Existence of dissipative solutions to the compressible Navier-Stokes system with potential temperature transport

We introduce dissipative solutions to the compressible Navier-Stokes system with potential temperature transport motivated by the concept of Young measures. We prove their global-in-time existence by means of convergence analysis of a mixed finite element-finite volume method. If a classical solution to the compressible Navier-Stokes system with potential temperature transport exists, we prove the strong convergence of numerical solutions. Our results hold for the full range of adiabatic indices including the physically relevant cases in which the existence of global-in-time weak solutions is open.


Introduction
We consider a compressible viscous Newtonian fluid that is confined to a bounded domain Ω ⊂ R d , d ∈ {2, 3}. Its time evolution is governed by the following system:

2)
∂ t (̺θ) + div x (̺θu) = 0 in (0, T ) × Ω. (1.3) Here ̺ ≥ 0, u, p and θ ≥ 0 stand for the fluid density, velocity, pressure, and potential temperature, respectively. The viscous stress tensor S(∇ x u) is given by where µ and λ are viscosity constants satisfying µ > 0 and λ ≥ − 2 d µ . Denoting by γ > 1 the adiabatic index, the pressure state equation reads p(̺θ) = a(̺θ) γ , a > 0 . (1.5) This type of Navier-Stokes equations is often used in meteorological applications; see, e.g., [1] and the references therein. System (1.1)-(1.5) governs the motion of viscous compressible fluids with potential temperature, where diabatic processes and the influence of molecular transport on potential temperature are excluded. Only potential entropy stratification in the initial data is imposed. We refer a reader to Feireisl et al. [2], where the singular limit in the low Mach/Froude number regime of the above Navier-Stokes system with γ > 3/2 was analyzed. For γ > 9/5, Bresch et al. [3] showed that the low Mach number limit for the considered system is the compressible isentropic Navier-Stokes equation. In [4] Lukáčová-Medvid'ová et al. use a slightly more complex version of the above system as the basis for their cloud model; see also Chertock et al. [5], where the uncertainty quantification was investigated. Due to the link between potential temperature and entropy, system (1.1)-(1.5) is often reported in the literature as the Navier-Stokes system with entropy transport. To avoid any misunderstanding, we call it in the present paper the Navier-Stokes system with potential temperature transport.
In literature we can find several existence results for the Navier-Stokes system (1.1)-(1.5). The question of stability of weak solutions for γ > 3/2, d = 3 was analyzed by Michálek [6]; see also [7], where the stability of weak solutions for the compressible Navier-Stokes equations with a scalar transport was studied for γ > 9/5 by Lions. Under the assumption γ ≥ 9/5 in the case d = 3 and γ > 1 in the case d = 2, system (1.1)-(1.5) is known to admit global-in-time weak solutions; see Maltese et al. [8, Theorem 1 with T (s) = s γ ]. Note that in the aforementioned paper the authors work with the entropy s instead of the potential temperature θ. However, in their framework the specified choice of the function T yields s = θ. We point out that the physically relevant adiabatic indices γ lie in the interval (1,2] if d = 2 and in the interval ( A simpler model for viscous compressible fluid flow is the barotropic Navier-Stokes system with the state equation p = a̺ γ , a = const. The first global-in-time existence result for weak solutions of this system allowing general initial data was established in 1998 by Lions [7] for γ ≥ 3/2 if d = 2 and γ ≥ 9/5 if d = 3. In 2001, Feireisl, Novotný, and Petzeltová [9] extended Lions's result to the situation γ > 1 for d = 2 and γ > 3/2 for d = 3; see also Feireisl, Karper, Pokorný [10]. To date, the latter is the best available global-in-time existence result for weak solutions for the barotropic Navier-Stokes system. The main obstacle that hampers the derivation of the existence result for γ ≤ 3/2 in three space dimensions is the lack of suitable a priori estimates for the convective term ̺u ⊗ u. These difficulties are inherited by the full Navier-Stokes-Fourier system that includes an energy equation, too. In [11], Feireisl and Novotný obtained the existence of global-in-time weak solutions for the Navier-Stokes-Fourier system. However, their result holds only for a very restrictive class of state equations. In particular, the natural example of the perfect gas law p = ̺θ is still open for the existence of weak solutions. In this context, we refer a reader to [12], where the complete Navier-Stokes-Fourier system for the perfect gas was studied in the context of generalized solutions.
The question of uniqueness of weak solutions remains open in general. However, we have a weak-strong uniqueness principle for the barotropic Navier-Stokes equations. It means that weak and strong solutions to the Navier-Stokes system emanating from the same initial data coincide; see, e.g., Feireisl, Jin, Novotný [13] or Feireisl [14].
In [15], Feireisl et al. introduced a new concept of generalized solutions to the barotropic Navier-Stokes system. They work with the so-called dissipative measure-valued (DMV) solutions that are motivated by the concept of Young measures. In this context, a DMV-strong uniqueness principle was established and the existence of global-in-time DMV solutions for a class of pressure state equations including the barotropic case with γ ≥ 1 was achieved. In our recent work [16], we have extended the DMV-strong uniqueness result to the Navier-Stokes system with potential temperature transport (1.1)- (1.5).
In [17,Chapter 13], Feireisl et al. give a constructive existence proof and demonstrate that DMV solutions to the barotropic Navier-Stokes system can also be obtained by means of a convergent numerical method that was originally developed by Karlsen and Karper [18], [19], [20], [21]. However, their result is based on the assumption that γ > 6/5 if d = 3 and γ > 8/7 if d = 2; for the three-dimensional case see also Feireisl and Lukáčová -Medvid'ová [22].
The goal of this paper is to introduce a concept of DMV solutions to the Navier-Stokes system with potential temperature transport and prove the global-in-time existence of such generalized solutions for all γ > 1 by analyzing the convergence of a suitable numerical scheme. To this end, we propose a new version of the mixed finite element-finite volume method of Karlsen and Karper [18]; see also [10], [17,Chapter 13], [22].
The paper is organized as follows: In Section 2, we introduce our notion of DMV solutions to the Navier-Stokes system with potential temperature transport and present our main result. Section 3 is devoted to the numerical method and the collection of its basic properties. In Section 4, we state a discrete energy equality for our method which serves as a basis for several stability estimates. The consistency of the numerical method is established in Section 5 and in Section 6 we conclude that any Young measure generated by the solutions to our numerical method represents a DMV solution to the Navier-Stokes system with potential temperature transport. In particular, we show that the numerical solutions converge weakly to the expected values with respect to the Young measure. The convergence of numerical solutions is strong as long as a strong solution of (1.1)-(1.5) exists.

