ON THE EXISTENCE OF SOLUTIONS TO GENERALIZED NAVIER–STOKES–FOURIER SYSTEM WITH DISSIPATIVE HEATING

. We consider a ﬂow of non-Newtonian incompressible heat conducting ﬂuids with dissipative heating. Such system can be obtained by scaling the classical Navier–Stokes–Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck–Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck–Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.


Introduction
The Rayleigh-Bénard problem of thermal convection is one of the most canonical examples of instability in the fluid flow and it has attracted lot of attention not only in the modelling and physical understanding of such phenomenon but also in the mathematical community.Indeed, this became a source of difficult mathematical problems and has driven the study of the so-called singular limits during last decades.Such singular limits may have many forms depending on the scaling, and the most classical model (the asymptotic limit) is the so-called Oberbeck-Boussinesq system, that is used exactly for Rayleigh-Bénard convection.This is the case when the fluid is heated below with the bottom temperature θ b and with prescribed lower temperature θ t on the top of two parallel plates and when the fluid can be understood as mechanically incompressible and all changes in the density are just because of variations in the temperature.It appears that such model is the very accurate approximation of real fluid in case that the temperature gradient and consequently (θ b − θ t ) is not large.On the other hand, for large temperature gradient, the Oberbeck-Boussinesq may significantly fail in giving the correct predictions and therefore there are many attempts how to generalize the approximative model.One of such attempt is to add a dissipative heating into the system.This approach was successfully used in [23] and [16], where the authors formally derived a generalization of the Oberbeck-Boussinesq system in the following form Here, Gr is the Grashof number, Pr denotes the Prandtl number, and the new additional scaling parameter is the dissipative number Di.Further, v means the velocity field, the body force is just the gravity, i.e. it has direction in the d-th component and is constant f := (−1)e d and the auxiliary functions Θ scales the difference between the temperature on the top and on the bottom of plates Here θ is the real physical temperature.Note here that assuming Di = 0, we arrive to the classical Oberbeck-Boussinesq system without dissipative heating.This system with Di = 0 is rigorously derived and treated in [6] as the singular limit of compressible Navier-Stokes-Fourier system with certain nonlocal boundary conditions.The existence analysis for such problem is then established in [4].It should be mentioned here that all considered models in [6,4] are Newtonian, i.e. the Cauchy stress S is linear with respect velocity gradient.
1.1.Beyond Oberbeck-Boussinesq system.Here, in this paper, we want to deal with Di > 0 and with possibly nonlinear S, and as a starting point for further analysis, we want to establish the existence of a solution.To simplify the notation and also computation (but without any essential impact on the analysis), we denote a new unknown ϑ := θ + Θ, consider general body force f and scale the equations such that Gr = Pr = Di = 1 and we obtain the final system div v = 0, (1.2a) which is supposed to be satisfied in Q := (0, T ) × Ω ⊂ (0, +∞) × R d with Ω being a Lipschitz domain.Note here, that (1.2a) is the incompressibility constraint, (1.2b) is the balance of linear momentum and (1.2c) is the balance of internal energy.Here v : Q → R d denotes the velocity field, D := (∇v + (∇v) T )/2 is the symmetric part of the velocity gradient ∇v, π : Q → R is the pressure, ϑ : Q → R is the temperature; S : Q → R d×d sym denotes the viscous part of the Cauchy stress tensor and q : Q → R d is the heat flux.
We prescribe the following Navier's slip (α ≥ 0) boundary condition for the velocity and the Neumann type boundary condition for the temperature (1.8) q • n = 0, on ∂Ω.
Here, n is the unit outward normal and v τ stands for the projection of the velocity field to the tangent plane, i.e. v τ := v − (v • n)n.The first condition in (1.7) expresses the fact that the solid boundary is impermeable, the second condition in (1.7) is Navier's slip boundary condition, and the condition in (1.8) states that there is no heat flux across the boundary.In fact, this is not the correct boundary condition and one should consider here either the Dirichlet boundary condition or inspired by [6] some version of nonlocal boundary condition.However, since we want to present the first large-data result for the problems with dissipative heating, we assume the simplest boundary conditions.However, we are sure that the similar result can be obtained for more realistic boundary conditions and it will be a part of the forthcoming paper about the stability, where the Dirichlet boundary condition plays the crucial role.

Definition of solution and main theorem
We assume that the initial data v 0 , ϑ 0 and the external body force f satisfy and that completed by a weak formulation of the internal energy inequality (2.4) and satisfying the boundary conditions and the initial conditions The inequality (2.4) can be also replaced by (this is usually used in the setting of compressible fluids, where the a priori estimates are not sufficient to define (2.4) in sense of distributions) , where the entropy η is defined as η := log ϑ.
Below we give a precise formulation of the notion of weak solution but before that we introduce some notation that will be needed in what follows.

Basic definitions and function spaces.
