On solutions for a generalized Navier–Stokes–Fourier system fulfilling the entropy equality

We consider a flow of a non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the Dirichlet boundary condition for the temperature. In three dimensions, for a power-law index greater or equal to 11/5, we show the existence of a solution fulfilling the entropy equality. The entropy equality can be formally deduced from the energy equality by renormalization. However, such a procedure can be justified by the DiPerna–Lions theory only for p>5/2. The main novelty is that we do not renormalize the temperature equation, but rather construct a solution which fulfils the entropy equality. This article is part of the theme issue ‘Non-smooth variational problems and applications’.


Formulation of the problem
We study the generalized Navier-Stokes-Fourier system and ∂ t (c ν ϑ) + div(c ν ϑv) + div q = S : Dv, (1.1c) 2022 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. Here, v : Q → R 3 denotes the velocity field, Dv := (∇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 3×3 sym denotes the viscous part of the Cauchy stress tensor and q : Q → R 3 is the heat flux.
The key motivation of this article is the qualitative analysis of the system (1.1). More precisely, in a three-dimensional setting, the existence of a weak solution to (1.1) was firstly proved for p ≥ 11/5 in [1], see also the related work [2]. Later, for p ∈ (9/5, 11/5) and slightly different boundary conditions, the existence of a weak solution was established in [3], and see also [4] for a more complicated model, with one proviso. The identity (1.1c) was replaced by the inequality, which in terms of the entropy η = log ϑ (1.8) can be rewritten into the so-called entropy inequality: A natural question, one may ask, is how regular is the solution (ϑ, v). However, looking on equation (1.1c), we see that system (1.1) has the so-called critical growth on the right-hand side, and there is no a priori regularity theory available. The main reason is that the righthand side of (1.1c) is just integrable. On the other hand, it seems to be extremely important to justify certain renormalization of equation (1.1c) since the renormalized solution can typically have better properties on one hand and on the other hand one may use the renormalization to rigorously justify arguments based on the notion of classical solution, e.g. when studying the stability theory, see [5]. Unfortunately, the classical technique of DiPerna and Lions, see ( [6], Lemma II.1 and Theorem II.1), can be used only for p > 5 2 here. Therefore, for p ∈ [11/5, 5/2], we need to change the methods and results used in [1,3,7] significantly. Thus, our main goal is to prove the existence of a weak solution, which satisfies (1.9) with the equality sign and also to show that the temperature is continuous with respect to time into the topology of L 1 (Ω), which is the natural function space. Then, inspired by [5], we know that one can renormalize (1.9) by a properly chosen set of functions. Indeed, in the standard approach of renormalization, based just on (1.1c), one needs (due to the commutator lemma) that ϑ ∈ L p (Q). Unfortunately, this is true only for p > 5 2 . Therefore, we introduce (1.9) with equality sign, and then to renormalize it, one just requires η ∈ L p (Q), which is true for any p > 1. Our result can be understood as a starting point and the cornerstone of further analysis, which in principle requires very special test functions, e.g. the regularity theory, the stability theory and so on, in various models of incompressible heat conducting fluids. Note that the bound p ≥ 11 5 then does not come from the heat equation but is a consequence of the structure of (1.1a) and is closely related to the difference of a notion of solution introduced in [1,7]. More precisely, for our theory, it is necessary to obtain at least weak sequential compactness of the term S : Dv arising on the right-hand side of (1.1c), which is known only in the case p ≥ 11 5 .

Rigorous statement of the main result
In what follows, we use the standard notation for the Lebesgue, the Sobolev and the Bochner spaces and endow them with standard norms. The symbol C ∞ 0 is reserved for smooth compactly supported functions, and the function spaces related to the incompressible setting are denoted by W 1,p 0,div := {v ∈ W 1,p 0 (Ω; R 3 ); div v = 0} and L 2 0,div denotes the closure of W 1,2 0,div in L 2 topology. Duality pairing between W 1,p 0,div and their duals is denoted ·, · . Next, we postulate the assumptions on data. Recall, we assume that q and S * satisfy (1.4)-(1.6). For initial and boundary data, we consider whereθ is the solution to (1.7). Hence, we transferred all assumptions on the boundary behaviour of ϑ b to the uniquely definedθ. Finally, we suppose that μ := min{ess inf x∈Ωθ (x), ess inf x∈Ω ϑ 0 (x)} > 0. (2. 2) The main result of this article is presented as follows.

