Abstract
It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill-posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to the well-posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables—the mass density, the linear momentum, and the energy—in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.
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1 Introduction
The motion of a compressible isentropic fluid in the Eulerian reference frame is described by the time evolution of the mass density \(\varrho = \varrho (t,x)\), \(t \geqq 0\), \(x \in Q \subset R^N\), \(N=1,2,3,\) and the momentum \(\mathbf{m}= \mathbf{m}(t,x)\) solving the Euler system as follows:
where \(\gamma > 1\) is the adiabatic constant. The problem is closed by prescribing the initial data
as well as appropriate boundary conditions. For the sake of simplicity, we eliminate possible problems connected with the presence of a kinematic boundary by considering the space–periodic flows, for which the physical domain can be identified with the flat torus,
It is well-known that solutions of (1.1) develop singularities—shock waves—in finite time no matter how smooth or small the initial data are. Accordingly, the concept of weak (distributional) solutions has been introduced to study global-in-time behavior of system (1.1). The existence of weak solutions in the simplified monodimensional geometry has been established for a rather general class of initial data, see Chen and Perepelitsa [8], DiPerna [16], Lions, Perthame and Souganidis [23], among others. More recently, the theory of convex integration has been used to show existence of weak solutions for \(N=2,3\) again for a rather vast class of data, see Chiodaroli [9], De Lellis and Székelyhidi [15], Luo, Xie and Xin [24].
Uniqueness and stability with respect to the initial data in the framework of weak solutions is a more delicate issue. Apparently, the Euler system is ill-posed in the class of weak solutions and explicit examples of multiple solutions emanating from the same initial state have been constructed, see for example the monograph of Smoller [26]. An admissibility criterion must be added to the weak formulation of (1.1) in order to select the physically relevant solutions. To this end, consider the total energy e given by
Admissible solutions satisfy, in addition to the weak version of (1.1), the total energy balance
or at least its integrated form,
Note that (1.5) follows directly form (1.4) thanks to the periodic boundary conditions; the same holds, of course, under suitable conservative boundary conditions, for instance,
Note that the inequality in (1.4) is needed to select the physically relevant discontinuous shock-wave solutions.
Even if (1.4) is imposed as an extra selection criterion, weak solutions are still not unique, see Chiodaroli, De Lellis and Kreml [10], Markfelder and Klingenberg [25]. The initial data giving rise to infinitely many admissible solutions are termed wild data. As shown in [10], this class includes certain Lipschitz initial data. Recently, this result has been extended to smooth initial data by Chiodaroli et al [12]. Furthermore, even if additional selection criteria as, for instance, maximality of the energy dissipation, are imposed, the problem remains ill-posed, see Chiodaroli and Kreml [11].
An important feature of systems with uniqueness is their semiflow property: 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. In other words, the knowledge of the whole past up to time s provides no more useful information about the outcome at time t than knowing the state of the system at time s only. For systems where the uniqueness is unknown or not valid, a natural question is whether a solution semiflow can be constructed anyway.
Therefore, inspired by the recent work of Cardona and Kapitanski [7], we propose a different approach to well-posedness of the Euler system based on the theory of Markov selection in stochastic analysis, see for example Krylov [22], Stroock and Varadhan [27], Flandoli and Romito [19], or [4]. More specifically, we establish the existence of a semiflow selection for the Euler system (1.1)–(1.3), that is, a mapping
enjoying the semigroup property
where \([\varrho , \mathbf{m}]\) represents a generalized solution to (1.1)–(1.3) with the energy E. More specifically, the triple \([\varrho , \mathbf{m}, E]\) termed dissipative solution will coincide with the expected value of suitable measure-valued solution satisfying the Euler system (1.1), together with the energy inequality (1.5), satisfied in a generalized sense. The precise definitions can be found in Sect. 2. In addition to the semigroup property (1.6), the semiflow we shall construct enjoys the following properties, which provides further justification of the physical relevance of our construction:
Stability of strong solutions. Let the Euler system (1.1)–(1.3) admit a strong \(W^{1,\infty }\) solution \({\widehat{\varrho }}\), \({\widehat{\mathbf{m}}}\), with the associated energy
$$\begin{aligned} E_0 = \int _{Q} \left[ \frac{|\mathbf{m}_0|^2}{\varrho _0} + \frac{a}{\gamma - 1} \varrho _0^\gamma \right] \,\mathrm{d} {x}, \end{aligned}$$defined on a maximal time interval \([0, T_{\mathrm{max}})\).
Then we have
$$\begin{aligned} U[t, \varrho _0, \mathbf{m}_0, E_0] = [{\widehat{\varrho }}, {\widehat{\mathbf{m}}}, E_0](t) \ \text{ for } \text{ all }\ t \in [0,T_{\mathrm{max}}). \end{aligned}$$This reflects the fact that dissipative solutions satisfy the weak-strong uniqueness principle.
Maximal dissipation. Let the Euler system (1.1)–(1.3) admit a dissipative solution \({\widehat{\varrho }}\), \({\widehat{\mathbf{m}}}\), with the associated energy \({\widehat{E}}\) such that
$$\begin{aligned} {\widehat{E}}(t) \leqq E(t) \ \text{ for } \text{ all }\ t \geqq 0, \end{aligned}$$where E is the energy of the solution semiflow \(U[t, \varrho _0, \mathbf{m}_0, E_0]\).
Then we have
$$\begin{aligned} E(t) = {\widehat{E}}(t) \ \text{ for } \text{ all }\ t \geqq 0. \end{aligned}$$In other words, our search for physically relevant solutions respects the ideas of Dafermos [13] who introduced the selection criterion based on the maximization of the energy dissipation for hyperbolic systems of conservation laws.
Stability of stationary states. Let \(\overline{\varrho } > 0\), \(\mathbf{m}\equiv 0\) be a stationary solution of the Euler system (1.1)–(1.3). Suppose that
$$\begin{aligned} \varrho (T, \cdot ) = \overline{\varrho },\ \mathbf{m}(T, \cdot ) = 0 \ \text{ for } \text{ some }\ T \geqq 0, \end{aligned}$$where \(\varrho \), \(\mathbf{m}\) are the density and the momentum components of a solution semiflow \(U[t, \varrho _0, \mathbf{m}_0, E_0]\).
Then we have
$$\begin{aligned} \varrho (t, \cdot ) = \overline{\varrho },\ \mathbf{m}(t, \cdot ) = 0 \ \text{ for } \text{ all }\ t \geqq T. \end{aligned}$$Hence, if the system reaches a stationary state where the density is constant and the momentum vanishes, it remains in this state for all future times.
The fact that a certain form of energy inequality has to be included as an integral part of the definition of solution is pertinent to the analysis of problems in fluid mechanics. One of the main novelties of our approach is including the total energy E as a third variable in the construction of the semiflow. Intuitively speaking, the knowledge of the initial state for the density and the momentum does not provide sufficient information to restart the semiflow. We have already observed a similar phenomenon in the context of Markov selection for stochastic compressible Navier–Stokes system in [4].
Our definition of a dissipative solution (see Definition 2.1) is motivated by the notion of dissipative measure-valued solutions known for example from [18, 20]. However, we chose a different formulation which in our opinion reflects better the nature of the system and is more suitable for the construction of the solution semiflow. Similarly to the notion of dissipative measure-valued solution, our definition permits to establish the weak-strong uniqueness principle. Consequently, strong solutions are always contained in the selected semiflow as long as they exist.
Note that this desirable property is not granted for the semiflow of weak solutions to the incompressible Navier–Stokes system presented in [7]. More precisely, even for the incompressible Navier–Stokes system, where global existence of unique solutions has not yet been excluded for smooth initial data, the semiflow in [7] may “select” completely pathological solutions like those that start from zero but have positive energy at later times (such solutions may exist thanks to the recent work by Buckmaster and Vicol [6]).
To conclude this introduction, we remark that our method applies mutatis mutandis to the incompressible Navier–Stokes and Euler system as well as to the isentropic Navier–Stokes system. We have chosen the isentropic Euler system for this paper as it is the system where uniqueness seems to be the most out of reach. However, it would be interesting to investigate whether for one of the “easier” systems one could understand further properties of the solution semiflow such as the dependence on the initial data. Moreover, uniqueness of the solution semiflow is also an open problem.
The paper is organized as follows: in Sect. 2 we introduce the concept of dissipative solutions and state the main result concerning the semiflow selection. Section 3 is devoted to the proof of existence and stability of dissipative solutions. In Sect. 4, we present the abstract setting and in Sect. 5, we show the existence of the semiflow selection. Section 6 contains concluding discussions concerning refined properties of the constructed semiflow.
