamma }{\gamma +1}}(\Omega ; {\mathbb {R}}^N))\\&\quad \times BV_\mathrm{loc}([0,\infty )), \end{aligned}$$

  with $\varrho \ge 0$;

• (ii) weak formulation of the renormalized continuity equation: for any $\tau >0$ and any functions

  $$\begin{aligned}&B\in C[0,\infty ) \cap C^1(0,\infty ), \ b\in C[0,\infty ) \text{ bounded } \text{ on } [0,\infty ),\\&B(0)=b(0)=0 \quad \text{ and } \quad b(z)=zB'(z)-B(z) \text{ for } \text{ any } z>0, \end{aligned}$$

  the integral identity

  $$\begin{aligned} \left[ \int \nolimits _{\Omega } B(\varrho ) \varphi (t,\cdot ) \mathrm{d}x\right] _{t=0}^{t=\tau }= & {} \int \nolimits _{0}^{\tau } \int \nolimits _{\Omega } [B(\varrho ) \partial _t\varphi \nonumber \\&+ B(\varrho )\mathbf{u }\cdot \nabla _x\varphi + b(\varrho ){{\,\mathrm{div}\,}}_x \mathbf{u } \varphi ] dxdt, \end{aligned}$$

  (8)

  holds for any $\varphi \in C_c^1([0,\infty )\times \Omega )$;

• (iii) weak formulation of the balance of momentum: for any $\tau >0$ the integral identity

  $$\begin{aligned} \left[ \int \nolimits _{\Omega } \varrho \mathbf{u } \cdot \varvec{\varphi }(t,\cdot ) \mathrm{d}x\right] _{t=0}^{t=\tau }= & {} \int \nolimits _{0}^{\tau } \int \nolimits _{\Omega } [\varrho \mathbf{u }\cdot \partial _t\varvec{\varphi } + (\varrho \mathbf{u } \otimes \mathbf{u }): \nabla _x \varvec{\varphi }\nonumber \\&+ p(\varrho ){{\,\mathrm{div}\,}}_x \varvec{\varphi }- {\mathbb {S}}(\nabla _x \mathbf{u }): \nabla _x \varvec{\varphi }] dxdt, \end{aligned}$$

  (9)

  holds for any $\varvec{\varphi } \in C_c^1([0,\infty )\times \Omega ; {\mathbb {R}}^N)$, where $(\varrho \mathbf{u })(0,\cdot )= (\varrho \mathbf{u })_0$;

• (iv) energy inequality: for a.e. $\tau \ge 0$ we have

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

  (10)

  where the pressure potential P is chosen as a solution of

  $$\begin{aligned} \varrho P'(\varrho )-P(\varrho )=p(\varrho ); \end{aligned}$$

  we also require $E=E(\tau )$ to be a non-increasing function of $\tau$:

  $$\begin{aligned}{}[E\psi ]_{t=\tau _1-}^{t=\tau _2+}-\int \nolimits _{\tau _1}^{\tau _2} E(t)\psi '(t) \mathrm{d}t + \int \nolimits _{\tau _1}^{\tau _2} \psi \int \nolimits _{\Omega } {\mathbb {S}}(\nabla _x \mathbf{u }): \nabla _x\mathbf{u } dxdt\le 0, \end{aligned}$$

  (11)

  for any $0\le \tau _1 \le \tau _2$, $\psi \in C_c^1[0,\infty ), \ \psi \ge 0$, where $E(0-)=E_0$.

Remark 1.2

Condition (ii) can be considered as a simple rescaling of the state variables in the continuity equation (1); it is necessary in order to prove the weak sequential stability and the existence of dissipative weak solutions. In particular, choosing $B(z)=z$ we get the standard weak formulation of the continuity equation.

Remark 1.3

At this stage, similarly to [3], the total energy is considered as an additional phase variable—a non-increasing function of time possessing one sided limits at any time. In contrast with [3], the energy can be determined in terms of $\varrho$ and $\mathbf{u }$, see (10), with the exception of a zero measure set of times.

Remark 1.4

The condition $E(0-) = E_0$ comes naturally from the assumption that the total energy is bounded at the initial time $t=0$, specifically

$$\begin{aligned} E(0+) \le E_0. \end{aligned}$$

1.2 Admissible solution

From now on, it is more convenient to work with the momentum $\mathbf{m }=\varrho \mathbf{u }$. Following [3], we introduce a subclass of dissipative solutions that reflect the physical principle of minimization of the total energy. At the present state, we retain the total energy E as an integral part of the solution so we work with the triples $[\varrho , \mathbf{m },E]$. Finally, in Sect. 5, we pass to the natural state variables $[\varrho , \mathbf{m }]$. To this end, let $[\varrho ^i, \mathbf{m }^i, E^i]$, $i=1,2$, be two dissipative solutions starting from the same initial data $[\varrho _0, \mathbf{m }_0, E_0]$. We introduce the relation

$$\begin{aligned}{}[\varrho ^1, \mathbf{m }^1, E^1] \prec [\varrho ^2, \mathbf{m }^2, E^2] \ \Leftrightarrow \ E^1(\tau \pm ) \le E^2(\tau \pm ) \text{ for } \text{ any } \tau \in (0,\infty ). \end{aligned}$$

Definition 1.5

We say that a dissipative weak solution $[\varrho , \mathbf{m }, E]$ starting from the initial data $[\varrho _0, \mathbf{m }_0, E_0]$ is admissible if it is minimal with respect to the relation $\prec$. Specifically, if

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

where $[{\tilde{\varrho }}, \tilde{\mathbf{m }}, {\tilde{E}}]$ is another dissipative solution starting from $[\varrho _0, \mathbf{m }_0, E_0]$, then

$$\begin{aligned} E={\tilde{E}}\ \text{ in } [0,\infty ). \end{aligned}$$

In particular, such selection criterion guarantees that equilibrium states belong to the class of dissipative weak solutions (see [3], Section 6.3).

2 Setup

We start by introducing suitable topologies on the space of initial data and the space of dissipative weak solutions. Fix $\ell > \frac{N}{2}+1$ and for simplicity consider the Hilbert space

$$\begin{aligned} X= W^{-\ell ,2}(\Omega ) \times W^{-\ell ,2}(\Omega ; {\mathbb {R}}^N) \times {\mathbb {R}}, \end{aligned}$$

together with its subset containing the initial data

$$\begin{aligned} D= & {} \left\{ \left[ \varrho _0,\mathbf{m }_0, E_0]\in X: \ \varrho _0 \in L^1(\Omega ), \ \varrho _0\ge 0, \ \mathbf{m }_0 \in L^1(\Omega ; {\mathbb {R}}^N), \right. \right. \\&\left. \int \nolimits _{\Omega }\left[ \frac{1}{2}\frac{|\mathbf{m }_0|^2}{\varrho _0}+ P(\varrho _0)\right] \mathrm{d}x \le E_0 \right\} . \end{aligned}$$

