Semiflow selection to models of general compressible viscous fluids

We prove the existence of a semiflow selection with range the space of c\`agl\`ad, i.e. left--continuous and having right--hand limits functions defined on $[0,\infty)$ and taking values in a Hilbert space. Afterwards, we apply this abstract result to the system arising from a compressible viscous fluid with a barotropic pressure of the type $a\varrho^{\gamma}$, $\gamma \geq 1$, with a viscous stress tensor being a nonlinear function of the symmetric velocity gradient.


Introduction
First developed by Krylov [16] and later adapted by Flandoli and Romito [13], Breit, Feireisl and Hofmanová [7] in the context of the Navier-Stokes system, the semiflow selection is an important stochastic tool when studying systems that lack uniqueness: it allows to identify a solution satisfying at least the semigroup property.
Inspired by the deterministic adaptation of Cardona and Kapitanski [9], in the first part of this work we will prove the existence of a semiflow selection in an abstract setting. More precisely, denoting with H a Hilbert space and with T = D([0, ∞); H) the Skorokhod space of càglàd functions defined on [0, ∞) and taking values in H, we will show the existence of a Borel measurable map u : D ⊆ H → T such that for any x ∈ D and any t 1 , t 2 ≥ 0 u(x)(t 1 + t 2 ) = u [u(x)(t 1 )] (t 2 ).
One could think that a more natural choice for T would be the space C([0, ∞); H) of continuous functions, as in [9]. However, this option can be too strong: if we want to apply this abstract setting to the typical systems arising from fluid dynamics, where [0, ∞) is the set of times and T represents the trajectory space, then it is difficult to ensure the energy of the system to be continuous, since it is at most a non-increasing quantity with possible jumps. For the aforementioned reason, in the context of the compressible Euler system, Breit, Feireisl and Hofmanová [6], [8] considered the energy in the L 1 -space. But the choice T = L 1 ([0, ∞); H) is still not optimal as it is better to work with a space whose elements are well-defined at any point.
In the second part of this work, we will apply the abstract machinery previously achieved to a general model of compressible viscous fluids, described by the following pair of equations: ∂ t ̺ + div x (̺u) = 0, ∂ t (̺u) + div x (̺u ⊗ u) + ∇ x p = div x S, (1.1) where the unknown variables are the density ̺ and the velocity u. In this context, the viscous stress tensor S is connected to the symmetric velocity gradient Du through the relation where F is a proper convex l.s.c. (lower semi-continuous) function and F * denotes its conjugate; moreover, the barotropic pressure will be of the type p(̺) = a̺ γ , γ ≥ 1. We will deal with the concept of dissipative solutions, i.e. solutions that satisfy our problem in the weak sense but with an extra defect term in the balance of momentum, arising from possible concentrations and/or oscillations in the convective and pressure terms; for more details see Definition 4.1. It is interesting to note that for γ = 1, the defect in the momentum equation vanishes and the latter is satisfied in the sense of distributions. Thus our approach represents an alternative to the "standard" measure-valued framework applied in this context by Matušů-Nečasová and Novotný [18]. Introducing the set-valued mapping U : D ⊆ H → P(T ) that associated to every initial data x ∈ D the family of dissipative solutions U (x) arising from x ∈ D, the key point in order to get the existence of semiflow selection will be to show that U satisfies five properties: non-emptiness, compactness, Borel-measurability, shift invariance and continuation.
As we will see, in order to verify the validity of the above mentioned properties, there are two main difficulties we have to overcome: the existence of dissipative solutions and the weak sequential stability of the family of dissipative solutions, arising from a fixed initial data. Luckily, the first problem was recently solved by Abbatiello, Feireisl and Novotný in [2] for γ > 1; following the same strategy, the case γ = 1 can be done as well replacing Lemma 8.1 in [2] with Lemma 5.2 below. Conversely, larger part of this work will be dedicated to solve the second issue.
The paper is organized as follows.
• In Section 2 we will define the Skorokhod space of càglàd functions, constructing a proper metric on it and giving a characterization of convergence.
• In Section 3 we will prove the existence of a semiflow selection in the abstract setting, cf. Theorem 3.2.
• In Section 4 we will focus on system (1.1), first clarifying the concept of dissipative solution (Section 4.1), fixing a proper setting (Section 4.2) and finally proving in Section 4.3 our main result, i.e., the existence of a semiflow selection, cf. Theorem 4.3.
• Section 5 will be entirely dedicated to prove the weak sequential stability of the solution set of (1.1) for a fixed initial data. (i) for t > 0, Φ(t−) = lim s↑t Φ(s) exists and Φ(t−) = Φ(t);
Unlike in the case of continuous functions, the topology on the space of càglàd functions on an unbounded interval cannot be built up by simply considering functions being càglàd on any compact, see e.g. Jakubowski [15]. Instead, we proceed as follows.