Let Ω ⊂ R d be a bounded domain with Lipschitz boundary ∂Ω, i.e.Ω ∈ C 0,1 .We say Ω ∈ C 1,1 if the mappings that locally describe the boundary ∂Ω belong to C 1,1 .
We consider the standard Lebesgue, Sobolev and Bochner spaces endowed with the classical norms.For our purposes, we introduce for arbitrary q ∈ [1, ∞) the subspaces of vector-valued Sobolev functions given by Similarly, we consider the classical Sobolev space W 1,q (Ω) and use the standard abbreviation for its dual space W −1,q ′ := (W 1,q (Ω)) * .Also, in what follows whenever there is v ∈ X * , u ∈ X, the symbol v, u means the duality paring in X.In case there was possible ambiguity, we would write v, u X .Notice that all above mentioned space are Banach spaces that are in addition separable provided that p, q < ∞.In addition, they are reflexive whenever p, q ∈ (1, ∞).Further, we introduce few inequalities used and needed in the text.Since we deal only with the symmetric gradient, we need some form of the Korn inequality.Since we want to deal with general boundary conditions, we use the following form (see [11,Lemma 1.11] or [5,Theorem 11]) for all v ∈ W 1,p n , which is valid for all p ∈ (1, ∞) provided that Ω is Lipschitz.Further, we also frequently use in the paper the following interpolation inequality 2.2.Definition of weak and suitable weak solutions and main theorem.Here, we introduce the notion of weak and suitable weak solution to (2.2)-(2.5).We consider only the dimension d = 3.
The global energy inequality holds in the following sense (2.16) . The initial conditions are attained in the following sense The above definition fulfills the basic assumption on the consistency, i.e. if we have a weak solution that is in addition smooth then it is also the classical solution, we refer here e.g. to the classical book [14], where such approach is justified.On the other hand, if we want to study further properties of the solution, for example the stability, we usually require more refined notion of the solution, namely the suitable weak solution.However, it also requires more assumption on the growth parameter p. Definition 2.2 (Suitable weak solution).Let Ω ⊂ R 3 be a bounded domain of class C 1,1 and let (0, T ) with T > 0 be the time interval.Let v 0 , ϑ 0 and f be given functions satisfying (2.1) and let p ∈ (9/5, +∞) be given.We say that a triplet (v, ϑ, π) is a suitable weak solution to the problem (2.2)-(2.5)if Definition 2.1 is satisfied with (2.16) replaced by ) . Moreover, we require that (2.4) is satisfied in the following sense Note that in above Theorem 2.1 we assume a stronger assumption on the boundary, namely Ω ∈ C 1,1 .The reason is the necessity of having a priori estimates of the pressure π that appears in (2.16) and cannot be omitted by using divergence-free functions as test functions as it is usual in studying Navier-Stokes equations without the temperature.At this, we would like to discuss the main novelty of the paper.It seems that the only relevant existence result are due to [21,20], where the authors treated the same system but with S * being linear with respect to the velocity gradient, i.e. the case p = 2, so the result of this paper is much more general.Second, in [20], the authors treated only the steady case and in [21], the authors were not able to show the validity of (2.19), i.e. they did not show the existence of a suitable weak solution, which is the main weak point in their result.We also refer to [18,17], which is an extension of [20] to more general boundary conditions, but deal only with the steady case.In our setting, we are able to prove the existence of a suitable weak solution.In addition, for p ≥ 11/5, we obtain the internal energy equality.Furthermore, in spirit of results [1,2,3], we see that the existence result obtained in this paper is the starting point for studying the stability analysis for the underlying problem.We would like to remind that there are naturally appearing numbers like 6/5, 8/5, 9/5, 11/5, which are dictated by the nature of the problem, and these borderline are usual in the theory for non-Newtonian models of heat conducting incompressible fluids.Adding the dissipative heating to the system does not bring any change in these borderlines.The only change, but rather essential, is the way how the uniform estimates are obtained, which makes the result highly nontrivial extension of works [1,2] (see also further references therein).In addition, (2.21) as well as energy equalities valid for p ≥ 11/5 seem to be the essential assumption to obtain the stability result for arbitrary weak solution, see [2,3] or [1] where the same system but without dissipative heating and in dimension two is treated.
The proof is split into several steps.In Section 3, we introduce an approximation, where the nonlinear convective terms are truncated by an auxiliary cut-off function.The existence of solutions for the k-approximation, is for the sake of completeness and clarity included in Appendix A. Then, in Section 4.1, we derive estimates that are uniform with respect to k-parameter.Finally, letting k → +∞ in Section 4.2, we complete the proof of Theorem 2.1.