Here, the convex function $[\varrho ,\mathbf{m }] \mapsto \frac{|\mathbf{m }|^2}{\varrho }$ is defined for $\varrho \ge 0$, $\mathbf{m }\in {\mathbb {R}}^N$ as

$$\begin{aligned} \frac{|\mathbf{m }|^2}{\varrho }= {\left\{ \begin{array}{ll} 0 &{}\text{ if } \mathbf{m }=0, \\ \frac{|\mathbf{m }|^2}{\varrho } &{}\text{ if } \varrho >0, \\ \infty &{}\text{ otherwise }. \end{array}\right. } \end{aligned}$$

If $[\varrho _0, \mathbf{m }_0, E_0]\in D$, applying Hölder inequality, we can also deduce that $\varrho _0 \in L^{\gamma }(\Omega )$ and $\mathbf{m }_0\in L^{\frac{2\gamma }{\gamma +1}}(\Omega ; {\mathbb {R}}^N)$; accordingly, the set of the data can be seen as a closed convex subset of the Banach space $L^{\gamma }(\Omega ) \times L^{\frac{2\gamma }{\gamma +1}}(\Omega ; {\mathbb {R}}^N) \times {\mathbb {R}}$. Indeed, denoting with $L^1_+(\Omega )$ the space of non-negative integrable functions, we can write

$$\begin{aligned} D= \left\{ [\varrho _0,\mathbf{m }_0, E_0]\in L^1_{+}(\Omega ) \times L^1(\Omega ; {\mathbb {R}}^N) \times {\mathbb {R}}: \ f([\varrho _0, \mathbf{m }_0]) \le E_0 \right\} , \end{aligned}$$

so that it coincides with the epigraph of the function $f:L^1_{+}(\Omega ) \times L^1(\Omega ; {\mathbb {R}}^N) \rightarrow [0,+\infty ]$ such that

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

Since f is lower semi-continuous and convex, we obtain that its epigraph is closed and convex.

We consider the trajectory space

$$\begin{aligned} Q= C_\mathrm{loc}([0,\infty ); W^{-\ell ,2}(\Omega )) \times C_\mathrm{loc}([0,\infty ); W^{-\ell ,2}(\Omega ; {\mathbb {R}}^N)) \times L^1_\mathrm{loc}[0,\infty ), \end{aligned}$$

which is a separable space. Dissipative weak solutions $[\varrho , \mathbf{m }, E]$, as defined in Definition 1.1, belong to this class—since $\ell >\frac{N}{2}$, we have that $L^p$ with $p\ge 1$ is compactly embedded in $W^{-\ell ,2}$ so in particular it holds for $p=\gamma$ and $p=\frac{2\gamma }{\gamma +1}$—while Equations (8) and (9) give an information on the time regularity of the density and the momentum. Moreover, for any initial data $[\varrho _0, \mathbf{m }_0, E_0]\in D$, it follows that a dissipative weak solution $[\varrho ,\mathbf{m }, E]$ evaluated at every $t\ge 0$ also belongs to the set D, due to the energy inequality. Finally, for initial data $[\varrho _0, \mathbf{m }_0, E_0]\in D$, we introduce the solution set

$$\begin{aligned}&{\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \\&\quad = \left\{ [\varrho , \mathbf{m }, E] \in Q \ \big | \ [\varrho ,\mathbf{m }=\varrho \mathbf{u }, E] \text{ is } \text{ a } \text{ dissipative } \text{ weak } \text{ solution } \text{ with } \text{ initial } \text{ data } \right. \\&\qquad \left. [\varrho _0, \mathbf{m }_0, E_0]\right\} . \end{aligned}$$

2.1 Semiflow selection: main result

We are now ready to define a semiflow selection to (1)–(6).

Definition 2.1

A semiflow selection in the class of dissipative weak solutions for the compressible Navier–Stokes system (1)–(6) is a mapping

$$\begin{aligned} U: D \rightarrow Q, \quad U\{ \varrho _0, \mathbf{m }_0, E_0 \} \in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \text{ for } \text{ any } [\varrho _0, \mathbf{m }_0, E_0]\in D \end{aligned}$$

enjoying the following properties:

• (i) Measurability. The mapping $U: D \rightarrow Q$ is Borel measurable.

• (ii) Semigroup property. We have

  $$\begin{aligned} U\{ \varrho _0,\mathbf{m }_0, E_0 \}(t_1+t_2) = U \{ \varrho (t_1), \mathbf{m }(t_1), E(t_1-) \}(t_2), \end{aligned}$$

  where $[\varrho , \mathbf{m }, E]= U\{ \varrho _0, \mathbf{m }_0, E_0 \}$ for any $[\varrho _0, \mathbf{m }_0, E_0]\in D$ and any $t_1, t_2 \ge 0$.

We are now ready to state our main result; the proof is postponed to Sect. 4.1.

Theorem 2.2

The compressible Navier–Stokes system (1)–(6) admits a semiflow selection U in the class of dissipative weak solutions in the sense of Definition 2.1. Moreover, we have that $U\{ \varrho _0, \mathbf{m }_0, E_0 \}$ is admissible in the sense of Definition 1.5, for any $[\varrho _0,\mathbf{m }_0, E_0]\in D$.

Theorem 2.2 is stated in terms of the three state variables $[\varrho , \mathbf{m },E]$. In Sect. 5, we state a version of this result in terms of the natural state variables $[\varrho , \mathbf{m }]$, see Theorem 5.1. The price to pay is validity of the semigroup property for any time with the exception of a zero measure set.

3 Properties of ${\mathcal {U}}$

3.1 Existence and sequential stability

First of all, we aim to show:

• 1. existence of a dissipative weak solution for any initial data $[\varrho _0, \mathbf{m }_0, E_0]\in D$, meaning

  $$\begin{aligned} {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \ne \emptyset ; \end{aligned}$$

• 2. weak sequential stability of the solution set, meaning

  $$\begin{aligned} {\mathcal {U}}[\varrho _0,\mathbf{m }_0, E_0]\subset Q \text{ is } \text{ compact } \end{aligned}$$

  and the multivalued mapping

  $$\begin{aligned}{}[\varrho _0,\mathbf{m }_0, E_0] \in D \subset X \rightarrow {\mathcal {U}}[\varrho _0,\mathbf{m }_0, E_0] \in 2^Q \end{aligned}$$

  has closed graph; in particular, this implies that it is continuous with respect to the Hausdorff complementary topology and thus Borel measurable, see [3] for details.

Both results can be found in [6], see Theorems 6.1, 6.2 and 7.1 for details.

Proposition 3.1

Suppose that $\{ \varrho _{0,\varepsilon }, \mathbf{m }_{0,\varepsilon }, E_{0,\varepsilon } \}_{\varepsilon >0}\subset D$ is a sequence of data giving rise to a family of dissipative weak solutions $\{ \varrho _{\varepsilon }, \mathbf{m }_{\varepsilon }, E_{\varepsilon } \}_{\varepsilon >0}$, that is, $[\varrho _{\varepsilon }, \mathbf{m }_{\varepsilon }, E_{\varepsilon }]\in {\mathcal {U}}[\varrho _{0,\varepsilon },\mathbf{m }_{0,\varepsilon }, E_{0,\varepsilon }]$. Moreover, we assume that the initial densities converge strongly

$$\begin{aligned} \varrho _{0,\varepsilon } \rightarrow \varrho _0 \quad \text{ in } L^{\gamma }(\Omega ) \end{aligned}$$

and there exists a constant ${\overline{E}}>0$ such that $E_{0,\varepsilon }\le {\overline{E}}$ for all $\varepsilon >0$.