Dissipative measure-valued solutions
Before defining dissipative measure-valued solutions to the Navier-Stokes system with potential temperature transport, we fix the initial and boundary conditions. The Navier-Stokes system with potential temperature transport (1.1)-(1.5) is endowed with the initial data and the no-slip boundary condition We henceforth write Ω t = (0, t) × Ω whenever t > 0. Furthermore, P : [0, ∞) → R, ∈ Ω T is a parametrized probability measure (Young measure) acting on R d+2 , we write whenever g ∈ C(R d+2 ). Moreover, we tend to write out the function g in terms of the integration variables ( ̺, θ, u) ∈ R × R × R d ∼ = R d+2 : if, for example, g( ̺, θ, u) = ̺ u, then we also write We proceed by defining dissipative measure-valued solutions to the Navier-Stokes system with potential temperature transport (1.1)-(1.5).
and for which there exists a constant c ⋆ > 0 such that is called a dissipative measure-valued (DMV) solution to the Navier-Stokes system with potential temperature transport (1.1)-(1.5) with initial and boundary conditions (2.1) and (2.2) if it satisfies: [1] P(R d+2 ) denotes the space of probability measures on R d+2 .
We proceed by defining the relevant discrete function spaces. The space of piecewise constant functions is denoted by Q h = v ∈ L 2 (Ω) v| Ω\Ω h = 0 and v| K ∈ P 0 (K) for all K ∈ T h [4] .
The Crouzeix-Raviart finite element spaces are denoted by [4] P n (K) denotes the set of all restrictions of polynomial functions R d → R of degree at most n to the set K.
With these spaces we associate the projection Π V,h : Additionally, we agree on the notation