Definition of approximating systems and their solutions
We start this part with definition of auxiliary cut-off functions.For any arbitrary natural number k ≥ 1, we define T k (z) = sign(z) min{k, |z|} for any z ∈ R, and a function g k : R + → [0, 1] such that it is continuous and satisfies Finally, we introduce an auxiliary function η, which is used in the proof of attainment of the initial data.Let T > 0 be given, and let 0 < ε ≪ 1 and t ∈ (0, T − ε) be arbitrary.Consider η ∈ C 0,1 ([0, T ]) as a piece-wise linear function of three parameters, such that We define an approximative problem P k (for simplicity we write (v, π, ϑ) instead of v k , π k , ϑ k ) such that we truncate the convective term (in order to be able to use the Minty method), we truncate the source term (in order to have proper estimates at the beginning) and we also modify the boundary conditions (to avoid problem with low integrability).More precisely, we consider the problem: and the following initial conditions For this problem, we have the following existence result, which is formulated in any dimension d.Note that the result is dimension-independent due to the presence of truncation functions.Lemma 3.1.Let Ω ⊂ R d be a bounded domain with C 1,1 boundary.Assume that S * and κ satisfy (1.4) and (1.5) with p > 2d/(d + 2).Then for any k ∈ N and any data v 0 , ϑ 0 , f fulfilling (2.1), there exists attaining the initial conditions (3.8) in the following sense satisfying equation (3.5) in the following sense: for any ϕ ∈ W 1,p n and for a.a.t ∈ (0, T ) there holds and satisfying (3.6) in the following sense: for any and for a.a.t ∈ (0, T ) there holds Proof.The complete proof is presented at the Appendix A for the most important case d = 3.For other dimensions the proof is however almost identical.
We would like to emphasize here, that the above existence result is in fact very strong.Although the equation (3.5) is satisfied in the classical weak sense (3.17), the equation (3.5) is satisfied in the renormalized weak sense as (3.18).This enables us to deduce the proper uniform estimates rigorously.

Limit in the approximating system
In the previous section we established the existence of a weak solution to the k-approximating system (3.4)-(3.6).Key k-uniform estimates and the limits as k → +∞ are derived in this section.We focus only on dimension d = 3, the proof for d = 2 is in fact even easier.4.1.Uniform estimates.For (v, ϑ, π) = (v k , ϑ k , π k ) we derive estimates that are uniform with respect to k-parameter (the relevant quantities are then bounded by a generic constant C, where We set ϕ = v in (3.17) and in (3.18) we set ϕ = 1 and f (ϑ) = ϑ.Summing both identities and using the fact that div v = 0 to eliminate convective terms 1 we deduce 1 Compare it with very similar computations in Appendix.
Integration with respect the time variable then leads to Next, for any ε > 0 we set f (ϑ) := log(ϑ + ε) and ϕ := 1 in (3.18), to deduce Integrating this identity over the time interval (0, t), it follows and taking the supremum over t ∈ (0, T ), we have Then employing (4.1), the fact that log ϑ 0 ∈ L 1 (Ω) and taking the limit as ε → 0 we get the following k-independent estimate In order to improve the uniform bound for the temperature, we fix arbitrary σ ∈ (0, 1) and set f (ϑ) = ϑ σ and ϕ = 1 in (3.18) to obtain Integrating it over time and using the fact that σ ∈ (0, 1) and the uniform estimate (4.1), we deduce To bound the term on the right hand side, we recall the interpolation inequality (here we consider d = 3) and using also the Hölder inequality, the a priori bound (4.1) and the assumption (1.4), we get the estimate Consequently, we deduced that for arbitrary σ ∈ (0, 1) there holds . Next, we derive the final estimates for ϑ.Using the interpolation inequality (4.7) z on the function z := ϑ σ , with σ := 3q 5 for q < 5/3, we obtain from (4.6) that Similarly, combining (4.5) and (4.8) and using the Hölder inequality, we get (4.9) The last bound can be viewed by setting Finally, we use that ϑ σ 2 is uniformly bounded in L 2 (0, T ; L 6 (Ω)) and consequently ϑ σ is uniformly bounded in L 1 (0, T ; L 3 (Ω)), combine it with the interpolation inequality (valid for some σ ∈ (0, 1) and λ ∈ (0, 1)) ϑ 2 ≤ C ϑ λ 1 ϑ 1−λ 3σ with λ = λ(σ) ∈ (0, 1) and use the uniform estimates (4.1) and (4.6), we see that Having the estimate (4.10), we can now proceed with further bounds on the velocity field.Setting ϕ := v in (3.17) and integrating in time, it yields that As consequence it follows from (1.5b) that (4.12) The interpolation inequality 2 ∇v 3 5 p + v 2 together with the Korn inequality (2.7) and the uniform