2 Set-up and Main Results
In this section we present several definitions of generalized solutions to the compressible Euler system. In particular, we introduce dissipative solutions and explain the concept of admissibility. Finally, we present our main result on semiflow selection in Sect. 2.4.
2.1 Weak Solutions
Weak solutions of the Euler system (1.1) on the time interval \([0,\infty )\) satisfy the integral identities
for any \(\varphi \in C^1_c([0, \infty ) \times {T^N})\), and
for any \(\varvec{\varphi }\in C^1_c([0, \infty ) \times {T^N}; R^N)\). A weak solution to (1.1) is a pair of measurable functions \([\varrho ,\mathbf{m}]\) such that all integrals in (2.1) and (2.2) are well-defined. In accordance with the energy inequality (1.5), we suppose additionally that
Consistently with (2.1), (2.2), this can be put in a variational form,
for any \(\psi \in C^1_c ([0, \infty ))\), \(\psi \geqq 0\).
2.2 Dissipative Solutions
If \(N=2,3\), the energy inequality (2.3) seems to be the only source of a priori bounds. However, as indicated by the numerous examples of “oscillatory” solutions (cf. [9, 15]) the set of all admissible weak solutions emanating from given initial data is not closed with respect to the weak topology on the trajectory space associated with the energy bounds (2.3). There are two potential sources of difficulties:
non-controllable oscillations due to the accumulation of singularities;
blow-up type collapse due to possible concentration points.
To accommodate the above mentioned singularities in the closure of the set of weak solutions, two kinds of tools are used: (i) the Young measures describing the oscillations, (ii) concentration defect measures for concentrations, see for example Brenier, De Lellis, Székelyhidi [5].
Let
be the phase space associated to the Euler system. Let \({\mathcal {P}}({\mathcal {S}})\) denote the set of probability measures on \({\mathcal {S}}\) and let \({\mathcal {M}}^{+}(T^{N})\) and \({\mathcal {M}}^{+}(T^{N}\times S^{N-1})\), respectively, denote the set of positive bounded Radon measures on \(T^{N}\) and \(T^{N}\times S^{N-1}\), respectively, where \(S^{N-1}\subset R^{N}\) denotes the unit sphere. A dissipative solution is defined via the following quantities:
the Young measure:
$$\begin{aligned} (t,x) \mapsto \nu _x(t) \in L^\infty _{\mathrm{weak - (*)}}((0, \infty ) \times T^N; {\mathcal {P}}({\mathcal {S}})); \end{aligned}$$(2.4)the kinetic and internal energy concentration defect measures:
$$\begin{aligned} t&\mapsto {\mathfrak {C}}_{\mathrm{kin}}(t) \in L^\infty _\mathrm{weak - (*)}(0, \infty ; {\mathcal {M}}^+(T^N)), \end{aligned}$$(2.5)$$\begin{aligned} t&\mapsto {\mathfrak {C}}_{\mathrm{int}}(t) \in L^\infty _\mathrm{weak - (*)}(0, \infty ; {\mathcal {M}}^+(T^N)); \end{aligned}$$(2.6)the convective and pressure concentration defect measures:
$$\begin{aligned} t&\mapsto {\mathfrak {C}}_{\mathrm{conv}} (t) \in L^\infty _{\mathrm{weak - (*)}} {\left( 0,\infty ; {\mathcal {M}}^+\left( T^N\times S^{N-1} \right) \right) }, \end{aligned}$$(2.7)$$\begin{aligned} t&\mapsto {\mathfrak {C}}_{\mathrm{press}} (t) \in {L^\infty _\mathrm{weak - (*)}\left( 0,\infty ; {\mathcal {M}}^+(T^N) \right) . } \end{aligned}$$(2.8)
The constitutive relations
enforce natural compatibility conditions
where \(r_x(t)\in {\mathcal {P}} (S^{N-1})\) are the measures associated to disintegration of \({\mathfrak {C}}_{\mathrm{conv}}(t)\) on the product \(T^N \times S^{N-1}\), see for example Ambrosio, Fusco, and Palara [2, Theorem 2.28].
Hereafter, we denote by \([{\tilde{\varrho }},{\tilde{\mathbf{m}}}]\) the dummy variables in phase space \({\mathcal {S}}\) whereas \(\xi \) is a dummy variable in \(S^{N-1}\). We are now in the position to present the basic building block for the semiflow selection—a dissipative solution of the Euler system (1.1).
Definition 2.1
(Dissipative solution) The triple of functions \([\varrho , \mathbf{m}, E]\) with
is called dissipative solution of the Euler system (1.1) with the initial data
if there exists a family of parametrized measures specified through (2.4)–(2.9) such that
- (a)
for a.a \(\tau >0\) we have
$$\begin{aligned} \begin{aligned} \varrho (\tau ,x)&= \left\langle \nu _{x}(\tau ); \tilde{\varrho }\right\rangle \geqq 0, \ \mathbf {m}(\tau ,x) = \left\langle \nu _{x}(\tau ); {\tilde{\mathbf {m}}}\right\rangle \ \text { for } \text { a.a }\ x \in T^N, \\ E(\tau )&= \int _{{T^N}} \left\langle \nu _{x}(\tau ); \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} + \frac{a}{\gamma -1} \tilde{\varrho }^\gamma \right\rangle \ \,\mathrm {d} {x}\\ {}&\qquad + \int _{T^N} \ \mathrm {d}{\mathfrak {C}}_{\mathrm {kin}}(\tau ) + \int _{T^N} \ \mathrm {d}{\mathfrak {C}}_{\mathrm {int}}(\tau )\ ; \end{aligned} \end{aligned}$$(2.10) - (b)
for any \(\tau > 0\) the integral identity
$$\begin{aligned} \left[ \int _{{T^N}} \varrho \varphi \ \,\mathrm {d} {x} \right] _{t = 0}^{t = \tau } = \int _{0}^{\tau } \int _{{T^N}} \Big [ \varrho \partial _t \varphi + \mathbf {m}\cdot \nabla _x\varphi \Big ] \ \,\mathrm {d} {x} \,\mathrm {d} t\end{aligned}$$(2.11)holds for any \(\varphi \in C^1_c([0, \infty ) \times {T^N})\), where \(\varrho (0, \cdot ) = \varrho _0\);
- (c)
for any \(\tau > 0\) the integral identity
$$\begin{aligned} \begin{aligned}&\left[ \int _{T^N} \mathbf {m}\cdot \mathbf {\varphi }\,\mathrm {d} {x}\right] _{t = 0}^{t = \tau }- \int _{0}^{\tau } \int _{{T^N}} \mathbf {m}\cdot \partial _t \mathbf {\varphi } \ \,\mathrm {d} {x}\,\mathrm {d} t \\ {}&= \int _{0}^{\tau } \int _{{T^N}} \left[ \left\langle \nu _{x}(t); 1_{\tilde{\varrho }> 0}\frac{{\tilde{\mathbf {m}}}\otimes {\tilde{\mathbf {m}}}}{\tilde{\varrho }} \right\rangle : \nabla _x\mathbf {\varphi }+ \left\langle \nu _{x}(t); a \tilde{\varrho }^\gamma \right\rangle \mathrm {div}_x\mathbf {\varphi }\right] \ \,\mathrm {d} {x} \,\mathrm {d} t \\ {}&+ { 2 \int _{0}^{\tau } \int _{T^N} \left\langle r_{x}(t); \xi \otimes \xi \right\rangle : \nabla _x\mathbf {\varphi }\ \ \mathrm {d} {\mathfrak {C}}_{\mathrm {kin}} \,\mathrm {d}t }\\ {}&+ (\gamma - 1)\int _0^\tau \int _{T^N} \mathrm {div}_x\mathbf {\varphi }\ \mathrm {d}{\mathfrak {C}}_{\mathrm {int}} \,\mathrm {d}t \end{aligned} \end{aligned}$$(2.12)holds for any \(\varvec{\varphi }\in C^1_c([0, \infty ) \times {T^N}; R^N)\), where \(\mathbf{m}(0, \cdot ) = \mathbf{m}_0\);
- d)
for any \(0 \leqq \tau _1 \leqq \tau _2\) the inequality
$$\begin{aligned} \begin{aligned} \Big [ E \psi \Big ]_{t = \tau _1-}^{t = \tau _2+} - \int _{\tau _1}^{\tau _2} E \partial _t \psi \ \,\mathrm{d} t \leqq 0 , \end{aligned} \end{aligned}$$(2.13)holds for any \(\psi \in C^1_c([0, \infty ))\) with \(\psi \geqq 0\), where \(E(0-) = E_0\).
