1 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 (e.g. [9, 14, 16]). Indeed, this became a source of difficult mathematical problems (e.g. [17, 20]) and has driven the study of the so-called singular limits during last decades [6]. 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 \(\theta ^{b}\) and with prescribed lower temperature \(\theta ^{t}\) 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 \((\theta ^b - \theta ^t\)) 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 [21, 28], where the authors formally derived a generalization of the Oberbeck–Boussinesq system in the following form

$$\begin{aligned}&{\text {div}}\textbf{v}= 0,\\&\partial _t\textbf{v}+ {\text {div}}(\textbf{v}\otimes \textbf{v}) - \frac{1}{\sqrt{\text {Gr}}}{\text {div}}\textbf{S} +\nabla p= - \theta \textbf{f}, \\&\tilde{c}_v(\theta )\left( \partial _t\theta +{\text {div}}(\theta \textbf{v})\right) \\&+\frac{1}{\text {Pr}\sqrt{\text {Gr}}}{\text {div}}\textbf{q} = \frac{2\text {Di}}{\sqrt{\text {Gr}}}\textbf{S} :\textbf{D}{\textbf{v}} +\text {Di}(\theta +\Theta ) (\textbf{v}\cdot \textbf{f}). \end{aligned}$$

Here, \(\text {Gr}\) is the Grashof number, \(\text {Pr}\) denotes the Prandtl number, and the new additional scaling parameter is the dissipative number \(\text {Di}\). Further, \(\textbf{v}\) means the velocity field, the body force is just the gravity, i.e. it has direction in the d-th component and is constant \(\textbf{f}:=(-1)\textbf{e}_d\) and the auxiliary functions \(\Theta\) scales the difference between the temperature on the top and on the bottom of plates

$$\begin{aligned} \Theta :=\frac{1}{2}\frac{\theta ^{b}+\theta ^{t}}{\theta ^{b}-\theta ^{t}}. \end{aligned}$$

Here \(\theta\) is the real physical temperature. Note here that assuming \(\text {Di}=0\), we arrive to the classical Oberbeck–Boussinesq system without dissipative heating. This system with \(\text {Di}=0\) 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 [4, 6] are Newtonian, i.e. the Cauchy stress \(\mathbf{S}\) is linear with respect velocity gradient.

1.1 Beyond Oberbeck–Boussinesq system

Here, in this paper, we want to deal with \(\text {Di}>0\) and with possibly nonlinear \(\mathbf{S}\), 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 \(\vartheta :=\theta +\Theta\), consider general body force \(\textbf{f}\) and scale the equations such that \(\text {Gr}=\text {Pr}=\text {Di}=1\) and we obtain the final system

$$\begin{aligned} {\text {div}}\textbf{v}&= 0, \end{aligned}$$
(1a)
$$\begin{aligned} \partial _t\textbf{v}+ {\text {div}}(\textbf{v}\otimes \textbf{v}) - {\text {div}}\textbf{S} + \nabla \pi&= \vartheta \textbf{f}, \end{aligned}$$
(1b)
$$\begin{aligned} \partial _t\vartheta +{\text {div}}(\vartheta \textbf{v}) + {\text {div}}\textbf{q}&= \textbf{S} :\textbf{D}-\vartheta (\textbf{v}\cdot \textbf{f}), \end{aligned}$$
(1c)

which is supposed to be satisfied in \(Q:=(0,T) \times \Omega \subset (0, +\infty )\times {{\mathbb {R}}}^d\) with \(\Omega\) being a Lipschitz domain. Note here, that (1a) is the incompressibility constraint, (1b) is the balance of linear momentum and (1c) is the balance of internal energy. Here \(\textbf{v}: Q\rightarrow {{\mathbb {R}}}^d\) denotes the velocity field, \(\textbf{D}:= (\nabla \textbf{v}+ (\nabla \textbf{v})^{T})/2\) is the symmetric part of the velocity gradient \(\nabla \textbf{v}\), \(\pi :Q\rightarrow {{\mathbb {R}}}\) is the pressure, \(\vartheta :Q\rightarrow {{\mathbb {R}}}\) is the temperature; \(\textbf{S} : Q\rightarrow {{\mathbb {R}}}^{d\times d}_\textrm{sym}\) denotes the viscous part of the Cauchy stress tensor and \(\textbf{q}:Q \rightarrow {{\mathbb {R}}}^d\) is the heat flux.

1.2 Cauchy stress tensor and heat flux

The heat flux \(\textbf{q}\) is represented by the Fourier law

$$\begin{aligned} \textbf{q} = \textbf{q}^*(\vartheta ,\nabla \vartheta ):= - \kappa (\vartheta ) \nabla \vartheta , \end{aligned}$$
(2)

with the heat conductivity \(\kappa : {{\mathbb {R}}}\rightarrow (0, +\infty )\) being a continuous function of the temperature satisfying, for all \(\vartheta \in (0, +\infty )\) and for some \(0<\underline{\kappa }\le \overline{\kappa } <+\infty ,\)

$$\begin{aligned} 0<\underline{\kappa }\le \kappa (\vartheta ) \le \overline{\kappa }<+\infty . \end{aligned}$$
(3)

We also assume that the Cauchy stress is given as \(\textbf{S} =\textbf{S} ^*(\vartheta , \textbf{D}\textbf{v})\), where \(\textbf{S} ^*: (0,\infty )\times {{\mathbb {R}}}^{d\times d}_\textrm{sym} \rightarrow {{\mathbb {R}}}^{d\times d}_\textrm{sym}\) is a continuous mapping fulfilling for some \(p>2d/(d+2)\), some \(0< \underline{\nu }, \overline{\nu }<+\infty\) and for all \(\vartheta \in {{\mathbb {R}}}_+\), \(\textbf{D}_1,\textbf{D}_2 \in {{\mathbb {R}}}^{d\times d}_\textrm{sym}\)

$$\begin{aligned}&(\textbf{S} ^*(\vartheta , \textbf{D}_1)-\textbf{S} ^*(\vartheta , \textbf{D}_2)): (\textbf{D}_1-\textbf{D}_2)\ge 0, \end{aligned}$$
(4a)
$$\begin{aligned}&\textbf{S} ^*(\vartheta ,\textbf{D}_1):\textbf{D}_1\ge \underline{\nu }|\textbf{D}_1|^p - \overline{\nu },\nonumber \\&\quad |\textbf{S} ^*(\vartheta , \textbf{D}_1)|\le \overline{\nu }(1+|\textbf{D}_1|)^{p-1}, \quad \textbf{S} ^*(\vartheta , 0)=0. \end{aligned}$$
(4b)

The prototypic relation \(\textbf{S} \sim \nu (\vartheta ) |\textbf{D}\textbf{v}|^{p-2}\textbf{D}\textbf{v}\) falls into the class (4).

1.3 Boundary and initial conditions

The system (1a)–(1c) is completed by the following initial conditions

$$\begin{aligned} \textbf{v}(0) = \textbf{v}_0, \ \ \vartheta (0) = \vartheta _0> 0 \qquad \text{ in } \Omega . \end{aligned}$$
(5)

We prescribe the following Navier’s slip (\(\alpha \ge 0\)) boundary condition for the velocity

$$\begin{aligned} \textbf{v}\cdot \textbf{n}=0, \qquad {\alpha }\textbf{v}_{\tau }=-[\textbf{S} \textbf{n}]_{\tau } \qquad \text {on } \partial \Omega , \end{aligned}$$
(6)

and the Neumann type boundary condition for the temperature

$$\begin{aligned} \textbf{q}\cdot \textbf{n}=0, \qquad \text {on } \partial \Omega . \end{aligned}$$
(7)

Here, \(\textbf{n}\) is the unit outward normal and \(\textbf{v}_{\tau }\) stands for the projection of the velocity field to the tangent plane, i.e. \(\textbf{v}_{\tau }:= \textbf{v}-(\textbf{v}\cdot \textbf{n})\textbf{n}\). The first condition in (6) expresses the fact that the solid boundary is impermeable, the second condition in (6) is Navier’s slip boundary condition, and the condition in (7) 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.

2 Definition of solution and main theorem

We assume that the initial data \(\textbf{v}_0, \vartheta _0\) and the external body force \(\textbf{f}\) satisfy

$$\begin{aligned}&\textbf{v}_0\in L^2_{\textbf{n},{\text {div}}},\quad \vartheta _0\in L^1(\Omega ), \quad \log \vartheta _0\in L^1(\Omega ), \nonumber \\&\textbf{f}\in L^{\infty }((0,T)\times \Omega ), \end{aligned}$$
(8a)

and that

$$\begin{aligned}&\vartheta _0\ge 0 \text{ for } \text{ a.a. } x\in \Omega . \end{aligned}$$
(8b)

We look for \((\textbf{v},\vartheta ,\pi ):[0,T]\times \Omega \rightarrow {{\mathbb {R}}}^d\times {{\mathbb {R}}}^{+}\times {{\mathbb {R}}}\) solving the following set of equations in \((0,T)\times \Omega\)

$$\begin{aligned} \partial _t\textbf{v}+ {\text {div}}(\textbf{v}\otimes \textbf{v}) - {\text {div}}\textbf{S} + \nabla \pi = \vartheta \textbf{f},\qquad {\text {div}}\textbf{v}= 0, \end{aligned}$$
(9)
$$\begin{aligned} \partial _t\left( \frac{|\textbf{v}|^2}{2}+\vartheta \right) +{\text {div}}\left( \textbf{v}\left( \frac{|\textbf{v}|^2}{2}+\vartheta +\pi \right) \right) + {\text {div}}\textbf{q} = {\text {div}}(\textbf{S} \textbf{v}), \end{aligned}$$
(10)

completed by a weak formulation of the internal energy inequality

$$\begin{aligned} \partial _t\vartheta +{\text {div}}(\vartheta \textbf{v}) + {\text {div}}\textbf{q} \ge \textbf{S} :\textbf{D}-\vartheta (\textbf{v}\cdot \textbf{f}), \end{aligned}$$
(11)

and satisfying the boundary conditions

$$\begin{aligned} \textbf{v}\cdot \textbf{n}=0, \quad \alpha \textbf{v}_{\tau }+[\textbf{S} \textbf{n}]_{\tau }=0, \quad \nabla \vartheta \cdot \textbf{n}=0 \qquad \text {on } (0,T)\times \partial \Omega , \end{aligned}$$
(12a)

and the initial conditions

$$\begin{aligned} \textbf{v}(0, \cdot )=\textbf{v}_0, \quad \vartheta (0, \cdot )=\vartheta _0. \end{aligned}$$
(12b)

The inequality (11) 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 (11) in sense of distributions)

$$\begin{aligned} \partial _t\eta +{\text {div}}(\eta \textbf{v}) + {\text {div}}\left( \frac{\textbf{q}}{\vartheta }\right) \ge \frac{\textbf{S} :\textbf{D}}{\vartheta }-\frac{\textbf{q}\cdot \nabla \vartheta }{\vartheta ^2} - \textbf{v}\cdot \textbf{f}, \end{aligned}$$
(13)

where the entropy \(\eta\) is defined as \(\eta :=\log \vartheta\).