Semiflow selection
In this section we will focus on proving the existence of a semiflow selection. We start by fixing our setting; from now on, P(X) will denote the family of all subsets of a space X. Let • H be a separable Hilbert space with a basis {e k } k∈N ; • D a closed convex subset of H; • U : D → P(T ) satisfy the following properties.
(P1) Non-emptiness: for every x ∈ D, U (x) is a non-empty subset of T . (P5) Continuation: introducing the continuation operator Φ 1 ∪ T Φ 2 for every T > 0 and for all t ≥ 0, then, for any T > 0, x ∈ D, Φ 1 ∈ U (x) and Φ 2 ∈ U (Φ 1 (T )), we have Remark 3.1. It is worth noticing that, in order to guarantee the validity of the shift invariance and continuation properties, it is necessary to verify that for any x ∈ D and any Φ ∈ U (x), Φ evaluated at any t ≥ 0 also belongs to set D.
We are now ready to state and prove the following result.
satisfying the semigroup property: for any x ∈ D and any t 1 , t 2 ≥ 0 Proof. The idea of the proof is to reduce iteratively the set U (x) for a fixed x ∈ D, selecting the minimum points of particular functionals in order to obtain finally a single point in T , which will define u(x). The procedure has been proposed by Cardona and Kapitanski [9] in the context of continuous trajectories and later adapted to more general setting in [8] and [3].
We introduce the functionals I λ,k : T → R defined for every Φ ∈ T as where λ > 0, {e k } k∈N is a basis in H and f : R → R is a fixed smooth, bounded and strictly increasing function; this choice is justified by the fact that for a fixed k ∈ N we can see I λ,k as the Laplace transform of the function f ( Φ(·); e k ), an useful interpretation for the proof of the existence of the semiflow u. Let us first show the continuity of I λ,k for every λ > 0 and k ∈ N fixed. Let as n → ∞. Due to continuity and boundedness of f , we get i.e., what we wanted to prove. We define the selection mapping for every x ∈ D as Notice, in particular, that the minimum exists since I λ,k is continuous on T and the set U (x) is compact. From the fact that U satisfies properties (P1)-(P5), it is not difficult to show that the set-valued mapping I λ,k • U : D → P(T ) satisfies properties (P1)-(P5) as well, cf. Proposition 5.1 in [8].
We are now ready to prove the existence of the semiflow selection u. Fixing a countable set {λ j } j∈N dense in (0, ∞), we can define the functionals Choosing an enumeration {j(i), k(i)} ∞ i=1 of the all involved combinations of indeces, we define the maps and It is easy to show that the set-valued map satisfies properties (P1)-(P5) as well; for details, see [8], Theorem 2.5 and [3], Theorem 2.2.
We now claim that for every for all i = 1, 2, . . . . Since the integrals I j(i),k(i) can be seen as Laplace transforms of the functions f ( Φ(·); e k ), we can apply Lerch's theorem to deduce that for all k ∈ N and a.e. t ∈ (0, ∞). Since the function f is strictly increasing, we obtain that for all k ∈ N and a.e. t ∈ (0, ∞); in particular, from (2.2) we get that d ∞ (ω 1 , ω 2 ) = 0 and thus Finally, we define the semiflow selection u for all x ∈ D as mesurability follows from the property (P3) for U ∞ , while the semigroup property follows from property (P4): for any x ∈ D and any t 1 , t 2 ≥ 0