Note that our definition is slightly different from the one used by Gwiazda, Świerczewska-Gwiazda, and Wiedemann [20] based on the concentration defect measures introduced by Alibert and Bouchitté [1]. We believe that the present setting based on the energy defects rather than the recession functions reflects better the underlying system of PDEs. It is also worth noting that the present definition contains definitely more information on the dissipative solutions than its counterpart introduced in [18] in the context of the compressible Navier–Stokes system. The class of solutions considered in [18] is apparently larger but still guarantees the weak–strong uniqueness principle. Indeed, the corresponding proof in [18] adapts easily to the Euler setting. In particular, we obtain the following result that can be proved exactly as [18] (see also Gwiazda et al. [20] and Sect. 6.5 below):
Proposition 2.2
(Weak-strong uniqueness) Let \([\varrho , \mathbf{m}, E]\) be a dissipative solution to (1.1) in the sense of Definition 2.1 starting from the initial state \([\varrho _0,\ \mathbf{m}_0, \ E_0]\), \(\varrho _0 > 0\). Let \([{\widehat{\varrho }}, \widehat{\mathbf{m}}]\) be a strong solutionFootnote 1 to (1.1) in \([0, T_{\mathrm{max}}) \times T^N\) starting from the same initial data \({\widehat{\varrho }}_0 = \varrho _0\), \(\widehat{\mathbf{m}}_0 = \mathbf{m}_0\), with
Then we have
2.3 Admissible Dissipative Solutions
Finally, we introduce a subclass of dissipative solutions that reflect the physical principle of maximization of the energy dissipation. 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
Definition 2.3
(Admissible dissipative solution) We say that a dissipative 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
where \([\tilde{\varrho }, {\tilde{\mathbf{m}}}, {\tilde{E}}]\) is another dissipative solution starting from \([\varrho _0, \mathbf{m}_0, E_0]\), then
Maximizing the energy dissipation or, equivalently, minimizing the total energy of the system is motivated by a similar selection criterion proposed by Dafermos [14]. In view of the arguments discussed in Sect. 6, such a selection criterion
rules out a large part of wild solutions obtained via “available” methods;
guarantees stability of equilibrium states in the class of dissipative solutions.
2.4 Semiflow Selection: Main Result
We start by introducing suitable topologies on the space of the initial data and the space of dissipative solutions. Fix \(\ell >N/2+1\) and consider the Hilbert space
together with its subset containing the initial data
Here the convex function \([\varrho , \mathbf{m}] \mapsto \frac{|\mathbf{m}|^2}{\varrho }\) is defined for \(\varrho \geqq 0\), \(\mathbf{m}\in R^N\) as
Note that D is a closed convex subset of X. We consider the trajectory space
which is a separable metric space. Dissipative solutions \([\varrho , \mathbf{m}, E]\), as defined in Definition 2.1, belong to this class. Indeed, equations (2.11) and (2.12) give an information on the time regularity of the density and the momentum whereas the energy can be controlled by (2.13). Moreover, for initial data \([\varrho _0, \mathbf{m}_0, E_0] \in D\) it follows from (2.10) and Jensen’s inequality that a dissipative solution \([\varrho ,\mathbf{m},E]\) evaluated at a.a. times \(t\geqq 0\) also belongs to the set D. Finally, for initial data \([\varrho _0, \mathbf{m}_0, E_0] \in D\), we introduce the solution set
We are now ready to define a semiflow selection to (1.1).
Definition 2.4
(Semiflow selection) A semiflow selection in the class of dissipative solutions for the compressible Euler system (1.1) is a mapping
enjoying the following properties:
- (a)
Measurability. The mapping \(U: D \rightarrow \Omega \) is Borel measurable.
- (b)
Semigroup property. We have
$$\begin{aligned} U \left\{ \varrho _0, \mathbf{m}_0, E_0 \right\} (t_1 + t_2) = U \left\{ \varrho (t_1), \mathbf{m}(t_1), E(t_1-) \right\} (t_2) \end{aligned}$$for any \([\varrho _0, \mathbf{m}_0, E_0] \in D\) and any \(t_1, t_2\geqq 0\), where \([\varrho , \mathbf{m}, E] = U \left\{ \varrho _0, \mathbf{m}_0, E_0 \right\} \).
Our main result reads as follows:
Theorem 2.5
The isentropic Euler system (1.1) admits a semiflow selection U in the class of dissipative solutions in the sense of Definition 2.4. Moreover, we have that
for any \([\varrho _0, \mathbf{m}_0, E_0] \in D\).
In the next section we prove the existence of at least one dissipative solution for given initial data and the sequential stability of the solution set. The abstract setting for the selection principle is presented in Sect. 4 and the proof of Theorem 2.5 can be found in Sect. 5. The additional regularity properties of the selection mentioned in Sect. 1 will be discussed in Sect. 6.
3 Existence and Sequential Stability
We aim to show
The existence of a dissipative solution for any initial data \([\varrho _0, \mathbf{m}_0, E_0] \in D\), meaning
$$\begin{aligned} {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0] \ne \emptyset ; \end{aligned}$$The sequential stability of the solution set, meaning
$$\begin{aligned} {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0] \subset \Omega \ \text{ is } \text{ compact } \end{aligned}$$and the multivalued mapping
$$\begin{aligned}{}[\varrho _0, \mathbf{m}_0, E_0] \in D \subset X \rightarrow {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0] \in 2^\Omega \end{aligned}$$has closed graph; whence by Lemma 12.1.8 in [27] it is (strongly) Borel measurable.
We note that if \({\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0]\) is a compact subset of the separable metric space \(\Omega \) for any \([\varrho _0, \mathbf{m}_0, E_0] \in D\), then the (Borel) measurability of the multivalued mapping
corresponds to measurability with respect to the Hausdorff metric on the space of all compact subsets of \(\Omega \).
3.1 Sequential Stability
We first address the issue of sequential stability as the existence proof leans basically on identical arguments.
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 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 there exists \(\overline{E}>0\) such that \( E_{0,\varepsilon } \leqq \overline{E}\) for all \(\varepsilon > 0\).
Then, at least for suitable subsequences,
and
where
Proof
We proceed via several steps.
First of all, observe that the convergence (3.1) follows immediately from the fact that the energy \(E_{0,\varepsilon }\) is bounded uniformly for \(\varepsilon \rightarrow 0\).
It follows from Jensen’s inequality that
$$\begin{aligned} \frac{1}{2} \frac{|\mathbf {m}_\varepsilon |^2}{\varrho _\varepsilon } + \frac{a}{\gamma - 1} \rho _\varepsilon ^\gamma \leqq \left\langle \nu ^\varepsilon _x (t) ; \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\rho }} + \frac{a}{\gamma - 1} \tilde{\rho }^\gamma \right\rangle \end{aligned}$$(3.2)a.a. in \((0,\infty )\times T^N\), where \(\nu ^{\varepsilon }\) is the Young measure associated with the solution \([\varrho _{\varepsilon },\mathbf{m}_{\varepsilon },E_{\varepsilon }]\). Consequently, as \(E_{0,\varepsilon } \rightarrow E_0\), we deduce from the energy inequality (2.13), (2.10), and equations (2.11), (2.12) that (up to a subsequence)
$$\begin{aligned} \begin{aligned} \varrho _\varepsilon&\rightarrow \varrho \ \text{ in }\ C_{\mathrm{weak, loc}}([0, \infty ); L^\gamma (T^N)), \ \varrho \geqq 0,\\ \mathbf{m}_\varepsilon&\rightarrow \mathbf{m}\ \text{ in }\ C_{\mathrm{weak, loc}}([0, \infty ); L^\frac{2\gamma }{\gamma +1}(T^N; R^N)), \end{aligned} \end{aligned}$$where
$$\begin{aligned} \varrho (0, \cdot ) = \varrho _0, \ \mathbf{m}(0, \cdot ) = \mathbf{m}_0. \end{aligned}$$In addition, by (2.13) and (2.10) the energy is non-increasing and non-negative; whence its total variation can be bounded by the initial value and the latter one is uniformly bounded by assumption. Hence by Helly’s selection theorem, we have
$$\begin{aligned} E_\varepsilon (\tau ) \rightarrow E(\tau ) \ \text{ for } \text{ any }\ \tau \in [0, \infty ) \ \text{ and } \text{ in }\ L^{1}_{\mathrm{loc}}(0,\infty ),\ E(0+) \leqq E_0. \end{aligned}$$In view of the above observations, it is easy to perform the limit in the equation of continuity (2.11) to obtain
$$\begin{aligned} \left[ \int _{{T^N}} \varrho \varphi \ \,\mathrm{d} {x} \right] _{t = 0}^{t = \tau } = \int _{0}^{\tau } \int _{{T^N}} \Big [ \varrho \partial _t \varphi + \mathbf{m}\cdot \nabla _x\varphi \Big ] \ \,\mathrm{d} {x} \,\mathrm{d} t \end{aligned}$$for any \(\varphi \in C^1_c([0, \infty ) \times {T^N})\), as well as in the energy balance (2.13): we get
$$\begin{aligned} \begin{aligned} \int _0^\infty E(t) \partial _t \psi \ \,\mathrm{d} t \geqq 0 \ \text{ for } \text{ any }\ \psi \in C^1_c((0, \infty )),\ \psi \geqq 0, \ E(0+) \leqq E_0, \end{aligned} \end{aligned}$$from which we deduce (2.13). Moreover, we have
$$\begin{aligned} \left[ \int _{{T^N}} \mathbf{m}_\varepsilon \cdot \varvec{\varphi } \ \,\mathrm{d} {x} \right] _{t = 0}^{t = \tau } \rightarrow \left[ \int _{{T^N}} \mathbf{m}\cdot \varvec{\varphi } \ \,\mathrm{d} {x} \right] _{t = 0}^{t = \tau }, \end{aligned}$$and
$$\begin{aligned} \int _0^\tau \int _{{T^N}} \mathbf{m}_\varepsilon \cdot \partial _t \varvec{\varphi } \ \,\mathrm{d} {x}\,\mathrm {d}t \rightarrow \int _0^\tau \int _{{T^N}} \mathbf{m}\cdot \partial _t \varvec{\varphi } \ \,\mathrm{d} {x}\,\mathrm {d}t \end{aligned}$$for any test function admissible in the momentum balance (2.12).