estimates (4.11), (4.12) ensure that (4.13) We finish this part by introducing the estimates on the pressure.It follows from (3.17) that for all ϕ ∈ W 2,p ′ (Ω) satisfying ∇ϕ • n = 0 on ∂Ω, and for almost all time t ∈ (0, T ) there holds (see also (A.59)) In addition, recall that Then, we use the fact that Ω ∈ C 1,1 and the theory for Laplace equation (see also [11,9,12,10] for details) and obtain that Applying the z ′ -power and integrating the result over (0, T ) we finally get where the last bound follows from the estimates (4.1), (4.5), (4.12) and (4.13).Finally, we recall the estimates on the time derivative.By using the very classical procedure, we can deduce from (3.17) with the help of above uniform estimates (4.1), (4.5), (4.12) and (4.13) that (4.15) where z ′ is defined in (4.14).Similarly, considering (3.18) with f (s) := log s, we can use the above estimates (4.1), (4.5), (4.12) and (4.13) to observe that for any ω > 5, we have (see e.g.[12] for similar estimate) 5p−6 } in (3.17), integrate it over the time interval (0, T ), then after the integration by parts in the time derivative term we get where we abbreviate 18), integrate it over the time interval (0, T ), and after the integration by parts with respect to the time variable, we get Now, we make two special choices of f , namely we consider f (s) = s and then f (s) = log s (such choices can be rigorously justified taking the mollification with compactly supported functions).With the first choice, we deduce With the second choice, we also use the abbreviation η k = log ϑ k and we see that Finally, to get the energy identity, we consider ϕ ∈ C ∞ ((0, T ) × Ω) fulfilling ϕ(T ) = 0 and use v k ϕ and ϕ as test functions in (4.17) and (4.19) respectively.Doing this and then taking the sum of the outcome we deduce that (using also the fact that div v k = 0 and integration by parts 2 ) where In particular, for any ϕ ∈ C ∞ 0 ([0, T )), i.e. ϕ independent of spatial variable, there holds (4.22) 2 To evaluate the convective term we proceed as follows We want to discuss the limit in formulations (4.17) and (4.19)- (4.22).By virtue of the uniform estimates (4.11), (4.12) we can extract a subsequence (v k , ϑ k , π k , S k , η k ) such that the following convergence results hold weakly in L p (0, T ; W 1,p n,div ), (4.24) weakly in L 2 (0, T ; L 2 (∂Ω; R 3 )), (4.26) }, (4.28) weakly in L 2 (0, T ; W 1,2 (Ω)) for any σ ∈ (0, 1/2), (4.29) weakly in L q (0, T ; W 1,s 0 (Ω)) for any q ∈ [1, 5/4).(4.30) Moreover, employing the Aubin-Lions compactness Lemma we deduce that strongly in L q (Q; R 3 ) for any q ∈ [1, 5p/3) and a.e. in Q, (4.32) strongly in L p ((0, T ; L 1 (∂Ω)) and a.e. in (0, T ) × ∂Ω, (4.33) strongly in L q (Q) for any q ∈ [1, 5/3) and a.e. in Q. (4.34) Consequently, we have that η = log ϑ and (ϑ) σ = (ϑ) σ .The above strong and weak convergence results are sufficient to pass to the limit in most term.
However, it is not sufficient for identification of S = S * (ϑ, Dv).This identification follows from the procedure developed in [13], see also [10].Indeed, there is shown that there exists a nondecreasing sequence of measurable sets {Q n } ∞ n=1 fulfilling lim n→∞ |Q \ Q n | = 0 such that for every n ∈ N we have Further, using the growth assumption (1.5b), the convergence result (4.34), the fact that Dv ∈ L p ′ (Q; R 3×3 ) and the Lebesgue dominated convergence theorem, we obtain Thus, using the monotonicity assumption (1.5a), the weak convergence result (4.24) and (4.36), we observe that for any n ∈ N (4.37) Hence, we have that Thus, it is a simple consequence of (4.36) and (4.24) that for every n ∈ N Finally, using (4.24), (4.27) and (4.38), we have for arbitrary Hence, using the Minty method, i.e. setting B := Dv ± εC in the above inequality, dividing by ε > 0 and then letting ε → 0 + , we have Since C is arbitrary and |Q \ Q n | → 0 as n → ∞, we observe from the above inequality that S = S * (ϑ, Dv).
Having identified the nonlinearity S, we may now focus on the limiting procedure in desired equations.Indeed, it is now easy to use (4.23)-(4.34)and to let k → ∞ in (4.17) to deduce (2.15).Here, it is essential that v k converges strongly in L 2+ε (Q), which is due to the assumption p > 6/5.Next, we let k → ∞ in (4.22).Using (4.32) and (4.34), we can pass to the limit in the first term on the left hand side.For the second term on the left hand side we use (4.33) and the Fatou lemma to obtain the inequality (2.16).In order to obtain the equality sign in (2.16), we can use the result in [11], where it is shown that 3 (4.39) provided that p > 8/5.Consequently, we obtain (2.16) with the equality sign.
