Unsteady non-Newtonian fluid flow with heat transfer and Tresca’s friction boundary conditions

Abstract

We consider an unsteady non-isothermal flow problem for a general class of non-Newtonian fluids. More precisely the stress tensor follows a power law of parameter p, namely $\sigma = 2 \mu ( \theta , \upsilon , \| D(\upsilon ) \|) \|D( \upsilon ) \|^{p-2} D(\upsilon ) - \pi \mathrm{Id}$ where θ is the temperature, π is the pressure, υ is the velocity, and $D(\upsilon )$ is the strain rate tensor of the fluid. The problem is then described by a non-stationary p-Laplacian Stokes system coupled to an $L^{1}$-parabolic equation describing thermal effects in the fluid. We also assume that the velocity field satisfies non-standard threshold slip-adhesion boundary conditions reminiscent of Tresca’s friction law for solids. First, we consider an approximate problem $(P_{\delta })$, where the $L^{1}$ coupling term in the heat equation is replaced by a bounded one depending on a small parameter $0 < \delta \ll 1$, and we establish the existence of a solution to $(P_{\delta })$ by using a fixed point technique. Then we prove the convergence of the approximate solutions to a solution to our original fluid flow/heat transfer problem as δ tends to zero.

1 Introduction

Motivated by lubrication or injection/extrusion industrial processes, we consider in this paper an unsteady incompressible non-isothermal flow problem with non-linear boundary conditions of friction type for a general class of non-Newtonian fluids. More precisely, we assume that the stress tensor is given by

$$\begin{aligned} \sigma = 2 \mu \bigl( \theta , \upsilon , \bigl\Vert D( \upsilon ) \bigr\Vert \bigr) \bigl\Vert D( \upsilon ) \bigr\Vert ^{p-2} D(\upsilon ) - \pi \mathrm{Id}_{\mathbb{R}^{3}}, \end{aligned}$$

(1.1)

where μ is a given mapping, θ is the temperature, π is the pressure, υ is the velocity, $D(\upsilon )$ is the strain rate tensor, and $p \in (1, + \infty )$ is a real parameter.

When $p>2$, this non-linear power-law models the behavior of dilatant (or shear thickening) fluids like colloidal fluids, while the case $p \in (1,2)$ gives a description of pseudo-plastic (or shear thinning) fluids like molten polymers [1, 3, 25, 30, 33, 37].

When $p=2$, the relationship between the stress tensor, the strain rate tensor and the pressure is still non-linear since the viscosity mapping μ depends on θ, υ, and $\vert D(\upsilon )\vert$, and we obtain a constitutive law that allows considering non-Newtonian fluids like oils [22].

Let us mention that (1.1) corresponds to a quasi-linear version of the Newton constitutive law and is also called the generalized Newtonian fluid model.

Several experimental studies have shown that such complex fluids exhibit a non-standard behavior at the boundary with threshold slip-adhesion phenomena reminiscent of Tresca’s friction law for solids [4, 13, 21, 26, 27, 35]. The first existence results for this kind of boundary conditions have been obtained by H. Fujita in [14–19, 31, 32] for stationary Newtonian Stokes flows and developed later on for steady and unsteady Newtonian fluid flows [7, 8, 23, 24, 36].

The case of stationary non-Newtonian fluids satisfying the general power law (1.1) is considered in [9], and thermal effects lead to a coupled fluid flow/heat transfer problem.

This paper aims to extend this result to the non-stationary case. More precisely, we consider the fluid flow domain

$$\begin{aligned} \Omega = \bigl\{ \bigl(x' , x_{3} \bigr) \in \mathbb{R}^{2}\times \mathbb{R} : x' \in \omega , 0< x_{3} < h \bigl(x' \bigr) \bigr\} , \end{aligned}$$

where ω is a non-empty bounded domain of $\mathbb{R}^{2}$ with a Lipschitz continuous boundary, and h is a Lipschitz continuous function bounded from above and below by some positive real numbers.

The conservation of mass and momentum and the energy conservation law yield the following p-Laplacian Stokes system:

$$\begin{aligned} \textstyle\begin{cases} \frac{\partial \upsilon }{\partial t}-2\operatorname{div} ( \mu (\theta , \upsilon , \Vert D(\upsilon ) \Vert ) \Vert D( \upsilon ) \Vert ^{p-2}D(\upsilon ) ) +\nabla \pi =f \quad \text{in } (0,T) \times \Omega, \\ \operatorname{div}(\upsilon )=0\quad \text{in } (0,T) \times \Omega, \end{cases}\displaystyle \end{aligned}$$

(1.2)

where $(0,T)$ is a non-trivial time interval, and f describes the external forces coupled to the following heat equation:

$$\begin{aligned} c \frac{\partial \theta }{\partial t} - \operatorname{div}(K \nabla \theta ) = 2\mu \bigl(\theta ,\upsilon , \bigl\Vert D(\upsilon ) \bigr\Vert \bigr) \bigl\Vert D( \upsilon ) \bigr\Vert ^{p} +r(\theta ) \quad \text{in } (0,T) \times \Omega , \end{aligned}$$

(1.3)

where c is the heat capacity, K is the thermal conductivity tensor, and r is a real function.

Let us observe that we take into account only two kinds of coupling effects in this description of the fluid flow/heat transfer problem. More precisely, in (1.2), the viscosity mapping μ depends on the temperature, and in (1.3), the right-hand side contains the heat source term $2\mu (\theta ,\upsilon , \Vert D(\upsilon ) \Vert ) \Vert D(\upsilon ) \Vert ^{p}$, which describes the heat generation due to inner friction. Indeed, in this first attempt for an existence result generalizing [9] to the non-stationary case, we focus in this paper on the difficulty due to the $L^{1}$ right-hand side in (1.3), which will be dealt with through the truncation technique introduced by L. Boccardo and T. Gallouët in [6].

Hence, we choose to neglect convective effects both in the momentum equation and in the heat equation. A more complete description, including the convective terms, could be handled with the same proof strategy (with more technicalities) under some restrictive conditions on the value of the parameter p and some compatibility conditions between the regularity properties of the fluid velocity and temperature fields. Similarly, we choose to neglect the semilinear temperature-dependent term modeling the buoyancy force in the momentum equation.