Next, we denote by \({\mathfrak {C}}^{\varepsilon }_{\mathrm{kin}}\) and \({\mathfrak {C}}^{\varepsilon }_{\mathrm{int}}\) the kinetic and internal energy concentration defect measure associated with \([\varrho _{\varepsilon },\mathbf{m}_{\varepsilon },E_{ \varepsilon }]\). Using again the energy inequality (2.13) and (2.10), we deduce (up to a subsequence) the convergence of the concentration measures:
$$\begin{aligned} \begin{aligned} {\mathfrak {C}}^\varepsilon _{\mathrm{kin}}&\rightarrow {\mathfrak {C}}^{\infty ,1}_{\mathrm{kin}}\ \text{ weakly-(*) } \text{ in }\ L^\infty (0, \infty ; {\mathcal {M}}^+(T^N)),\\ {\mathfrak {C}}^\varepsilon _{\mathrm{int}}&\rightarrow {\mathfrak {C}}^{\infty ,1}_{\mathrm{int}}\ \text{ weakly-(*) } \text{ in }\ L^\infty (0, \infty ; {\mathcal {M}}^+(T^N)). \end{aligned} \end{aligned}$$In view of (2.9) we denote by
$$\begin{aligned} {\mathfrak {C}}^{\varepsilon ,1}_{\mathrm{conv}}(t) \equiv 2r^\varepsilon (t) \otimes {\mathfrak {C}}^{\varepsilon }_{\mathrm{kin}}(t), \end{aligned}$$(3.3)the convective concentration defect measure associated with \([\varrho _{\varepsilon },\mathbf{m}_{\varepsilon },E_{\varepsilon }]\) and deduce
$$\begin{aligned} {\mathfrak {C}}^{\varepsilon ,1}_{\mathrm{conv}} \rightarrow {\mathfrak {C}}^{\infty ,1}_\mathrm{conv} \ \text{ weakly-(*) } \text{ in }\ L^\infty (0, \infty ; {\mathcal {M}}^+(T^N\times S^{N-1})). \end{aligned}$$(3.4)We remark that the final convective concentration defect measure will be constructed below as a sum of \({\mathfrak {C}}^{\infty ,1}_\mathrm{conv}\) and another measure obtained from the concentrations of the Young measures \(\nu ^{\varepsilon }_{x}(t)\).
With the above convergences at hand, we are able to pass to the limit in the kinetic as well as internal energy concentration defect measure in (2.10) and also in the pressure concentration defect measure in (2.12). Furthermore, we can pass to the limit in the integrals related to the convective term. More precisely, in view of (3.3), we have
$$\begin{aligned} \begin{aligned}&2 \int _{0}^{\infty } \int _{T^N} \left\langle r^\varepsilon _{x}(t); \xi \otimes \xi \right\rangle : \nabla _x\varvec{\varphi }\ {\mathfrak {C}}^\varepsilon _{\mathrm{kin}}(t,\mathrm {d}x) \,\mathrm {d}t\\&\quad = \int _0^\infty \int _{T^N} \int _{S^{N-1}}(\xi \otimes \xi ): \nabla _x\varvec{\varphi }\ {\mathfrak {C}}_{\mathrm{conv}}^{\varepsilon ,1} (t,\mathrm {d}x,\mathrm {d}\xi )\,\mathrm {d}t \end{aligned} \end{aligned}$$and the right hand side converges by (3.4) to
$$\begin{aligned} \int _0^\infty \int _{T^N} \int _{S^{N-1}} (\xi \otimes \xi ): \nabla _x\varvec{\varphi }\ {\mathfrak {C}}_{\mathrm{conv}}^{\infty ,1} (t,\mathrm {d}x,\mathrm {d}\xi ) \,\mathrm {d}t. \end{aligned}$$Finally, we realize that \(2 {\mathfrak {C}}^{\infty ,1}_{\mathrm {kin}}(t,\mathrm {d}x)\) is the marginal of \({\mathfrak {C}}_{\mathrm {conv}}^{\infty ,1}(t,\mathrm {d}x,\mathrm {d}\xi )\) corresponding to the variable x, that is,
$$\begin{aligned} 2 {\mathfrak {C}}^{\infty ,1}_{\mathrm {kin}}(t,\mathrm {d}x)={\mathfrak {C}}_{\mathrm {conv}}^{\infty ,1}(t,\mathrm {d}x,S^{N-1}). \end{aligned}$$(3.5)Indeed, by (3.3), this is true on the approximate level and the property is preserved through the passage to the limit as \(\varepsilon \rightarrow 0\).
Finally, it remains to handle the terms containing the Young measure. First, we deduce from the energy inequality (2.13) together with (2.10) and (3.2) that the Young measures \(\nu ^{\varepsilon }_{x}(t)\) have uniformly bounded first moments. This implies their (relative) compactness leading to
$$\begin{aligned} \nu ^\varepsilon _{x}(t) \rightarrow \nu _{x}(t) \ \text{ weakly-(*) } \text{ in }\ L^\infty \left( (0, \infty ) \times T^N; {\mathcal {P}}({\mathcal {S}}) \right) . \end{aligned}$$(3.6)Note that the fact that the limit is again a (parametrized) probability measure follows from the finiteness of the first moments, see for example Ball [3].