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.

2.1 Basic definitions and function spaces

Let \(\Omega \subset {{\mathbb {R}}}^d\) be a bounded domain with Lipschitz boundary \(\partial \Omega\), i.e. \(\Omega \in \mathcal {C}^{0,1}\). We say \(\Omega \in \mathcal {C}^{1,1}\) if the mappings that locally describe the boundary \(\partial \Omega\) belong to \(\mathcal {C}^{1,1}\).

We consider the standard Lebesgue, Sobolev and Bochner spaces endowed with the classical norms. For our purposes, we introduce for arbitrary \(q\in [1,\infty )\) the subspaces of vector-valued Sobolev functions given by

$$\begin{aligned} W_{\textbf{n}}^{1,q}:=\overline{\{\textbf{v}\in \mathcal {C}^{\infty }(\Omega ;{{\mathbb {R}}}^d)\cap \mathcal {C}(\overline{\Omega };{{\mathbb {R}}}^d): \text {tr}(\textbf{v})\cdot \textbf{n} =0 \text { on } \partial \Omega \}}^{\Vert \cdot \Vert _{1,q}}, \qquad W_{\textbf{n}}^{-1,q'}:=\left( W_{\textbf{n}}^{1,q}\right) ^{*}, \end{aligned}$$

and

$$\begin{aligned} W_{\textbf{n},{\text {div}}}^{1,q}:=\{\textbf{v}\in W_{\textbf{n}}^{1,q}: {\text {div}}(\textbf{v}) =0 \}, \qquad W_{\textbf{n},{\text {div}}}^{-1,q'}:=\left( W_{\textbf{n},{\text {div}}}^{1,q}\right) ^{*}, \\ L_{\textbf{n},{\text {div}}}^q:=\overline{\{\textbf{v}\in W_{\textbf{n},{\text {div}}}^{1,q}\}}^{\Vert \cdot \Vert _{q}}. \end{aligned}$$

Similarly, we consider the classical Sobolev space \(W^{1,q}(\Omega )\) and use the standard abbreviation for its dual space \(W^{-1,q'}:=(W^{1,q}(\Omega ))^*\). Also, in what follows whenever there is \(v\in X^*\), \(u\in X\), the symbol \(\langle v,u\rangle\) means the duality paring in X. In case there was possible ambiguity, we would write \(\langle v,u\rangle _X\). Notice that all above mentioned space are Banach spaces that are in addition separable provided that \(p,q <\infty\). In addition, they are reflexive whenever \(p,q\in (1,\infty )\). 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 [12, Lemma 1.11] or [5, Theorem 11])

$$\begin{aligned} \Vert \textbf{v}\Vert _{1,p}\le C(p) (\Vert \textbf{D}(\textbf{v})\Vert _p+\Vert \textbf{v}\Vert _2) \qquad \text {for all } \textbf{v}\in W_{\textbf{n}}^{1,p}, \end{aligned}$$
(14)

which is valid for all \(p\in (1,\infty )\) provided that \(\Omega\) is Lipschitz. Further, we also frequently use in the paper the following interpolation inequality

$$\begin{aligned} \Vert u\Vert _{\frac{p(d+2)}{d}}^{\frac{p(d+2)}{d}}\le C(\Omega ,p) \Vert u\Vert _{2}^{\frac{2p}{d}} \Vert u\Vert _{1,p}^{p}. \end{aligned}$$
(15)

2.2 Definition of weak and suitable weak solutions and main theorem

Here, we introduce the notion of weak and suitable weak solution to (9)–(12). We consider only the dimension \(d=3\).

Definition 1

(Weak solution) Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded domain of class \(C^{1,1}\) and let (0, T) with \(T>0\) be the time interval. Let \(\textbf{v}_0,\vartheta _0\) and \(\textbf{f}\) be given functions satisfying (8) and let p be given in the interval \((6/5, +\infty )\). We say that a triplet \((\textbf{v},\vartheta ,\pi )\) is a weak solution to the problem (9)–(12) if

$$\begin{aligned} \textbf{v}&\in \mathcal {C}_\textrm{weak}(0,T;L^2(\Omega ))\cap L^p(0,T;W_{\textbf{n},{\text {div}}}^{1,p})\cap L^2(0,T; L^2(\partial \Omega )^3), \end{aligned}$$
(16)
$$\begin{aligned} \mathbf{S}&\in L^{p'}(Q) \text{ and } \textbf{S} =\textbf{S} ^*(\vartheta , \textbf{D}\textbf{v}) \text{ for } \text{ a.a. } (t,x), \end{aligned}$$
(17)
$$\begin{aligned} \vartheta&\in L^\infty (0, T; L^1(\Omega ))\cap L^q(0, T; W^{1,q}(\Omega )) \text{ for } \text{ any } q\in \left[ 1, \frac{5}{4}\right) , \end{aligned}$$
(18)
$$\begin{aligned} \vartheta&\in L^{q}(Q) \text{ for } \text{ any } q\in \left[ 1, \frac{5}{3}\right) , \ \vartheta (t, x)\ge 0 \ \text {for a.a. } (t, x), \end{aligned}$$
(19)
$$\begin{aligned} \eta&\in L^\infty (0, T; L^1(\Omega )) \cap L^2(0, T; W^{1,2}(\Omega )) \text{ and } \eta =\log \vartheta \text{ for } \text{ a.a. } (t,x), \end{aligned}$$
(20)
$$\begin{aligned} \pi&\in L^{q'}(Q) \text{ with } q= \max \left\{ p, \frac{5p}{5p-6}\right\} \text { and } \int _{\Omega }\pi (t,x)\,\textrm{d} {x}=0 \quad \text {for a.a. } t, \end{aligned}$$
(21)

fulfills the following weak formulations: The linear momentum equation (9) is satisfied in the following sense

$$\begin{aligned} \begin{aligned}&-\int _0^T \int _\Omega \textbf{v}\cdot \partial _t\varvec{\varphi }\,\textrm{d} {x}\,\textrm{d} t \\&+\int _0^T\int _\Omega (\textbf{S} - \textbf{v}\otimes \textbf{v}):\textbf{D}\varvec{\varphi }\,\textrm{d} {x}\,\textrm{d} t \\&+\alpha \int _0^T\int _{\partial \Omega }\textbf{v}\cdot \varvec{\varphi }\,\textrm{d}\sigma _x \,\textrm{d} t \\&=\int _0^T\int _\Omega \pi {\text {div}}\varvec{\varphi }+ \vartheta \, \textbf{f}\cdot \varvec{\varphi }\,\textrm{d} {x}\,\textrm{d} t \\&+ \int _\Omega \textbf{v}_0\cdot \varvec{\varphi }(0)\,\textrm{d} {x}\end{aligned} \end{aligned}$$
(22)

for any \(\varvec{\varphi }\in \mathcal {C}_0^\infty ([0, T); W_{\textbf{n}}^{1, q}\cap L^{\infty }(\Omega ))\cap L^2((0,T)\times \partial \Omega )\) with \(q= \max \{p, \frac{5p}{5p-6}\}\); The global energy inequality holds in the following sense

$$\begin{aligned}{} & {} -\int _0^T \int _\Omega \left( \frac{|\textbf{v}|^2}{2}+\vartheta \right) \,\partial _t \varphi \,\textrm{d} {x}\,\textrm{d} t \nonumber \\{} & {} + \alpha \int _0^T \int _{\partial \Omega } |\textbf{v}|^2 \varphi \,\textrm{d} \sigma _x \,\textrm{d} t \le \int _\Omega \left( \frac{|\textbf{v}_0|^2}{2}+\vartheta _0\right) \, \varphi (0)\,\textrm{d} {x}\end{aligned}$$
(23)

for any nonnegative \(\varphi \in \mathcal {C}_0^\infty ([0,T))\);