Semiflow selection for compressible viscous fluids
Let us consider a general mathematical model of compressible viscous fluids, represented by the following system here ̺ = ̺(t, x) denotes the density, u = u(t, x) the velocity, p = p(̺) the barotropic pressure and S the viscous stress tensor, which we suppose to be connected to the symmetric velocity gradient where, denoting with R d×d sym the space of d-dimensional real symmetric tensors, and F * is its conjugate, defined for every A ∈ R d×d sym as Notice that conditions (4.4) guarantees that Furthermore, we will suppose F to satisfy relation for some µ > 0 and q > 1. Notice that condition (4.3) is equivalent in requiring where ∂ denotes the subdifferential of a convex function. Regarding pressure, we will consider the standard isentropic case with a a positive constant; however, more general EOS preserving the essential features of (4.7) can be considered, cf. [2]. The pressure potential P , satisfying the ODE will be of the form in particular, this implies that P is a strictly convex superlinear continuous function on [0, ∞). (4.9) We will study the system on the set (0, ∞)×Ω, where the physical domain Ω ⊂ R d is assumed to be bounded and Lipschitz, on the boundary of which we impose the no-slip condition u| ∂Ω = 0. (4.10) Finally, we fix the initial conditions Our goal is to apply the abstract machinery introduced in the previous section in order to show the existence of a semiflow selection for system (4.1)-(4.11). More precisely, we aim to prove Theorem 4.3 below, clarifying first the concept of solution we will work with and fixing a proper setting.

Dissipative solution
As already mentioned in the introduction, inspired by the recent work of Abbatiello, Feireisl and Novotný [2], we will refer to the concept of dissipative solutions. From now on, it is better to consider the density ̺ and the momentum m = ̺u as state variables, since they are at least weakly continuous in time.
if the following holds: holds for any τ > 0 and any for a.e. τ > 0, such that the energy inequality holds for any 0 ≤ τ 1 ≤ τ 2 and any ψ ∈ C 1 which can also be identified as the dual space of L 1 (0, ∞; C(Ω)).

Short remark on the Reynolds stress
The concentration measure R, that we may call Reynolds stress, appearing in the weak formulation of the balance of momentum (4.13) arises from possible oscillations and/or concentrations in the convective and pressure terms when γ > 1, while for γ = 1, i.e. when the pressure is a linear function of the density ̺, it is only the convective term that contributes to R. By consistency, as clearly explained in [2], we should have introduced the dissipation defect E ∈ L ∞ weak (0, ∞; M + (Ω)) of the total energy arising from possible concentrations and/or oscillations in the kinetic and potential energy terms Instead of (4.14) we would then have satisfying the energy inequality (4.15). Choosing a positive constant λ > 0 such that In this sense, our choice of the energy (4.14) makes the problem more general and easier to handle with only one free quantity instead of two; however, it reduces to (4.16) simply choosing a dissipation defect E of the type E(τ ) := 1 λ Tr[R(τ )] for a.e. τ > 0.

Set-up
First of all, we must fix the space H, the subset D ⊆ H and the map U introduced at the beginning of Section 3. In this context where the natural number k > d 2 + 1 is fixed; • D represents the space of initial data; it can be chosen as Notice that everything is well-defined; indeed, denoting with L 1 + (Ω) the space of nonnegative integrable functions on Ω, we can rewrite D as so that it coincides with the epigraph of the function g : From (4.9) and the fact that we get that the function g is lower semi-continuous and convex and thus its epigraph is a closed convex subset of L γ (Ω) × L 2γ γ+1 (Ω; R d ) × R for all γ ≥ 1. From our choice of k, we can use the Sobolev embedding L r (Ω) ֒→֒→ W −k,2 (Ω) for every r ≥ 1 (4.19) to conclude that for every r ≥ 1. Furthermore, due to the weak continuity of the density ̺ and the momentum m, for every fixed T > 0 and every t ∈ [0, T ], from the energy inequality we can deduce that Finally, from condition (i) of Definition 4.1 we also have that ̺(t, ·) ≥ 0 for all t ≥ 0, while relationˆΩ holds for all t ≥ 0 since the energy is convex and ̺ and m are weakly continuous in time. In particular, we have that for every t ≥ 0 [̺(t, ·), m(t, ·), E(t)] ∈ D.