Next, let \(\chi _k\) and and \(\psi _k\), \(k\in {\mathbb {N}}\), be cut-off functions satisfying
$$\begin{aligned} \chi _k&\in C^\infty (R),\ 0 \leqq \chi _k(Z) \leqq 1,\\ \chi _k(Z)&= 0 \text { if } Z \leqq k - 1,\ 0 \leqq \chi _k(Z) \leqq 1,\ \chi _k(Z) = 1 \text { if } Z \geqq k;\\ \psi _k&\in C^\infty (R), 0 \leqq \psi _k(Z) \leqq 1, \\ \psi _k(Z)&= 0\ \text { for }\ Z \leqq 0,\ 0 \leqq \psi _k(Z) \leqq 1 \text { if } 0 \leqq Z \le \frac{1}{k},\ \psi _k(Z) = 1 \text { if } Z \geqq \frac{1}{k}. \end{aligned}$$We consider the two families of measures (here \(b \in C(S^{N-1})\))
$$\begin{aligned} {\mathfrak {C}}^{\varepsilon ,k, 2}_{\mathrm{conv}} \in L^\infty _{\mathrm{weak-(*)}}(0, \infty ; {\mathcal {M}}^+ (T^N \times S^{N-1})), \end{aligned}$$$$\begin{aligned} \int _{S^{N-1}}b(\xi ){\mathfrak {C}}^{\varepsilon ,k, 2}_{\mathrm{conv}}(t,\mathrm {d}x,{\mathrm {d}\xi })= & {} 2\left\langle \nu ^\varepsilon _x(t) ; b\left( \frac{{\tilde{\mathbf{m}}}}{|{\tilde{\mathbf{m}}}|}\right) \psi _k(\widetilde{\varrho }) {\widetilde{\chi }}_k \left( \frac{1}{2} \frac{ |{\tilde{\mathbf{m}}} |^2 }{\tilde{\varrho }} \right) \frac{1}{2} \frac{ |{\tilde{\mathbf{m}}} |^2 }{{\tilde{\varrho }}} \right\rangle \,\mathrm{d} {x}\\&+ 2\left\langle \nu _{t,x}; b\left( \frac{{\tilde{\mathbf{m}}}}{|{\tilde{\mathbf{m}}}|}\right) 1_{\tilde{\varrho }> 0} (1 - \psi _k (\tilde{\varrho })) \frac{1}{2} \frac{ |{\tilde{\mathbf{m}}} |^2 }{{\tilde{\varrho }}} \right\rangle dx , \end{aligned}$$and
$$\begin{aligned} {\mathfrak {C}}^{\varepsilon ,k, 2}_{\mathrm{press}} \in L^\infty _{\mathrm{weak-(*)}}(0, \infty ; {\mathcal {M}}^+ (T^N)),\ {\mathfrak {C}}^{\varepsilon ,k, 2}_\mathrm{press}(t,\mathrm {d}x) = \left\langle \nu ^\varepsilon _x(t) ; \chi _k \left( a {\tilde{\varrho }}^\gamma \right) a {{\tilde{\varrho }}}^\gamma \right\rangle \,\mathrm{d} {x}. \end{aligned}$$Due to (2.13) and (2.10) they are bounded uniformly in \(\varepsilon ,k\) and hence passing to the limit, first for \(\varepsilon \rightarrow 0\) then \(k \rightarrow \infty \) we obtain
$$\begin{aligned}&{\mathfrak {C}}^{\varepsilon ,k, 2}_{\mathrm{conv}} \rightarrow {\mathfrak {C}}^{\infty , 2}_\mathrm{conv} \ \text{ weakly-(*) } \text{ in }\ L^\infty _{\mathrm{weak-(*)}}(0, \infty ; {\mathcal {M}}^+ (T^N \times S^{N-1})), \nonumber \\&\quad {\mathfrak {C}}^{\varepsilon ,k, 2}_{\mathrm{press}} \rightarrow {\mathfrak {C}}^{\infty , 2}_{\mathrm{press}} \ \text{ weakly-(*) } \text{ in }\ L^\infty _{\mathrm{weak-(*)}}(0, \infty ; {\mathcal {M}}^+ (T^N)). \end{aligned}$$(3.7)We set
$$\begin{aligned} 2{\mathfrak {C}}^{\infty ,2}_{\mathrm{kin}}(t) = {\mathfrak {C}}^{\infty ,2}_\mathrm{conv}(t,\,\mathrm{d} {x}, S^{N-1}),\ {\mathfrak {C}}^{\infty ,2}_{\mathrm{int}}(t) = \frac{1}{\gamma -1} {\mathfrak {C}}^{\infty ,2}_{\mathrm{press}}. \end{aligned}$$(3.8)Accordingly, the convective term in the momentum equation (2.12) can be decomposesd as
$$\begin{aligned}&\left\langle \nu ^\varepsilon _{x}(t); 1_{\tilde{\varrho }> 0} \frac{{\tilde{\mathbf {m}}}\otimes {\tilde{\mathbf {m}}}}{\tilde{\varrho }} \right\rangle \,\mathrm {d} {x}\\ {}&=\left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf {m}}}\otimes {\tilde{\mathbf {m}}}}{\tilde{\varrho }} \right) \psi _k(\tilde{\varrho }) (1 - \chi _k) \left( \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} \right) \right\rangle \,\mathrm {d} {x}\\ {}&+ \left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf {m}}}\otimes {\tilde{\mathbf {m}}}}{\tilde{\varrho }} \right) \psi _k (\tilde{\varrho }) \chi _k \left( \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} \right) \right\rangle \,\mathrm {d} {x}\\ {}&+\left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf {m}}}\otimes {\tilde{\mathbf {m}}}}{\tilde{\varrho }} \right) 1_{\tilde{\varrho }> 0}(1 - \psi _k(\tilde{\varrho })) \right\rangle \,\mathrm {d} {x}\\ {}&=\left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf {m}}}\otimes {\tilde{\mathbf {m}}}}{\tilde{\varrho }} \right) \psi _k (\tilde{\varrho })(1 - \chi _k) \left( \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} \right) \right\rangle \,\mathrm {d} {x}\\ {}&+2\left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf {m}}}}{|{\tilde{\mathbf {m}}}|} \otimes \frac{{\tilde{\mathbf {m}}}}{|{\tilde{\mathbf {m}}}|} \right) \psi _k(\tilde{\varrho }) \chi _k \left( \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} \right) \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }}\right\rangle \,\mathrm {d} {x}\\ {}&+2\left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf {m}}}}{|{\tilde{\mathbf {m}}}|} \otimes \frac{{\tilde{\mathbf {m}}}}{|{\tilde{\mathbf {m}}}|} \right) 1_{\tilde{\varrho }> 0}(1 - \psi _k(\tilde{\varrho }))\frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} \right\rangle \,\mathrm {d} {x}.\end{aligned}$$Performing successively the limits \(\varepsilon \rightarrow 0\), \(k \rightarrow \infty \) we obtain
$$\begin{aligned} \left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf{m}}}\otimes {\tilde{\mathbf{m}}}}{\tilde{\varrho }} \right) \psi _k(\tilde{\varrho }) (1 - \chi _k) \left( \frac{1}{2} \frac{|{\tilde{\mathbf{m}}}|^2}{\tilde{\varrho }} \right) \right\rangle \,\mathrm{d} {x}\rightarrow \left\langle \nu _{x}(t); 1_{\tilde{\varrho }> 0}\left( \frac{{\tilde{\mathbf{m}}}\otimes {\tilde{\mathbf{m}}}}{\tilde{\varrho }} \right) \right\rangle \,\mathrm{d} {x}. \end{aligned}$$Indeed, the passage to the limit as \(\varepsilon \rightarrow 0\) is follows from (3.6) since the Young measures are applied to continuous and bounded functions, whereas the passage to the limit \(k\rightarrow \infty \) is a consequence of dominated convergence together with the energy inequality (2.13) and (2.10). On the other hand, by definition of \({\mathfrak {C}}^{\varepsilon ,k,2}_{\mathrm{conv}}\) and (3.7) we obtain
$$\begin{aligned}&2\left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf{m}}}}{|{\tilde{\mathbf{m}}}|} \otimes \frac{{\tilde{\mathbf{m}}}}{|{\tilde{\mathbf{m}}}|} \right) \psi _k (\tilde{\varrho }) \chi _k \left( \frac{1}{2} \frac{|{\tilde{\mathbf{m}}}|^2}{\tilde{\varrho }} \right) \frac{1}{2} \frac{|{\tilde{\mathbf{m}}}|^2}{\tilde{\varrho }}\right\rangle \,\mathrm{d} {x}\\&\qquad + 2\left\langle \nu ^\varepsilon _{x}(t); \left( \frac{{\tilde{\mathbf{m}}}}{|{\tilde{\mathbf{m}}}|} \otimes \frac{{\tilde{\mathbf{m}}}}{|{\tilde{\mathbf{m}}}|} \right) 1_{\tilde{\varrho }> 0}(1 - \psi _k(\tilde{\varrho }))\frac{1}{2} \frac{|{\tilde{\mathbf{m}}}|^2}{\tilde{\varrho }} \right\rangle \,\mathrm{d} {x}\\&\quad =\int _{S^{N-1}}(\xi \otimes \xi ){\mathfrak {C}}^{\varepsilon ,k,2}_\mathrm{conv}(t,\,\mathrm{d} {x}, \mathrm {d}\xi )\rightarrow \int _{S^{N-1}}(\xi \otimes \xi ){\mathfrak {C}}^{\infty ,2}_\mathrm{conv}(t,\,\mathrm{d} {x}, \mathrm {d}\xi ). \end{aligned}$$The pressure term can be handled in a similar manner. Finally, we set
$$\begin{aligned} {\mathfrak {C}}_{\mathrm{kin}} = {\mathfrak {C}}_{\mathrm{kin}}^{\infty ,1} + {\mathfrak {C}}^{\infty ,2}_{\mathrm{kin}},\ {\mathfrak {C}}_{\mathrm{int}} = {\mathfrak {C}}_{\mathrm{int}}^{\infty ,1} + {\mathfrak {C}}_\mathrm{int}^{\infty ,2} = {\mathfrak {C}}_{\mathrm{int}}^{\infty ,1} + \frac{1}{\gamma - 1} {\mathfrak {C}}_{\mathrm{press}}^{\infty ,2} \end{aligned}$$and use relations (3.5), (3.8) to obtain, after final disintegration
$$\begin{aligned} {\mathfrak {C}}_{\mathrm{conv}} = {\mathfrak {C}}^{ \infty ,1}_{\mathrm{conv}} + {\mathfrak {C}}^{\infty ,2}_{\mathrm{conv}} = 2 r_x(t) \otimes \left( {\mathfrak {C}}_{\mathrm{kin}}^{\infty ,1} + {\mathfrak {C}}^{\infty ,2}_\mathrm{kin} \right) \end{aligned}$$for some measures \(r_x(t) \in {\mathcal {P}} (S^{N-1})\).