Mesh-related operators
Next, we define some mesh-related operators. We start by introducing the discrete counterparts of the differential operators ∇ x and div x . They are determined by the stipulations respectively. We continue by defining several trace operators. For arbitrary The convective terms will be approximated by means of a dissipative upwind operator. For where ε > 0 is a given constant, Remark 3.1. In the sequel, we tend to omit the letter σ in the subscripts and superscripts of the operators defined in Sections 3.2 and 3.3. In some places, we also suppress the letter h and the superscript in in the notation if no confusion arises.

Time discretization
In order to approximate the time derivatives, we apply the backward Euler method, i.e., the time derivative is represented by where ∆t > 0 is a given time step and s k−1 h and s k h are the numerical solutions at the time levels t k−1 = (k − 1)∆t and t k = k∆t, respectively. For the sake of simplicity, we assume that ∆t is constant and that there is a number N T ∈ N such that N T ∆t = T .

Numerical scheme
We are now ready to formulate our mixed finite element-finite volume (FE-FV) method. where Remark 3.3. We note that our FE-FV method is a generalization of the scheme presented in [17,Chapter 13]. New ingredients are a modified upwind operator and the artificial pressure terms The latter are added to ensure the consistency of our method for values of γ close to 1, see Sections 4, 5.

Initial data
The initial data for the FE-FV method (3.2)-(3.4) are given as As a consequence of this stipulation, we observe that

Properties of the numerical method
We proceed by summarizing several properties of the FE-FV method (3.2)-(3.4).
If, in addition, there are constants 0 < c < c such that Proof. For the proof we refer the reader to Appendix A.3.
From Lemma 3.4 we easily deduce the following corollary.
starting from the discrete initial data (3.5) has the following properties:

Stability
We continue by discussing the stability of the FE-FV method (3.2)-(3.4) that follows from a discrete energy balance. For its derivation, we rely on the concept of (discrete) renormalization.
The same technique will be used to establish a discrete entropy inequality.

Discrete renormalization
In the sequel, we shall state renormalized versions of (3.2) and (3.3) that describe the evolution Together with suitable choices for the function b, the first two renormalized equations will help us to handle the pressure terms when deriving the discrete energy balance. The last equation will be used to establish the discrete entropy inequality.
Proof. The proof of assertion (i) can be found in [19,Lemma 5.1]. The main idea is to take 3) and to rewrite the results by means of basic algebraic manipulations, Gauss's theorem, and Taylor expansions.

Discrete energy balance
We now have all necessary tools at hand to establish the energy balance for our numerical method.
starting from the discrete initial data (3.5) and P the pressure potential introduced in (2.3). Denoting the discrete energy at the time level k ∈ N 0 by we deduce that Proof. The proof can be done following the arguments in [10, Chapter 7.5]. Therefore, we depict only the most important steps. First, taking Next, we observe that Moreover, by applying Lemma 4. (4.10) Plugging (4.6)-(4.10) into (4.5), we see that we have almost arrived at (4.4). Indeed, it only remains to show that which follows by direct calculations. This completes the proof.

Time-dependent numerical solutions and energy estimates
Next, we formulate appropriate stability estimates for the time-dependent numerical solutions introduced below.
that are piecewise constant in time by setting The most important stability estimates that can be obtained from the discrete energy balance (4.4) read as follows.

Corollary 4.4 (Stability estimates). Any solution
starting from the initial data (3.5) has the following properties: (4.14) Proof. The proof is provided in Appendix A.4.

Discrete entropy inequality
We conclude this section by stating a discrete entropy inequality. It is obtained by taking b = χ in Lemma 4.1(ii).

