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 strong 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: 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 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 with potential temperature transport 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. We note that the artificial pressure approach was used independently in the recent work of Kwon and Novotný [23] also for the Navier-Stokes equations. The paper is organized as follows: In Sect. 2, we introduce our notion of DMV solutions to the Navier-Stokes system with potential temperature transport and present our main result. Sect. 3 is devoted to the numerical method and the collection of its basic properties. Subsequently, we follow the strategy delineated in Figure 1 to prove the convergence of the numerical scheme: In Sect. 4, we state a discrete energy balance for our method which serves as a basis for several stability estimates. The consistency of the numerical method is established in Sect. 5 and in Sect. 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 and that the convergence of the numerical solutions is strong as long as a strong solution of (1.1)-(1.5) exists. The mesh-related estimates can be found in Appendix A.1.
We proceed by defining dissipative measure-valued solutions to the Navier-Stokes system with potential temperature transport (1.1)-(1.5).
We are ready to formulate the main result of this paper: the existence of DMV solutions to the Navier-Stokes system with potential temperature transport.

Numerical Scheme
In this section, we present our numerical method, the mixed finite element-finite volume method. For v ∈ Q h and K ∈ T h we set v K = v(x K ), where x K denotes the center of mass of K. The projection The Crouzeix-Raviart finite element spaces are denoted by 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.
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 Sects 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 Δt > 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)Δt and t k = kΔt, respectively. For the sake of simplicity, we assume that Δt is constant and that there is a number N T ∈ N such that N T Δt = 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 h δ ( k h ) 2 , h δ ( k h θ k h ) 2 . The latter are added to ensure the consistency of our method for values of γ close to 1, see Sects 4, 5.
3.5.1. 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 ( 0

Properties of the Numerical Method.
We proceed by summarizing several properties of the FE-FV method (3.2)-(3.4). (3.4) has the following properties: . Proof. For the proof we refer the reader to Appendix A.3.
From Lemma 3.4 we easily deduce the following corollary. (3.4) starting from the discrete initial data (3.5) has the following properties:

Corollary 3.5. Any solution
for all k ∈ N.

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 of 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.

solution to the FE-FV method (3.2)-(3.4) 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
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. 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 ( h , θ h , u h ) to the FE-FV method (3.2)-(3.4) starting from the initial data (3.5) has the following properties:
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). (3.4) starting from the discrete initial data (3.5) and χ ∈ C 2 (0, ∞) a concave function. Then

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: .
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. and For the proof of the Lemmata 5.
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.
and make the following introductory observations: • Due to the construction of the family  Applying the first estimate in Lemma 5.2 with (r h , φ) = ( h , ϕ) as well as (A.15), the second estimate in (4.11), and the fact that Δt ≈ h, we obtain h .
Next, let us consider the second term on the left-hand side of (5.12). Using Lemma 5.
These terms can be further estimated as follows.
• Term |I 2,h |. Due to (4.19), we obtain • Term |I 3,h |. By means of Hölder's inequality, the second estimate in (A.2), the first estimate in (A.1), the second estimate in (4.13), and the first estimate in (4.12), we derive • Term |I 4,h |. Employing Hölder's inequality, the second estimate in (4.12), and the second estimate in (4.13), we conclude that Applying the first estimate in (A.1) and the second estimate in (4.11), we get Consequently, with α 1 = min ε, 1−δ 2 > 0 as h ↓ 0. Next, using Hölder's inequality, the first estimate in (A.2), the second estimate in (4.13), and the first estimate in (4.12), we see that Therefore, we may rewrite (5.13) as The potential temperature equation.
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. The momentum equation.
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) We continue by estimating the above terms.
Using the first estimate in (A.1) and the third estimate in (4.11), we deduce that with α 2 = min ε, 1−2δ 4 > 0 as h ↓ 0. Then, using Hölder's inequality, the first estimate in (A.2), the second estimate in (4.14), and the first estimate in (4.12), we deduce that Hence, we may rewrite (5.19) as

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.

Proof of Theorem 2.3. Let {( h , θ h , u h )} h ↓ 0 be a sequence of solutions to the FE-FV method (3.2)-(3.4) starting from the initial data
Here we suppose that the parameters satisfy (5.1).
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.Here we have added a w to the standard bar notation for the weak limits to avoid any confusion with the projection operator · ≡ Π Q,h . 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 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.

Poincaré's inequality.
Let τ ∈ [0, T ] and ε > 0 be arbitrary. Further, let {ϑ k } k ∈ N ⊂ C c (R d+2 ) be the sequence of functions defined by and 0 e l s e , and C ≥ 1 a constant such that and ||v|| Clearly, such a constant exists due to (A.10) and the usual Poincaré inequality. Due to (6.19), we observe that Using the monotone convergence theorem, Lemma A.3, Lemma A.2(ii), (iii), the first estimate in (A.2), the first estimate in (4.12), and (6.3), we deduce that for almost all τ ∈ (0, T ). Consequently, choosing
From the proof of Theorem 2.3 it follows that any Young measure generated by a sequence represents a DMV solution to the Navier-Stokes system with potential temperature transport (1.1)- 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 From the proof of Theorem 2.3 and the DMV-strong uniqueness principle established in [16] we deduce that there is a

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.
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A.1 Mesh-related estimates
We summarize several important mesh-related estimates; see, e.g., [17] and the references therein. We begin with the discrete trace and inverse inequalities. We have and ||r|| for all r ∈ Q h , all K ∈ T h , all σ ∈ E h (K), and all 1 ≤ q ≤ p ≤ ∞. In addition, and 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 .
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.  Then there is f ∈ W such that F (f , 1) = (0, 0).

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 . To show that F satisfies assumption (i) of Lemma A.4, we suppose that for all φ h ∈ Q h and φ h ∈ V 0,h . Adapting and repeating the arguments from Sect. 4 to derive the energy estimates, we deduce that Consequently, k h ≥ ( k h ) K ≥ 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). Next, applying Hölder's inequality, we observe that 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 ||u h || L 2 (0, T ; L 6 (Ω h ) d ) .
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.