Finally, we can pass to the limit in (2.10). Arguing as for the convective term we obtain
$$\begin{aligned}&\int _{{T^N}} \left\langle \nu ^\varepsilon _{x}(\tau ); \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} + \frac{a}{\gamma -1} \tilde{\varrho }^\gamma \right\rangle \ \,\mathrm {d} {x}\\ {}&\quad \rightarrow \int _{{T^N}} \left\langle \nu _{x}(\tau ); \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} + \frac{a}{\gamma -1} \tilde{\varrho }^\gamma \right\rangle \ \,\mathrm {d} {x} \\ {}&\,\quad + \int _{T^N} \ \mathrm {d}{{\mathfrak {C}}_{\mathrm {kin}}^{\infty ,2}}(\tau ) + \int _{T^N} \ \mathrm {d}{\mathfrak {C}}_{\mathrm {int}}^{\infty ,2}(\tau ) \end{aligned}$$weakly-(*) in \(L^\infty (0,T)\) such that
$$\begin{aligned} E(\tau )&= \int _{{T^N}} \left\langle \nu _{x}(\tau ); \frac{1}{2} \frac{|{\tilde{\mathbf {m}}}|^2}{\tilde{\varrho }} + \frac{a}{\gamma -1} \tilde{\varrho }^\gamma \right\rangle \ \,\mathrm {d} {x} \\ {}&+ \int _{T^N} \ \mathrm {d}{\mathfrak {C}}_{\mathrm {kin}}(\tau ) + \int _{T^N} \ \mathrm {d}{\mathfrak {C}}_{\mathrm {int}}(\tau ). \end{aligned}$$
The proof is hereby complete. \(\quad \square \)
3.2 Existence
The sequential stability from the previous part combined with a suitable approximation implies the existence of a dissipative solution. The precise statement is the content of the following proposition:
Proposition 3.2
Let \([\varrho _0,\mathbf{m}_0,E_0]\in D\) be given. Then the isentropic Euler system (1.1) admits a dissipative solution in the sense of Definition 2.1 with the initial data \([\varrho _0, \mathbf{m}_0, E_0]\).
Proof
We adapt the method of Kröner and Zajaczkowski [21] adding an artificial viscosity term of higher order to the momentum equation. First observe that the definition of D implies
with the respective bounds in terms of \(E_{0}\). It is a routine matter to construct approximating sequences satisfying,
and
as \(\varepsilon \rightarrow 0\), where \(\varrho _{0,\varepsilon } > 0\) and the velocity \(\mathbf{u}_{0,\varepsilon }\) are smooth functions.
We consider the “multipolar fluid” type approximation of the Euler system (1.1):
where \(\varepsilon >0\), \(m\in {\mathbb {N}}\), and the initial data is chosen as
It is well known that for \(m\in {\mathbb {N}}\) large enough, see for example [21], problem (3.9), (3.10) admits a unique smooth solution \([\varrho _\varepsilon , \mathbf{u}_\varepsilon ]\) in the time interval \((0, \infty )\). Moreover, we have the total energy balance,
Using the arguments of the preceding section, it is easy to perform the limit \(\varepsilon \rightarrow 0\) in the sequence of approximate solutions
to obtain the desired dissipative solution as long as we control the artificial viscosity terms. However, this is standard as (3.11) yields
Accordingly, the corresponding term in the weak formulation of the momentum equation \((3.9)_2\) can be handled as
and vanishes asymptotically. \(\quad \square \)
4 Abstract Setting
Our goal is to adapt the abstract machinery developed by Cardona and Kapitanski [7] to the family \({\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0]\), \([\varrho _0, \mathbf{m}_0, E_0] \in D\). The following statement is a direct consequence of Propositions 3.1 and 3.2:
Lemma 4.1
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 \(\Omega \). Moreover, \([\varrho (T), \mathbf{m}(T), {\mathcal {E}} ] \in D\) for any \(T > 0\), and for arbitrary \({\mathcal {E}} \geqq E(T+)\).
4.1 Shift and Continuation Operations
Two main ingredients for the construction of the semiflow are the shift invariance property and the continuation property of the set of solutions (this corresponds to the disintegration and reconstruction property in the probabilistic setting of Markov selections). For \(\omega \in \Omega \), we define the positive shift operator
Lemma 4.2
(Shift invariance property) 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
for any \(T > 0\), and any \({\mathcal {E}} \geqq E(T+)\).
Proof
Obviously, a dissipative solution on the time interval \((0, \infty )\) solves also the same problem on \((T, \infty )\) with the initial data \([\varrho (T, \cdot ), \mathbf{m}(T, \cdot ), E(T+)]\). Moreover, the energy is non-increasing; whence
The rest follows by shifting the test functions in the integrals. \(\quad \square \)
For \(\omega _1, \omega _2 \in \Omega \) we define the continuation operator \(\omega _1 \cup _T \omega _2\) by
Lemma 4.3
(Continuation property) Let \([\varrho _0, \mathbf{m}_0, E_0] \in D\) and
where \(\varepsilon \le E^{1}(T-)\) Then
Proof
We have only to realize that the energy of the solution \([\varrho ^1, \mathbf{m}^1, E^1] \cup _T [\varrho ^2, \mathbf{m}^2, E^2]\) indeed remains non-increasing on \((0, \infty )\). \(\quad \square \)
4.2 General Ansatz
Summarizing the previous part of this section and the results of Sect. 3, we have shown the existence of a set-valued mapping
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 \(\Omega \).
- (A2)
Measurability: The mapping
$$\begin{aligned} D\ni [\varrho _0, \mathbf{m}_0, E_0] \mapsto {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0] \in 2^\Omega \end{aligned}$$is Borel measurable, where the range of \({\mathcal {U}}\) is endowed with the Hausdorff metric on the subspace of compact sets in \(2^\Omega \).
- (A3)
Shift invariance: For any
$$\begin{aligned}{}[\varrho , \mathbf{m}, E] \in {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0], \end{aligned}$$we have
$$\begin{aligned} S_T \circ [\varrho , \mathbf{m}, E] \in {\mathcal {U}}[\varrho (T), \mathbf{m}(t), E(T-)]\ \text{ for } \text{ any }\ T > 0. \end{aligned}$$ - (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
$$\begin{aligned}{}[\varrho ^1, \mathbf{m}^1, E^1] \cup _T [\varrho ^2, \mathbf{m}^2, E^2] \in {\mathcal {U}} [\varrho _0, \mathbf{m}_0, E_0]. \end{aligned}$$
The conditions (A1)–(A4) have been introduced in Cardona and Kapitanski [7]. In what follows, we will adopt their method based on the ideas of Krylov [22] and Strook and Varadhan [27] to select the desired solution semiflow. We remark that the value \(E(T-)\) in (A3) and (A4) can be replaced by
where \(\eta \in [0,1]\) is given.
5 Semiflow Selection
Following the general method by Krylov [22], we consider the family of functionals
where
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
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 \(\Omega \) and the set \({\mathcal {U}}[\varrho _{0},\mathbf{m}_{0},E_{0}]\) is compact in \(\Omega \). We obtain the following result for the set \(I_{\lambda , F} \circ {\mathcal {U}}\).
Proposition 5.1
Let \(\lambda > 0\) and F be a bounded continuous functional on X. Let
be a multivalued mapping having the properties (A1)–(A4). Then the map \(I_{\lambda ,F} \circ {\mathcal {U}}\) enjoys (A1)–(A4) as well.
Proof
Apart from the proof of (A2), we follow the lines of the proof of Cardona and Kapitanski [7, Section 2], which in turn relies on the classical approach by Krylov [22] for stochastic differential equations. As a matter of fact, Cardona and Kapitanski [7] consider \(\Omega \) as a space of continuous functions on a separable complete metric space X. This is not true in our case since, due to the possibility of energy sinks, the energy E lacks continuity. We therefore present the details of the proof also for the convenience of the reader.