The entropy inequality (13) is satisfied as

$$\begin{aligned} \begin{aligned}&-\int _0^T \int _{\Omega } \eta \, \partial _t\varphi \ \,\textrm{d} {x}\,\textrm{d} t - \int _0^T \int _{\Omega } \eta \,\textbf{v}\cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&+ \int _0^T \int _{\Omega } \kappa (\vartheta )\nabla \eta \cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&\qquad \ge \int _0^T \int _{\Omega } \frac{\textbf{S} :\textbf{D}{\textbf{v}}}{\vartheta }\,\varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&+\int _0^T \int _{\Omega } \kappa (\vartheta )\,|\nabla \eta |^2\, \varphi \ \,\textrm{d} {x}\,\textrm{d} t + \int _{\Omega } (\log \vartheta _0)\, \varphi (0) \ \,\textrm{d} {x} \end{aligned} \end{aligned}$$
(24)

for any nonnegative \(\varphi \in \mathcal {C}_0^\infty ([0, T); W^{1,\infty }({\Omega }))\). The initial conditions are attained in the following sense

$$\begin{aligned}&\lim _{t\rightarrow 0_+}\left( \Vert \textbf{v}(t)-\textbf{v}_0\Vert _2 + \Vert \vartheta (t) -\vartheta _0\Vert _1\right) =0. \end{aligned}$$
(25)

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 [18], 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

[Suitable weak solution] Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded domain of class \(C^{1,1}\) and let (0, T) with \(T>0\) be the time interval. Let \(\textbf{v}_0,\vartheta _0\) and \(\textbf{f}\) be given functions satisfying (8) and let \(p\in (9/5, +\infty )\) be given. We say that a triplet \((\textbf{v},\vartheta ,\pi )\) is a suitable weak solution to the problem (9)–(12) if Definition 1 is satisfied with (23) replaced by

$$\begin{aligned} \begin{aligned} \int _0^T \int _\Omega -\left( \frac{|\textbf{v}|^2}{2}+\vartheta \right) \,\partial _t \varphi -\left( \textbf{v}\left( \frac{|\textbf{v}|^2}{2}+\vartheta +\pi \right) +\textbf{q}-\textbf{S} \textbf{v}\right) \, \cdot \nabla \varphi \,\textrm{d} {x}\,\textrm{d} t \\ + \alpha \int _0^T \int _{\partial \Omega } |\textbf{v}|^2 \varphi \,\textrm{d} \sigma _x \,\textrm{d} t = \int _\Omega \left( \frac{|\textbf{v}_0|^2}{2}+\vartheta _0\right) \, \varphi (0)\,\textrm{d} {x}, \end{aligned} \end{aligned}$$
(26)

which is valid for any \(\varphi \in \mathcal {C}_0^\infty ([0,T);W^{1,\infty }({\Omega }))\). Moreover, we require that (11) is satisfied in the following sense

$$\begin{aligned}{} & {} \int _0^T \int _{\Omega } -\vartheta \, \partial _t \varphi -(\vartheta \textbf{v}+\textbf{q})\cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \ge \int _0^T \int _{\Omega } \textbf{S} :\textbf{D}\textbf{v}\,\varphi - \vartheta (\textbf{v}\cdot \textbf{f}) \, \varphi \ \,\textrm{d} {x}\,\textrm{d} t \nonumber \\{} & {} + \int _{\Omega } \vartheta _0\,\varphi (0) \ \,\textrm{d} {x}, \end{aligned}$$
(27)

for any nonnegative \(\varphi \in \mathcal {C}_0^\infty ([0,T);W^{1,\infty }({\Omega }))\).

Next, we formulate the main theorem of this paper.

Theorem 1

Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded domain with \(\mathcal {C}^{1,1}\) boundary. Assume that \(\textbf{S} ^*\) and \(\kappa\) satisfy (3) and (4) with \(p>6/5\). Then for any data \(\textbf{v}_0,\vartheta _0, \textbf{f}\) fulfilling (8), there exists a weak solution to (9)–(12) in the sense of Definition 1. Moreover, if \(p>8/5\) then (23) holds with the equality sign. In addition, if \(p>9/5\) then there exists a suitable weak solution in sense of Definition 2. Furthermore, if \(p\ge 11/5\), then (24) and (27) holds with equality sign and the following is true

$$\begin{aligned} \limsup _{m\rightarrow \infty }\int _{Q}\frac{m|\nabla \vartheta |^2}{\vartheta ^2} \chi _{\{\vartheta >m\}}\,\textrm{d} {x}\,\textrm{d} t =0. \end{aligned}$$
(28)

Note that in above Theorem 1 we assume a stronger assumption on the boundary, namely \(\Omega \in \mathcal {C}^{1,1}\). The reason is the necessity of having a priori estimates of the pressure \(\pi\) that appears in (23) 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 [25, 26], where the authors treated the same system but with \(\textbf{S} ^*\) 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 [25], the authors treated only the steady case and in [26], the authors were not able to show the validity of (26), 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 [22, 23], which is an extension of [25] 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 \(p\ge 11/5\), 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, (28) as well as energy equalities valid for \(p\ge 11/5\) 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 Sect. 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 Appendix A. Then, in Sect. 4.1, we derive estimates that are uniform with respect to k-parameter. Finally, letting \(k\rightarrow +\infty\) in Sect. 4.2, we complete the proof of Theorem 1.

3 Definition of approximating systems and their solutions

We start this part with definition of auxiliary cut-off functions. For any arbitrary natural number \(k\ge 1\), we define

$$\begin{aligned} \mathcal {T}_k(z) = \textrm{sign} (z) \min \{k, |z|\} \text{ for } \text{ any } z\in {{\mathbb {R}}}, \end{aligned}$$
(29)

and a function \(g_k:\mathbb {R}^{+}\rightarrow [0,1]\) such that it is continuous and satisfies

$$\begin{aligned} g_k(z)={\left\{ \begin{array}{ll} 1 &{} \text{ if } z< k,\\ 0 &{} \text{ if } z> 2k. \end{array}\right. } \end{aligned}$$
(30)

Finally, we introduce an auxiliary function \(\eta\), which is used in the proof of attainment of the initial data. Let \(T > 0\) be given, and let \(0 < \varepsilon \ll 1\) and \(t \in (0, T-\varepsilon )\) be arbitrary. Consider \(\eta \in \mathcal {C}^{0,1}([0, T])\) as a piece-wise linear function of three parameters, such that

$$\begin{aligned} \eta (\tau )={\left\{ \begin{array}{ll} 1 &{} \text {if }\tau \in [0,t),\\ 1+\frac{t-\tau }{\varepsilon } &{} \text {if } \tau \in [t,t+\varepsilon ),\\ 0 &{} \text {if } \tau \in [t+\varepsilon ,T]. \end{array}\right. } \end{aligned}$$
(31)

We define an approximative problem \(\mathcal {P}^{k}\) (for simplicity we write \((\textbf{v},\pi ,\vartheta )\) instead of \(\textbf{v}^{k},\pi ^{k},\vartheta ^{k}\)) 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:

$$\begin{aligned}&{\text {div}}\textbf{v}= 0, \end{aligned}$$
(32)
$$\begin{aligned}&\partial _t\textbf{v}+ {\text {div}}(\textbf{v}\otimes \textbf{v}\, g_{k}(|\textbf{v}|^2)) - {\text {div}}\textbf{S} + \nabla \pi = \mathcal {T}_k(\vartheta )\textbf{f}, \end{aligned}$$
(33)
$$\begin{aligned}&\partial _t\vartheta +{\text {div}}(\mathcal {T}_k(\vartheta ) \, \textbf{v}) + {\text {div}}\textbf{q} = \textbf{S} :\textbf{D}-\mathcal {T}_k(\vartheta ) (\textbf{v}\cdot \textbf{f}), \end{aligned}$$
(34)

in \((0,T)\times \Omega\) complemented with the boundary conditions

$$\begin{aligned} \textbf{v}\cdot \textbf{n}=0, \qquad \alpha \textbf{v}_{\tau }g_{k}(|\textbf{v}_{\tau }|)+[\textbf{S} \textbf{n}]_{\tau }=0,\qquad \nabla \vartheta \cdot \textbf{n}=0 \qquad \text {on } \partial \Omega , \end{aligned}$$
(35)

and the following initial conditions

$$\begin{aligned} \textbf{v}(0) = \textbf{v}_0, \qquad \vartheta (0) = \vartheta _0>0 \qquad \text {in } \Omega . \end{aligned}$$
(36)

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 1

Let \(\Omega \subset {{\mathbb {R}}}^d\) be a bounded domain with \(\mathcal {C}^{1,1}\) boundary. Assume that \(\textbf{S} ^*\) and \(\kappa\) satisfy (3) and (4) with \(p>2d/(d+2)\). Then for any \(k\in \mathbb {N}\) and any data \(\textbf{v}_0,\vartheta _0, \textbf{f}\) fulfilling (8), there exists a triplet \((\textbf{v},\vartheta ,\pi )=(\textbf{v}^k,\vartheta ^k,\pi ^k)\) satisfying

$$\begin{aligned}&\textbf{v}\in \mathcal {C}(0,T;L_{\textbf{n},{\text {div}}}^2)\cap L^p(0,T;W_{\textbf{n},{\text {div}}}^{1,p}), \end{aligned}$$
(37)
$$\begin{aligned}&\partial _t \textbf{v}\in L^{p'}(0,T;W_{\textbf{n}}^{-1,p}), \end{aligned}$$
(38)
$$\begin{aligned}&\vartheta \in L^\infty (0, T; L^1(\Omega )) \quad \text { and } \quad \vartheta (t,x)\ge 0 \quad \text {for a.a. } (t, x)\in (0, T)\times \Omega , \end{aligned}$$
(39)
$$\begin{aligned}&\log \vartheta \in L^\infty (0, T; L^1(\Omega )), \end{aligned}$$
(40)
$$\begin{aligned}&\pi \in L^{p'}(0,T;L^{p'}(\Omega )) \quad \text { and } \quad \int _{\Omega }\pi (t,x)\,\textrm{d} {x}=0 \quad \text {for a.a. } t\in (0,T), \end{aligned}$$
(41)
$$\begin{aligned}&(1+\vartheta )^{\frac{1-\varepsilon }{2}}\in L^2(0,T; W^{1,2}(\Omega )) \text{ for } \text{ any } \varepsilon >0 \end{aligned}$$
(42)