Consistency
The goal of this section is to establish the consistency of the FE-FV method (3.2)-(3.4).
The structure of the proof of Theorem 5.1 is essentially the same as that of [17,Theorem 13.2]. In particular, we will use similar tools. Apart from the estimates listed in Appendix A.1, we will need the following results.
Then the subsequent relations hold: Then [5] (5.8) Then (5.9) [5] In integrals of the form E(K) we consider the the vector n σ in the definition of the trace operators (·) in,σ and (·) out,σ to be replaced by n K .
Remark 5.5. The formula in Lemma 5.3 also holds true when the dissipative upwind term is replaced by the usual upwind term and the last term on the right-hand side of the identity is canceled. The same applies to Corollary 5.4. Then and which follows from the fact that r ∈ Q h . Corollary 5.4 can be proven by applying Lemma 5.3 with Having all necessary tools at our disposal, we can approach the proof of Theorem 5.1. • Recall that the elements of Q h and V h vanish outside Ω h . This allows us to replace Ω h by Ω when appropriate.

The continuity equation.
Next, let us consider the second term on the left-hand side of (5.12). Using Lemma 5.3 with These terms can be further estimated as follows.
The proof of (5.3) can be done by repeating the proof of (5.2) with ̺ h and ̺ 0 h replaced by ̺ h θ h and ̺ 0 h θ 0 h , respectively.

From (3.4) we deduce that
Let us consider the first term on the left-hand side of (5.15). Due to the second estimate in . . , d}, as well as Remark A.1, Hölder's inequality, the third estimate in (4.11), and the fact that ∆t ≈ h, we have Next, we turn to the last three terms on the left-hand side of (5.15). It follows from Lemma 5.6 that Finally, let us examine the second term on the left-hand side of (5.15). Applying Corollary 5.4 with (s, w, g, ψ) = (̺ h u h , u h , ϕ h , ϕ)(t, ·), t ∈ [0, T ], as well as the estimates (A.7)-(A.9) and (A.12), we deduce that We continue by estimating the above terms.

The entropy inequality.
Taking χ = ln in Lemma 4.5, we deduce that where Now we may rewrite the first two integrals in (5.20) following the procedure used to handle the continuity equation. We arrive at Moreover, combining c ⋆ ≤ θ h ≤ c ⋆ with Hölder's inequality, the first estimate in (A.1), the second estimate in (4.11) and the first estimate in (4.13), we deduce that Finally, seeing that (by a computation similar to that in (5.14)) we have we may rewrite (5.21) as where α 3 = min{α 1 , ε − 1}. In particular, we can choose β = min{α 1 , α 2 , α 3 } = min ε − 1, 1−2δ 4 .

Convergence
We proceed by proving our main result, namely Theorem 2.3. (3.5). Here we suppose that the parameters satisfy (5.1).

Proof of Theorem
Due to the second estimate in (4.11), the first estimate in (4.13), the third estimate in (4.12), (A.15), and Corollary 3.
Taking into account the remaining estimates in (4.11)-(4.14) as well as the first estimate in (A.2) and passing to a subsequence as the case may be, we obtain that as h ↓ 0. Following the arguments given in [ Moreover, using Hölder's inequality, (A.15), the first estimate in (A.2), the assumption on the initial data, and Lemma A.2, we easily verify that (Ω) and f ∈ C 1 (0, ∞), (6.9) and as h ↓ 0.
• Applying measure-theoretic arguments to the viscous terms, we conclude that • Using the density of C ∞ c (Ω) in W 1,2 0 (Ω) as well as Gauss's theorem, we easily verify that In particular, we may rewrite (6.12) in the form

Potential temperature equation.
The potential temperature equation can be handled in the same manner as the continuity equation.
In particular, Consequently, (6.16) can be rewritten as It is easy to see that (6.17) also holds for test functions ϕ of the class Accordingly, we may use the dominated convergence theorem to extend the validity of (6.17) to test functions ϕ ∈ C 1 (Ω T ) d satisfying ϕ| [0,T ]×∂Ω = 0.