We decompose the boundary of Ω as $\partial \Omega = \Gamma _{0}\cup \Gamma _{L}\cup \Gamma _{1}$ with

$$\begin{aligned} \Gamma _{0} = \bigl\{ \bigl(x' , x_{3} \bigr) \in \overline{\Omega } : x_{3} =0 \bigr\} , \quad\quad \Gamma _{1}= \bigl\{ \bigl(x' , x_{3} \bigr) \in \overline{\Omega } : x_{3} = h \bigl(x' \bigr) \bigr\} , \end{aligned}$$

and $\Gamma _{L}$ is the lateral part of ∂Ω. We introduce a function $g : \partial \Omega \to \mathbb{R}^{3}$ such that

$$\begin{aligned} \int _{\Gamma _{L}} g\cdot n\,dY = 0,\quad\quad g=0\quad \text{on }\Gamma _{1}, \quad\quad g\cdot n=0 \quad \text{on } \Gamma _{0}, \quad\quad g\neq 0 \quad \text{on } \Gamma _{L}, \end{aligned}$$

(1.4)

where $n= (n_{1} , n_{2} , n_{3} )$ is the unit outward normal vector to ∂Ω, and $g \cdot n$ denotes the Euclidean inner product of the vectors g and n in $\mathbb{R}^{3}$. We define by $\upsilon _{n}= \upsilon \cdot n$ and $\upsilon _{\tau } = \upsilon - \upsilon _{n} n$ the normal and the tangential velocities on ∂Ω. The normal and tangential components of the stress vector on ∂Ω are given by $\sigma _{n}$ and $\sigma _{\tau }$ with

$$\begin{aligned} \sigma _{n} = \sum_{i,j=1}^{3} \sigma _{ij} n_{j}n_{i}, \qquad \sigma _{\tau } = \Biggl( \sum_{j=1}^{3} \sigma _{ij} n_{j} -\sigma _{n}n_{i} \Biggr)_{1 \le i \le 3}. \end{aligned}$$

As usual in lubrication or extrusion/injection problems, the upper part of the boundary is a fixed wall, while the lower part is a moving device. Hence, we assume that the fluid is subjected to the non-homogeneous Dirichlet boundary conditions on $\Gamma _{1} \cup \Gamma _{L}$ and to non-linear slip boundary conditions of friction type on $\Gamma _{0}$, i.e.,

$$\begin{aligned} \upsilon =0 \quad \text{on } (0,T) \times \Gamma _{1} ,\qquad \upsilon =g\xi \quad \text{on } (0,T) \times \Gamma _{L}, \end{aligned}$$

(1.5)

where ξ is a function depending only on the time variable such that

$$\begin{aligned} \xi (0)=1 \end{aligned}$$

(1.6)

and

$$\begin{aligned}& \upsilon _{n}=0\quad \text{on } (0,T) \times \Gamma _{0} \quad \text{(slip condition)}, \end{aligned}$$

(1.7)

$$\begin{aligned}& \left . \textstyle\begin{array}{l} \Vert \sigma _{\tau } \Vert =k\quad \Rightarrow \quad \exists \lambda \geq 0 \quad \upsilon _{\tau }=s-\lambda \sigma _{\tau }, \\ \Vert \sigma _{\tau } \Vert < k\quad \Rightarrow \quad \upsilon _{\tau }=s \end{array}\displaystyle \right \} \quad \text{on } (0,T) \times \Gamma _{0} \quad \text{(Tresca's law)}, \end{aligned}$$

(1.8)

where s is the sliding velocity of the lower part of the boundary, and k is the positive friction threshold.

Moreover, we assume the mixed Dirichlet–Neumann boundary conditions on $\Gamma _{1} \cup \Gamma _{L}$ and $\Gamma _{0}$ for the temperature, i.e.,

$$\begin{aligned} \theta =0\quad \text{on } (0,T) \times ( \Gamma _{1}\cup \Gamma _{L}), \qquad ( K\nabla \theta ) \cdot n =\theta ^{b} \quad \text{on } (0,T) \times \Gamma _{0}, \end{aligned}$$

(1.9)

where $\theta ^{b}$ is a given heat flux on $\Gamma _{0}$.

The paper is organized as follows: In Sect. 2, we introduce the functional framework and derive the mathematical formulation of the problem as a non-linear parabolic variational inequality for the velocity and pressure fields coupled to a non-linear parabolic equation for the temperature. By observing that the right-hand side of the heat equation belongs to $L^{1} ( 0,T; L^{1}(\Omega ) )$, we introduce in Sect. 3 an approximate problem $(P_{\delta })$, where the $L^{1}$ coupling term is replaced by a bounded one depending on a small parameter $0< \delta \ll 1$, and we establish the existence of a solution to $(P_{\delta })$ by using a fixed point technique. Finally, in Sect. 4, we prove that the approximate solutions $(\upsilon _{\delta }, \pi _{\delta }, \theta _{\delta })$ converge to a solution to our original fluid flow/heat transfer problem as δ tends to zero.

2 Mathematical formulation of the problem

Throughout the paper, we will denote by X the functional space $X^{3}$.

In order to describe the fluid flow problem, we introduce the following subspaces of $\mathbf{W}^{1,p}(\Omega )$:

$$\begin{aligned}& V^{p}_{\Gamma _{1}}= \bigl\{ \varphi \in \mathbf{W}^{1,p}( \Omega ); \varphi =0 \text{ on } \Gamma _{1} \bigr\} , \\& V^{p}_{0}= \bigl\{ \varphi \in \mathbf{W}^{1,p}( \Omega ); \varphi =0 \text{ on }\Gamma _{1}\cup \Gamma _{L} \text{ and } \varphi \cdot n=0 \text{ on }\Gamma _{0} \bigr\} \end{aligned}$$

and

$$\begin{aligned} V^{p}_{0.\operatorname{div}}= \bigl\{ \varphi \in V^{p}_{0} ; \operatorname{div}(\varphi )=0 \text{ in } \Omega \bigr\} \end{aligned}$$

for all $p>1$ endowed with the norm

$$\begin{aligned} \Vert \upsilon \Vert _{1.p}= \biggl( \int _{\Omega } \Vert \nabla \upsilon \Vert ^{p} \,dx \biggr)^{1/p}. \end{aligned}$$