Then, at least for suitable subsequences,

$$\begin{aligned} \mathbf{m }_{0,\varepsilon } \rightharpoonup \mathbf{m }_0 \quad \text{ in } L^{\frac{2\gamma }{\gamma +1}}(\Omega ; {\mathbb {R}}^N), \quad E_{0,\varepsilon } \rightarrow E_0, \end{aligned}$$

and

$$\begin{aligned} \varrho _{\varepsilon } \rightarrow \varrho \quad&\text{ in } C_\mathrm{weak,loc}([0,\infty ); L^{\gamma }(\Omega )) \\ \mathbf{m }_{\varepsilon } \rightarrow \mathbf{m } \quad&\text{ in } C_\mathrm{weak,loc}([0,\infty ); L^{\frac{2\gamma }{\gamma +1}}(\Omega ; {\mathbb {R}}^N))\\ E_{\varepsilon }(\tau ) \rightarrow E(\tau ) \quad&\text{ for } \text{ every } \tau \in [0,\infty ) \text{ and } \text{ in } L^1_\mathrm{loc}(0,\infty ), \end{aligned}$$

where

$$\begin{aligned} {[\varrho , \mathbf{m }, E]} \in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]. \end{aligned}$$

Proposition 3.2

Let $[\varrho _0, \mathbf{m }_0, E_0] \in D$ be given; then, the Navier–Stokes system (1)– (6) admits a dissipative weak solution in the sense of Definition 1.1 with the initial data $[\varrho _0, \mathbf{m }_0, E_0]$.

The following statement is a direct consequence of Propositions 3.1 and 3.2 .

Lemma 3.3

For any $[\varrho _0, \mathbf{m }_0, E_0]\in D$, the set ${\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$ is a non-empty, compact subset of Q. Moreover, $[\varrho (T), \mathbf{m }(T), {\mathcal {E}}]\in D$ for any $T>0$, and for arbitrary ${\mathcal {E}} \ge E(T+)$.

3.2 Shift and continuation operations

In order to construct the semiflow, two main ingredients are needed: the shift invariance property and the continuation. For $\omega \in Q$, we define the positive shift operator

$$\begin{aligned} S_T \circ \omega , \ S_T\circ \omega (t)=\omega (T+t), \ t\ge 0. \end{aligned}$$

Lemma 3.4

Let $[\varrho _0, \mathbf{m }_0, E_0]\in D$ and $[\varrho , \mathbf{m },E]\in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$. Then, we have

$$\begin{aligned} S_T \circ [\varrho , \mathbf{m }, E] \in {\mathcal {U}}[\varrho (T), \mathbf{m }(T), {\mathcal {E}}] \end{aligned}$$

for any $T>0$, and any ${\mathcal {E}} \ge E(T+)$.

Proof

Obviously, a dissipative weak solution on the time interval $(0,\infty )$ solves also the same problem on $(T,\infty )$ with the initial data $[\varrho (T), \mathbf{m }(T), E(T+)]$. Shifting the test functions in the integrals, this implies

$$\begin{aligned} S_T \circ [\varrho , \mathbf{m }, E] \in {\mathcal {U}}[\varrho (T), \mathbf{m }(T), E(T+)]. \end{aligned}$$

Since the energy is non-increasing, we can choose every ${\mathcal {E}}\ge E(T+)$ as initial energy; indeed, everything will be well defined

$$\begin{aligned} S_T\circ E(0-) = {\mathcal {E}} \ge E(T+). \end{aligned}$$

$\square$

Remark 3.5

Since the energy is non-increasing, one can choose ${\mathcal {E}}=E(T-)\ge E(T+)$.

For $\omega _1, \omega _2 \in Q$ we define the continuation operator $\omega _1 \cup _T \omega _2$ by

$$\begin{aligned} \omega _1 \cup _T \omega _2(t)= {\left\{ \begin{array}{ll} \omega _1(t) &{} \text{ for } 0\le t \le T, \\ \omega _2(t-T) &{}\text{ for } t>T. \end{array}\right. } \end{aligned}$$

Lemma 3.6

Let $[\varrho _0, \mathbf{m }_0, E_0]\in D$ and

$$\begin{aligned} {[\varrho ^1, \mathbf{m }^1, E^1]} \in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0], \ [\varrho ^2, \mathbf{m }^2, E^2] \in {\mathcal {U}}[\varrho ^1(T), \mathbf{m }^1(T), {\mathcal {E}}], \end{aligned}$$

for some ${\mathcal {E}} \le E^1(T-)$. Then,

$$\begin{aligned} {[\varrho ^1, \mathbf{m }^1, E^1]} \cup _T [\varrho ^2, \mathbf{m }^2, E^2] \in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]. \end{aligned}$$

Proof

We are simply pasting two solutions together at the time T, letting the second start from the point reached by the first one at the time T; thus, the integral identities remain satisfied. Choosing the initial energy for $[\varrho ^2, \mathbf{m }^2, E^2]$ less or equal $E^1(T-)$, the energy of the solution $[\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2, \mathbf{m }^2, E^2]$ remains non-increasing on $(0,\infty )$. $\square$

3.3 General ansatz

Summarizing the results achieved in this section, we have shown the existence of a set-valued mapping

$$\begin{aligned} D \ni [\varrho _0, \mathbf{m }_0, E_0] \mapsto {\mathcal {U}}[\varrho _0, \mathbf{m }_0,E_0] \in 2^Q \end{aligned}$$

enjoying the following properties.

(A1):

Compactness: for any $[\varrho _0, \mathbf{m }_0, E_0]\in D$, the set ${\mathcal {U}}[\varrho _0, \mathbf{m }_0,E_0]$ is a non-empty compact subset of Q.

(A2):

Measurability: the mapping

$$\begin{aligned} D \ni [\varrho _0, \mathbf{m }_0, E_0] \mapsto {\mathcal {U}}[\varrho _0, \mathbf{m }_0,E_0] \in 2^Q \end{aligned}$$

is Borel measurable, where the range of ${\mathcal {U}}$ is endowed with the Hausdorff metric on the subspace of compact sets in $2^Q$.

(A3):

Shift invariance: for any $[\varrho , \mathbf{m }, E]\in {\mathcal {U}}[\varrho _0, \mathbf{m }_0,E_0]$, we have

$$\begin{aligned} S_T \circ [\varrho , \mathbf{m }, E] \in {\mathcal {U}}[\varrho (T), \mathbf{m }(T), E(T-)], \end{aligned}$$

for any $T>0$.