and

$$\begin{aligned} \lim _{m\rightarrow \infty } \int _{(0,T)\times \Omega \cap \{\vartheta \ge m\}} \frac{m|\nabla \vartheta |^2}{\vartheta ^2} \,\textrm{d} {x}\,\textrm{d} t =0; \end{aligned}$$
(43)

attaining the initial conditions (36) in the following sense

$$\begin{aligned} \lim _{t\rightarrow 0_+}\left( \Vert \textbf{v}-\textbf{v}_0\Vert _2+ \Vert \vartheta -\vartheta _0\Vert _1\right) =0; \end{aligned}$$
(44)

satisfying equation (33) in the following sense: for any \(\varvec{\varphi }\in W^{1,p}_{\textbf{n}}\) and for a.a. \(t\in (0, T)\) there holds

$$\begin{aligned} \begin{aligned}&\langle \partial _t\textbf{v}, \varvec{\varphi }\rangle - \int _{\Omega } g_k(|\textbf{v}|^2)\, (\textbf{v}\otimes \textbf{v}): \nabla \varvec{\varphi } \ \,\textrm{d} {x} + \int _{\Omega } \textbf{S} :\textbf{D}\varvec{\varphi } \ \,\textrm{d} {x} \\&+ \alpha \int _{\partial \Omega } \alpha \textbf{v}_{\tau }g_{k}(|\textbf{v}_{\tau }|)\cdot \varvec{\varphi }\,\textrm{d}\sigma _x\\&= \int _{\Omega } \mathcal {T}_k(\vartheta ) \,\textbf{f}\cdot \varvec{\varphi }+ \pi {\text {div}}\varvec{\varphi } \ \,\textrm{d} {x}, \end{aligned} \end{aligned}$$
(45)

and satisfying (34) in the following sense: for any \(f\in \mathcal {C}^2({{\mathbb {R}}})\) satisfying \(f''\in \mathcal {C}_0({{\mathbb {R}}})\), for any \(\varphi \in W^{1,2}(\Omega )\cap L^\infty (\Omega )\) and for a.a. \(t\in (0, T)\) there holds

$$\begin{aligned} \begin{aligned}&\langle \partial _t f(\vartheta ), \varphi \rangle - \int _{\Omega } f(\mathcal {T}_k(\vartheta ))\, \textbf{v}\cdot \nabla \varphi \ \,\textrm{d} {x} \\&+ \int _{\Omega } f'(\vartheta )\kappa (\vartheta )\nabla \vartheta \cdot \nabla \varphi \ \,\textrm{d} {x} + \int _{\Omega } f''(\vartheta )\kappa (\vartheta )|\nabla \vartheta |^2\, \varphi \ \,\textrm{d} {x}\\&= \int _{\Omega } f'(\vartheta )\,\textbf{S} :\textbf{D}\textbf{v}\,\varphi \ \,\textrm{d} {x} - \int _{\Omega } \mathcal {T}_k(\vartheta )f'(\vartheta )\,\textbf{v}\cdot \textbf{f} \,\varphi \ \,\textrm{d} {x}. \end{aligned} \end{aligned}$$
(46)

Proof

The complete proof is presented at the Appendix Appendix A for the most important case \(d=3\). For other dimensions the proof is however almost identical. \(\square\)

We would like to emphasize here, that the above existence result is in fact very strong. Although the equation (33) is satisfied in the classical weak sense (45), the equation (33) is satisfied in the renormalized weak sense as (46). This enables us to deduce the proper uniform estimates rigorously.

4 Limit in the approximating system

In the previous section we established the existence of a weak solution to the k-approximating system (32)–(34). Key k-uniform estimates and the limits as \(k\rightarrow +\infty\) 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 \((\textbf{v},\vartheta ,\pi )=(\textbf{v}^k,\vartheta ^k,\pi ^k)\) we derive estimates that are uniform with respect to k-parameter (the relevant quantities are then bounded by a generic constant C, where \(C(\Vert \textbf{f}\Vert _{\infty }, \Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1, \Vert \log \vartheta _0\Vert _1)\)).

We set \(\varvec{\varphi }=\textbf{v}\) in (45) and in (46) we set \(\varphi =1\) and \(f(\vartheta )=\vartheta \). Summing both identities and using the fact that \({\text {div}}\textbf{v}= 0\) to eliminate convective termsFootnote 1 we deduce

$$\begin{aligned} \frac{1}{2}\frac{\textrm{d}}{\,\textrm{d} t }\Vert \textbf{v}\Vert _2^2 + \frac{\textrm{d}}{\,\textrm{d} t } \Vert \vartheta \Vert _1 + \alpha \Vert \textbf{v}\sqrt{g_k(|\textbf{v}|)}\Vert _{2,\partial \Omega }^2 = 0. \end{aligned}$$

Integration with respect the time variable then leads to

$$\begin{aligned}{} & {} \sup _{t\in (0, T)} \left( \Vert \textbf{v}(t)\Vert _2^2 + \Vert \vartheta (t)\Vert _1\right) \nonumber \\{} & {} +\alpha \int _0^T \Vert \textbf{v}\sqrt{g_k(|\textbf{v}|)}\Vert _{2,\partial \Omega }^2\,\textrm{d} t \le C(\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1). \end{aligned}$$
(47)

Next, for any \(\varepsilon >0\) we set \(f(\vartheta ):= \log (\vartheta + \varepsilon )\) and \(\varphi := 1\) in (46), to deduce

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\,\textrm{d} t } \int _{\Omega } \log (\vartheta + \varepsilon ) \ \,\textrm{d} {x}= \int _{\Omega } \frac{\kappa (\vartheta )|\nabla \vartheta |^2}{(\vartheta + \varepsilon )^2} \ \,\textrm{d} {x} \\{} & {} +\int _{\Omega } \frac{\textbf{S} :\textbf{D}\textbf{v}}{\vartheta + \varepsilon } \ \,\textrm{d} {x} - \int _{\Omega } \frac{\mathcal {T}_k(\vartheta )}{\vartheta + \varepsilon } \textbf{v}\cdot \textbf{f} \ \,\textrm{d} {x}. \end{aligned}$$

Integrating this identity over the time interval (0, t), it follows

$$\begin{aligned} \begin{aligned}&\int _0^t\int _{\Omega } \frac{\kappa (\vartheta )|\nabla \vartheta |^2}{(\vartheta + \varepsilon )^2} \ \,\textrm{d} {x}\,\textrm{d} t \\&+ \int _0^t\int _{\Omega } \frac{\textbf{S} :\textbf{D}\textbf{v}}{ \vartheta + \varepsilon } \ \,\textrm{d} {x}\,\textrm{d} t + \int _{\{\vartheta (t)+\varepsilon \le 1\}} |\log (\vartheta (t) +\varepsilon )|\,\textrm{d} {x}\\&= \int _0^t\int _{\Omega } \frac{\mathcal {T}_k(\vartheta )}{\vartheta + \varepsilon }\textbf{v}\cdot \textbf{f} \ \,\textrm{d} {x}\,\textrm{d} t + \int _{\{\vartheta (t)+\varepsilon >1\}}\log (\vartheta (t) +\varepsilon )\,\textrm{d} {x}\ - \int _{\Omega } \log (\vartheta _0 +\varepsilon ) \ \,\textrm{d} {x}, \end{aligned} \end{aligned}$$

and taking the supremum over \(t\in (0, T)\), we have

$$\begin{aligned} \begin{aligned}&\int _0^T \int _{\Omega } \frac{\kappa (\vartheta )|\nabla \vartheta |^2}{(\vartheta + \varepsilon )^2} \ \,\textrm{d} {x}\,\textrm{d} t + \int _0^T \int _{\Omega } \frac{\textbf{S} :\textbf{D}(\textbf{v})}{ \vartheta + \varepsilon } \ \,\textrm{d} {x}\,\textrm{d} t \\&+ \sup _{t\in (0,T)}\Vert \log {(\vartheta (t) + \varepsilon )}\Vert _1 \\ \le C\int _0^T \int _{\Omega } |\textbf{v}\cdot \textbf{f}| \ \,\textrm{d} {x}\,\textrm{d} t +C\int _{\Omega } |\vartheta (t)| \ \,\textrm{d} {x} +\varepsilon C\, \text {meas}(\Omega )+ \Vert \log \vartheta _0\Vert _1. \end{aligned} \end{aligned}$$

Then employing (47), the fact that \(\log \vartheta _0\in L^1(\Omega )\) and taking the limit as \(\varepsilon \rightarrow 0\) we get the following k-independent estimate

$$\begin{aligned} \begin{aligned}&\sup _{t\in (0,T)}\Vert \log {(\vartheta (t))}\Vert _1+ \int _0^T \int _{\Omega } \frac{\kappa (\vartheta )|\nabla \vartheta |^2}{\vartheta ^2} \ \,\textrm{d} {x}\,\textrm{d} t + \int _0^T \int _{\Omega } \frac{\textbf{S} :\textbf{D}\textbf{v}}{ \vartheta } \ \,\textrm{d} {x}\,\textrm{d} t \\&\qquad \le C(\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1, \Vert \log \vartheta _0\Vert _1, \Vert \textbf{f}\Vert _{\infty ,Q}).\end{aligned} \end{aligned}$$
(48)