By using the convexity of the mapping $z \mapsto z^{p}$ on $\mathbb{R}^{+}_{*}$, we obtain

$$\begin{aligned} \biggl( \int _{\Omega } \bigl\Vert D(u) \bigr\Vert ^{p} \,dx \biggr)^{1/p} = \bigl\Vert D(u) \bigr\Vert _{(L^{p}(\Omega ))^{3 \times 3}} \le \Vert u \Vert _{1.p} \quad \forall u \in \mathbf{W}^{1,p}( \Omega ), \end{aligned}$$

(2.1)

where $D(u) = ( d_{ij} (u) )_{1 \le i, j \le 3} = ( \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} ) )_{1 \le i,j \le 3}$, and with Korn’s inequality [38], we have

$$\begin{aligned} \biggl( \int _{\Omega } \bigl\Vert D(u) \bigr\Vert ^{p} \,dx \biggr)^{1/p} = \bigl\Vert D(u) \bigr\Vert _{(L^{p}(\Omega ))^{3 \times 3}} \ge C_{ \mathrm{Korn}, p} \Vert u \Vert _{1.p} \quad \forall u \in V^{p}_{ \Gamma _{1 }}, \end{aligned}$$

(2.2)

where $C_{\mathrm{Korn}, p}>0$. Moreover, let ${\mathcal{Y}} = \{ \psi \in {\mathbf{L}}^{2}(\Omega ); \operatorname{div}(\psi ) \in L^{2}(\Omega ) \}$ endowed with its canonical norm

$$\begin{aligned} \Vert \psi \Vert _{\mathcal{Y}} = \bigl( \Vert \psi \Vert ^{2}_{{ \mathbf{L}}^{2}( \Omega )} + \bigl\Vert \operatorname{div}(\psi ) \bigr\Vert ^{2}_{{ L}^{2}(\Omega )} \bigr)^{1/2} \quad \forall \psi \in {\mathcal{Y}}, \end{aligned}$$

and let H be its subspace given by $H= \{ \psi \in {\mathbf{L}}^{2}(\Omega ); \operatorname{div}( \psi ) = 0 \text{ in } \Omega , \psi \cdot n = 0 \text{ on } \partial \Omega \}$. Owing to that ${\mathbf{W}}^{1,p} (\Omega )$ is continuously embedded into ${\mathbf{L}}^{2}(\Omega )$ if and only if $p \ge \frac{6}{5}$ in a 3D setting and that the space ${\mathcal{V}} = \{ \Phi \in ( {\mathcal{D}}(\Omega ) )^{3}; \operatorname{div}(\Phi ) =0 \text{ in } \Omega \}$ is dense in H (see Chap. 1, Theorem 2.8 in [20]), we obtain that the embedding of $V_{0.\operatorname{div}}^{p}$ into H is continuous and dense if and only if $p \ge \frac{6}{5}$, and $( V_{0.\operatorname{div}}^{p}, H, (V_{0.\operatorname{div}}^{p})' )$ is a Gelfand triplet. Let us also recall that the trace operator is compact from ${\mathbf{W}}^{1,p} (\Omega )$ into ${\mathbf{L}}^{p}(\partial \Omega )$ for all $p>1$ (see Theorem 1.23 in [29]), which will allow us to deal with the boundary friction term (see (2.13) and the definition of the mapping J).

Similarly, we let

$$\begin{aligned} W^{1, q}_{ \Gamma _{1}\cup \Gamma _{L}}(\Omega ) = \bigl\{ \varphi \in W^{1, q}(\Omega ) : \varphi =0 \text{ on }\Gamma _{1} \cup \Gamma _{L} \bigr\} \end{aligned}$$

for all $q>1$ endowed with the norm

$$\begin{aligned} \Vert \varphi \Vert _{W^{1, q}_{ \Gamma _{1}\cup \Gamma _{L}}(\Omega )} = \biggl( \int _{\Omega } \Vert \nabla \varphi \Vert ^{q} \,dx \biggr)^{1/q}. \end{aligned}$$

Let us now introduce the assumptions on the data.

The heat capacity and the thermal conductivity tensor satisfy

$$\begin{aligned}& c \in W^{1, q'} (\Omega ), \quad\quad K \in \bigl( L^{\infty } (\Omega ) \bigr)^{3 \times 3} \end{aligned}$$

(2.3)

$$\begin{aligned}& \begin{gathered} \text{there exists $(c_{0}, c_{1}) \in \mathbb{R}^{2}$ such that} \\ 0< c_{0} \le c(x) \le c_{1} \quad \text{for a.e. } x \in \Omega, \end{gathered} \end{aligned}$$

(2.4)

where $q' = \frac{q}{q-1}$ is the conjugate number of q, and

$$\begin{aligned} \begin{gathered} \text{there exists $ k_{0} >0$ such that $ \sum_{i,j=1}^{3} K_{i j} (x) \gamma _{i} \gamma _{j} \ge k_{0} \sum_{i=1}^{3} \vert \gamma _{i} \vert ^{2}$} \\ \text{for all } \gamma = (\gamma _{i})_{1 \le i \le 3} \in { \mathbb{R}}^{3} \text{ for a.e. } x \in \Omega . \end{gathered} \end{aligned}$$

(2.5)

We also assume that

$$\begin{aligned} \text{the mapping $r : {\mathbb{R}} \to {\mathbb{R}}$ is continuous}, \end{aligned}$$

(2.6)

and

$$\begin{aligned} \text{there exists $r_{1} \in \mathbb{R}$ such that } \bigl\vert r(z) \bigr\vert \le r_{1} \quad \text{for all } z \in \mathbb{R}. \end{aligned}$$

(2.7)

The viscosity mapping $\mu : \mathbb{R} \times \mathbb{R}^{3} \times \mathbb{R}_{+} \to \mathbb{R}$ satisfies

$$\begin{aligned}& (o,e,d)\mapsto \mu (o,e,d)\text{ is continuous on } \mathbb{R} \times \mathbb{R}^{3}\times \mathbb{R}_{+}, \end{aligned}$$

(2.8)

$$\begin{aligned}& d\mapsto \mu (\cdot ,\cdot ,d) \text{ is monotone increasing on } \mathbb{R}_{+}, \end{aligned}$$