(A4):

Continuation: if $T>0$, and

$$\begin{aligned}{}[\varrho ^1, \mathbf{m }^1, E^1] \in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0], \ [\varrho ^2, \mathbf{m }^2, E^2] \in {\mathcal {U}}[\varrho ^1(T), \mathbf{m }^1(T), E^1(T-)], \end{aligned}$$

then

$$\begin{aligned}{}[\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2, \mathbf{m }^2, E^2] \in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]. \end{aligned}$$

We remark that the value $E(T-)$ in (A3) and (A4) can be replaced by

$$\begin{aligned} {\mathcal {E}}= \eta E(T-) +(1-\eta ) E(T+), \end{aligned}$$

where $\eta \in [0,1]$ is given, since in this case $E(T+) \le {\mathcal {E}} \le E(T-)$.

4 Semiflow selection

The arguments presented below are the same as in [3]. We repeat them here in detail for reader’s convenience. We consider the family of functionals

$$\begin{aligned} I_{\lambda ,F}[\varrho , \mathbf{m },E]= \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda t}F(\varrho (t),\mathbf{m }(t),E(t)) \mathrm{d}t, \quad \lambda >0, \end{aligned}$$

where $F: X= W^{-\ell ,2}(\Omega )\times W^{-\ell ,2}(\Omega ;{\mathbb {R}}^N) \times {\mathbb {R}}\rightarrow {\mathbb {R}}^N$ is a bounded and continuous functional.

Given $I_{\lambda ,F}$ and a set-valued mapping ${\mathcal {U}}$ we define a selection mapping $I_{\lambda ,F}\circ {\mathcal {U}}$, by

$$\begin{aligned}&I_{\lambda ,F}\circ {\mathcal {U}}[\varrho _0,\mathbf{m }_0, E_0] \\&\quad = \{ [\varrho , \mathbf{m },E] \in {\mathcal {U}}[\varrho _0,\mathbf{m }_0, E_0] \ | \ I_{\lambda ,F}[\varrho , \mathbf{m },E] \\&\quad \le I_{\lambda ,F}[{\tilde{\varrho }}, \tilde{ \mathbf{m }},{\tilde{E}}] \ \text{ for } \text{ all } [{\tilde{\varrho }}, \tilde{ \mathbf{m }},{\tilde{E}}] \in {\mathcal {U}}[\varrho _0,\mathbf{m }_0, E_0] \}. \end{aligned}$$

In other words, the selection is choosing minima of the functional $I_{\lambda ,F}$. Note that a minimum exists since $I_{\lambda ,F}$ is continuous on Q and the set ${\mathcal {U}}[\varrho _0,(\varrho \mathbf{u })_0, E_0]$ is compact in Q. We obtain the following result for the set $I_{\lambda ,F}\circ {\mathcal {U}}$.

Proposition 4.1

Let $\lambda >0$ and F be a bounded continuous functional on X. Let

$$\begin{aligned} {\mathcal {U}}: [\varrho _0, \mathbf{m }_0, E_0] \in D \mapsto {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \subset 2^Q \end{aligned}$$

be a multi-valued mapping having the properties (A1)–(A4). Then, the map $I_{\lambda ,F} \circ {\mathcal {U}}$ enjoys (A1)–(A4) as well.

Proof

(A1):

We proved that ${\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \ne \emptyset$ for every $[\varrho _0, \mathbf{m }_0, E_0] \in D$, and thus $I_{\lambda , F} \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]\ne \emptyset$. Since $I_{\lambda ,F}: {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \rightarrow {\mathbb {R}}$ is continuous and since the set of minima of a continuous function is closed (it is the counterpart of a point), we get that $I_{\lambda , F} \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]\subseteq {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$ is closed in a compact set and hence compact itself.

(A2):

Notice that, since $I_{\lambda ,F} \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$ is a compact subset of the separable metric space Q for any $[\varrho _0, \mathbf{m }_0, E_0]\in D$, the Borel measurability of the multivalued mapping

$$\begin{aligned} {[\varrho _0, \mathbf{m }_0, E_0]} \in D \mapsto I_{\lambda , F} \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \in {\mathcal {K}} \subset 2^Q \end{aligned}$$

corresponds to measurability with respect to the Hausdorff metric on the space of all compact subsets of Q. In other words, let $d_H$ be the Hausdorff metric on the subspace ${\mathcal {K}} \subset 2^Q$ of all the compact subsets of Q:

$$\begin{aligned} d_H(K_1,K_2) = \inf _{\varepsilon \ge 0} \{ K_1 \subset V_{\varepsilon }(K_2) \text{ and } K_2 \subset V_{\varepsilon }(K_1) \} \quad \text{ for } \text{ all } K_1,K_2 \in {\mathcal {K}}, \end{aligned}$$

where $V_{\varepsilon }(A)$ is the $\varepsilon$-neighborhood of the set A in the topology of Q; then, it is enough to show that the mapping defined for all $K \in {\mathcal {K}}$ as

$$\begin{aligned} {\mathcal {I}}_{\lambda , F}[K] = \{ z\in K | I_{\lambda , F}(z) \le I_{\lambda , F}({\tilde{z}}) \text{ for } \text{ all } {\tilde{z}}\in K \} = \left\{ z\in K | \min _{z\in K} I_{\lambda , F}(z) \right\} \end{aligned}$$

is continuous as a mapping on ${\mathcal {K}}$ endowed with the Hausdorff metric $d_H$. In particular, we want to show that if $K_n \overset{d_H}{\longrightarrow } K$ with $K_n,K \in {\mathcal {K}}$ then ${\mathcal {I}}_{\lambda , F}[K_n] \overset{d_H}{\longrightarrow } {\mathcal {I}}_{\lambda , F}[K]$ for $n \rightarrow \infty$. More precisely, it is enough to show that for every $\varepsilon >0$ there exists $n_0=n_0(\varepsilon )$ such that

$$\begin{aligned} {\mathcal {I}}_{\lambda , F}[K_n] \subset V_{\varepsilon }({\mathcal {I}}_{\lambda , F}[K]) \quad \text{ and } \quad {\mathcal {I}}_{\lambda , F}[K] \subset V_{\varepsilon }({\mathcal {I}}_{\lambda , F}[K_n]) \end{aligned}$$

(12)

for all $n\ge n_0$. First of all, notice that by the continuity of $I_{\lambda , F}$ we have

$$\begin{aligned} \min _{K_n} I_{\lambda , F} \rightarrow \min _{K} I_{\lambda , F} \quad \text{ for } n\rightarrow \infty . \end{aligned}$$

(13)

We start proving the first inclusion of (12). By contradiction, suppose that exists a sequence $\{z_n\}_{n\in {\mathbb {N}}}$ such that

$$\begin{aligned} z_n \in K_n, \quad I_{\lambda , F}(z_n) = \min _{K_n} I_{\lambda , F}, \quad z_n \rightarrow z\in K \setminus V_{\varepsilon } ({\mathcal {I}}_{\lambda , F}[K]); \end{aligned}$$

in particular, $I_{\lambda , F}(z) > \min _{K} I_{\lambda , F}$. By the continuity of $I_{\lambda , F}$ we have

$$\begin{aligned} \min _{K_n} I_{\lambda , F} = I_{\lambda ,F} (z_n) \rightarrow I_{\lambda , F}(z) > \min _{K} I_{\lambda ,F} \quad \text{ for } n\rightarrow \infty ; \end{aligned}$$

but this contradicts (13). Interchanging the roles of $K_n$ and K we get the opposite inclusion in (12). We get the claim.