In order to improve the uniform bound for the temperature, we fix arbitrary \(\sigma \in (0, 1)\) and set \(f(\vartheta )=\vartheta ^{\sigma }\) and \(\varphi =1\) in (46) to obtain

$$\begin{aligned}{} & {} \frac{1}{\sigma }\frac{\textrm{d}}{\,\textrm{d} t } \int _{\Omega } \vartheta ^{\sigma } \ \,\textrm{d} {x} + (\sigma -1)\int _{\Omega } \kappa (\vartheta )\frac{|\nabla \vartheta |^2}{\vartheta ^2}\vartheta ^{\sigma } \ \,\textrm{d} {x}\\{} & {} = \int _{\Omega } \vartheta ^{\sigma -1} \textbf{S} :\textbf{D}\textbf{v} \ \,\textrm{d} {x} - \int _{\Omega } \frac{\mathcal {T}_k(\vartheta )}{\vartheta }\, \vartheta ^{\sigma }\,\textbf{v}\cdot \textbf{f} \ \,\textrm{d} {x}. \end{aligned}$$

Integrating it over time and using the fact that \(\sigma \in (0,1)\) and the uniform estimate (47), we deduce

$$\begin{aligned} \begin{aligned}{} & {} \int _0^T \int _{\Omega } \vartheta ^{\sigma -1} \textbf{S} :\textbf{D}\textbf{v} \ \,\textrm{d} {x}\,\textrm{d} t + \int _0^T \int _{\Omega } \kappa (\vartheta )\frac{|\nabla \vartheta |^2}{\vartheta ^2}\vartheta ^{\sigma } \ \,\textrm{d} {x}\,\textrm{d} t \\{} & {} \le C(\sigma , \Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q})\left( 1+\int _0^T \int _{\Omega } \vartheta ^{\sigma } |\textbf{v}| \ \,\textrm{d} {x}\,\textrm{d} t \right) . \end{aligned} \end{aligned}$$
(49)

To bound the term on the right hand side, we recall the interpolation inequality (here we consider \(d=3\))

$$\begin{aligned} \Vert \vartheta ^{\frac{\sigma }{2}}\Vert _4^2 \le C(\Omega )\Vert \vartheta ^{\frac{\sigma }{2}}\Vert _2^{\frac{1}{2}} \Vert \vartheta ^{\frac{\sigma }{2}}\Vert _{1,2}^{\frac{3}{2}} \end{aligned}$$

and using also the Hölder inequality, the a priori bound (47) and the assumption (3), we get the estimate

$$\begin{aligned} \begin{aligned}&C(\sigma , \Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q})\left( 1+\int _0^T\Vert \textbf{v}\Vert _2\Vert \vartheta ^{2\sigma }\Vert _1^{\frac{1}{2}}\,\textrm{d} t \right) \\&\le C\left( 1+\int _0^T\Vert \vartheta ^{\frac{\sigma }{2}}\Vert _4^{2}\,\textrm{d} t \right) \\&\quad \le C\left( 1+\int _0^T\Vert \vartheta ^{\frac{\sigma }{2}}\Vert _2^{\frac{1}{2}} \Vert \vartheta ^{\frac{\sigma }{2}}\Vert _{1,2}^{\frac{3}{2}}\,\textrm{d} t \right) \le \left( C+ \int _0^T\int _{\Omega } C(\sigma ,\underline{\kappa })\vartheta ^{\sigma }\right. \\&\left. + \frac{\underline{\kappa }}{2}|\nabla \vartheta ^{\frac{\sigma }{2}}|^2 \,\textrm{d} {x}\,\textrm{d} t \right) \\&\quad \le C(\sigma ,\Omega , \underline{\kappa }, \Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}) +\int _0^T\int _{\Omega } \frac{\kappa (\vartheta )|\nabla \vartheta |^2}{2\vartheta ^{2-\sigma }}\,\textrm{d} {x}\,\textrm{d} t . \end{aligned} \end{aligned}$$
(50)

Collecting (49) and (50) we deduce the following estimates that are uniform with respect to \(k\in {\mathbb {N}}\)

$$\begin{aligned} \int _0^T \int _{\Omega } \kappa (\vartheta )\frac{|\nabla \vartheta |^2}{\vartheta ^2}\vartheta ^{\sigma } \ \,\textrm{d} {x}\,\textrm{d} t \le C(\sigma , \underline{\kappa },\Omega ,\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}). \end{aligned}$$
(51)

Consequently, we deduced that for arbitrary \(\sigma \in (0,1)\) there holds

$$\begin{aligned} \vartheta ^{\frac{\sigma }{2}} \text{ is } \text{ uniformly } \text{ bounded } \text{ with } \text{ respect } \text{ to } k\in {\mathbb {N}} \text{ in } L^2(0, T; W^{1,2}(\Omega ))\cap L^\infty (0, T; L^2(\Omega )). \end{aligned}$$
(52)

Next, we derive the final estimates for \(\vartheta \). Using the interpolation inequality

$$\begin{aligned} \Vert z\Vert _{\frac{10}{3}}^{\frac{10}{3}}\le C(\Omega )\Vert z\Vert _2^{\frac{4}{3}}\Vert z\Vert _{1,2}^{2} \end{aligned}$$
(53)

on the function \(z:=\vartheta ^{\sigma }\), with \(\sigma :=\frac{3q}{5}\) for \(q<5/3\), we obtain from (52) that

$$\begin{aligned}{} & {} \int _0^T \int _{\Omega } \vartheta ^q \ \,\textrm{d} {x}\,\textrm{d} t = \int _0^T \int _{\Omega } \left| \vartheta ^{\frac{\sigma }{2}}\right| ^{\frac{10}{3}} \ \,\textrm{d} {x}\,\textrm{d} t \le C(q, \underline{\kappa },\Omega ,\Vert \textbf{v}_0\Vert _2,\nonumber \\{} & {} \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}) \text{ for } \text{ any } q\in [1, 5/3). \end{aligned}$$
(54)

Similarly, combining (51) and (54) and using the Hölder inequality, we get

$$\begin{aligned} \begin{aligned}&\int _0^T \int _{\Omega } |\nabla \vartheta |^r \ \,\textrm{d} {x}\,\textrm{d} t =\int _0^T \int _{\Omega } \left( \frac{|\nabla \vartheta |^2}{\vartheta ^{2-\sigma }}\right) ^{\frac{r}{2}}\vartheta ^{\frac{r(2-\sigma )}{2}} \ \,\textrm{d} {x}\,\textrm{d} t \\&\le \left( \int _0^T \int _{\Omega } \frac{|\nabla \vartheta |^2}{\vartheta ^{2-\sigma }} \ \,\textrm{d} {x}\,\textrm{d} t \right) ^{\frac{r}{2}} \left( \int _0^T \int _{\Omega } \vartheta ^{\frac{r(2-\sigma )}{(2-r)}} \ \,\textrm{d} {x}\,\textrm{d} t \right) ^{\frac{2-r}{2}} \\&\le C(r, \underline{\kappa },\Omega ,\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}) \text{ for } \text{ any } r\in [1, 5/4). \end{aligned} \end{aligned}$$
(55)

The last bound can be viewed by setting

$$\begin{aligned} q:=\frac{5-r}{3(2-r)}<\frac{5}{3} \qquad \sigma :=1-\frac{5-4r}{3r}<1 \text { provided that }r<\frac{5}{4}. \end{aligned}$$

Finally, we use that \(\vartheta ^{\frac{\sigma }{2}}\) is uniformly bounded in \(L^2(0, T; L^6(\Omega ))\) and consequently \(\vartheta ^\sigma \) is uniformly bounded in \(L^1(0, T; L^3(\Omega ))\), combine it with the interpolation inequality (valid for some \(\sigma \in (0,1)\) and \(\lambda \in (0,1)\))

$$\begin{aligned} \Vert \vartheta \Vert _2\le C\Vert \vartheta \Vert _1^\lambda \Vert \vartheta \Vert _{3\sigma }^{1-\lambda } \text{ with } \lambda =\lambda (\sigma )\in (0,1) \end{aligned}$$

and use the uniform estimates (47) and (52), we see that

$$\begin{aligned}{} & {} \int _0^T\Vert \vartheta \Vert _2\,\textrm{d} t \le C \int _0^T \Vert \vartheta \Vert _{3\sigma }^{1-\lambda }\,\textrm{d} t = C \int _0^T \Vert \vartheta ^{\frac{\sigma }{2}}\Vert _{6}^\frac{2(1-\lambda )}{\sigma }\,\textrm{d} t \le \nonumber \\{} & {} C(\underline{\kappa },\Omega ,\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}). \end{aligned}$$
(56)

Having the estimate (56), we can now proceed with further bounds on the velocity field. Setting \(\varvec{\varphi }:=\textbf{v}\) in (45) and integrating in time, it yields that

$$\begin{aligned} \begin{aligned}&\sup _{t\in (0, T)}\Vert \textbf{v}(t)\Vert ^2_2+\int _0^T \int _{\Omega } \textbf{S} :\textbf{D}\textbf{v} \ \,\textrm{d} {x}\,\textrm{d} t \le C\left( \Vert \textbf{v}_0\Vert ^2_2 + \int _0^T \int _{\Omega } \vartheta |\textbf{v}| \ \,\textrm{d} {x}\,\textrm{d} t \right) \le \\&C\left( 1+\int _0^T \Vert \vartheta \Vert _2\,\textrm{d} t \right) \\&\le C(\underline{\kappa },\Omega ,\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}). \end{aligned} \end{aligned}$$
(57)

As consequence it follows from (4b) that

$$\begin{aligned}{} & {} \int _0^T \int _{\Omega } |\textbf{S} |^{p'} \ \,\textrm{d} {x}\,\textrm{d} t + \int _0^T \int _{\Omega } |\textbf{D}\textbf{v}|^p \ \,\textrm{d} {x}\,\textrm{d} t \le C(\underline{\kappa },\nonumber \\{} & {} \underline{\nu }, \overline{\nu },\Omega ,\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}). \end{aligned}$$
(58)