The map \(I_{\lambda , F}: {\mathcal {U}}[ \varrho _0, \mathbf{m}_0, E_0] \subset \Omega \rightarrow R \) is continuous. As the set \({\mathcal {U}}[ \varrho _0, \mathbf{m}_0, E_0]\) is non-empty and compact, the set \(I_{\lambda , F}\circ {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0]\) is a non-empty compact subset of \(\Omega \), which completes the proof of (A1).
Let \(d_H\) be the Hausdorff metric on the subspace \({\mathcal {K}} \subset 2^\Omega \) of compact sets, specifically,
$$\begin{aligned} d_H(K_1, K_2) = \inf _{\varepsilon \geqq 0} \left\{ K_1 \subset V_\varepsilon (K_2) \ \text{ and } \ K_2 \subset V_\varepsilon (K_1) \right\} ,\ K_1,K_2 \in {\mathcal {K}}, \end{aligned}$$where \(V_\varepsilon (A)\) denotes the \(\varepsilon \)-neighborhood of a set A in the topology of \(\Omega \). To show Borel measurability of the multivalued mapping
$$\begin{aligned}{}[\varrho _0, \mathbf{m}_0, E_0] \in D \mapsto I_{\lambda , F} \circ {\mathcal {U}}[ \varrho _0, \mathbf{m}_0, E_0] \in {\mathcal {K}} \subset 2^\Omega , \end{aligned}$$it is enough to show that the mapping \({\mathcal {I}}_{\lambda , F}\) defined for any \(K \in {\mathcal {K}}\) as
$$\begin{aligned} {\mathcal {I}}_{\lambda , F} [K] = \left\{ z \in K \ \Big | \ I_{\lambda , F}(z) \leqq I_{\lambda , F}({\tilde{z}}) \ \text{ for } \text{ all }\ {\tilde{z}} \in K \right\} \in {\mathcal {K}}, \end{aligned}$$is continuous as a mapping on \({\mathcal {K}}\) endowed with the Hausdorff metric \(d_H\). Suppose
$$\begin{aligned} K_n {\mathop {\rightarrow }\limits ^{d_H}} K,\ K_n, K \in {\mathcal {K}}. \end{aligned}$$As \(I_{\lambda , F}\) is continuous, we easily observe that
$$\begin{aligned} \min _{K_n} I_{\lambda , F} \rightarrow \min _K I_{\lambda ,F}. \end{aligned}$$(5.1)Consider the \(\varepsilon \)-neighborhood \(V_\varepsilon ({\mathcal {I}}_{\lambda , F} [K])\) of the compact set \({\mathcal {I}}_{\lambda , F} [K]\). Our goal is to show that
$$\begin{aligned} {\mathcal {I}}_{\lambda , F}[K_n] \subset V_\varepsilon ({\mathcal {I}}_{\lambda , F} [K]) \ \text{ for } \text{ all }\ n \geqq n_0(\varepsilon ). \end{aligned}$$Arguing by contradiction, we construct a sequence such that
$$\begin{aligned} z_n \in K_n,\ I_{\lambda , F}(z_n) = \min _{K_n}I_{\lambda , F},\ z_n \rightarrow z \in K \setminus V_\varepsilon ({\mathcal {I}}_{\lambda , F} [K]). \end{aligned}$$Continuity of \(I_{\lambda , F}\) yields
$$\begin{aligned} I_{\lambda , F}(z_n) \rightarrow I_{\lambda , F}(z) > \min _K I_{\lambda , F} \end{aligned}$$in contrast to (5.1). Interchanging the roles of \(K_n\) and K we get the opposite inclusion
$$\begin{aligned} {\mathcal {I}}_{\lambda , F}[K] \subset V_\varepsilon ({\mathcal {I}}_{\lambda , F} [K_n]) \ \text{ for } \text{ all }\ n \geqq n_0(\varepsilon ) \end{aligned}$$by a similar argument. This implies that \({\mathcal {I}}_{\lambda ,F}[K_{n}]\overset{d_{H}}{\rightarrow }{\mathcal {I}}_{\lambda ,F}[K]\) and completes the proof of (A2).
For the shift invariance let us consider some \([\varrho , \mathbf{m}, E]\in I_{\lambda , F}\circ {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0]\) for some \([\varrho _0, \mathbf{m}_0, E_0]\in D\). We aim to show that the shift \(S_T\circ [\varrho , \mathbf{m}, E]\) belongs to the set \(I_{\lambda , F}\circ {\mathcal {U}}[\varrho (T), \mathbf{m}(T), E(T-)]\) for \(T>0\) arbitrary. Indeed, for any \([\varrho ^T, \mathbf{m}^T, E^T]\in I_{\lambda , F}\circ {\mathcal {U}}[\varrho (T), \mathbf{m}(T), E(T-)]\) we obtain
$$\begin{aligned}{}[\varrho , \mathbf{m}, E]\cup _T[\varrho ^T, \mathbf{m}^T, E^T]\in {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0] \end{aligned}$$by (A4) and hence
$$\begin{aligned} I_{\lambda , F}(S_T\circ [\varrho , \mathbf {m}, E])&=\int _0^\infty e^{-\lambda t}F(S_T\circ [\varrho , \mathbf {m}, E](t))\,\mathrm {d} t \\ {}&=\int _0^\infty e^{-\lambda t} F([\varrho , \mathbf {m}, E](T+t))\,\mathrm {d} t \\ {}&=e^{\lambda T}\int _T^\infty e^{-\lambda t} F([\varrho , \mathbf {m}, E](t))\,\mathrm {d} t \\ {}&=e^{\lambda T}\bigg (I_{\lambda , F}[\varrho , \mathbf {m}, E]-\int _0^T e^{-\lambda t} F([\varrho , \mathbf {m}, E](t))\,\mathrm {d} t \bigg )\\ {}&\leqq e^{\lambda T}\bigg (I_{\lambda , F}([\varrho , \mathbf {m}, E]\cup _T[\varrho ^T, \mathbf {m}^T, E^T])\\ {}&\quad \qquad -\int _0^T e^{-\lambda t} F([\varrho , \mathbf {m}, E](t))\,\mathrm {d} t \bigg )\\ {}&=e^{\lambda T}\int _{T}^{\infty }e^{-\lambda t} F([\varrho ^{T},\mathbf {m}^{T},E^{T}](t-T))\,\,\mathrm {d} t \\ {}&=I_{\lambda , F}[\varrho ^T, \mathbf {m}^T, E^T], \end{aligned}$$where the inequality follows from the fact that \([\varrho , \mathbf{m}, E]\) minimizes \(I_{\lambda , F}\) on \({\mathcal {U}}[\varrho _{0},\mathbf{m}_{0},E_{0}]\) by assumption. This implies that \(S_T\circ [\varrho , \mathbf{m}, E]\) minimizes \(I_{\lambda , F}\) and consequently belongs to \(I_{\lambda , F}\circ {\mathcal {U}}[\varrho (T), \mathbf{m}(T), E(T-)]\). We have shown property (A3).