(A3):

We want to prove the shift invariance: for every $[\varrho _0, \mathbf{m }_0, E_0] \in D$ and $[\varrho ,\mathbf{m }, E] \in I_{\lambda ,F} \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$,

$$\begin{aligned} S_T \circ [\varrho ,\mathbf{m }, E] \in I_{\lambda , F} \circ {\mathcal {U}}[\varrho (T), \mathbf{m }(T), E(T-)] \quad \text{ for } \text{ any } T>0. \end{aligned}$$

Let $[\varrho ^T, \mathbf{m }^T, E^T] \in I_{\lambda , F} \circ {\mathcal {U}}[\varrho (T), \mathbf{m }(T), E(T-)]$; then, since in particular

$$\begin{aligned}{}[\varrho , \mathbf{m }, E]&\in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0], \\ [\varrho ^T, \mathbf{m }^T, E^T]&\in {\mathcal {U}}[\varrho (T), \mathbf{m }(T), E(T-)], \end{aligned}$$

and since ${\mathcal {U}}$ satisfies property (A4), we get

$$\begin{aligned}{}[\varrho ,\mathbf{m }, E] \cup _T [\varrho ^T, \mathbf{m }^T, E^T] \in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]. \end{aligned}$$

From the choice of $[\varrho ,\mathbf{m }, E]$, which minimize $I_{\lambda , F}$ on ${\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$, we obtain

$$\begin{aligned} I_{\lambda , F} [\varrho ,\mathbf{m }, E] \le I_{\lambda , F} ([\varrho ,\mathbf{m },E]\cup _T [\varrho ^T, \mathbf{m }^T, E^T]). \end{aligned}$$

(14)

Hence, using (14) in the fifth line and the definition of $\cup _T$ in the sixth line

$$\begin{aligned}&I_{\lambda , F} (S_T \circ [\varrho , \mathbf{m }, E]) = \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda t} F(S_T \circ [\varrho , \mathbf{m }, E](t)) \mathrm{d}t \\&= \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda t} F([\varrho , \mathbf{m }, E](t+T)) \mathrm{d}t \\&= \mathrm{e}^{\lambda T} \int \nolimits _{T}^{\infty } \mathrm{e}^{-\lambda s} F([\varrho , \mathbf{m }, E](s)) \mathrm{d}s \\&= \mathrm{e}^{\lambda T} \left( I_{\lambda , F}[\varrho , \mathbf{m }, E] - \int \nolimits _{0}^{T} \mathrm{e}^{-\lambda s} F([\varrho , \mathbf{m }, E](s)) \mathrm{d}s\right) \\&\le \mathrm{e}^{\lambda T} \left( I_{\lambda , F} ([\varrho , \mathbf{m },E]\cup _T [\varrho ^T, \mathbf{m }^T, E^T]) \right. \\&\quad \left. - \int \nolimits _{0}^{T}\mathrm{e}^{-\lambda s} F([\varrho , \mathbf{m }, E](s)) \mathrm{d}s \right) \\&= \mathrm{e}^{\lambda T} \int \nolimits _{T}^{\infty } \mathrm{e}^{-\lambda s}F([\varrho ^T, \mathbf{m }^T, E^T](s-T)) \mathrm{d}s \\&= \mathrm{e}^{\lambda T} \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda (t+T)} F([\varrho ^T, \mathbf{m }^T, E^T](t)) \mathrm{d}t \\&= I_{\lambda , F} [\varrho ^T,\mathbf{m }^T, E^T]. \end{aligned}$$

This implies that $S_T \circ [\varrho , \mathbf{m }, E]$ minimizes $I_{\lambda ,F}$ and consequently belongs to

$I_{\lambda , F} \circ {\mathcal {U}}[\varrho (T), \mathbf{m }(T), E(T-)]$ for any $T>0$.

(A4):

We want to prove the continuation: if $T>0$ and $[\varrho ^1,\mathbf{m }^1,E^1]\in I_{\lambda ,F}\circ {\mathcal {U}}[\varrho _0,\mathbf{m }_0, E_0]$, $[\varrho ^2,\mathbf{m }^2,E^2]\in I_{\lambda , F}\circ {\mathcal {U}}[\varrho ^1(T),\mathbf{m }^1(T), E^1(T-)]$, then

$$\begin{aligned}{}[\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2,\mathbf{m }^2,E^2] \in I_{\lambda , F} \circ {\mathcal {U}}[\varrho _0,\mathbf{m }_0, E_0]. \end{aligned}$$

Using the shift invariance for ${\mathcal {U}}$, we obtain

$$\begin{aligned} S_T \circ [\varrho ^1,\mathbf{m }^1, E^1] \in {\mathcal {U}}[\varrho ^1(T),\mathbf{m }^1(T), E^1(T-)]; \end{aligned}$$

since $[\varrho ^2,\mathbf{m }^2, E^2]$ is a minimum of $I_{\lambda ,F}$ on ${\mathcal {U}}[\varrho ^1(T),\mathbf{m }^1(T), E^1(T-)]$, we get

$$\begin{aligned} I_{\lambda , F} [\varrho ^2,\mathbf{m }^2, E^2] \le I_{\lambda ,F} (S_T \circ [\varrho ^1,\mathbf{m }^1, E^1]). \end{aligned}$$

(15)

Hence, using (15) in the fourth line

$$\begin{aligned}&I_{\lambda ,F} ([\varrho ^1, \mathbf{m }^1, E^1] \cup _T {[}\varrho ^2,\mathbf{m }^2,E^2] ) \\&\quad = \int \nolimits _{0}^{T} \mathrm{e}^{-\lambda t} F([\varrho ^1,\mathbf{m }^1, E^1](t)) \mathrm{d}t + \int \nolimits _{T}^{\infty } \mathrm{e}^{-\lambda t} F([\varrho ^2,\mathbf{m }^2, E^2] (t-T)) \mathrm{d}t \\&\quad = \int \nolimits _{0}^{T} \mathrm{e}^{-\lambda t} F([\varrho ^1,\mathbf{m }^1, E^1](t)) \mathrm{d}t + \mathrm{e}^{-\lambda T} \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda s} F([\varrho ^2,\mathbf{m }^2, E^2] (s)) \mathrm{d}s \\&\quad = \int \nolimits _{0}^{T} \mathrm{e}^{-\lambda t} F([\varrho ^1,\mathbf{m }^1, E^1](t)) \mathrm{d}t + \mathrm{e}^{-\lambda T} I_{\lambda ,F} [\varrho ^2,\mathbf{m }^2, E^2] \\&\quad \le \int \nolimits _{0}^{T} \mathrm{e}^{-\lambda t} F([\varrho ^1,\mathbf{m }^1, E^1](t)) \mathrm{d}t + \mathrm{e}^{-\lambda T} I_{\lambda ,F} (S_T \circ [\varrho ^1,\mathbf{m }^1, E^1]) \\&\quad = \int \nolimits _{0}^{T} \mathrm{e}^{-\lambda t} F([\varrho ^1,\mathbf{m }^1, E^1](t)) \mathrm{d}t + \mathrm{e}^{-\lambda T} \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda s} F ([\varrho ^1,\mathbf{m }^1, E^1](s+T)) \mathrm{d}s \\&\quad = \int \nolimits _{0}^{T} \mathrm{e}^{-\lambda t} F([\varrho ^1,\mathbf{m }^1, E^1](t)) \mathrm{d}t + \int \nolimits _{T}^{\infty } \mathrm{e}^{-\lambda t} F([\varrho ^1, \mathbf{m }^1, E^1](t)) \mathrm{d}t \\&\quad = I_{\lambda ,F} [\varrho ^1,\mathbf{m }^1, E^1]. \end{aligned}$$