Next, we focus on the energy and the entropy (in)equalities (2.17) and (2.20).To do so, we first show how to pass to the limit with possibly inequality signs in highest order terms.Let us consider arbitrary nonnegative ϕ ∈ L ∞ (Q).Then using (1.5a), (1.5b) These convergence results are sufficient to take the limit in (4.20) and to obtain (2.17).Then we can proceed to the limit in the equation (4.19) to deduce (2.20), provided we show To show the above convergence result, we recall the strong convergence results (4.32) and (4.34).Thus, it is sufficient to show that ϑ k v k is uniformly bounded in L 1+ε (Q) for some ε > 0. Using the Hölder inequality and the classical interpolation, we have (here σ ∈ (0, 1) will be specified later, but typically 3 It follows from the following interpolation (assuming p ≤ 2, for p > 2 it is obvious)

Since 2p
5p−6 < p for p > 8/5, we see that the desired compactness follows from the a priori estimates.
it will be almost equal to one) provided that (we are using the uniform bounds coming from (4.24) and (4.29)) Note that if then we can always find σ ∈ (0, 1) such that the inequality (4.44) is satisfied.As a consequence, we deduce (4.43).Thus, we may proceed to the limit also in (4.19) to obtain (2.20).Next, we want to let k → ∞ in (4.21) to get (2.19).We already discuss almost all terms except the one |v k | 2 v. Thus, for identification of the limit also in this term, we need that v k is compact at least in L 3 (Q).Comparing it with the strong convergence result (4.32), we see that the condition p > 9/5 is exactly the one guaranteeing the compactness of the velocity in L 3 (Q).
Finally, in case that p ≥ 11/5, one may follow the standard monotone operator theory and to conclude that (4.38) holds true even in L 1 (Q) (and not only in a subset Q n ⊂ Q).Thus, we have the weak convergence on the whole set Q and therefore we are able to identify the limits in terms containing S k : Dv k with equality signs and consequently, we may consider equalities in (2.17We also omit the proof of attaining of the initial conditions (2.18) and refer the reader e.g. to [12].The proofs of both main theorems are thus complete.
Appendix A. Solvability of the k-approximation -Lemma 3.1 This appendix is devoted to the proof of Lemma 3.1, i.e. we assume that k > 0 is given and fixed.We provide the complete rigorous proof that is however very much inspired by [12], which can be used also as a tool for interested reader.We refer also to [2,3], where the existence, the stability and the convergence to the equilibria are studied in details.
We proceed as follows.First, we introduce the two-level Galerkin approximation: one for the velocity field and the second one for the temperature.Then, we pass to the limit in the temperature equation and derive all necessary a priori estimates.Finally, we pass to the limit also in the equation for the velocity and complete the proof.Note that because of the presence of the cut-off functions, the proof is relatively standard and we use just the monotone operator theory together with the relatively classical approach for parabolic equations with L 1 data.
For the sake of simplicity, we set d = 3 from the very beginning (since it is physically most relevant case) and to shorten the proof we consider only the case α ≡ 0 here.However, the general dimension d and also the case when α > 0 is treated similarly.We also recall that in what follows the constant C denotes some universal constant depending only on the data of the problem, i.e. on v 0 , θ 0 and f .If there is any dependence on different quantities, it will be clearly denoted.[19,Appendix A.4] how to construct such basis.Similarly, let {w j } ∞ j=1 be a basis of W 1,2 (Ω) which is again orthonormal in the space L 2 (Ω).Moreover, defining the subspaces

A.1. Galerkin approximations. We start by considering an orthogonal basis {w
and the associated projections we are in position to consider the usual Galerkin approximations.Now, we construct Galerkin approximations {v n,m , ϑ n,m } ∞ n,m=1 of the form for all i ∈ {1, . . ., n} and (A.2) for all j ∈ {1, . . ., m}.Here, we have used the following abbreviations In addition, we assume that v n,m and ϑ n,m satisfy the following initial conditions where v n,m 0 is independent of m-parameter and ϑ n,m 0 has the following meaning.First, we use the convention that Then, we compute the standard regularization of an integrable function η 0 with the kernel r 1/n having the support in a ball of radii 1/n.It means, we define η n 0 := r 1/n * η 0 .Then, since ϑ 0 and ln ϑ 0 are assumed to belong to L 1 (Ω), we have Finally, we apply the projection onto the linear hull of {w j } m j=1 to get P m (ϑ n 0 ).Note that as an immediate consequence of the properties of the projectors we have In addition, since log ϑ 0 ∈ L 1 (Ω) by hypothesis, we also have that We focus on solvability of (A.1)-(A.2).Defining the auxiliary vector-valued functions Then, the system (A.1)-(A.2) can be rewritten as where F is a Carathéodory function.Thus, using the classical Carathéodory theory, see [24, Chapter 1], we deduce the existence of solution to (A.1)-(A.2) at least for a short time interval.The uniform estimates derived in the next subsection enable us to extend the solution onto the whole time interval (0, T ).Then, we set m → ∞ and then n → ∞.Note that some of the estimates are independent of the order of approximation and are frequently used later (after using weak lower semicontinuity of norm in a reflexive space).
A.2. Estimates independent of m.In this part, we start by assuming that n ∈ N is arbitrary, but fixed, and we let m → ∞.