The interpolation inequality

$$\begin{aligned} \Vert \textbf{v}\Vert _{\frac{5p}{3}} \le C \Vert \textbf{v}\Vert _2^{\frac{2}{5}}\Vert \nabla \textbf{v}\Vert _{p}^{\frac{3}{5}} +\Vert \textbf{v}\Vert _2 \end{aligned}$$

together with the Korn inequality (14) and the uniform estimates (57), (58) ensure that

$$\begin{aligned} \int _0^T\Vert \textbf{v}\Vert _{\frac{5p}{3}}^{\frac{5p}{3}}\,\textrm{d} t \le C(\underline{\kappa }, \underline{\nu }, \overline{\nu },\Omega ,\Vert \textbf{v}_0\Vert _2, \Vert \vartheta _0\Vert _1,\Vert \textbf{f}\Vert _{\infty ,Q}). \end{aligned}$$
(59)

We finish this part by introducing the estimates on the pressure. It follows from (45) that for all \(\varphi \in W^{2,p'}(\Omega )\) satisfying \(\nabla \varphi \cdot \textbf{n} =0\) on \(\partial \Omega \), and for almost all time \(t\in (0,T)\) there holds (see also (A.59))

$$\begin{aligned} \begin{aligned}&\int _{\Omega } \pi \, \Delta \varphi \ \,\textrm{d} {x}= \int _{\Omega } \textbf{S} : \nabla ^2 \varphi \ \,\textrm{d} {x} - \int _{\Omega } g_k(|\textbf{v}|^2) \, \textbf{v}\otimes \textbf{v}: \nabla ^2 \varphi \ \,\textrm{d} {x} \\&\quad + \alpha \int _{\partial \Omega } g_k(|\textbf{v}|)\textbf{v}\cdot \nabla \varphi \textrm{d}\sigma _x \\&- \int _{\Omega } \mathcal {T}_k(\vartheta )\textbf{f}\cdot \nabla \varphi \ \,\textrm{d} {x}. \end{aligned} \end{aligned}$$

In addition, recall that

$$\begin{aligned} \int _{\Omega } \pi \,\textrm{d} {x}=0 \qquad \text { for a.a. } t\in (0,T). \end{aligned}$$

Then, we use the fact that \(\Omega \in \mathcal {C}^{1,1}\) and the theory for Laplace equation (see also [10,11,12,13] for details) and obtain that

$$\begin{aligned}{} & {} \Vert \pi \Vert _{z'}\le C(\Vert \textbf{S} \Vert _{p'} + \Vert |\textbf{v}|^2\Vert _{\frac{5p}{6}} + \Vert \textbf{v}g_k(|\textbf{v}|)\Vert _{2, \partial \Omega } \\{} & {} +\Vert \textbf{f}\Vert _\infty \Vert \vartheta \Vert _{\frac{5}{4}}) \text{ with } z':=\min \left\{ p', \frac{5p}{6}, \frac{5}{3} \right\} . \end{aligned}$$

Applying the \(z'\)-power and integrating the result over (0, T) we finally get

$$\begin{aligned} \begin{aligned} \int _0^T\Vert \pi \Vert _{z'}^{z'}\,\textrm{d} t&\le C\left( \int _0^T 1+\Vert \textbf{S} \Vert _{p'}^{p'} + \Vert \textbf{v}\Vert _{\frac{5p}{3}}^{\frac{5p}{3}} + \Vert \textbf{v}\sqrt{g_k(|\textbf{v}|)}\Vert ^2_{2, \partial \Omega } \right. \\&\left. + \Vert \textbf{f}\Vert ^{\frac{5}{3}}_\infty \Vert \vartheta \Vert ^{\frac{5}{3}}_{\frac{5}{4}}\right) \\&\le C, \qquad \text{ for } z':=\min \left\{ p', \frac{5p}{6}, \frac{5}{3} \right\} , \end{aligned} \end{aligned}$$
(60)

where the last bound follows from the estimates (47), (51), (58) and (59).

Finally, we recall the estimates on the time derivative. By using the very classical procedure, we can deduce from (45) with the help of above uniform estimates (47), (51), (58) and (59) that

$$\begin{aligned} \int _0^T \Vert \partial _t \textbf{v}^k\Vert ^{z'}_{W_{\textbf{n}}^{-1,z'}}\,\textrm{d} t \le C, \end{aligned}$$
(61)

where \(z'\) is defined in (60). Similarly, considering (46) with \(f(s):=\log s\), we can use the above estimates (47), (51), (58) and (59) to observe that for any \(\omega >5\), we have (see e.g. [13] for similar estimate)

$$\begin{aligned} \int _0^T \Vert \partial _t \eta ^k\Vert _{W^{-1,\omega '}}\,\textrm{d} t \le C. \end{aligned}$$
(62)

4.2 Limit as \(k\rightarrow +\infty\)

Let us consider \(\varvec{\varphi }\in C_0^\infty ([0, T); W_{\textbf{n}}^{1, q}\cap L^{\infty }(\Omega ))\cap L^2((0,T)\times \partial \Omega )\) with \(q= \max \{p, \frac{5p}{5p-6}\}\) in (45), integrate it over the time interval (0, T), then after the integration by parts in the time derivative term we get

$$\begin{aligned} \begin{aligned}&-\int _0^T \int _{\Omega } \textbf{v}^k\cdot \partial _t \varvec{\varphi } \ \,\textrm{d} {x}\,\textrm{d} t \\&- \int _0^T \int _{\Omega } g_k(|\textbf{v}^k|^2)\, (\textbf{v}^k\otimes \textbf{v}^k): \nabla \varvec{\varphi } \ \,\textrm{d} {x}\,\textrm{d} t + \int _0^T \int _{\Omega } \textbf{S} ^k:\textbf{D}\varvec{\varphi } \ \,\textrm{d} {x}\,\textrm{d} t \\&\qquad + \alpha \int _0^T\int _{\partial \Omega } g_k(|\textbf{v}^k|)\textbf{v}^k \cdot \varvec{\varphi }\,\textrm{d}\sigma _x\,\textrm{d} t = \int _0^T \int _{\Omega } \mathcal {T}_k(\vartheta ^k) \,\textbf{f}\cdot \varvec{\varphi }\\&+ \pi ^k {\text {div}}\varvec{\varphi } \ \,\textrm{d} {x}\,\textrm{d} t + \int _{\Omega } \textbf{v}_0^k\cdot \varvec{\varphi }(0) \ \,\textrm{d} {x}, \end{aligned} \end{aligned}$$
(63)

where we abbreviate \(\textbf{S} ^k=\textbf{S} ^*(\vartheta ^k, \textbf{D}\textbf{v}^k).\) Next, we consider \(\varphi \in \mathcal {C}_0^\infty ([0,T);\mathcal {C}^{\infty }({\Omega }))\) in (46), integrate it over the time interval (0, T), and after the integration by parts with respect to the time variable, we get

$$\begin{aligned} \begin{aligned}&-\int _0^T \int _{\Omega } f(\vartheta ^k) \partial _t \varphi \ \,\textrm{d} {x}\,\textrm{d} t - \int _0^T \int _{\Omega } f(\mathcal {T}_k(\vartheta ^k))\, \textbf{v}^k\cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&\qquad + \int _0^T \int _{\Omega } f'(\vartheta ^k)\kappa (\vartheta ^k)\nabla \vartheta ^k \cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&+ \int _0^T \int _{\Omega } f''(\vartheta ^k)\kappa (\vartheta ^k)|\nabla \vartheta ^k|^2\, \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&= \int _0^T \int _{\Omega } f'(\vartheta ^k)\,\textbf{S} ^k:\textbf{D}\textbf{v}^k \,\varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&-\int _0^T \int _{\Omega } \mathcal {T}_k(\vartheta ^k)f'(\vartheta ^k)\,\textbf{v}^k\cdot \textbf{f} \,\varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&+\int _{\Omega } f(\vartheta ^k_0) \varphi (0) \ \,\textrm{d} {x}. \end{aligned} \end{aligned}$$
(64)

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

$$\begin{aligned} \begin{aligned}&-\int _0^T \int _{\Omega } \vartheta ^k \partial _t \varphi \ \,\textrm{d} {x}\,\textrm{d} t - \int _0^T \int _{\Omega } \mathcal {T}_k(\vartheta ^k)\, \textbf{v}^k\cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&+ \int _0^T \int _{\Omega } \kappa (\vartheta ^k)\nabla \vartheta ^k \cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&\qquad = \int _0^T \int _{\Omega } \textbf{S} ^k:\textbf{D}\textbf{v}^k \,\varphi \ \,\textrm{d} {x}\,\textrm{d} t - \int _0^T \int _{\Omega } \mathcal {T}_k(\vartheta ^k)\,\textbf{v}^k\cdot \textbf{f} \,\varphi \ \,\textrm{d} {x}\,\textrm{d} t + \int _{\Omega } \vartheta ^k_0\, \varphi (0) \ \,\textrm{d} {x}. \end{aligned} \end{aligned}$$
(65)

With the second choice, we also use the abbreviation \(\eta ^k=\log \vartheta ^k\) and we see that

$$\begin{aligned} \begin{aligned}&-\int _0^T \int _{\Omega } \eta ^k \, \partial _t \varphi \ \,\textrm{d} {x}\,\textrm{d} t - \int _0^T \int _{\Omega } \log (\mathcal {T}_k(\vartheta ^k))\, \textbf{v}^k\cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&\qquad + \int _0^T \int _{\Omega } \kappa (\vartheta ^k)\nabla \eta ^k \cdot \nabla \varphi \ \,\textrm{d} {x}\,\textrm{d} t - \int _0^T \int _{\Omega } \kappa (\vartheta ^k)|\nabla \eta ^k|^2\, \varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&\quad = \int _0^T \int _{\Omega } \frac{1}{\vartheta ^k}\,\textbf{S} ^k:\textbf{D}\textbf{v}^k \,\varphi \ \,\textrm{d} {x}\,\textrm{d} t \\&-\int _0^T \int _{\Omega } \frac{\mathcal {T}_k(\vartheta ^k)}{\vartheta ^k}\,\textbf{v}^k\cdot \textbf{f} \,\varphi \ \,\textrm{d} {x}\,\textrm{d} t + \int _{\Omega } \eta ^k_0 \varphi (0) \ \,\textrm{d} {x}. \end{aligned} \end{aligned}$$
(66)