(2.9)

$$\begin{aligned}& \begin{gathered} \text{there exists $(\mu _{0}, \mu _{1}) \in {\mathbb{R}}^{2}$ such that } \\ 0< \mu _{0}\leq \mu (o,e,d)\leq \mu _{1} \quad \text{for all $(o,e,d)\in \mathbb{R}\times \mathbb{R}^{3}\times \mathbb{R}_{+}$}, \end{gathered} \end{aligned}$$

(2.10)

and we define $\mathcal{F}: \mathbb{R}\times \mathbb{R}^{3}\times \mathbb{R}^{3 \times 3}\rightarrow \mathbb{R}^{3\times 3}$ by

$$\begin{aligned} \textstyle\begin{cases} \mathcal{F}(\lambda _{0}, \lambda _{1},\lambda _{2})= 2 \mu (\lambda _{0} ,\lambda _{1}, \Vert \lambda _{2} \Vert ) \Vert \lambda _{2} \Vert ^{p-2}\lambda _{2} \quad \text{if } \lambda _{2} \neq 0_{ \mathbb{R}^{3\times 3}}, \\ \mathcal{F}(\lambda _{0}, \lambda _{1},\lambda _{2})= 0_{ \mathbb{R}^{3\times 3}} \quad \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

(2.11)

With (2.10), we obtain immediately

$$\begin{aligned} \bigl\vert \mathcal{F}(\lambda _{0}, \lambda _{1},\lambda _{2}) \bigr\vert \leq 2\mu _{1} \Vert \lambda _{2} \Vert ^{p-1} \quad \forall (\lambda _{0}, \lambda _{1},\lambda _{2})\in \mathbb{R}\times \mathbb{R}^{3}\times \mathbb{R}^{3\times 3}. \end{aligned}$$

Let $\tilde{p}>1$ and $p>1$ such that $\tilde{p}-p+1>0$. Then for any $\theta \in L^{\tilde{q}} ( 0,T; L^{q} (\Omega ) )$ with $\tilde{q} \ge 1$ and $q \ge 1$, and for any $u \in L^{\tilde{p}} ( 0,T; \mathbf{W}^{1,p}(\Omega ) )$, we have

$$\begin{aligned} \mathcal{F} \bigl( \theta , u , D( u ) \bigr) \in L^{ \frac{\tilde{p}}{p-1}} \bigl( 0,T; \bigl(L^{p'}(\Omega ) \bigr)^{3 \times 3} \bigr), \end{aligned}$$

where $p' = \frac{p}{p-1}$ is the conjugate number of p. Hence,

$$\begin{aligned} \mathcal{F} \bigl( \theta , u , D( u ) \bigr) : D( \overline{\varphi }) \in L^{1} \bigl( 0,T; L^{1}(\Omega ) \bigr) \end{aligned}$$

for all $\overline{\varphi } \in L^{\frac{\tilde{p}}{\tilde{p} -p+1}} ( 0,T; {\mathbf{W}}^{1,p} (\Omega ) )$, and the right-hand side of the heat equation is well defined in $L^{1} ( 0,T; L^{1}(\Omega ) )$ if and only if $\tilde{p} \ge \frac{\tilde{p}}{\tilde{p} - p +1}$, i.e., $\tilde{p} \ge p$. Then we may expect $\theta \in L^{q} ( 0,T; W^{1}_{\Gamma _{1} \cup \Gamma _{L}} ( \Omega ) )$ with $1< q < \frac{5}{4}$ (see [6]).

Let us introduce the operator $\mathcal{A}: L^{\tilde{p}} (0,T;V^{p}_{\Gamma _{1}} ) \rightarrow ( L^{\tilde{p}} (0,T;V^{p}_{\Gamma _{1}} ) )'$ defined by

$$\begin{aligned} \bigl[\mathcal{A} (u) ,\overline{\varphi } \bigr] = \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \theta , u , D( u ) \bigr) : D( \overline{\varphi }) \,dx\,dt \quad \forall (u, \overline{\varphi }) \in \bigl( L^{\tilde{p} } \bigl(0,T;V^{p}_{\Gamma _{1}} \bigr) \bigr)^{2}, \end{aligned}$$

where $[\cdot ,\cdot ]$ denotes the duality product between the space $L^{\tilde{p}} (0,T;V^{p}_{\Gamma _{1}} )$ and its dual $( L^{\tilde{p}} (0,T;V^{p}_{\Gamma _{1}}) )'$. With (2.10), we have

$$\begin{aligned} \begin{aligned} \bigl\vert \bigl[\mathcal{A} (u) ,\overline{ \varphi } \bigr] \bigr\vert &\le 2 \mu _{1} \bigl\Vert D( u ) \bigr\Vert _{ L^{\tilde{p}} (0,T;L^{p} ( \Omega ) ) }^{p-1} \bigl\Vert D( \overline{\varphi }) \bigr\Vert _{ L^{ \frac{\tilde{p}}{\tilde{p} - p +1} } (0,T;L^{p} (\Omega ) ) } \\ &\le 2 \mu _{1} T^{\frac{\tilde{p} -p}{\tilde{p}}} \Vert u \Vert _{ L^{ \tilde{p}} (0,T;V^{p}_{\Gamma _{1}}) }^{p-1} \Vert \overline{\varphi } \Vert _{ L^{\tilde{p} } (0,T;V^{p}_{\Gamma _{1}}) } \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \bigl[\mathcal{A}( u),u - \overline{\varphi } \bigr] \ge 2(C_{\mathrm{Korn}, p})^{p}\mu _{0} \Vert u \Vert _{L^{p}(0,T;V_{ \Gamma _{1}}^{p})}^{p} - 2\mu _{1} \Vert u \Vert _{L^{ \tilde{p}}(0,T;V_{\Gamma _{1}}^{p})}^{p-1} \Vert \overline{\varphi } \Vert _{L^{\frac{\tilde{p}}{\tilde{p}-p+1}} (0,T;V_{\Gamma _{1}}^{p})} \end{aligned}$$

for any $(u, \overline{\varphi }) \in ( L^{\tilde{p}} (0,T;V_{ \Gamma _{1}}^{p} ) )^{2}$. It follows that ${\mathcal{A}}$ is a bounded operator and is coercive when $\tilde{p} = p$.