Proof of theorem 2.1
The existence proof relies on the methods developed in [3], and a large part of the proof is identical. Therefore, we omit unnecessary details and focus mainly on the new aspects of the proof, i.e. on the proof of entropy equality (2.11). Hence, following [3] (compare also with [1], where a different approach is used), we introduce , appendix A.4). Next, for given initial conditions v 0 and ϑ 0 , we denote v n 0 the projection of v 0 onto the subspace [w 1 , . . . , w n ], and ϑ n 0 ∈ L 2 (Ω), fulfilling ϑ n 0 ≥ μ a.e. in Ω, is the regularization of ϑ 0 such that v n 0 → v 0 strongly in L 2 0,div as n → +∞ (3.1) and ϑ n 0 → ϑ 0 strongly in L 1 (Ω) as n → +∞.
Then, for every n ∈ N, we can find a triple (v n , ϑ n , S n ), such that v n ∈ W 1,2 ((0, T), T); (W 1,2 0 (Ω)) * ) and S n ∈ L ∞ (Q) and such that for a.e. t ∈ (0, T) and for all ψ ∈ L 2 (0, T; In addition, v n and S n are given by c n i (t)w i (x) and S n = S * (ϑ n , Dv n ), (3.5) the initial conditions v n (0, ·) = v n 0 , ϑ n (0, ·) = ϑ n 0 are satisfied (note v n and ϑ n are continuous into the topology of L 2 , and thus, it makes sense to talk about the initial value) and ϑ n attains the boundary conditions, i.e. ϑ n |∂Ω =θ and fulfils the minimum principle, i.e. ϑ n ≥ μ a.e. in Q. Then, following [3] and definingθ n := ϑ n −θ, it is rather standard to deduce the following n-independent a priori estimates valid for all r ∈ [1, 5/3), s ∈ [1, 5/4) and α ∈ (0, 1/2) : and This is the starting point of the proof. It is rather sketchy, but it does not contain any essentially new information.
(a) Limit in momentum and energy equations as n → +∞ By virtue of the established uniform estimates (3.6)-(3.8) and employing the Aubin-Lions compactness lemma, we can extract a subsequence that we do not relabel, and we can find , and a.e. in Q, In addition, by using (3.14) and the first part of the uniform estimate (3.7), we have that ϑ ∈ L ∞ (0, T; L 1 (Ω)). (3.16) Now we are in the position to analyze the limit of formulations (3.3)-(3.4). Indeed, by virtue of the convergence results (3.10)-(3.12), we may follow [9] to take the limit in formulation (3.3) to deduce that for all w ∈ L p (0, T; W (3.17) In addition, we have v ∈ C([0, T]; L 2 0,div ) and v(0, ·) = v 0 . To complete the proof of (2.9), it remains to show S = S * (ϑ, Dv) a.e. in Q. (3.18) Next, thanks to (3.12)-(3.15), one may follow a standard procedure (see, e.g. [7]) and let n → ∞ in (3.4) to show that for all ψ ∈ C ∞ 0 ((−∞, T) × Ω), there holds (compare with (2.10)) where the last identity follows from (3.17) with setting w := v. Consequently, using this estimate, the monotonicity and the growth assumption (1.6), the strong convergence result (3.14), the weak convergence (3.10) and the Lebesgue dominated convergence theorem, we deduce that for all

(b) Limit in entropy equation as n → +∞
In this section, we show the validity of (2.11). To see this, we first set ψ := ϕ/ϑ n in (3.4) with arbitrary ϕ ∈ C ∞ 0 ((−∞, T) × Ω) to derive the following identity for approximated entropy η n := ln ϑ n : where we set η n 0 := ln ϑ n 0 . Next, we want to let n → ∞. The identification of the limit in the terms on the left-hand side is rather standard and follows from the convergence results (3.10), (3.12), (3.14) and (3.15). Similarly, to pass to the limit in the first term on the right-hand side of (3.24) is straightforward thanks to (3.14), (3.20) and the Egorov and Dunford-Pettis theorems. Also the limit passage in the last term is obvious. The most problematic term is however the second term on the right-hand side since κ(ϑ n )|∇ϑ n | 2 /(ϑ n ) 2 is uniformly bounded only in L 1 ((0, T) × Ω), and so we cannot even a priori extract an L 1 weakly convergent subsequence. We overcome this in two steps. First, the point-wise convergence of ∇ϑ n is shown, and then the strong convergence of κ(ϑ n )|∇η n | 2 in L 1 ((0, T) × Ω) is deduced.  Its primitive function attaining zero at zero is denoted G k , i.e. G k = T k , G k (0) = 0. Note that |G k (s)| ≤ k|s| for all s ∈ R. Next, we also introduce a mollification of T k . For arbitrary δ ∈ (0, k) (typically δ 1), we denote by T k,δ ∈ C 2 (R) a mollification of T k , which is given by a convolution with a symmetric, positive kernel of radius δ. Such a mollification then has the following properties: We fix m, n, k ∈ N, k > 2 θ L ∞ (∂Ω) and ε, δ > 0, ε < k and define w m,n δ = T k+ε,δ (ϑ n ) − T k,δ (ϑ m ). We set ψ := T k+ε,δ (ϑ n )T ε (w m,n δ ) in (3.4) for ϑ n and ψ := T k,δ (ϑ m )T ε (w m,n δ ) in (3.4) for ϑ m . Note that it is allowed since T k+ε,δ (ϑ n )T ε (w m,n δ ) and also T k,δ (ϑ m )T ε (w m,n δ ) belong to L 2 ((0, T); W 1,2 0 (Ω)). Then, we subtract the so obtained equations to obtain where we denoted Our first goal is to let δ → 0 + in (3.26). Note that such convergence procedure is very standard in all terms except the term G m,n involving the second derivative of T ·,δ . To obtain a proper δindependent bound, we consider M ≥ θ L ∞ (∂Ω) and δ ∈ (0, M/2). Since ϑ n =θ on ∂Ω, we can deduce that ψ := 1 − T M,δ (ϑ n ) ∈ L 2 (0, T; W 1,2 0 (Ω)), and therefore, it can be used in (3.4). Such a choice then leads tô S n : Dv n dx dt ≤ C, (3.27) where we exploited that div v n = 0, used the properties of T M,δ (in particular the concavity) and (3.6) and (3.2). In a very similar manner, we can estimate the term with time derivative in (3.26).