Entropy inequality.
Due to (6.1), (6.2), and (6.9), we may take the limit h ↓ 0 in (5.5). We obtain for all ψ ∈ C ∞ c ([0, T ) × Ω), ψ ≥ 0. By an approximation argument similar to that in the case of the continuity equation, the validity of (6.21) can be extended to test functions ψ ≥ 0 of the class C c ([0, T ) × Ω) ∩ W 1,∞ (Ω T ). In particular, we may consider test functions of the form ψ = φ τ,δ η, Consequently, The entropy inequality (2.8) follows by performing the limit δ ↓ 0 in the above inequality. For the limit process, we rely on Lebesgue's differentiation theorem as well as the dominated convergence theorem. This completes the proof of Theorem 2.3.
From the proof of Theorem 2.3 it follows that any Young measure generated by a sequence 4) represents a DMV solution to the Navier-Stokes system with potential temperature transport (1.1)-(1.5). Moreover, If there is a strong solution to system (1.1)-(1.5) for given initial data (̺ 0 , θ 0 , u 0 ), then we may use the DMV-strong uniqueness result established in [16] to strengthen the aforementioned convergence statement as follows.
Proof. Let (̺ h , θ h , u h ) h ↓ 0 be a sequence as described above. To prove Theorem 6.1, it suffices to show that every subsequence (̺ h⋆ , θ h⋆ , From the proof of Theorem 2.3 and the DMV-strong uniqueness principle established in [16] we deduce that there is a subsequence

Conclusions
In the present paper, we introduced DMV solutions to the Navier-Stokes system with potential temperature transport (1.1)-(1.5) and proved their existence. For the existence proof we examined the convergence properties of solutions to a mixed FE-FV method that is a generalization of the method developed for the barotropic Navier-Stokes equations; see [22], [17,Chapter 13], [10,Chapter 7]. In particular, we showed that any Young measure generated by a sequence represents a DMV solution to the Navier-Stokes system with potential temperature transport (1.1)-(1.5).
In order to ensure the validity of our existence result for all physically relevant values of the adiabatic index γ -that is, γ ∈ (1, 2] if d = 2 and γ ∈ (1, 5/3] if d = 3 -we added two artificial pressure terms to our method. In the case of values of γ close to 1, these terms provided us with sufficiently good stability estimates for the limit process. In the limit process itself, we profited from the generality of DMV solutions that allowed us to hide the terms arising from the artificial pressure terms in the energy concentration defect and the Reynolds concentration defect, respectively. The strategy of adding artificial pressure terms points out a flexibility of the DMV concept. Indeed, it would not work in the framework of weak solutions. In spite of the generality of DMV solutions to system (1.1)-(1.5), we can show DMV-strong uniqueness, i.e., provided there is a strong solution, we can show that in a suitable sense any DMV solution on the same time interval coincides with it. We will present the detailed result in our upcoming paper [16]. Here, we made use of this result to prove the strong convergence of the solutions to our FE-FV method to the strong solution of the system. are valid for all p ∈ [1, ∞], all v ∈ V 0,h , all K ∈ T h , and all σ ∈ E h (K). Moreover, given φ ∈ C 1 (Ω), an application of Taylor's theorem yields Next, combining [ for all q ∈ [1, ∞], all φ ∈ W 1,q (Ω), and all ψ ∈ W 2,q (Ω). The latter estimates are also known as the Crouzeix-Raviart estimates.
Remark A.1. Clearly, the operators Π Q,h and Π V,h are linear. Furthermore, we may use (A.12) and the triangle inequality to deduce that there exists an h-independent constant C > 0 such that (A.14) Consequently, Π V,h is continuous.
Next, we prove the following auxiliary result that is needed in the proof of Theorem 2.3.
provided h is sufficiently small. Therefore, an application of Lemma A.2(ii), (iii) yields Since ε > 0 was chosen arbitrarily, the desired result follows.