Main result
Keeping in mind the notation introduced in the previous section, we are now ready to state our main result.
satisfying the semigroup property: for any [̺ 0 , m 0 , E 0 ] ∈ D and any t 1 , t 2 ≥ 0 • Properties (P2) and (P3) hold true if we manage to prove the weak sequential stability of the solution set U [̺ 0 , m 0 , E 0 ] for every [̺ 0 , m 0 , E 0 ] ∈ D fixed, since it will in particular imply compactness and the closed-graph property of the mapping and thus the Borel-measurality of U , cf. Lemma 12.1.8 in [19].
In conclusion, we are done if we show the weak sequential stability of the solution set U [̺ 0 , m 0 , E 0 ] for every [̺ 0 , m 0 , E 0 ] ∈ D fixed. Being the proof quite elaborated, it is postponed to the next section.
Remark 4.4. As already done for the Euler and Navier-Stokes systems, cf. [8], [3], among all the dissipative solutions emanating from the same initial data it is possible to select only the admissible ones, i.e., satisfying the physical principal of minimizing the total energy or equivalently, that are minimal with respect to relation ≺ defined as where [̺ i , m i , E i ], i = 1, 2 are two dissipative solutions sharing the same initial data. Indeed, it is sufficient to start the selection considering in (3.1) the functional I 1,k with the function f such that f ( Φ(t); e k ) = f (E(t)) for all t ≥ 0, where Φ(t) = [̺(t), m(t), E(t)]; see [8], Lemma 5.2 for more details.

Weak sequential stability
This section will be entirely dedicated to the proof of the following result.
then, at least for suitable subsequences, where the natural number k > d 2 + 1 is fixed and [̺, m] is another dissipative solution of the same problem with total energy E.
The proof will be divided in four steps: 1. in Section 5.1 we will first deduce a family of uniform bounds and convergences, including the limits ̺ of the densities, m of the momenta and u of the velocities; 2. in Section 5.2 we will pass to the limit in the weak formulation of the continuity equation and the balance of momentum; 3. in order to show that m can be written as the product ̺u, in Section 5.3 we will state and prove Lemma 5.2; 4. finally, in Section 5.4 we will focus on finding the limit E of the energies.

Uniform bounds and limits establishment
Our first goal is to show the following convergences, passing to suitable subsequences as the case may be: From our hypothesis, all the initial energies are uniformly bounded by a positive constant E independent of n; specifically,ˆΩ From (4.14) and the energy inequality (4.15) it is easy to deduce the following uniform bounds ess sup t>0 m n √ ̺ n (t, ·)