A.2.1.Estimates independent of m for the velocity: Multiplying the i-th equation in (A.1) by c n,m i , then taking the sum over i = 1, . . ., n we get Integrating the result over time interval (0, t), we obtain where we have used the identity which follows 4 from the fact div v n,m = 0 and integration by parts.Next, we apply (1.5b) 1 to the second term on the left-hand side of (A.7).For the right hand side, we use the assumption that f is bounded and apply the L 2 − L 2 Hölder's inequality and the definition of T k , see (3.1) and the fact that v n,m 0 2 ≤ v 0 2 to obtain for all t ∈ (0, T ) First, the use of the Gronwall inequality directly leads to the L ∞ − L 2 estimate for v n,m .Using this information and the Korn inequality (2.7) we deduce from (A.9) the following m-independent estimate (A.10) A.2.2.Estimates independent of m for the temperature: Next, multiplying the j-th equation in (A.2) by d n,m j , taking the sum over j = 1, . . ., k we get 4 The computation goes as follows: and G k denotes the primitive function to g k .and integrating the result over time we arrive at where we have used the abbreviation (A.3) and the fact that (A.12) ˆΩ T k (ϑ n,m )v n,m • ∇ϑ n,m dx = 0, which follows 5 from the fact that div(v n,m ) = 0 and integration by parts.Proceeding as before, we are going to study each term in (A.11).For the left hand side, we use the lower bound (1.4) on the second term.For the right hand side, we use the fact that Consequently, since the velocity field is for almost all time t ∈ (0, T ) taken from the finite dimensional space W n , can deduce from (1.5b) that , where the second inequality follows from (A.10).Combining both (recall that ϑ n,m and applying the Gronwall lemma and recalling (A.10), we have Finally, using the three-dimensional version of the interpolation inequality (2.8), it follows from (A.14) and from the embedding W 3,2 ֒→ W A.2.3.Estimates independent of m for time derivatives: In order to deduce the compactness of the velocity and the temperature, we also estimate the norms of their time derivative.More specifically, our goal in this section is to prove that We start with the velocity field.Since Therefore, multiplying the i-equation in (A.1) by ċn,m i (t), summing over i = 1, . . ., n and integrating it over time, we obtain (after using the estimate (A.14)) that 5 It is a consequence of the following computation where T k denotes a primitive function to T k .Now, we are going to study each term separately.Using the fact that {w i } ∞ i=1 forms a basis of W 3,2 n,div , the fact that W n is finite dimensional, the Hölder inequality and the δ-Young inequality and the a priori bound (A.14), we have Proceeding similarly and using (1.5b) 2 we get Combining it all, taking δ small enough and applying (A.17) and (A.14), we obtain which gives the first part of (A. 16).
Next, we focus on the estimates for ∂ t ϑ n,m .Using the orthogonality of the basis of W m and the regularity of ∂ t ϑ n,m , we have that for all t ∈ (0, T ) ˆΩ ∂ t ϑ n,m (t, x)P m (ϕ)(x) dx.Now, using (A.2) we get (we omit writing (t, x)) and proceeding as before, we obtain Since, the properties of the basis W m gives that P m (ϕ) 1,2 ≤ ϕ 1,2 , we can use (A.20) in (A.19), apply the second power and then integrate over t ∈ (0, T ) to get and consequently, using (A.14), we deduce that In this part, we let m → ∞ but keep n ∈ N fixed.Our goal is to identify limits in the equations (A.1)-(A.2) as well as in the constitutive relations (A.3).
A.3.3.Attainment of the initial condition for (v n , ϑ n ).We start with the initial condition for the velocity.Since v n,m 0 := P n (v 0 ) we have that v n,m 0 ≡ v n 0 is independent of m-parameter.Equivalently, we have c n,m 0 = c n 0 , for all m ∈ N. Due to (A.23b) we have c n,m (t) → c n (t) strongly in C([0, T ]) and consequently we get c n (0) = c n 0 .Now, from the definition of v n (t, x) and v n 0 (x) it is clear that v n (0, x) = v n 0 (x) for all x ∈ Ω.It remains to show that ϑ n (0, x) = ϑ n 0 (x).First, note that a priori estimates (following from (A.22d) and (A.22e)) together with the standard parabolic embedding imply that Thus, it makes a good sense to define an initial condition.To prove our goal we integrate the equation (A.2) over time (0, t) to get Here, we have used the fact that ϑ n,m 0 = ϑ n,m (0).Now, using the previous convergence results (A.22)-(A.27)and the convergence of the initial condition (A.5b), we can let m → ∞ and to obtain for almost all t ∈ (0, T ) that Since ϑ n ∈ C(0, T ; L 2 (Ω)), the above identity can be extended for all t ∈ (0, T ).Consequently, letting t → 0 + in the above identity, we observe that But as ϑ n ∈ C(0, T ; L 2 (Ω)) and weak limit as time tends to zero is ϑ n 0 , we have that lim t→0+ ϑ n (t) − ϑ n 0 2 = 0.
