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, ∞) × , where ⊂ R N , N = 2, 3 is a bounded domain with ∂ 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 well posed, see Kazhikhov [8].
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 be a general non-monotone function of the density, besides the standard case p( ) = a γ with a > 0 and γ > N 2 . Still, as we shall see below, the problem admits global-intime 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 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 [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 [ , 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 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 0 , the momentum 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 0 and the momentum m 0 .

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 , 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: with ≥ 0; (ii) weak formulation of the renormalized continuity equation: for any τ > 0 and any functions holds for any ϕ ∈ C 1 c ([0, ∞) × ); (iii) weak formulation of the balance of momentum: for any τ > 0 the integral identity holds for any ϕ ∈ C 1 c ([0, ∞) × ; R N ), where ( u)(0, ·) = ( u) 0 ; (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 [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 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 [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 [ , m, E]. Finally, in Sect. 5, we pass to the natural state variables [ , m]. To this end, let [ i , m i , E i ], i = 1, 2, be two dissipative solutions starting from the same initial data [ 0 , m 0 , E 0 ]. We introduce the relation In particular, such selection criterion guarantees that equilibrium states belong to the class of dissipative weak solutions (see [3], Section 6.3).

Setup
We start by introducing suitable topologies on the space of initial data and the space of dissipative weak solutions. Fix > N 2 + 1 and for simplicity consider the Hilbert space together with its subset containing the initial data Here, the convex function [ , m] → |m| 2 is defined for ≥ 0, m ∈ R N as applying Hölder inequality, we can also deduce that 0 ∈ L γ ( ) and m 0 ∈ L 2γ γ +1 ( ; R N ); accordingly, the set of the data can be seen as a closed 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. We consider the trajectory space which is a separable space. Dissipative weak solutions [ , m, E], as defined in Definition 1.1, belong to this class-since > N 2 , we have that L p with p ≥ 1 is compactly embedded in W − ,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. Moreover, for any initial data [ 0 , m 0 , E 0 ] ∈ D, it follows that a dissipative weak solution [ , m, E] evaluated at every t ≥ 0 also belongs to the set D, due to the energy inequality. Finally, for initial data [ 0 , m 0 , E 0 ] ∈ D, we introduce the solution set

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

Existence and sequential stability
First of all, we aim to show: 2. weak sequential stability of the solution set, meaning 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.
Moreover, we assume that the initial densities converge strongly Then, at least for suitable subsequences,

Shift and continuation operations
In order to construct the semiflow, two main ingredients are needed: the shift invariance property and the continuation. For ω ∈ Q, we define the positive shift operator Since the energy is non-increasing, we can choose every E ≥ E(T +) as initial energy; indeed, everything will be well defined For ω 1 , ω 2 ∈ Q we define the continuation operator ω 1 ∪ T ω 2 by for some E ≤ E 1 (T −). Then, 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 [ 2 , m 2 , E 2 ] less or equal E 1 (T −), the energy of the solution [ 1 , m 1 , E 1 ] ∪ T [ 2 , m 2 , E 2 ] remains non-increasing on (0, ∞).

General ansatz
Summarizing the results achieved in this section, we have shown the existence of a set-valued mapping enjoying the following properties.
We remark that the value E(T −) in (A3) and (A4) can be replaced by where η ∈ [0, 1] is given, since in this case E(T +) ≤ E ≤ E(T −).

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 Given I λ,F and a set-valued mapping U we define a selection mapping I λ,F • U, by In other words, the selection is choosing minima of the functional I λ,F . Note 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. 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 d H −→ K with K n , K ∈ K then 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 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. (A3) We want to prove the shift invariance: for 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) 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 Vol. 21 (2021) Semiflow selection for the compressible Navier-Stokes system 289

Selection sequence
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 we consider the functional I 1,β with β( , m, E) = β(E), β : R → R smooth, bounded, and strictly increasing.
Proof. We proceed by contradiction.
Using the monotonicity of the integral, we obtain The only possibility is to have the equality in both the integral relations above and thus β(E(t)) = β(Ẽ(t)) for a.e. t ∈ (0, ∞); since β is strictly increasing, this implies E =Ẽ a.e. in (0, ∞).
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∈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 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∈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. Selecting I 1,β • U from U we know from Lemma 4.2 that the new selection contains only admissible solutions for any [ 0 , m 0 , E 0 ] ∈ D.
Next, we choose a countable basis {e n } n∈N in L 2 ( ; R N ), and a countable set {λ k } k∈N which is dense in (0, ∞). We consider a countable family of functionals, We define The set-valued mapping enjoys the properties (A1)-(A4). Indeed: (A1) 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,β • 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.
for every j. By Proposition 4.1, we can deduce that I 1,β • 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 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 β is strictly increasing, we must in particular have E 1 (t) = E 2 (t), m 1 (t, ·); e n L 2 ( ;R N ) = m 2 (t, ·); e n L 2 ( ;R N ) , for all n ∈ N and for a.e. t ∈ (0, ∞). Since {e n } n∈N form a basis in L 2 ( ; 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 (A2), while the semigroup property follows from (A3): for t 1 , t 2 ≥ 0 it holds This completes the proof.
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 : 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).

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 ) = 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 ±), while 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 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.