Finally, to get the energy identity, we consider \(\varphi \in \mathcal {C}^\infty ((0,T)\times \Omega )\) fulfilling \(\varphi (T)=0\) and use \(\textbf{v}^k \varphi\) and \(\varphi\) as test functions in (63) and (65) respectively. Doing this and then taking the sum of the outcome we deduce that (using also the fact that \({\text {div}}\textbf{v}^k=0\) and integration by partsFootnote 2)

$$\begin{aligned} \begin{aligned}&\int _0^T \int _\Omega -\left( \frac{|\textbf{v}^k|^2}{2}+\vartheta \right) \,\partial _t \varphi \,\textrm{d} {x}\,\textrm{d} t \\&+ \alpha \int _0^T \int _{\partial \Omega } g_k(|\textbf{v}^k|)|\textbf{v}^k|^2 \varphi \,\textrm{d} \sigma _x \,\textrm{d} t \\&\quad +\int _0^T \int _{\Omega } \left( -\textbf{v}^k\left( \frac{2g_k(|\textbf{v}^k|^2)\, |\textbf{v}^k|^2 -\mathcal {G}_k(|\textbf{v}^k|^2)}{2}+\mathcal {T}_k(\vartheta ^k)+\pi ^k\right) +\kappa (\vartheta ^k)\nabla \vartheta ^k+\textbf{S} ^k \textbf{v}^k\right) \, \cdot \nabla \varphi \,\textrm{d} {x}\,\textrm{d} t \\&= \int _\Omega \left( \frac{|\textbf{v}_0^k|^2}{2}+\vartheta _0\right) \, \varphi (0)\,\textrm{d} {x}, \end{aligned} \end{aligned}$$
(67)

where \(\mathcal {G}_k\) is such that \(\mathcal {G}_k'(s)=g_k(s)\). In particular, for any \(\varphi \in \mathcal {C}_0^\infty ([0,T))\), i.e. \(\varphi\) independent of spatial variable, there holds

$$\begin{aligned}&-\int _0^T \int _\Omega \left( \frac{|\textbf{v}^k|^2}{2}+\vartheta ^k\right) \,\partial _t \varphi \,\textrm{d} {x}\,\textrm{d} t \nonumber \\&+ \alpha \int _0^T \int _{\partial \Omega } g_k(|\textbf{v}^k|)|\textbf{v}^k|^2 \varphi \,\textrm{d} \sigma _x \,\textrm{d} t = \int _\Omega \left( \frac{|\textbf{v}_0^k|^2}{2}+\vartheta _0^k\right) \, \varphi (0)\,\textrm{d} {x}. \end{aligned}$$
(68)

We want to discuss the limit in formulations (63) and (65)–(68). By virtue of the uniform estimates (57), (58) we can extract a subsequence \((\textbf{v}^k,\vartheta ^k, \pi ^k, \textbf{S} ^k, \eta ^k)\) such that the following convergence results hold

$$\begin{aligned} \textbf{v}^k&\rightharpoonup ^* \textbf{v} \text{ weakly-* } \text{ in } L^\infty (0, T; L^2(\Omega ; {{\mathbb {R}}}^3)), \end{aligned}$$
(69)
$$\begin{aligned} \textbf{v}^k&\rightharpoonup \textbf{v} \text{ weakly } \text{ in } L^p(0, T; W^{1,p}_{\textbf{n},{\text {div}}}), \end{aligned}$$
(70)
$$\begin{aligned} \partial _t \textbf{v}^k&\rightharpoonup \partial _t \textbf{v} \text{ weakly } \text{ in } L^{q'}(0, T; W^{-1,q'}_{\textbf{n}}) \text{ with } q'=\min \left\{ p', \frac{5p}{6}, \frac{5}{3} \right\} , \end{aligned}$$
(71)
$$\begin{aligned} g_k(|\textbf{v}^k|)\textbf{v}^k&\rightharpoonup \textbf{v} \text{ weakly } \text{ in } L^2(0,T; L^2(\partial \Omega ; {{\mathbb {R}}}^3)), \end{aligned}$$
(72)
$$\begin{aligned} \textbf{S} ^k&\rightharpoonup \textbf{S} \text{ weakly } \text{ in } L^{p'}(Q; {{\mathbb {R}}}^{3\times 3}), \end{aligned}$$
(73)
$$\begin{aligned} \pi ^k&\rightharpoonup \pi \text{ weakly } \text{ in } L^{q'}(Q) \text{ with } q'=\min \{p', \frac{5p}{6}, \frac{5}{3} \}, \end{aligned}$$
(74)
$$\begin{aligned} (\vartheta ^k)^{\sigma }&\rightharpoonup \overline{(\vartheta )^{\sigma }} \text{ weakly } \text{ in } L^2(0, T; W^{1,2}(\Omega )) \text{ for } \text{ any } \sigma \in (0, {1}/{2}), \end{aligned}$$
(75)
$$\begin{aligned} \vartheta ^k&\rightharpoonup \vartheta \text{ weakly } \text{ in } L^q(0, T; W_0^{1, s}(\Omega )) \text{ for } \text{ any } q\in [1, 5/4). \end{aligned}$$
(76)
$$\begin{aligned} \eta ^k&\rightharpoonup \eta \text{ weakly } \text{ in } L^{2}(0,T; W^{1,2}(\Omega )). \end{aligned}$$
(77)

Moreover, employing the Aubin–Lions compactness Lemma we deduce that

$$\begin{aligned}&\textbf{v}^k\rightarrow \textbf{v} \text{ strongly } \text{ in } L^q(Q; {{\mathbb {R}}}^3) \text{ for } \text{ any } q\in [1, {5p}/{3}) \text{ and } \text{ a.e. } \text{ in } \text{ Q }, \end{aligned}$$
(78)
$$\begin{aligned}&\textbf{v}^k\rightarrow \textbf{v} \text{ strongly } \text{ in } L^p((0, T; L^1(\partial \Omega )) \nonumber \\&\text{ and } \text{ a.e. } \text{ in } (0, T)\times \partial \Omega , \end{aligned}$$
(79)
$$\begin{aligned}&\vartheta ^k \rightarrow \vartheta \text{ strongly } \text{ in } L^q(Q) \text{ for } \text{ any } q\in [1, 5/3) \text{ and } \text{ a.e. } \text{ in } \text{ Q }. \end{aligned}$$
(80)

Consequently, we have that \(\eta =\log \vartheta\) and \(\overline{(\vartheta )^{\sigma }}=(\vartheta )^{\sigma }\). 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 \(\textbf{S} =\textbf{S} ^*(\vartheta , \textbf{D}\textbf{v})\). This identification follows from the procedure developed in [15], see also [11]. Indeed, there is shown that there exists a nondecreasing sequence of measurable sets \(\{Q_n\}_{n=1}^{\infty }\) fulfilling \(\lim _{n\rightarrow \infty }|Q{\setminus } Q_n|=0\) such that for every \(n\in \mathbb {N}\) we have

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _{Q_n} \textbf{S} ^k: (\textbf{D}\textbf{v}^k - \textbf{D}\textbf{v}) \,\textrm{d} {x}\,\textrm{d} t = 0. \end{aligned}$$
(81)

Further, using the growth assumption (4b), the convergence result (80), the fact that \(\textbf{D}\textbf{v}\in L^{p'}(Q; {{\mathbb {R}}}^{3\times 3})\) and the Lebesgue dominated convergence theorem, we obtain

$$\begin{aligned} \textbf{S} ^*(\vartheta ^k, \textbf{D}\textbf{v})&\rightarrow \textbf{S} ^*(\vartheta , \textbf{D}\textbf{v}){} & {} \text{ strongly } \text{ in } L^{p'}(Q; {{\mathbb {R}}}^{3\times 3}). \end{aligned}$$
(82)

Thus, using the monotonicity assumption (4a), the weak convergence result (70) and (82), we observe that for any \(n\in \mathbb {N}\)

$$\begin{aligned} \begin{aligned}&\lim _{k\rightarrow \infty }\int _{Q_n}|(\textbf{S} ^k - \textbf{S} ^*(\vartheta ^k, \textbf{D}\textbf{v})):(\textbf{D}\textbf{v}^k - \textbf{D}\textbf{v})| \,\textrm{d} {x}\,\textrm{d} t \\&\quad =\lim _{k\rightarrow \infty }\int _{Q_n}(\textbf{S} ^k - \textbf{S} ^*(\vartheta ^k, \textbf{D}\textbf{v})):(\textbf{D}\textbf{v}^k - \textbf{D}\textbf{v}) \,\textrm{d} {x}\,\textrm{d} t \overset{(4.35),(4.36)}{=}0. \end{aligned} \end{aligned}$$
(83)

Hence, we have that

$$\begin{aligned} (\textbf{S} ^k - \textbf{S} ^*(\vartheta ^k, \textbf{D}\textbf{v})):(\textbf{D}\textbf{v}^k - \textbf{D}\textbf{v})&\rightarrow 0 \text{ strongly } \text{ in } L^{1}(Q_n). \end{aligned}$$