A.4. Estimates independent of n.In this part, we derive estimates that are n-independent and that help us to pass to the limit n → ∞.Some of the estimates are also independent of k-approximation but if there is dependence on k, it will be clearly denoted.
We consider ψ(t, x) := χ [0,s] (t) min{0, ϑ n (t, x)} ≤ 0 as a test function in (A.29).Integrating it over time t ∈ (0, T ) we obtain Next, we show that the right hand side is non-positive.Indeed, using the definition of q n , see (A.27), and of ψ we have that and consequently, since ψ is non-positive, we have that almost everywhere in (0, T ) × Ω.Further, since ϑ n * = max{0, ϑ n }, it directly follows from the definition of ψ that T k (ϑ n * )v n • f ψ ≡ 0. Finally, to estimate the convective term, we introduce the primitive function and we get after integration by parts (compare with (A.12)), that where we used the fact that div v n = 0 in Ω and v n • n = 0 on ∂Ω.Hence, we arrive at Finally, the fact that ψ(x, 0) = 0 a.a.x ∈ Ω, implies that ψ(s) 2 = 0 for all s ∈ (0, T ).Then one can easily obtain the desired conclusion (A.30).Consequently, we can now replace ϑ n * by ϑ n everywhere.
A.4.2.Renormalization of the temperature equation.For our result, the very special choice of the test functions in the temperature equation plays an essential role.Therefore, we state already at this point the renormalized version of the temperature equation, which then helps us to get the a priori estimates and even more, will be important for proving the entropy (in)equality.For future use, it is convenient to record a renormalized version of the approximate equation (A.29).Using (A.22d) we have that ϑ n ∈ L 2 (0, T ; W 1,2 (Ω)) and moreover we know θ n ≥ 0. Therefore, if we consider for almost all t ∈ (0, T ).Consequently, such ϕ can be used in (A.29) and we deduce and for almost all t ∈ (0, T ).Please notice here that for the first term, we used the regularization approach as follows (the computation is done for almost all t and ϑ n ε denotes the classical regularization with respect to the t-variable) A.4.3.Energy and entropy estimates independent of n and k.First, we derive the estimates based on the kinetic and internal energy.It is noticeable that they are independent of n and also of k.We set ψ ≡ 1 ∈ W 1,2 (Ω) in (A.29) and deduce the identity which is valid for almost all t ∈ (0, T ).Then, we multiply the i-the equation in (A.28) by c n i , sum over i = 1, . . ., n and we obtain for a.a.t ∈ (0, T ).Here, we have used the symmetry of S n , the fact that div v n = 0 and the integration by parts (see the similar computation for the convective term in (A.8)).Summing the above identities and using the nonnegativity of the temperature ϑ n , we get after integration with respect to time the energy equality 2 ≤ C, where the second inequality follows from the assumptions on ϑ 0 and v 0 and from the properties of the projection P n .Therefore, we have Next, we show the uniform bound on the entropy.We fix 0 < ǫ ≪ 1, and consider f (s) = ln(ǫ + s) and φ = 1 in (A.31).Note that due to the nonnegativity of the temperature ϑ n , such choice of f is admissible.Using the following inequalities and moving all terms with positive sign to one side, we observe that for almost all t ∈ (0, T ) we have Thus, using the fact that f is bounded, see (2.1a), we deduce after using the Hölder inequality and after integration over time that Consequently, using the assumptions on ϑ n 0 , letting ǫ → 0 + , using the Fatou lemma, the simple algebraic inequality max{0, ln(ǫ + ϑ n (t))} ≤ C(1 + ϑ n (t)) and the uniform bounds (A.34) and (A.35), we deduce where C is a uniform constant depending only on data.
A.4.4.Gradient estimates independent of n possibly depending on k.Here, we derive the estimates for the velocity and the temperature gradients that are independent of n but may depend on the truncation k.We start with the velocity filed.Integrating in time the equation (A.33) and using the estimate (A.35), we obtain Thus, the assumption (1.5b) and the Korn inequality (2.7) and a priori estimate (A.35) imply that Next, we focus on the estimate for the temperature.The starting point is to show that for all σ ∈ (0, 1) there holds To show (A.38), we define the auxiliary function where 0 < σ < 1.Then we use (A.31) with this f and also set φ := 1. Doing so, we deduce Here, we used the definition of q n in (A.27).Noticing again that almost everywhere in (0, T )×Ω, we can integrate the above identity over time, use the Hölder inequality and the already obtained bounds (A.34) and (A.35) to deduce (A.38).