Using the continuation property for ${\mathcal {U}}$, we have that

$$\begin{aligned}{}[\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2,\mathbf{m }^2,E^2]\in {\mathcal {U}}[\varrho _0,\mathbf{m }_0,E_0], \end{aligned}$$

and since $[\varrho ^1,\mathbf{m }^1, E^1]$ is a minimum of $I_{\lambda , F}$, we must have

$$\begin{aligned} I_{\lambda ,F} ([\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2,\mathbf{m }^2,E^2] )= I_{\lambda ,F} [\varrho ^1,\mathbf{m }^1, E^1]; \end{aligned}$$

thus $[\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2,\mathbf{m }^2,E^2] \in I_{\lambda , F} \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$.

$\square$

4.1 Selection sequence

First of all, we will select only those solutions that are admissible, meaning minimal with respect to the relation $\prec$ introduced in Definition 1.5. To this end we consider the functional $I_{1, \beta }$ with

$$\begin{aligned} \beta (\varrho ,\mathbf{m }, E)=\beta (E), \quad \beta : {\mathbb {R}}\rightarrow {\mathbb {R}} \text{ smooth, } \text{ bounded, } \text{ and } \text{ strictly } \text{ increasing }. \end{aligned}$$

Lemma 4.2

Suppose that $[\varrho , \mathbf{m }, E]\in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$ satisfies

$$\begin{aligned} \int \nolimits _{0}^{\infty } \mathrm{e}^{-t} \beta (E(t)) \mathrm{d}t \le \int \nolimits _{0}^{\infty } \mathrm{e}^{-t} \beta ({\tilde{E}}(t)) \mathrm{d}t \end{aligned}$$

(16)

for any $[{\tilde{\varrho }}, \tilde{ \mathbf{m }}, {\tilde{E}}]\in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$. Then, $[\varrho , \mathbf{m }, E]$ is $\prec$ minimal, meaning, admissible.

Proof

We proceed by contradiction. Let $[{\tilde{\varrho }}, \tilde{\mathbf{m }}, {\tilde{E}}]\in {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$ be such that $[{\tilde{\varrho }}, \tilde{ \mathbf{m }}, {\tilde{E}}] \prec [\varrho , \mathbf{m }, E]$, that is, ${\tilde{E}} \le E$ in $[0,\infty )$. Then, since $\beta$ is strictly increasing, $\beta ({\tilde{E}}(t))\le \beta (E(t))$ for every $t\in [0,\infty )$, which implies that $\mathrm{e}^{-t}[\beta (E(t))- \beta ({\tilde{E}}(t))]\ge 0$. Using the monotonicity of the integral, we obtain

$$\begin{aligned} \int \nolimits _{0}^{\infty } \mathrm{e}^{-t} [\beta (E(t))- \beta ({\tilde{E}}(t))] \mathrm{d}t \ge 0; \end{aligned}$$

on the other side, condition (16) tells us that

$$\begin{aligned} \int \nolimits _{0}^{\infty } \mathrm{e}^{-t} [\beta (E(t))- \beta ({\tilde{E}}(t))] \mathrm{d}t \le 0. \end{aligned}$$

The only possibility is to have the equality in both the integral relations above and thus $\beta (E(t)) = \beta ({\tilde{E}}(t))$ for a.e. $t\in (0,\infty )$; since $\beta$ is strictly increasing, this implies $E={\tilde{E}}$ a.e. in $(0,\infty )$. $\square$

We will need this topological result, which is a variation of the Cantor’s intersection theorem.

Theorem 4.3

Let S be a Hausdorff space. A decreasing nested sequence of non-empty compact subsets of S is non-empty. In other words, supposing $\{ C_k \}_{k\in {\mathbb {N}}}$ is a sequence of non-empty compact subsets of S satisfying

$$\begin{aligned} C_0 \supseteq C_1 \supseteq C_2 \supseteq \dots \supseteq C_k \supseteq \dots \end{aligned}$$

it follows that

$$\begin{aligned} \bigcap _{k\in {\mathbb {N}}} C_k \ne \emptyset . \end{aligned}$$

Proof

By contradiction, assume $\bigcap _{k\in {\mathbb {N}}} C_k = \emptyset$. For each n, let $U_n=C_0 \setminus C_n$; since

$$\begin{aligned} \bigcup _{n\in {\mathbb {N}}} U_n = \bigcup _{n\in {\mathbb {N}}} (C_0\setminus C_n) = C_0 \setminus \left( \bigcap _{n\in {\mathbb {N}}} U_n\right) \end{aligned}$$

and $\bigcap _{n\in {\mathbb {N}}} C_n = \emptyset$, we obtain $\bigcup _{n\in {\mathbb {N}}}U_n = C_0$. Since $C_0 \subset S$ is compact and $\{ U_n \}_{n\in {\mathbb {N}}}$ is an open cover (on $C_0$) of $C_0$, we can extract a finite cover $\{ U_{n_1}, \dots , U_{n_m} \}$. Let $U_k$ be the largest set of this cover ($C_k$ the correspondent smallest set), which exists by the ordering hypothesis on the collection $\{ C_n \}_{n\in {\mathbb {N}}}$. Then,

$$\begin{aligned} C_0 \subset \bigcup _{j=1}^{m} U_{n_j} = \bigcup _{j=1}^{m} (C_0 \setminus C_{n_j}) = C_0 \setminus \bigcap _{j=1}^{m} C_{n_j}= C_0 \setminus C_k = U_k. \end{aligned}$$

Then, $C_k = C_0 \setminus U_k= \emptyset$, a contradiction since every set of the sequence $\{ C_n \}_{n\in {\mathbb {N}}}$ is non-empty by hypothesis. $\square$

We are now ready to prove Theorem 2.2.

Proof of Theorem 2.2

Selecting $I_{1,\beta } \circ {\mathcal {U}}$ from ${\mathcal {U}}$ we know from Lemma 4.2 that the new selection contains only admissible solutions for any $[\varrho _0, \mathbf{m }_0, E_0]\in D$.