A.3 Properties of the numerical scheme
In this section, we present a proof of Lemma 3.4 that is based on the following lemma.
Lemma A.4 ([27, Theorem A.1]). Let M, N be natural numbers, C 1 > α > 0 and C 2 > 0 real numbers, and Further, let F : V × [0, 1] → R N × R M be a continuous function that complies with the following conditions: (ii) The equation F (f , 0) = (0, 0) is a linear system with respect to f and admits a solution in W .

The proof of Lemma 3.4 is done in two steps.
Proof of Lemma 3.4(i). We start by showing that, given for all φ h ∈ Q h and φ h ∈ V 0,h . The proof of this fact is essentially identical to that of [17,Lemma 11.3]. In order to be able to apply Lemma A.4, we set where ||u k h || ≡ ||∇ h u k h || L 2 (Ω h ) d×d and the numbers α, C 1 , C 2 are yet to be determined. Clearly, we can construe Q 2 h as a subset of R 2N and V 0,h as a subset of R dM , where N is the number of tetrahedra (triangles) and M the number of inner faces (edges) of the mesh T h . Next, we define the continuous map for all φ h ∈ Q h and φ h ∈ V 0,h . Adapting and repeating the arguments from Section 4 to derive the energy estimates, we deduce that in Ω h , where L ∈ T h is chosen in such a way that (Z k h ) L = min R ∈ T h {(Z k h ) R }. In view of (A.22), we can find a constant α ≡ α( Thus, we have in Ω h and, analogously, Consequently, there is a constant C 1 ≡ C 1 (̺ k−1 h , Z k−1 h , u k−1 h ) > 0 such that ̺ k−1 h , Z k−1 h , ̺ k h , Z k h < C 1 in Ω h . Therefore, F fulfills assumption (i) of Lemma A.4. We proceed by proving that F satisfies assumption (ii) of Lemma A.4. To this end, we consider the equation F (((̺ k h , Z k h ), u k h ), 0) = (0, 0) that can be written as Obviously, this is a linear system for ((̺ k h , Z k h ), u k h ) with a positive definite matrix. Thus, the equation F (((̺ k h , Z k h ), u k h ), 0) = (0, 0) has a unique solution. Therefore, F also satisfies assumption (ii) of Lemma A. 4 is a solution to (3.2)-(3.4).

Proof of Lemma 3.4(ii). Suppose the triplet (r
Then [10, Chapter 7.6, Lemma 6] shows that r k h ∈ Q + h . The desired conclusions follow by applying this observation with

A.4 Stability estimates
The aim of this section is to provide the reader with a proof of Corollary 4.4.
Proof of Corollary 4.4. To begin with, we observe that 0 ≤ E k h ≤ E k−1 h for all k ∈ N. This follows from the fact that the second term on the left-hand side of (4.4) is nonnegative and all terms on the right-hand side are nonpositive. Here, the nonpositivity of the terms on the right-hand side is ensured by the convexity of the pressure potential P . Moreover, employing Hölder's inequality and Remark A.1, we see that ||̺ 0 || L ∞ (Ω) ||u 0 || 1.
Using this observation, it is easy to establish the first estimate in (4.11), the first two estimates in (4.12), the estimates in (4.13), and the estimates (4.15)-(4.18). Then, due to Corollary 3.5(i), the second estimate in (4.11) follows from the first estimate in (4.13 Consequently, the last estimate in (4.11) follows from the first two. Furthermore, an application of Poincaré's inequality (A.10) reveals that the last estimate in (4.12) is a consequence of the first. Due to Corollary 3.5(i), the validity of the first estimate in (4.14) results from the third estimate in (4.11). Using Hölder's inequality and the second estimate in (A.1), we deduce that Therefore, the second estimate in (4.14) follows from the third estimate in (4.12), the second estimate in (4.13), and (A.15). Finally, we combine Hölder's inequality, the estimates (A.3) and (4.15), the first estimate in (A.1), and the first and third estimate in (4.12) to conclude that We note in passing that estimate (4.20) can be proven in the same way.