Convergences of ̺ n and m n
For γ > 1, from (4.8) and (5.10) we can easily deduce, passing to a suitable subsequence as the case may be, Similarly, from (5.14), (5.9) and the fact that for a.e. t > 0 passing to a suitable subsequence, we obtain Since the L 1 -space is not reflexive, for γ = 1 a more detailed analysis is needed. If we consider the Young function Φ(z) = z log + z, the densities {̺ n } n∈N can be seen as uniformly bounded in L ∞ (0, ∞; L Φ (Ω)), where L Φ (Ω) is the Orlicz space associated to Φ; indeed, noticing that from (5.10), for a.e. τ > 0 we havê As the function Φ satisfies the ∆ 2 -condition, L Φ (Ω) can be seen as the dual space of the Orlicz space L Ψ (Ω), where Ψ denotes the complementary Young function of Φ, and hence, passing to a suitable subsequence, we get We are now able to prove the uniform integrability of the sequence {m n (t, ·)} n∈N ⊂ L 1 (Ω; R d ) for a.e. t > 0. More precisely, we want to show that for every ε > 0 there exists δ = δ(ε) such that for all n ∈ N and a.e. t > 0 M |m n |(t, ·)dx < ε for all M ⊂ Ω such that |M | < δ.
Next, to get (5.2) from (5.14) and ( while from the uniform boundedness of the momenta m n in L ∞ (0, ∞; L p (Ω; R d )) with p = 2γ γ+1 and γ ≥ 1, sup As a consequence of the Arzelà-Ascoli theorem, we get (5.2). A similar argument can be applied to get (5.3). Finally, notice that, from the Sobolev embedding (4.19), ̺ n : [0, ∞) → W −k,2 (Ω) is a continuous function for all n ∈ N, and thus by Proposition 2.1 showing where {e k } k∈N is an orthonormal basis of L 2 (Ω); but this easily follows from convergence (5.2) and Parseval's identity. The same argument can be applied to show that m n → m in D([0, ∞); W −k,2 (Ω; R d )).

Convergences of u n and S n
From (4.6) and (5.12) we can also deduce that Fixing a compact interval [a, b] ⊂ (0, +∞) and an open bounded interval I such that [a, b] ⊂ I, the previous inequality combined with the L q -version of the trace-free Korn's inequality (see for instance [5], Theorem 3.1) implies the standard Poincaré inequality ensures then u n L q (I;W 1,q 0 (Ω;R d )) ≤ c(E), and thus we get convergence (5.4).

Convergences of p(̺ n ), mn⊗mn
̺n and R n .

Limit passage
We are now ready to pass to the limit in the weak formulation of the continuity equation and the balance of momentum, obtaining that holds τ > 0 and any ϕ ∈ C 1 c ([0, ∞) × Ω), with ̺(0, ·) = ̺ 0 , and holds for any τ > 0 and any ϕ ∈ C 1 c ([0, ∞) × Ω; R d ), ϕ| ∂Ω = 0 with m(0, ·) = m 0 . The last integral identity can be rewritten as with We can prove the stronger conditioň R ∈ L ∞ weak (0, ∞; M + (Ω; R d×d sym )); (5.19) more precisely, we want to show that for all ξ ∈ R d , all open sets B ⊂ Ω and a.e. τ > 0 we can rewrite the term on the left-hand side aŝ Since the indicator function ½ [τ −d,τ +d]×B can be approximated by some non-negative test functions, it is enough to show thatˆ∞ 0ˆΩ ϕ (ξ ⊗ ξ) : dŘ(t)dt ≥ 0 holds for all ϕ ∈ C ∞ c ((0, T )×Ω), ϕ ≥ 0. We can notice that the first term on the right-hand side of (5.18) will obviously satisfy the above inequality since R belongs to L ∞ weak (0, ∞; M + (Ω; R d×d sym )), andˆ∞ since ̺ → p(̺) is convex and weakly lower semi-continuous in L 1 , which implies p(̺) ≥ p(̺), see for instance [11], Theorem 2.11. Finally, following the same idea developed in [12], Section 3.2, as a consequence of (5.6) we can writê (5.20) The Cauchy-Schwarz inequality allows to write |m · ξ| 2 ≤ |m| 2 |ξ| 2 , and thus by (5.9) we obtain ess sup t>0 ½ ̺n>0 |m n · ξ| 2 ̺ n (t, ·) it is possible then to find the limit ½ ̺n>0 and rewrite the first line in (5.20) aŝ as in the previous passage, (5.19) will now follow from the weak lower semi-continuity on D of the convex function [̺, m] → |m·ξ| 2 ̺ . We proved in particular that the pair of functions [̺, m] satisfies conditions (ii) and (iii) of Definition 4.1. However,Ř has to be slightly modified in order to get the energy (4.14), as we will see in Section 5.4.

Auxiliary lemma
In order to prove that m = ̺u a.e. in (0, ∞) × Ω, and in particular to show that [̺, m] satisfy condition (iv) of Definition 4.1, we need the following result. Proof.