Next, we choose a countable basis $\{ \mathbf{e }_n \}_{n\in {\mathbb {N}}}$ in $L^2(\Omega ; {\mathbb {R}}^N)$, and a countable set $\{ \lambda _k \}_{k\in {\mathbb {N}}}$ which is dense in $(0,\infty )$. We consider a countable family of functionals,

$$\begin{aligned} I_{k,0}[\varrho , \mathbf{m }, E]&= \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda _kt} \beta (E(t)) \mathrm{d}t, \nonumber \\ I_{k,n}[\varrho , \mathbf{m }, E]&= \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda _kt} \beta \left( \int \nolimits _{\Omega } \mathbf{m }(t,\cdot ) \cdot \mathbf{e }_n \mathrm{d}x \right) \mathrm{d}t; \end{aligned}$$

(17)

the functionals are well defined since $\mathbf{m }(t,\cdot )\in W^{-\ell , 2}(\Omega ; {\mathbb {R}}^N)$ for all t. Let $\{ (k(j),n(j)) \}_{j=1}^{\infty }$ be an enumeration of all the involved combinations of indices, that is, an enumeration of the countable set

$$\begin{aligned} ({\mathbb {N}} \times \{ 0 \}) \cup ({\mathbb {N}} \times {\mathbb {N}}). \end{aligned}$$

We define

$$\begin{aligned} {\mathcal {U}}^j = I_{k(j),n(j)} \circ \dots \circ I_{k(1),n(1)} \circ I_{1,\beta } \circ {\mathcal {U}}, \quad j=1,2, \dots , \end{aligned}$$

and

$$\begin{aligned} {\mathcal {U}}^{\infty } = \bigcap _{j=1}^{\infty } {\mathcal {U}}^j. \end{aligned}$$

The set-valued mapping

$$\begin{aligned}{}[\varrho _0, \mathbf{m }_0, E_0] \in D \mapsto {\mathcal {U}}^{\infty } [\varrho _0, \mathbf{m }_0, E_0] \end{aligned}$$

enjoys the properties (A1)–(A4). Indeed:

(A1):

first, notice that for every fixed initial data $[\varrho _0, \mathbf{m }_0, E_0]\in D$ the sets ${\mathcal {U}}^j[\varrho _0, \mathbf{m }_0, E_0]$ are nested:

$$\begin{aligned} I_{1,\beta } \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0] \supseteq {\mathcal {U}}^1[\varrho _0, \mathbf{m }_0, E_0] \supseteq \dots \supseteq {\mathcal {U}}^j[\varrho _0, \mathbf{m }_0, E_0] \supseteq \dots . \end{aligned}$$

By Proposition 4.1 we can deduce that $I_{1, \beta } \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$ is compact, and iterating this procedure we obtain that all ${\mathcal {U}}^j[\varrho _0, \mathbf{m }_0, E_0]$ are compact. Since Q is a Hausdorff space, every compact set is also closed and a countable intersection of closed set is closed. Since ${\mathcal {U}}^{\infty }[\varrho _0, \mathbf{m }_0, E_0] \subseteq I_{1,\beta } \circ {\mathcal {U}}[\varrho _0, \mathbf{m }_0, E_0]$, which is compact, we obtain that ${\mathcal {U}}^{\infty }[\varrho _0, \mathbf{m }_0, E_0]$ is compact. By Proposition 4.1 we can also deduce that every ${\mathcal {U}}^j[\varrho _0, \mathbf{m }_0, E_0]$ is non-empty; applying Theorem 4.3 we then get that ${\mathcal {U}}^{\infty }[\varrho _0, \mathbf{m }_0, E_0] \ne \emptyset$;

(A2):

as it is an intersection set-valued map obtained from measurable set-valued maps, it is also measurable;

(A3):

to prove the shift property, let $[\varrho _0, \mathbf{m }_0, E_0] \in D$ and $[\varrho , \mathbf{m }, E]\in {\mathcal {U}}^{\infty }[\varrho _0, \mathbf{m }_0, E_0]$; then, in particular $[\varrho , \mathbf{m }, E]\in {\mathcal {U}}^j[\varrho _0, \mathbf{m }_0, E_0]$ for every j. By Proposition 4.1, we can deduce that $I_{1, \beta } \circ {\mathcal {U}}$ satisfies the shift invariance property, and iterating this procedure we obtain that this holds for every ${\mathcal {U}}^j$. This implies

$$\begin{aligned} S_T \circ [\varrho , \mathbf{m }, E] \in {\mathcal {U}}^j[\varrho (T), \mathbf{m }(T), E(T-)], \text{ for } \text{ all } j \text{ and } \text{ all } T>0. \end{aligned}$$

Thus,

$$\begin{aligned} S_T \circ [\varrho , \mathbf{m }, E] \in {\mathcal {U}}^{\infty }[\varrho (T), \mathbf{m }(T), E(T-)], \text{ for } \text{ all } T>0; \end{aligned}$$

(A4):

to prove the continuation property, let $T>0$, $[\varrho ^1,\mathbf{m }^1,E^1]\in {\mathcal {U}}^{\infty }[\varrho _0,\mathbf{m }_0, E_0]$ and $[\varrho ^2,\mathbf{m }^2,E^2]\in {\mathcal {U}}^{\infty }[\varrho ^1(T), \mathbf{m }^1(T), E^1(T-)]$; then, in particular we have $[\varrho ^1,\mathbf{m }^1,E^1]\in {\mathcal {U}}^j[\varrho _0, \mathbf{m }_0, E_0]$ and $[\varrho ^2, \mathbf{m }^2,E^2]\in {\mathcal {U}}^j[\varrho ^1(T),\mathbf{m }^1(T), E^1(T-)]$ for every j. By Proposition 4.1, we can deduce that $I_{1, \beta } \circ {\mathcal {U}}$ satisfies the continuation property, and iterating this procedure we obtain that this holds for every ${\mathcal {U}}^j$. This implies

$$\begin{aligned}{}[\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2, \mathbf{m }^2,E^2] \in {\mathcal {U}}^j[\varrho _0, \mathbf{m }_0, E_0] \text{ for } \text{ all } j \text{ and } \text{ all } T>0. \end{aligned}$$

Thus,

$$\begin{aligned}{}[\varrho ^1, \mathbf{m }^1, E^1] \cup _T [\varrho ^2, \mathbf{m }^2,E^2] \in {\mathcal {U}}^{\infty }[\varrho _0,\mathbf{m }_0, E_0] \text{ for } \text{ all } T>0. \end{aligned}$$

We claim that for every $[\varrho _0, \mathbf{m }_0, E_0] \in D$ the set ${\mathcal {U}}^{\infty }$ is a singleton, meaning

$$\begin{aligned} {\mathcal {U}}^{\infty }[\varrho _0, \mathbf{m }_0, E_0]= U\{ \varrho _0, \mathbf{m }_0, E_0 \} \in Q. \end{aligned}$$

