Semiflow selection for the compressible Navier-Stokes system

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.


Introduction
Consider the compressible Navier-Stokes system where ̺ = ̺(t, x) denotes the density, u = u(t, x) the velocity, p = p(̺) the pressure and S = S(∇ x u) the viscous stress. We will consider the system on the set (t, x) ∈ (0, ∞) × Ω, where Ω ⊂ R N , N = 2, 3 is a bounded domain with ∂Ω of class C 2+ν for a certain ν > 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 be well posed, see Kazhikhov [7]. We impose the no-slip boundary condition for the velocity and we prescribe the initial conditions ̺(0, ·) = ̺ 0 , (̺u)(0, ·) = (̺u) 0 .
Finally, we assume a barotropic pressure p ∈ C[0, ∞) ∩ C 1 (0, ∞) such that p(0) = 0 and for certain constants a 1 > 0, a 2 and b, with γ > 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 with µ > 0 and λ ≥ 0. We would like to point out that (5) allows the pressure to a general non-monotone function of the density. Still, as we shall see below, the problem admits globalin-time weak solutions and retains other fundamental properties of the system, notable the weak-strong uniqueness, see [4]. We will consider dissipative weak solutions, i.e. solutions satisfying equations (1) and (2) in a distributional sense along with the energy inequality, see Section 1.1 below. Although the existence of global in time solutions has already been established for any finite energy initial data, see e.g. [9] and [5], 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 [1]. 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 [8] and later adapted by Flandoli and Romito [6], Cardona and Kapitanski [3] in the context of the incompressible Navier-Stokes system.
Breit, Feireisl and Hofmanová [2] used the deterministic version motivated by [3] 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 m = ̺u, we show the existence of a measurable mapping satisfying the semigroup property: where [̺, m = ̺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 [2]. The semigroup constructed for the Euler system in [2] 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 [̺, 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 [̺, u] at least for a.a. t ∈ [0, ∞), 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 Section 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 ̺ 0 , the momentum m 0 , and the energy E 0 . In Section 3 we analyze the properties (compactness, non-emptiness, the shift invariance and continuation properties) of the solution set for a given initial data while Section 4 is devoted to the proof of the existence of a semiflow selection. Finally, in Section 5 we study a new selection defined only in terms of the initial density ̺ 0 and the momentum m 0 .

Dissipative weak solution
Following [5], we can give the definition of a dissipative solution to the compressible Navier-Stokes system. Definition 1.1. The pair of functions ̺, u is called dissipative weak solution of the Navier-Stokes system (1)-(6) with the total energy E and initial data if the following holds: (i) regularity class: (ii) weak formulation of the renormalized continuity equation: for any τ > 0 and any functions holds for any ϕ ∈ C 1 c ([0, ∞) × Ω), where ̺(0, ·) = ̺ 0 ; (iii) weak formulation of the balance of momentum: for any τ > 0 the integral identity (iv) energy inequality: for a.e. τ ≥ 0 we have where the pressure potential P is chosen as a solution of we also require E = E(τ ) to be a non-increasing function of τ : 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 [2], 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 [2], the energy can be determined in terms of ̺ and 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

Admissible solution
From now on, it is more convenient to work with the momentum m = ̺u. Following [2], for a fixed initial data, we focus on a subclass of dissipative weak solutions consisting of the ones which minimize 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 [̺, m, E]. Finally, in Section 5 we pass to the natural state variables [̺, m]. We introduce the relation In particular, such selection criterion guarantees that equilibrium states belong to the class of dissipative weak solutions (see [2], Section 6.3).