Hence, from now on, we will assume that $\tilde{p} = p \ge \frac{6}{5}$ and $q \in (1, \frac{5}{4} )$.

Let $\upsilon ^{0} \in {\mathbf{W}}^{1,p} (\Omega )$ be a given initial velocity such that

$$\begin{aligned} \begin{gathered} \operatorname{div} \bigl(\upsilon ^{0} \bigr)=0\quad \text{in } \Omega ,\quad\quad \upsilon ^{0}=0 \quad \text{on } \Gamma _{1}, \quad\quad \upsilon ^{0}=g \quad \text{on }\Gamma _{L}, \\ \upsilon ^{0}\cdot n=0 \quad \text{on }\Gamma _{0}. \end{gathered} \end{aligned}$$

(2.12)

In order to deal with homogeneous boundary conditions on $(0,T) \times (\Gamma _{1}\cup \Gamma _{L})$, we set $\overline{\upsilon }=\upsilon -\upsilon ^{0}\xi$. We obtain the following weak formulation of the problem:

Problem $(P)$

Let c, K, r, and μ satisfy (2.3)–(2.10). Let $f \in L^{p'} (0,T;\mathbf{L}^{2}(\Omega ) )$, $k \in L^{p'} (0,T; L_{+}^{p'}(\Gamma _{0}) )$, $s\in L^{ p} (0,T;\mathbf{L}^{p}(\Gamma _{0}) )$, $\xi \in W^{1,p'}(0,T)$ satisfying (1.6), $\theta ^{b} \in L^{1} ( (0,T) \times \omega )$, $\theta ^{0} \in L^{1}(\Omega )$ and $\upsilon ^{0}\in \mathbf{W}^{1,p}(\Omega )$ satisfying (2.12).

Find $\theta \in L^{q} ( 0,T; W^{1,q}_{\Gamma _{1} \cup \Gamma _{L}} ( \Omega ) )$, $\overline{\upsilon }\in C ([0,T];\mathbf{L}^{2}(\Omega ) ) \cap L^{ p} (0,T;V^{p}_{0.\operatorname{div}} )$ with $\frac{\partial \overline{\upsilon }}{\partial t} \in L^{p'} (0,T; (V^{p}_{0.\operatorname{div}})' )$ and $\pi \in H^{-1} (0,T;L_{0}^{p'}(\Omega ) )$ satisfying the following parabolic variational coupled problem:

$$\begin{aligned} \begin{gathered} \biggl\langle \frac{\partial }{\partial t}( \overline{\upsilon } , \tilde{\vartheta })_{{\mathbf{L}}^{2}(\Omega )}, \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) : D( \tilde{\vartheta }) \zeta \,dx\,dt \\ \quad\quad {} - \biggl\langle \int _{\Omega } \pi \operatorname{div}( \tilde{\vartheta }) \,dx, \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} + J( \overline{\upsilon }+\tilde{ \vartheta }\zeta )-J( \overline{\upsilon }) \\ \quad \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t} , \tilde{\vartheta } \biggr)_{{ \mathbf{L}}^{2}( \Omega )}\zeta \,dt \quad \forall \tilde{\vartheta }\in V^{p}_{0}, \forall \zeta \in \mathcal{D}(0,T) \end{gathered} \end{aligned}$$

(2.13)

with the initial condition

$$\begin{aligned} \overline{\upsilon }(0, \cdot )=\upsilon ^{0}- \upsilon ^{0}\xi (0)=0 \quad \text{in } \Omega \end{aligned}$$

(2.14)

and

$$\begin{aligned} \begin{gathered} - \int _{0}^{T} \int _{\Omega } c \theta w \widetilde{\zeta }' \,dx\,dt + \int _{0}^{T} \int _{\Omega } (K \nabla \theta ) \cdot \nabla w \widetilde{ \zeta } \,dx \,dt \\ \quad = \int _{0}^{T} \int _{\Omega } \bigl( \mathcal{F} \bigl( \theta , \overline{ \upsilon } + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) : D \bigl( \overline{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) w \widetilde{\zeta }\,dx \,dt \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } r( \theta ) w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\omega } \theta ^{b} w \widetilde{\zeta } \,dx' \,dt + \int _{\Omega } c \theta ^{0} w \widetilde{\zeta }(0) \,dx \\ \quad \forall w \in W^{1, q'}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ), \forall \widetilde{\zeta }\in C^{\infty } \bigl([0,T] \bigr) \text{ s.t. } \widetilde{\zeta }(T) =0 , \end{gathered} \end{aligned}$$

(2.15)

where

$$\begin{aligned} J : \quad \textstyle\begin{cases} L^{p} ( 0,T; V^{p}_{0} ) \rightarrow \mathbb{R}, \\ \overline{\varphi } \mapsto {\int _{0}^{T} \int _{\Gamma _{0}}k \vert \overline{\varphi } -\tilde{s} \vert \,dx \,dt }, \qquad \tilde{s}=s-(\upsilon ^{0})_{\tau }\xi \end{cases}\displaystyle \end{aligned}$$

and $\langle \cdot ,\cdot \rangle _{\mathcal{D}'(0,T),\mathcal{D}(0,T)}$ and $(\cdot ,\cdot )_{{\mathbf{L}}^{2}(\Omega ) }$ (respectively $( \cdot , \cdot )_{L^{2}(\Omega )}$) denote the duality product between $\mathcal{D}(0,T)$ and $\mathcal{D}'(0,T)$ and the inner product in ${\mathbf{L}}^{2}(\Omega )$ (respectively $L^{2}(\Omega )$).