Thus, it is a simple consequence of (82) and (70) that for every \(n\in \mathbb {N}\)

$$\begin{aligned} \textbf{S} ^k:\textbf{D}\textbf{v}^k&\rightharpoonup \textbf{S} ^*(\vartheta , \textbf{D}\textbf{v})): \textbf{D}\textbf{v} \text{ weakly } \text{ in } L^{1}(Q_n). \end{aligned}$$
(84)

Finally, using (70), (73) and (84), we have for arbitrary \(\textbf{B}\in L^p(Q; {{\mathbb {R}}}^{3\times 3})\) that

$$\begin{aligned} 0\le (\textbf{S} ^k - \textbf{S} ^*(\vartheta ^k, \textbf{B})): (\textbf{D}\textbf{v}^k - \textbf{B})\rightharpoonup (\textbf{S} - \textbf{S} ^*(\vartheta , \textbf{B})): (\textbf{D}\textbf{v}- \textbf{B})\text { weakly in }L^1(Q_n). \end{aligned}$$

Hence, using the Minty method, i.e. setting \(\textbf{B}:=\textbf{D}\textbf{v}\pm \varepsilon \textbf{C}\) in the above inequality, dividing by \(\varepsilon >0\) and then letting \(\varepsilon \rightarrow 0_+\), we have

$$\begin{aligned} 0\le \pm \left( \textbf{S} - \textbf{S} ^*(\vartheta , \textbf{D}\textbf{v})\right) : \textbf{C} \text { a.e. in } Q_n. \end{aligned}$$

Since \(\textbf{C}\) is arbitrary and \(|Q\setminus Q_n|\rightarrow 0\) as \(n\rightarrow \infty\), we observe from the above inequality that \(\textbf{S} =\textbf{S} ^*(\vartheta , \textbf{D}\textbf{v})\).

Having identified the nonlinearity \(\mathbf{S}\), we may now focus on the limiting procedure in desired equations. Indeed, it is now easy to use (69)–(80) and to let \(k\rightarrow \infty\) in (63) to deduce (22). Here, it is essential that \(\textbf{v}^k\) converges strongly in \(L^{2+\varepsilon }(Q)\), which is due to the assumption \(p>6/5\). Next, we let \(k\rightarrow \infty\) in (68). Using (78) and (80), 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 (79) and the Fatou lemma to obtain the inequality (23). In order to obtain the equality sign in (23), we can use the result in [12], where it is shown thatFootnote 3

$$\begin{aligned} \textbf{v}^k \rightarrow \textbf{v}\text { strongly in } L^2(0,T; L^2(\partial \Omega )^3), \end{aligned}$$
(85)

provided that \(p>8/5\). Consequently, we obtain (23) with the equality sign.

Next, we focus on the energy and the entropy (in)equalities (24) and (27). 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 \(\varphi \in L^{\infty }(Q)\). Then using (4a), (4b) and the convergence result (84), we deduce that for any \(n\in \mathbb {N}\)

$$\begin{aligned} \begin{aligned}&\liminf _{k\rightarrow +\infty }\int _0^T \int _{\Omega } \textbf{S} ^k:\textbf{D}\textbf{v}^k\, \varphi \ \,\textrm{d} {x}\,\textrm{d} t \ge \liminf _{k\rightarrow +\infty }\\&\int _{Q_n}\textbf{S} ^k:\textbf{D}\textbf{v}^k\, \varphi \,\textrm{d} {x}\,\textrm{d} t =\int _{Q_n}\textbf{S} :\textbf{D}\textbf{v}\, \varphi \,\textrm{d} {x}\,\textrm{d} t . \end{aligned} \end{aligned}$$

Consequently, letting \(n\rightarrow \infty\) in the above inequality, we deduce

$$\begin{aligned} \begin{aligned}&\liminf _{k\rightarrow +\infty }\int _0^T \int _{\Omega } \textbf{S} ^k:\textbf{D}\textbf{v}^k\, \varphi \ \,\textrm{d} {x}\,\textrm{d} t \ge \\&\int _0^T \int _{\Omega } \textbf{S} :\textbf{D}\textbf{v}\, \varphi \ \,\textrm{d} {x}\,\textrm{d} t . \end{aligned} \end{aligned}$$
(86)

Similarly, using in addition (80) and the nonnegativity of \(\vartheta ^k\), we deduce

$$\begin{aligned} \begin{aligned} \liminf _{k\rightarrow +\infty }\int _0^T \int _{\Omega } \frac{\textbf{S} ^k:\textbf{D}\textbf{v}^k\, \varphi }{\vartheta ^k} \ \,\textrm{d} {x}\,\textrm{d} t \ge \int _0^T \int _{\Omega } \frac{\textbf{S} :\textbf{D}\textbf{v}\, \varphi }{\vartheta } \ \,\textrm{d} {x}\,\textrm{d} t . \end{aligned} \end{aligned}$$
(87)

Very similarly, we can use the weak lower semicontinuity, the weak convergence (77), the pointwise convergence of \(\vartheta ^k\) (80) and the boundedness of \(\kappa (\cdot )\), see (3), there holds

$$\begin{aligned}{} & {} \int _0^T \int _{\Omega } \kappa (\vartheta )|\nabla \eta |^2 \, \varphi \ \,\textrm{d} {x}\,\textrm{d} t \le \liminf _{k\rightarrow +\infty } \nonumber \\{} & {} \int _0^T \int _{\Omega } \kappa (\vartheta ^k)|\nabla \eta ^k|^2 \, \varphi \ \,\textrm{d} {x}\,\textrm{d} t . \end{aligned}$$
(88)

These convergence results are sufficient to take the limit in (66) and to obtain (24). Then we can proceed to the limit in the equation (65) to deduce (27), provided we show

$$\begin{aligned} \vartheta ^k \textbf{v}^k \rightarrow \vartheta \textbf{v}\text { stronlgy in }L^1(Q;{{\mathbb {R}}}^3). \end{aligned}$$
(89)

To show the above convergence result, we recall the strong convergence results (78) and (80). Thus, it is sufficient to show that \(\vartheta ^k \textbf{v}^k\) is uniformly bounded in \(L^{1+\varepsilon }(Q)\) for some \(\varepsilon >0\). Using the Hölder inequality and the classical interpolation, we have (here \(\sigma \in (0,1)\) will be specified later, but typically it will be almost equal to one)

$$\begin{aligned} \begin{aligned}&\int _0^T \int _{\Omega } |\vartheta ^k|^{1+\varepsilon }|\textbf{v}^k|^{1+\varepsilon }\,\textrm{d} {x}\,\textrm{d} t \le \int _{0}^T \Vert \vartheta ^k\Vert ^{1+\varepsilon }_{2(1+\varepsilon )}\Vert \textbf{v}^k\Vert ^{1+\varepsilon }_{2(1+\varepsilon )}\,\textrm{d} t \\&=\int _{0}^T \Vert (\vartheta ^k)^{\sigma }\Vert ^{\frac{1+\varepsilon }{\sigma }}_{\frac{2(1+\varepsilon )}{\sigma }}\Vert \textbf{v}^k\Vert ^{1+\varepsilon }_{2(1+\varepsilon )}\,\textrm{d} t \\&\le C\int _{0}^T \Vert (\vartheta ^k)^{\sigma }\Vert ^{\frac{1+\varepsilon }{\sigma }-\frac{3(2+2\varepsilon -\sigma )}{4\sigma }}_{1} \Vert (\vartheta ^k)^{\sigma }\Vert ^{\frac{3(2+2\varepsilon -\sigma )}{4\sigma }}_{3}\Vert \textbf{v}^k\Vert ^{1+\varepsilon -\frac{3\varepsilon p}{5p-6}}_{2} \Vert \textbf{v}^k\Vert ^{\frac{3\varepsilon p}{5p-6}}_{1,p}\,\textrm{d} t \\&\overset{(4.1)}{\le }C\int _{0}^T \Vert (\vartheta ^k)^{\sigma }\Vert ^{\frac{3(2+2\varepsilon -\sigma )}{4\sigma }}_{3} \left( \Vert \textbf{v}^k\Vert ^p_{1,p}\right) ^{\frac{3\varepsilon }{5p-6}}\,\textrm{d} t \le C, \end{aligned} \end{aligned}$$

provided that (we are using the uniform bounds coming from (70) and (75))

$$\begin{aligned} \frac{3\varepsilon }{5p-6}+\frac{3(2+2\varepsilon -\sigma )}{4\sigma } \le 1. \end{aligned}$$
(90)

Note that if

$$\begin{aligned} \varepsilon < \frac{5p-6}{12}, \end{aligned}$$

then we can always find \(\sigma \in (0,1)\) such that the inequality (90) is satisfied. As a consequence, we deduce (89). Thus, we may proceed to the limit also in (65) to obtain (27).

Next, we want to let \(k\rightarrow \infty\) in (67) to get (26). We already discuss almost all terms except the one \(|\textbf{v}^k|^2\textbf{v}\). Thus, for identification of the limit also in this term, we need that \(\textbf{v}^k\) is compact at least in \(L^3(Q)\). Comparing it with the strong convergence result (78), 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 \(p\ge 11/5\), one may follow the standard monotone operator theory and to conclude that (84) holds true even in \(L^1(Q)\) (and not only in a subset \(Q_n\subset Q\)). Thus, we have the weak convergence on the whole set Q and therefore we are able to identify the limits in terms containing \(\textbf{S} ^k:\textbf{D}\textbf{v}^k\) with equality signs and consequently, we may consider equalities in (24) and (27). The relation (28) is proved in the same way as (43), see the computation following (A.57).

We also omit the proof of attaining of the initial conditions (25) and refer the reader e.g. to [13]. The proofs of both main theorems are thus complete.