Set-up
First of all, we must choose suitable topologies on the space of the initial data and the space of dissipative weak solutions. For simplicity, we will consider the Hilbert space where the constant ℓ > N 2 + 1 is fixed, along with its subset containing the initial data applying Hölder inequality, we can also deduce that ̺ 0 ∈ L γ (Ω) and m 0 ∈ L 2γ γ+1 (Ω; R N ); accordingly, the set of the data can be seen as a closed convex subset of the Banach space so that it coincides with the epigraph of the function f : Since f is lower semi-continuous and convex, we obtain that its epigraph is closed and convex.
As trajectory space, we will consider the separable space The choice of such topologies is justified by the fact that we want any dissipative weak solution [̺, m, E], as defined in Definition 1.1, to belong to the class Q (since ℓ > N 2 , the L p -space with p ≥ 1 is compactly embedded in W −ℓ,2 so in particular it holds for p = γ and p = 2γ γ+1 while equations (8) and (9) give an information on the time regularity of the density and the momentum) but also to the set D (this easily follows from the energy inequality) when evaluated at any time t ≥ 0 in order to have the possibility to restart the system at a random time t.
Finally, for a fixed initial data [̺ 0 , m 0 , E 0 ] ∈ D, we introduce the solution set
Definition 2.1. A semiflow selection in the class of dissipative weak solutions for the compressible Navier-Stokes system (1)-(6) is a mapping enjoying the following properties: (i) Measurability. The mapping U : D → Q is Borel measurable.
(ii) Semigroup property. We have We are now ready to state our main result; the proof is postponed to Section 4.1.
introduced in the previous section, enjoys the following properties; in particular, the last two are the main tools we will need in order to construct the semiflow.
This statement is equivalent in proving the existence of a dissipative weak solution for any initial data [̺ 0 , m 0 , E 0 ] ∈ D; more precisely, we have the following result.
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 For the proof see [5], Theorem 7.1.
This statement is equivalent in showing the weak sequential stability of the solution set; specifically, the following result holds.
Moreover, we assume that the initial densities converge strongly and there exists a constant E > 0 such that E 0,ε ≤ E for all ε > 0.
Then, at least for suitable subsequences, For the proof see [5], Theorems 6.1 and 6.2.
Notice that, since U [̺ 0 , m 0 , E 0 ] is a compact subset of the separable space Q for any initial data [̺ 0 , m 0 , E 0 ] ∈ D, requiring the Borel measurability of U is equivalent in proving the measurability with respect to the Hausdorff metric on the space of all compact subsets of Q. Due to Proposition 3.2, it is sufficient to apply the following lemma with Y = D and X = Q.
Let Y be a metric space and B its Borel σ-field. Let y → K y be a map of Y into Comp(X) for some separable metric space X, with Comp(X) the set of all the compact subsets of X. Suppose for any sequence y n → y and x n ∈ K yn , it is true that x n has a limit point x in K y . Then the map y → K y is a Borel map of Y into Comp(X).
The proof can be found in [10], Lemma 12.1.8.
(P4) Shift invariance. Introducing the positive shift operator for every q ∈ Q as for any T > 0.
Instead of E(T −), we could choose any E ≥ E(T +) (recall the energy is non-increasing and thus in particular E(T −) ≥ E(T +)); indeed, this more general result holds.
Since the energy is non-increasing, we can choose every E ≥ E(T +) as initial energy; indeed, everything will be well-defined (P5) Continuation. Introducing the continuation operator for any q 1 , q 2 ∈ Q as then, if T > 0, and In this case, instead of E 1 (T −), we could choose any E ≤ E 1 (T −); indeed, this more general result holds.

Semiflow selection
Starting from the family U [̺ 0 , m, E 0 ] of dissipative weak solutions for a fixed initial data [̺ 0 , m 0 , E 0 ] ∈ D, the idea for the construction of the selection is to make this set smaller and smaller choosing the minima of particular functionals. More precisely, following the same arguments presented in [2], we consider the family of functionals where F : X = W −ℓ,2 (Ω) × W −ℓ,2 (Ω; R N ) × R → R is a bounded and continuous functional. This choice is justified by the fact that I λ,F can be seen as Laplace transform of the functional F , an useful interpretation in the proof of the existence of the semiflow, as we will see in the next section. Given I λ,F and a set-valued mapping U , we define a selection mapping I λ,F • U by Notice that a minimum exists since I λ,F is continuous on Q and the set U [̺ 0 , (̺u) 0 , E 0 ] is compact in Q. We obtain the following result for the set I λ,F • U .
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 K ⊂ 2 Q of all the compact subsets of Q: where V ε (A) is the ε-neighborhood of the set A in the topology of Q; then, it is enough to show that the mapping defined for all K ∈ K as is continuous as a mapping on K endowed with the Hausdorff metric d H . In particular we want to show that if K n for n → ∞. More precisely, it is enough to show that for every ε > 0 there exists n 0 = n 0 (ε) such that for all n ≥ n 0 . First of all, notice that by the continuity of I λ,F we have We start proving the first inclusion of (12). By contradiction, suppose that exists a sequence {z n } n∈N such that z n ∈ K n , I λ,F (z n ) = min in particular, I λ,F (z) > min K I λ,F . By the continuity of I λ,F we have but this contradicts (13). Interchanging the roles of K n and K we get the opposite inclusion in (12). We get the claim.
and since U satisfies property (A4), we get From the choice of [̺, m, E], which minimize I λ,F on U [̺ 0 , m 0 , E 0 ], we obtain Hence, using (14) in the fifth line and the definition of ∪ T in the sixth line Using the shift invariance for U we obtain Hence, using (15) in the fourth line Using the continuation property for U , we have that and since [̺ 1 , m 1 , E 1 ] is a minimum of I λ,F , we must have