For this non-linear fluid flow/heat transfer problem, a natural proof strategy applies a fixed point technique. Indeed, for any given temperature field $\theta \in L^{\tilde{q}} ( 0,T; L^{q}(\Omega ) )$ with $\tilde{q} \ge 1$ and $q \ge 1$, the fluid flow problem (2.13)–(2.14) admits a solution [10, 11]. Moreover, we know that parabolic problems with $L^{1}$ data given by

$$\begin{aligned} \textstyle\begin{cases} \frac{\partial \theta }{\partial t} - \operatorname{div} ( a(x, \nabla \theta ) )=g \quad \text{in $\Omega \times (0,T)$}, \\ \theta = 0 \quad \text{in $\partial \Omega \times (0,T)$} \end{cases}\displaystyle \end{aligned}$$

(2.16)

with $g \in L^{1} ( 0,T; L^{1}(\Omega ))$ and the coercivity property $a(x, \gamma ) \cdot \gamma \ge \alpha _{a} \Vert \gamma \Vert ^{2}$ ($\alpha _{a}>0$) for all $\gamma \in \mathbb{R}^{3}$ and for almost every $x \in \Omega$ admit a solution $\theta \in L^{q} ( 0,T; W^{1,q}_{0} (\Omega ) )$ with $1< q < \frac{5}{4}$ (see [6]). For a given fluid velocity $\upsilon = \overline{\upsilon } + \upsilon ^{0} \xi \in L^{p} ( 0,T; V^{p}_{ \Gamma _{1}})$, we can not apply this result directly to the heat transfer problem since we consider a nonconstant heat capacity and mixed Dirichlet–Neumann boundary conditions, but we may still expect the existence of a solution $\theta \in L^{q} ( 0,T; W^{1,q}_{\Gamma _{1} \cup \Gamma _{L}}( \Omega ) )$. Nevertheless, in both cases, the proofs of existence rely on compactness arguments and uniqueness of the solution to the flow problem (2.13)–(2.14) for a given temperature is not ensured, nor the uniqueness of the solution to $L^{1}$-parabolic problems (2.16).

In order to cope with this difficulty, we will consider in Sect. 3 an auxiliary approximate fluid flow/heat transfer problem $(P_{\delta })$, where the $L^{1}$ right-hand side in the heat equation (1.3) is replaced by a bounded one depending on a small parameter $0< \delta \ll 1$, and we will prove the existence of a solution $(\overline{\upsilon }_{\delta }, \pi _{\delta }, \theta _{\delta })$ to $(P_{\delta })$ by using a fixed point argument. Then in Sect. 4, we will prove that the sequence $(\overline{\upsilon }_{\delta }, \pi _{\delta }, \theta _{\delta })_{ \delta >0}$ converges to a solution of problem $(P)$.

3 The approximate problem $(P_{\delta })$

For any $\delta >0$, we consider the following approximate problem:

Problem $(P_{\delta })$

Let c, K, r, and μ satisfy (2.3)–(2.10). Let $f \in L^{p'} (0,T;\mathbf{L}^{2}(\Omega ) )$, $k \in L^{p'} (0,T; L_{+}^{p'}(\Gamma _{0}) )$, $s\in L^{ p} (0,T;\mathbf{L}^{p}(\Gamma _{0}) )$, $\xi \in W^{1,p'}(0,T)$ satisfying (1.6), $\theta ^{b} \in L^{1} ( (0,T) \times \omega )$, $\theta ^{0} \in L^{1}(\Omega )$ and $\upsilon ^{0}\in \mathbf{W}^{1,p}(\Omega )$ satisfying (2.12).

Find $\theta _{\delta } \in W^{1,2} ( (0,T) \times \Omega ) \cap L^{ \infty } ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} ( \Omega ) ) \cap C^{0} ( [0,T]; L^{2}(\Omega ) )$, $\overline{\upsilon }_{\delta }\in C ([0,T]; \mathbf{L}^{2}( \Omega ) )\cap L^{ p} (0,T;V^{p}_{0.\operatorname{div}} )$ with $\frac{\partial \overline{\upsilon }_{\delta }}{\partial t} \in L^{p'} (0,T; (V^{p}_{0.\operatorname{div}})' )$ and $\pi _{\delta } \in H^{-1} (0,T; L_{0}^{p'}(\Omega ) )$, satisfying the following parabolic variational coupled problem:

$$\begin{aligned} \begin{gathered} \biggl\langle \frac{\partial }{\partial t}( \overline{\upsilon }_{ \delta } , \tilde{\vartheta })_{{\mathbf{L}}^{2}( \Omega )}, \zeta \biggr\rangle _{ \mathcal{D}'(0,T),\mathcal{D}(0,T)} \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \theta _{\delta }, \overline{\upsilon }_{\delta } + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{\delta } + \upsilon ^{0} \xi \bigr) \bigr) : D( \tilde{\vartheta }) \zeta \,dx \,dt \\ \quad\quad {} - \biggl\langle \int _{\Omega } \pi _{\delta } \operatorname{div}( \tilde{ \vartheta })\,dx , \zeta \biggr\rangle _{ \mathcal{D}'(0,T),\mathcal{D}(0,T)} + J( \overline{\upsilon }_{ \delta } +\tilde{\vartheta }\zeta )-J( \overline{\upsilon }_{\delta }) \\ \quad \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t} , \tilde{\vartheta } \biggr)_{{ \mathbf{L}}^{2}( \Omega )}\zeta \,dt \quad \forall \tilde{\vartheta }\in V^{p}_{0}, \forall \zeta \in \mathcal{D}(0,T) \end{gathered} \end{aligned}$$

(3.1)

and

$$\begin{aligned} \begin{gathered} { \int _{0}^{T} \int _{\Omega } c \frac{\partial \theta _{\delta }}{\partial t} w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\Omega } (K \nabla \theta _{\delta }) \cdot \nabla w \widetilde{\zeta }\,dx \,dt} \\ \quad = \int _{0}^{T} \int _{\Omega } g_{\delta } (\theta _{ \delta }, \overline{\upsilon }_{\delta }) w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\Omega } r( \theta _{\delta }) w \widetilde{\zeta } \,dx \,dt \\ \quad\quad {} + \int _{0}^{T} \int _{\omega } \theta ^{b}_{\delta } w \widetilde{\zeta }\,dx' \,dt \quad \forall w \in W^{1, 2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ), \forall \widetilde{\zeta }\in \mathcal{D}(0,T) \end{gathered} \end{aligned}$$

(3.2)

with the initial conditions

$$\begin{aligned} \overline{\upsilon }_{\delta }(0, \cdot ) =0 \quad \text{in } \Omega \end{aligned}$$

(3.3)

and

$$\begin{aligned} \theta _{\delta } (0, \cdot ) = \theta ^{0}_{\delta } \quad \text{in } \Omega, \end{aligned}$$