Thus, using the interpolation inequality (2.8) (we use its three dimensional variant only here), we deduce the classical estimate for the temperature of the form which is valid for all σ ∈ (0, 1).Therefore, it directly follows from the above inequality that To derive an estimate on ∇ϑ n , we consider 1 ≤ r < 5/4.Combining the Hölder inequality and the previous estimate (A.39), we obtain provided that we can choose σ ∈ (0, 1) such that 4 ), we can always find σ ∈ (0, 1) so that the above inequality holds and therefore (A.40) ˆT 0 ϑ n r 1,r dt ≤ C(k, r) for all r ∈ 1, 5 4 .
A.4.5.Estimates for time derivatives independent of n.In order to deduce the compactness of the velocity and the temperature, we also need to get a bound on the norms of their time derivatives.More specifically, our goal in this section will be to prove the uniform bounds in the following spaces ∂ t v n ∈ L p ′ (0, T ; (W 3,2 n,div ) * ) and ∂ t ϑ n ∈ L 1 (0, T ; (W 1,z (Ω)) * ), for all z > 5. (A.41) We start with the velocity field.Assume that ϕ ∈ W 3,2 n,div is arbitrary and fulfills ϕ W 3,2 n,div ≤ 1.We also recall the orthogonality of the basis {w i } ∞ i=1 as well as the continuity of the projection P n in the space W 3,2 n,div .Then we have due to the regularity of ∂ t v n that Now, using (A.28) we get and proceeding as before and using the growth assumption (1.5b), we obtain that for all t ∈ (0, T ) .
Consequently, we have and raising this inequality to the power p ′ and integrating the result over (0, T ), using also the already obtained bound (A.37), we deduce Finally, we focus on the estimate for time derivative of ϑ n .Recall that (A.43) is valid for all ψ ∈ W 1,2 (Ω) and for a.a.t ∈ (0, T ).Let us consider ψ ∈ W 1,z (Ω) with z > 5 fulfilling ψ 1,z ≤ 1.Then using the fact that W 1,z (Ω) ֒→ L ∞ (Ω), we get by using the Hölder inequality that for almost all t ∈ (0, T ) the following inequality holds ∂ t ϑ n (t), ψ = ˆΩ(T k (ϑ n (t))v n (t) + q n (t)) • ∇ψ + S n (t) : Hence, using (1.5b), and the fact that z > 2, we deduce that for almost all t ∈ (0, T ) there holds ∂ t ϑ n (t), ψ ≤ C(k) 1 + v n (t) 2 + ∇ϑ n (t) z ′ + v n (t) p 1,p .Since z > 5 we have that z ′ < 5  4 and therefore integration of the above inequality over (0, T ) and applying the uniform bounds (A.35), (A.37) and (A.40) leads to ˆT 0 ∂ t ϑ n (W 1,z (Ω)) * dt ≤ C(k, z) for all z > 5. (A.44) A.5.3.Attainment of the initial condition for v.The arguments concerning the attainment of the initial conditions v 0 are standard.For sake of completeness, we included all details below.Since v ∈ C(0, T ; L 2 (Ω) 3 ), we get that v(t) → v(0) strongly in L 2 n,div as t → 0 + .In what follows, we show that v(t) ⇀ v 0 weakly in L 2 n,div as t → 0 + , and these convergence results together (due to the uniqueness of weak convergence) identify the limit (3.16) 1 , that we want to prove.
Let 0 < ε ≪ 1 and t ∈ (0, T − ε) be arbitrary.We consider the auxiliary function η defined in (3.3), multiply (A.28) by this η, and integrate the result over (0, T ) to obtain for every i = 1, . . ., n Next, we integrate by parts in the first term, use that η(T ) = 0, and the equality v n (0) = P n (v 0 ) to get This identity is ready for the use of the convergence results (A.45)-(A.46)as well as the convergence of the projection P n (v 0 ) to obtain for any i ∈ N that Using the definition of η, i.e. using that η(τ ) = 1 for τ ∈ [0, t) and η(τ ) = 0 for τ ∈ (t + ε, T ], and η ′ (τ ) = − 1 ε for τ ∈ (t, t + ε), we obtain 1 ε Next, since v ∈ C(0, T ; L 2 n,div ), we can easily let ε → 0 + to get ˆΩ v(t) • w i dx + ˆt 0 ˆΩ(S − v ⊗ v g k (|v| 2 )) : ∇w i η − T k (ϑ)f • w i η dx ds = ˆΩ v 0 • w i dx for all t ∈ (0, T ).Therefore, we see that This holds for every i ∈ N, and since {w i } ∞ i=1 is a basis of the space W 3,2 n,div , this is nothing more than v(t) ⇀ v 0 weakly in (W 3,2  n,div ) * as t → 0 + .Finally, since W 3,2 n,div is dense in L 2 n,div , the weak convergence result that we expected is also true, and it identifies the strong limit of the initial condition in L 2 n,div as required in (3.First, we notice that we can extend the solution to the interval (0, T + 1), e.g., by extending f by zero outside of (0, T ).Multiplying the i-th equation in (A.28) by c n i , summing the result over i = 1, . . ., n
we formulate the main theorem of this paper.