Selection sequence
In this section we will prove the existence of the semiflow selection for the compressible Navier-Stokes system. We will need the following topological result, which is a variation of the Cantor's intersection theorem.
Theorem 4.2. 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∈N is a sequence of non-empty compact subsets of S satisfying Proof. By contradiction, assume k∈N C k = ∅. For each n, let U n = C 0 \ C n ; since and n∈N C n = ∅, we obtain n∈N U n = C 0 . Since C 0 ⊂ S is compact and {U n } n∈N is an open cover (on C 0 ) of C 0 , we can extract a finite cover {U n 1 , . . . , U nm }. 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∈N . Then Then C k = C 0 \ U k = ∅, a contradiction since every set of the sequence {C n } n∈N is non-empty by hypothesis.
We are now ready to prove Theorem 2.2.
Proof of Theorem 2.2. First of all, we will select only those solutions that are admissible, meaning minimal with respect to the relation ≺ introduced in Definition 1.5. To this end, it is sufficient to consider the functional I λ,α with α(̺, m, E) = α(E), α : R → R smooth, bounded, and strictly increasing.
Using the monotonicity of the integral, we obtain on the other side, condition (17) tells us that The only possibility is to have the equality in both the integral relations above and thus α(E(t)) = α(Ẽ(t)) for a.e. t ∈ (0, ∞); since α is strictly increasing, this implies E =Ẽ a.e. in (0, ∞). Next, we choose a countable basis {e n } n∈N in L 2 (Ω; R N ), and a countable set {λ k } k∈N which is dense in (0, ∞). We consider a countable family of functionals, where again α satisfies condition (16); the functionals are well defined since m(t, ·) ∈ W −ℓ,2 (Ω; R N ) for all t. Let {(k(j), n(j))} ∞ j=1 be an enumeration of all the involved combinations of indices, that is, an enumeration of the countable set We define U j = I k(j),n(j) • · · · • I k(1),n(1) • I 1,α • U , j = 1, 2, . . . , The set-valued mapping enjoys the properties (P1)-(P5). Indeed: (P1)-(P2) first, notice that for every fixed initial data [̺ 0 , m 0 , E 0 ] ∈ D the sets U j [̺ 0 , m 0 , E 0 ] are nested: By Proposition 4.1 we can deduce that I 1,α • U [̺ 0 , m 0 , E 0 ] is compact, and iterating this procedure we obtain that all U j [̺ 0 , 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  for every j. By Proposition 4.1, we can deduce that I 1,α • U satisfies the continuation property, and iterating this procedure we obtain that this holds for every U j . This implies for all j and all T > 0. Thus We claim that for every [̺ 0 , m 0 , E 0 ] ∈ D the set U ∞ is a singleton, meaning To verify this, we observe that for all j = 1, 2, . . . ; from the choice of {k(j), n(j)} j∈N , we can see the integrals I k(j),n(j) as Laplace transforms of the functions 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 for all n ∈ N and for a.e. t ∈ (0, ∞). As α is strictly increasing we must in particular have E 1 (t) = E 2 (t), m 1 (t, ·); e n L 2 (Ω;R N ) = m 2 (t, ·); e n L 2 (Ω;R N ) , for all n ∈ N and for a.e. t ∈ (0, ∞). Since {e n } n∈N form a basis in L 2 (Ω; R N ) we conclude m 1 = m 2 , and E 1 = E 2 a.e. on (0, ∞).
It remains to prove that U is a semiflow selection: measurability follows from (P3) while the semigroup property follows from (P4): for t 1 , t 2 ≥ 0 it holds This completes the proof.

Restriction to semigroup acting only on the initial data
As a matter of fact, the semiflow selection U = U {̺ 0 , m 0 , E 0 } is determined in terms of the three state variables: the density ̺ 0 , the momentum m 0 , and the energy E 0 . Introduction of the energy might be superfluous; indeed, as pointed out in (10) E(τ ) =ˆΩ 1 2 |m| 2 ̺ + P (̺) (τ, ·)dx for a.e. τ ≥ 0.
The point is that the equality holds with the exception of a zero measure set of times. More specifically, the energy E(τ ) is a non-increasing function with well-defined right and left limits E(τ ±), whileˆΩ 1 2 is defined at any τ in terms of weakly continuous functions t → ̺(t, ·), t → m(t, ·). Due to the convexity of the superposition where equality holds with the exception of a set of time of measure zero. We may introduce a new selection defined only in terms of the initial data ̺ 0 , 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.
Defining V : D → Q such that for all t ∈ (0, ∞), then V will satisfy the semigroup property only almost everywhere; more precisely, calling T ⊂ (0, ∞) the set of times defined as holds for all t 1 , t 2 ∈ T .