(3.4)

where

$$\begin{aligned} g_{\delta }(\theta _{\delta }, \overline{\upsilon }_{\delta }) = \frac{2\mu (\theta _{\delta }, \overline{\upsilon }_{\delta } + \upsilon ^{0} \xi , \vert D( \overline{\upsilon }_{\delta } + \upsilon ^{0} \xi ) \vert ) \vert D(\overline{\upsilon }_{\delta } + \upsilon ^{0} \xi ) \vert ^{p} }{ 1 + 2\delta \mu (\theta _{\delta }, \overline{\upsilon }_{\delta } + \upsilon ^{0} \xi , \vert D( \overline{\upsilon }_{\delta } + \upsilon ^{0} \xi ) \vert ) \vert D( \overline{\upsilon }_{\delta } + \upsilon ^{0} \xi ) \vert ^{p}} \end{aligned}$$

(3.5)

and $\theta ^{b}_{\delta }$ and $\theta ^{0}_{\delta }$ are chosen as smooth approximations of $\theta ^{b}$ and $\theta ^{0}$, respectively, i.e., $\theta ^{b}_{\delta } \in {\mathcal{D}} ( (0,T) \times \omega )$ and $\theta ^{0}_{\delta } \in {\mathcal{D}} (\Omega )$ such that

$$\begin{aligned} \bigl\Vert \theta ^{b}_{\delta } - \theta ^{b} \bigr\Vert _{L^{1} ((0,T) \times \omega )} \le \delta , \qquad \bigl\Vert \theta ^{0}_{\delta } - \theta ^{0} \bigr\Vert _{L^{1} (\Omega )} \le \delta . \end{aligned}$$

(3.6)

In order to prove the existence of a solution to $(P_{\delta })$, we apply a fixed point technique. As a first step, we consider the fully decoupled fluid flow and heat transfer problems $(P^{\mathrm{flow}}_{ (\tilde{u}, \tilde{\theta })})$ and $(P^{\mathrm{heat}}_{\delta , (\tilde{u}, \tilde{\theta })})$:

Problem $(P^{\mathrm {flow}}_{ (\tilde {u}, \tilde {\theta })})$

Let $(\tilde{u}, \tilde{\theta })$ be given in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{ \tilde{q}_{1}} ( 0,T; L^{\tilde{q}_{2}}(\Omega ) )$ with $\tilde{q}_{1}>1$ and $\tilde{q}_{2} >1$. Find $\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}\in C ([0,T]; \mathbf{L}^{2}(\Omega ) )\cap L^{ p} (0,T;V^{p}_{0.\operatorname{div}} )$ with $\frac{\partial \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}}{\partial t} \in L^{p'} (0,T; (V^{p}_{0.\operatorname{div}})' )$ and $\pi _{ (\tilde{u}, \tilde{\theta })} \in H^{-1} (0,T;L_{0}^{p'}( \Omega ) )$, satisfying the following parabolic variational inequality

$$\begin{aligned} \begin{gathered} \biggl\langle \frac{\partial }{\partial t}( \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })} , \tilde{\vartheta })_{{\mathbf{L}}^{2}(\Omega )}, \zeta \biggr\rangle _{ \mathcal{D}'(0,T),\mathcal{D}(0,T)} \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D( \tilde{\vartheta }) \zeta \,dx \,dt \\ \quad\quad {} - \biggl\langle \int _{\Omega } \pi _{ (\tilde{u}, \tilde{\theta })} \operatorname{div}(\tilde{ \vartheta }) \,dx , \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} + J( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta })} + \tilde{\vartheta }\zeta )-J(\overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })}) \\ \quad \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t} , \tilde{\vartheta } \biggr)_{{ \mathbf{L}}^{2}( \Omega )}\zeta \,dt \quad \forall \tilde{\vartheta }\in V^{p}_{0}, \forall \zeta \in \mathcal{D}(0,T) \end{gathered} \end{aligned}$$

(3.7)

with the initial condition

$$\begin{aligned} \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}(0, \cdot ) =0 \quad \text{in } \Omega \end{aligned}$$

(3.8)

and

Problem $(P^{\mathrm {heat}}_{\delta , (\tilde {u}, \tilde {\theta })})$

Let $(\tilde{u}, \tilde{\theta })$ be given in $L^{p} ( 0,T; V^{p}_{0.\operatorname{div}} ) \times L^{\tilde{q}_{1}} ( 0,T; L^{\tilde{q}_{2}}(\Omega ) )$ with $\tilde{q}_{1}>1$ and $\tilde{q}_{2} >1$. Find $\theta _{\delta , (\tilde{u}, \tilde{\theta })} \in W^{1,2} ( (0,T) \times \Omega ) \cap L^{\infty } ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) ) \cap C^{0} ( [0,T]; L^{2}( \Omega ) )$, satisfying the following parabolic variational equality:

$$\begin{aligned} \begin{gathered} { \int _{0}^{T} \int _{\Omega } c \frac{\partial \theta _{\delta , (\tilde{u}, \tilde{\theta })} }{\partial t} w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\Omega } (K \nabla \theta _{\delta , (\tilde{u}, \tilde{\theta })}) \cdot \nabla w \widetilde{\zeta }\,dx \,dt} \\ \quad = \int _{0}^{T} \int _{\Omega } g_{\delta } ( \tilde{\theta }, \tilde{u}) w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\Omega } r( \tilde{\theta }) w \widetilde{\zeta }\,dx \,dt \\ \quad\quad {} + \int _{0}^{T} \int _{\omega } \theta ^{b}_{\delta } w \widetilde{\zeta }\,dx' \,dt \quad \forall w \in W^{1, 2}_{ \Gamma _{1} \cup \Gamma _{L}} (\Omega ), \forall \widetilde{\zeta } \in { \mathcal{D}}(0,T) \end{gathered} \end{aligned}$$

(3.9)

with the initial condition

$$\begin{aligned} \theta _{\delta , (\tilde{u}, \tilde{\theta })} (0, \cdot ) = \theta ^{0}_{ \delta } \quad \text{in } \Omega . \end{aligned}$$

(3.10)

Remark 3.1

Let us emphasize that problem $(P^{\mathrm{flow}}_{ (\tilde{u}, \tilde{\theta })})$ does not depend on δ.