On the other hand, let us consider \([\varrho ^1, \mathbf{m}^1, E^1] \in I_{\lambda , F}\circ {\mathcal {U}} [\varrho _0, \mathbf{m}_0, E_0]\) as well as \([\varrho ^2, \mathbf{m}^2, E^2] \in I_{\lambda , F}\circ {\mathcal {U}} [\varrho ^1(T), \mathbf{m}^1(T), E^1(T-)]\) where \([\varrho _0, \mathbf{m}_0, E_0]\in D\) and \(T>0\). We obtain for the continuation \([\varrho ^1, \mathbf{m}^1, E^1]\cup _T [\varrho ^2, \mathbf{m}^2, E^2]\) that
$$\begin{aligned}&I_{\lambda , F}([\varrho ^1, \mathbf {m}^1, E^1]\cup _T [\varrho ^2, \mathbf {m}^2, E^2])\\ {}&=\int _0^T e^{-\lambda t}F([\varrho ^1, \mathbf {m}^1, E^1](t))\,\mathrm {d} t +\int _T^\infty e^{-\lambda t} F([\varrho ^2, \mathbf {m}^2, E^2](t-T))\,\mathrm {d} t \\ {}&=\int _0^T e^{-\lambda t}F([\varrho ^1, \mathbf {m}^1, E^1](t))\,\mathrm {d} t +e^{-\lambda T}I_{\lambda ,F}[\varrho ^2, \mathbf {m}^2, E^2]\\ {}&\leqq \int _0^T e^{-\lambda t}F([\varrho ^1, \mathbf {m}^1, E^1](t))\,\mathrm {d} t +e^{-\lambda T}I_{\lambda ,F}(S_T\circ [\varrho ^1, \mathbf {m}^1, E^1])\\ {}&=I_{\lambda , F}[\varrho ^1, \mathbf {m}^1, E^1], \end{aligned}$$where the inequality follows from the fact that \([\varrho ^2, \mathbf{m}^2, E^2]\) is a minimizer of \(I_{\lambda ,F}\) in the set \({\mathcal {U}} [\varrho ^1(T), \mathbf{m}^1(T), E^1(T-)]\). As \([\varrho ^1, \mathbf{m}^1, E^1]\) is a minimizer in \({\mathcal {U}} [\varrho _0, \mathbf{m}_0, E_0]\) and \([\varrho ^1, \mathbf{m}^1, E^1]\cup _T [\varrho ^2, \mathbf{m}^2, E^2]\in {\mathcal {U}} [\varrho _0, \mathbf{m}_0, E_0]\) by (A4) we must have equality and \([\varrho ^1, \mathbf{m}^1, E^1]\cup _T [\varrho ^2, \mathbf{m}^2, E^2]\) is a minimizer too. This proves (A4) for \(I_{\lambda , F}\circ {\mathcal {U}}\) and the proof is complete.\(\quad \square \)
5.1 Selection Sequence
The first step is to select only those solutions that are admissible, meaning minimal with respect to the relation \(\prec \) introduced in Definition 2.3. To this end, we consider the functional \(I_{1, \beta }\) with
Lemma 5.2
Suppose that \([\varrho , \mathbf{m}, E] \in {\mathcal {U}}[\varrho _0, \mathbf{m}_0, E_0]\) satisfies
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, \( {\tilde{E}} \leqq E \) in \((0,\infty ).\) Then we get
whence \(E = {{\tilde{E}}}\) a.a. in \((0,\infty )\) since \(\beta \) is strictly increasing. \(\quad \square \)
Proof of Theorem 2.5
Selecting \(I_{1,\beta } \circ {\mathcal {U}}\) from \({\mathcal {U}}\) we know that the new selection contains only admissible solutions (minimal with respect to \(\prec \)) for any \([\varrho _0, \mathbf{m}_0, E_0] \in D\).
Next, we choose a countable basis \(\{ e_n \}_{n=1}^\infty \) in \(L^2(T^N)\) formed by trigonometric polynomials, its vector valued analogue \(\{ \mathbf{w}_m \}_{m=1}^\infty \) in \(L^2(T^N; R^N)\), and a countable set \(\{\lambda _k \}_{k=1}^{\infty }\) which is dense in \((0,\infty )\). We consider a countable family of functionals,
and let \(\{(k(j),n(j),m(j))\}_{j=1}^{\infty }\) be an enumeration of all the involved combinations of indices, that is, an enumeration of the countable set
We define
and
\(\quad \square \)
By Proposition 5.1 the set-valued mapping
enjoys the properties (A1)–(A4). Indeed, since \({\mathcal {U}}^{\infty }[\varrho _{0},\mathbf{m}_{0},E_{0}]\) is an intersection of countably many non-empty compact nested sets, it is non-empty and compact. As it is an intersection of set-valued maps obtained from measurable set-valued maps, it is also measurable. The shift property (A3) as well as the continuation property (A4) are straightforward.
Finally, we claim that for every \([\varrho _{0},\mathbf{m}_{0},E_{0}]\in D\) the set \({\mathcal {U}}^\infty [\varrho _{0},\mathbf{m}_{0},E_{0}]\) is a singleton, meaning there exists a single trajectory \(U\{\varrho _{0},\mathbf{m}_{0},E_{0}\}\in \Omega \) such that
for any \([\varrho _0, \mathbf{m}_0, E_0] \in D\), which completes the proof of Theorem 2.5. Indeed, the semigroup property follows from the definition of the shift property (A3); for all \(t_{1},t_{2}\geqq 0\) it holds that
To verify (5.2), we observe that
for all \(j = 1,2,\dots \). This implies, by means of Lerch’s theorem and the choice of \(\{(k(j),n(j),m(j))\}_{j=1}^{\infty }\), that
for all \(n,m\in {\mathbb {N}}\) and a.a. \(t\in (0,\infty )\). As \(\beta \) is strictly increasing and \(\{ e_n \}_{n=1}^\infty \) and \(\{ \mathbf{w}_m \}_{m=1}^\infty \) form a basis in \(L^2(T^N)\) and \(L^2(T^N; R^N)\), respectively, we conclude that
which finishes the proof. \(\quad \square \)
6 Concluding Remarks
Regularity of the constructed semiflow as well as possible dependence of the trajectories on the initial data represent major open issues that probably cannot be solved within the present abstract framework. In what follows, we discuss some simple observations that may shed some light on the complexity of the problem.
6.1 Energy Profile
The hypothetical possibility of “energetic sinks”–the times \(T > 0\) for which
implies the existence of solutions in the semiflow with positive jump of the initial energy
It is interesting to note that the existence proof presented in Proposition 3.2 does not provide solutions of this type. One may be tempted to say that these are exactly the solutions obtained via the method of convex integration, however, such a conclusion is not straightforward as shown in the next section.
6.2 Wild Weak Solutions
In the context of the recent results achieved by the method of convex integration, see [9, 15, 17], some of the solutions involved in the semiflow might be the so-called wild (weak) solutions producing energy. This seems particularly relevant for the initial data of the form
However, such a possibility seems to be ruled out by the available convex integration ansatz used in the context of compressible flow, cf. [17]. Indeed the weak solutions are “constructed” with prescribed energy profile \(e_{\mathrm{kin}}(t,x) + e_\mathrm{int}(t,x)\)—a given continuous function of t and x—as limits of subsolutions \([\varrho ^s, \mathbf{m}^s]\). The subsolutions \(\varrho ^s\), \(\mathbf{m}^s\) satisfy the strict inequality
for any \(t>0\), \(x\in T^N\). Consequently, the same method gives rise to another solution with the same initial data with a chosen energy profile
which rules out the former solution on the basis of the \(\prec \) minimality.
6.3 Total Mass Conservation and Stability of Equilibrium States
It follows directly from the continuity equation (2.11) that any dissipative solution conserves the total mass,
The equilibrium states
are global in time regular solutions; whence, in accordance with the weak-strong uniqueness principle stated in Proposition 2.2,
We claim that
meaning the energy cannot “jump up” for any dissipative solution in the selection starting from the equilibrium \([\varrho _M, \mathbf{m}_M]\). Indeed suppose that
In accordance with (6.1), the total mass is conserved, namely
On the other hand, the energy is weakly lower semi-continuous, whence
for any \(\tau > 0\). Finally, we use (6.2) and Jensen’s inequality to obtain
where the equality holds if and only if \(\varrho = \varrho _M\). Consequently \(E(\tau \pm ) \geqq E_M\) for any \(\tau > 0\), meaning \([\varrho , \mathbf{m}, E_0]\) can be \(\prec \) minimal if and only if \(\varrho = \varrho _M\), \(\mathbf{m}= 0\).
We have obtained the following:
Corollary 6.1
Let \([\varrho , \mathbf{m}, E] = U\{ \varrho _0, \mathbf{m}_0, E_0 \}\) belong to the semiflow constructed in Theorem 2.5. Suppose that
Then
6.4 General Equation of State
The results presented above can be extended in a straightforward manner to a more general barotropic equation of state provided the pressure \(p = p(\varrho )\) and the pressure potential \(P(\varrho )\) given by
satisfy the asymptotic “adiabatic law”
If \(\gamma = 1\), we need the extra hypothesis
6.5 Relative Energy Inequality
Let P be the pressure potential introduced in the previous section. We define the relative energy
Following [20], we can derive the relative energy inequality
which holds for any dissipative solution \([\varrho , \mathbf{m}, E]\) starting from the initial data \([\varrho _0, \mathbf{m}_0, E_0] \in D\), and any \(r \in W^{1,\infty }_{\mathrm{loc}}([0,\infty )\times T^N)\), \(\mathbf{U} \in W^{1,\infty }_{\mathrm{loc}}([0,\infty )\times T^N; R^N)\), \(r > 0\). In particular, we have, by Gronwall’s lemma,
for any strong solution \(r, \mathbf{M} = r \mathbf{U}\), \(r > 0\), of the Euler system, which yields the weak-strong uniqueness property stated in Proposition 2.2.
Notes
A strong solution belongs to the class \(W^{1,\infty }\) and satisfies (1.1) almost everywhere pointwise.
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The research of E.F. leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 18–05974S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
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Breit, D., Feireisl, E. & Hofmanová, M. Solution Semiflow to the Isentropic Euler System. Arch Rational Mech Anal 235, 167–194 (2020). https://doi.org/10.1007/s00205-019-01420-6
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DOI: https://doi.org/10.1007/s00205-019-01420-6