2.
Regularization. We claim that it is sufficient to suppose {u n } n∈N to be uniformly bounded in L q loc (0, ∞; W m,r (Ω; R d )) (5.27) with q > 1 and m, r arbitrarily large. Seeing all the quantities involved as embedded in R d with compact support, we consider regularization in the spatial variable by convolution with a family of regularizing kernels {θ δ } δ>0 , where θ is a bell-shaped function such that As in the previous step, writing ̺ n u n = ̺ n θ δ * u n + ̺ n (u n − θ δ * u n ), our goal is tho show that for every [a, b] ⊂ (0, ∞) b aˆΩ ̺ n |u n − θ δ * u n |dxdt → 0 as δ → 0, uniformly in n.
It remains to show smallness of the first integral for fixed M . To this end, denoting with Ψ the complementary Young function of Φ, we consider the Orlicz space L Ψ (Ω) that can be identified with the dual of L Φ (Ω) as Φ satisfies the ∆ 2 -condition. By Proposition 5.3 below, we recover the compact embedding which, combined with boundedness of convolution on L Ψ (Ω) (see [14], Lemma 4.4.3), gives 3. Conclusion. Using the Sobolev embedding L 1 (Ω) ֒→֒→ W −1,s ′ for any s > d, from (5.24) we get that and thus, to conclude the proof of the Lemma it is sufficient to choose m = 1 and r = s in (5.27).

Proposition 5.3.
Let Ω ⊂ R d be a bounded domain. Then, for a fixed q ≥ 1 where L Φ is the Orlicz space associated to the Young function Φ.
Proof. Let K be a bounded set of X and let Φ 1 be a Young function such that Φ ≺≺ Φ 1 , i.e. lim t→∞ Φ(t) Φ 1 (λt) = 0, for all λ > 0. Then, in particular, K is bounded in the Orlicz space L Φ 1 (Ω); indeed, denoting with Ψ the complementary Young function of Φ, we have that for every u ∈ K and every v belonging to the Orlicz class L Ψ (Ω) and thus where u X = max{ u W 1,q (Ω) , u L ∞ (Ω) } and the constant c is independent of the choice u ∈ K. Applying [17], Theorems 3.17.7 and 3.17.8 we get that where E Φ (Ω) is the closure of the set of all bounded measurable functions defined on Ω with respect to the Orlicz norm · L Φ , and that the functions in K have uniformly continuous L Φ -norms, i.e., for every ε > 0 there exists a δ = δ(ε) > 0 such that provided M ∈ Ω is measurable, |M | < δ and u ∈ K. Furthermore, since W 1,q (Ω) ֒→֒→ L 1 (Ω), the set K is relatively compact in L 1 (Ω) and consequently it is relatively compact with respect to the convergence in measure. Finally, it is sufficient to apply [17], Theorem 3.14.11, which we report for reader's convenience.
Theorem 5.4. Let K be a subset of E Φ (Ω) which is relatively compact in the sense of convergence in measure and such that the functions in K have uniformly continuous L Φ -norms. Then K is relatively compact in L Φ .

Limit of the energies
From (4.14) we can notice that the energies E n (τ ) are non-increasing and for γ > 1 they are also non-negative, while for γ = 1 we have so that {E n } n∈N is locally of bounded variation. We can then use Helly's selection theorem (compactness theorem for BV loc ): a sequence of functions that is locally of total bounded variation and uniformly bounded at a point has a convergent subsequence, pointwise and in L 1 loc . Passing to a suitable subsequence as the case may be, we obtain On the other side, from (5.9), (5.10) and (5.11) we get |m n | 2 ̺ n * ⇀ |m| 2 ̺ in L ∞ weak (0, ∞; M(Ω)) P (̺ n ) * ⇀ P (̺) in L ∞ weak (0, ∞; M(Ω)) 1 λ n Tr [R n ] * ⇀ E in L ∞ weak (0, ∞; M + (Ω)).