To verify this, we observe that

$$\begin{aligned} I_{k(j),n(j)} [\varrho ^1,\mathbf{m }^1, E^1] = I_{k(j),n(j)} [\varrho ^2,\mathbf{m }^2, E^2] \end{aligned}$$

for any $[\varrho ^1, \mathbf{m }^1, E^1], [\varrho ^2, \mathbf{m }^2, E^2]\in {\mathcal {U}}^{\infty }[\varrho _0, \mathbf{m }_0, E_0]$ for all $j=1,2, \dots$; from the choice of $\{ k(j), n(j) \}_{j\in {\mathbb {N}}}$, we can see the integrals $I_{k(j), n(j)}$ as Laplace transforms

$$\begin{aligned} F(\lambda _k)= \int \nolimits _{0}^{\infty } \mathrm{e}^{-\lambda _k t}f(t)\mathrm{d}t \end{aligned}$$

of the functions

$$\begin{aligned} f\in \left\{ \beta (E), \ \beta \left( \int \nolimits _{\Omega } \mathbf{m }\cdot \mathbf{e }_n \mathrm{d}x \right) \right\} . \end{aligned}$$

We can apply Lerch’s theorem: if a function F has the inverse Laplace transform f, then f is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). Then, we get that

$$\begin{aligned} \beta (E^1(t))&= \beta (E^2(t)), \\ \beta \left( \int \nolimits _{\Omega } \mathbf{m }^1(t,\cdot )\cdot \mathbf{e }_n \mathrm{d}x \right)&= \beta \left( \int \nolimits _{\Omega } \mathbf{m }^2(t,\cdot )\cdot \mathbf{e }_n \mathrm{d}x \right) , \end{aligned}$$

for all $n \in {\mathbb {N}}$ and for a.e. $t\in (0,\infty )$. As $\beta$ is strictly increasing, we must in particular have

$$\begin{aligned} E^1(t)= E^2(t), \quad \langle \mathbf{m }^1(t,\cdot ); \mathbf{e }_n \rangle _{L^2(\Omega ; {\mathbb {R}}^N)}= \langle \mathbf{m }^2(t,\cdot ); \mathbf{e }_n \rangle _{L^2(\Omega ; {\mathbb {R}}^N)}, \end{aligned}$$

for all $n \in {\mathbb {N}}$ and for a.e. $t\in (0,\infty )$. Since $\{ \mathbf{e }_n \}_{n\in {\mathbb {N}}}$ form a basis in $L^2(\Omega ; {\mathbb {R}}^N)$, we conclude

$$\begin{aligned} \mathbf{m }^1=\mathbf{m }^2, \text{ and } E^1=E^2 \text{ a.e. } \text{ on } (0,\infty ). \end{aligned}$$

From the continuity equation (1) and from the fact that $\varrho ^1(0,\cdot )= \varrho ^2(0,\cdot )$ it is easy to see that

$$\begin{aligned} \varrho ^1= \varrho ^2 \quad \text{ a.e. } (0,\infty ). \end{aligned}$$

It remains to prove that U is a semiflow selection: measurability follows from (A2), while the semigroup property follows from (A3): for $t_1,t_2 \ge 0$ it holds

$$\begin{aligned} U\{ \varrho _0, \mathbf{m }_0, E_0 \} (t_1+t_2)= S_{t_1} \circ U\{ \varrho _0, \mathbf{m }_0, E_0 \} (t_2) = U\{ \varrho (t_1), \mathbf{m }(t_1), E(t_1 -) \} (t_2). \end{aligned}$$

This completes the proof. $\square$

Remark 4.4

In comparison with [3], it is worth noticing that the family of functionals defined in (17) do not depend on the density $\varrho$: they are not necessary since, once the uniqueness of the momenta is achieved, the one of the densities easily follows from the continuity equation (1).

5 Restriction to semigroup acting only on the initial data

As a matter of fact, the semiflow selection $U=U\{\varrho _0,\mathbf{m }_0, E_0\}$ is determined in terms of the three state variables: the density $\varrho _0$, the momentum $\mathbf{m }_0$, and the energy $E_0$. Introduction of the energy might be superfluous; indeed, as pointed out in (10)

$$\begin{aligned} E(\tau ) = \int \nolimits _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m }|^2}{\varrho } +P(\varrho )\right] (\tau ,\cdot ) \mathrm{d}x \quad \text{ for } \text{ a.e. } \tau \ge 0. \end{aligned}$$

The point is that the equality holds with the exception of a zero measure set of times. More specifically, the energy $E(\tau )$ is a non-increasing function with well-defined right and left limits $E(\tau \pm )$, while

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

is defined at any $\tau$ in terms of weakly continuous functions $t\mapsto \varrho (t,\cdot )$, $t\mapsto \mathbf{m }(t,\cdot )$. Due to the convexity of the superposition

$$\begin{aligned}{}[\varrho , \mathbf{m }] \mapsto \frac{1}{2} \frac{|\mathbf{m }|^2}{\varrho } +P(\varrho ) \end{aligned}$$

the function

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

is lower semi-continuous in $\tau$. In particular,

$$\begin{aligned} \int \nolimits _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m }|^2}{\varrho } +P(\varrho )\right] (\tau ,\cdot ) \mathrm{d}x \le E(\tau \pm ) \quad \text{ for } \text{ any } \tau , \end{aligned}$$

where equality holds with the exception of a set of times of measure zero.

We may introduce a new selection defined only in terms of the initial data $\varrho _0$, $\mathbf{m }_0$; however, the price to pay is that the semigroup property will hold almost everywhere in time. More specifically, we can state this final result.

Theorem 5.1

Let $U=U\{ \varrho _0, \mathbf{m }_0, E_0 \}$ be the semiflow selection associated with the Navier–Stokes system in the sense of Definition 2.1. Consider the set of initial data

$$\begin{aligned} {\widetilde{D}} = \left\{ [\varrho _0, \mathbf{m }_0]: \ \left[ \varrho _0, \mathbf{m }_0, \int \nolimits _{\Omega } \left( \frac{1}{2} \frac{|\mathbf{m }_0|^2}{\varrho _0} +P(\varrho _0) \right) \mathrm{d}x \right] \in D\right\} . \end{aligned}$$

Defining $V:{\widetilde{D}} \rightarrow Q$ such that

$$\begin{aligned} V\{ \varrho _0, \mathbf{m }_0 \} (t) = U\left\{ \varrho _0, \mathbf{m }_0, \int \nolimits _{\Omega } \left[ \frac{1}{2} \frac{|\mathbf{m }_0|^2}{\varrho _0} +P(\varrho _0)\right] \mathrm{d}x \right\} (t) \end{aligned}$$

for all $t\in (0,\infty )$, then V will satisfy the semigroup property only almost everywhere; more precisely, calling ${\mathcal {T}} \subset (0,\infty )$ the set of times defined as

$$\begin{aligned} {\mathcal {T}} = \left\{ \tau \in (0,\infty ): \ E(\tau )= \int \nolimits _{\Omega } \left[ \frac{1}{2}\frac{|\mathbf{m }|^2}{\varrho } +P(\varrho )\right] (\tau ,\cdot ) \mathrm{d}x \right\} , \end{aligned}$$

then ${\mathcal {T}}$ is a set of full measure and

$$\begin{aligned} V\{ \varrho _0, \mathbf{m }_0 \} (t_1+t_2) = V\{ V\{\varrho _0, \mathbf{m }_0\} (t_1) \} (t_2) \end{aligned}$$

holds for all $t_1,t_2 \in {\mathcal {T}}$.