Then we have

Proposition 3.2

(Existence and uniqueness result for $(P^{\mathrm{flow}}_{ (\tilde{u}, \tilde{\theta })})$)

Let $(\tilde{u}, \tilde{\theta })$ be given in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{ \tilde{q}_{1}} ( 0,T; L^{\tilde{q}_{2}} (\Omega ) )$ with $\tilde{q}_{1}>1$ and $\tilde{q}_{2} >1$. Let μ satisfy (2.8)–(2.10), $f \in L^{p'} (0,T;{\mathbf{L}}^{2}(\Omega ) )$, $k \in L^{p'} (0,T;L_{+}^{p'}(\Gamma _{0}) )$, $s\in L^{ p} (0,T;{\mathbf{L}}^{p}(\Gamma _{0}) )$, $\xi \in W^{1,p'}(0,T)$ satisfying (1.6), and $\upsilon ^{0}\in {\mathbf{W}}^{1,p}(\Omega )$ satisfying (2.12). Then problem $(P^{\mathrm{flow}}_{ (\tilde{u}, \tilde{\theta })})$ admits a unique solution. Moreover, there exists a constant $C^{\mathrm{flow}}$, independent of $(\tilde{\theta }, \tilde{u} )$, such that

$$\begin{aligned}& \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p}(0,T; V^{p}_{0.\operatorname{div}})} \le C^{\mathrm{flow}}, \end{aligned}$$

(3.11)

$$\begin{aligned}& \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{ \infty } (0,T; {\mathbf{L}}^{2}(\Omega ) )} \le C^{\mathrm{flow}} \end{aligned}$$

(3.12)

and

$$\begin{aligned} \biggl\Vert \frac{\partial \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} }{\partial t} \biggr\Vert _{L^{p'} (0,T; (V^{p}_{0.\operatorname{div}})')} \le C^{\mathrm{flow}}. \end{aligned}$$

(3.13)

Proof

The existence and uniqueness of a solution to the problem $(P^{\mathrm{flow}}_{ (\tilde{u}, \tilde{\theta })})$ is an immediate consequence of Theorem 3.1 in [10] when $p \in [6/5, 2)$ and Theorem 3.1, Theorem 4.1 and Remark 4.1 in [11] when $p \ge 2$. Let us prove now (3.11) and (3.13). For any ${\overline{\varphi }}= \tilde{\vartheta }\zeta$ with $\tilde{\vartheta }\in V^{p}_{0.\operatorname{div}}$ and $\zeta \in \mathcal{D}(0,T)$, we have

$$\begin{aligned} \begin{gathered} \underbrace{ \biggl\langle \frac{\partial }{\partial t}(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} , \tilde{\vartheta })_{{\mathbf{L}}^{2}(\Omega )},\zeta \biggr\rangle _{\mathcal{D}'(0,T),\mathcal{D}(0,T)}}_{ = \int _{0}^{T} < \frac{\partial \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} }{\partial t}, {\overline{\varphi }} >_{ (V^{p}_{0.\operatorname{div}})', V^{p}_{0.\operatorname{div}} } \,dt } \\ \quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D( { \overline{\varphi }} ) \,dx \,dt \\ \quad {} + J( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + {\overline{\varphi }} ) - J( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t} , {\overline{\varphi }} \biggr)_{{ \mathbf{L}}^{2}(\Omega )} \,dt . \end{gathered} \end{aligned}$$

(3.14)

By density of $\mathcal{D}(0,T) \otimes V^{p}_{0.\operatorname{div}}$ into $L^{p} (0,T; V^{p}_{0.\operatorname{div}} )$, the same inequality is true for any ${\overline{\varphi }} \in L^{p} (0,T; V^{p}_{0.\operatorname{div}} )$.

Let us choose $\overline{\varphi } = - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} {\mathbf{1}}_{\vert [0,t]}$, where $t \in (0,T]$, and ${\mathbf{1}}_{[0,t]}$ is the indicatrix function of the time interval $[0,t]$. With (3.14), we have

$$\begin{aligned} \begin{gathered} \int _{0}^{t} \biggl\langle \frac{ \partial \overline{\upsilon }_{(\tilde{u}, \tilde{\theta })}}{\partial t} ,\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \biggr\rangle _{(V_{0.\operatorname{div}}^{p})',V_{0.\operatorname{div}}^{p}} \,d\tilde{t} \\ \quad\quad {} + \int _{0}^{t} \int _{\Omega } {\mathcal{F}} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}) \,dx \,d\tilde{t} \\ \quad \leq \int _{0}^{t} \biggl( f + \upsilon ^{0} \frac{ \partial \xi }{\partial t} , \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \biggr)_{{\mathbf{L}}^{2}(\Omega ) } \,d\tilde{t} + \int _{0}^{t} \int _{\Gamma _{0}} k \vert \tilde{s} \vert \,dx\,d\tilde{t}. \end{gathered} \end{aligned}$$

With (2.10) and (2.1)–(2.2), we get

$$\begin{aligned} \begin{gathered} { \int _{0}^{t} \int _{\Omega } {\mathcal{F}} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \,dx\,d\tilde{t} } \\ \quad \geq {2(C_{\mathrm{Korn}, p})^{p} \mu _{0} \int _{0}^{t} \bigl\Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} +\upsilon ^{0} \xi \bigr\Vert _{1.p}^{p} \,d\tilde{t}} \\ \quad\quad {} - {2\mu _{1} \int _{0}^{t} \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{1.p} \bigl\Vert \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0}\xi \bigr\Vert _{1.p}^{p-1}\,d\tilde{t}} \\ \quad \geq {2(C_{\mathrm{Korn}, p})^{p} \mu _{0} \bigl\vert \Vert \overline{\upsilon }_{(\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,t; V_{0.\operatorname{div}}^{p})} - \bigl\Vert \upsilon ^{0}\xi \bigr\Vert _{L^{p} (0,t; V^{p}_{ \Gamma _{1}})} \bigr\vert ^{p} } \\ \quad\quad {} - {2\mu _{1} \bigl( \Vert \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,t; V_{0.\operatorname{div}}^{p})} + \bigl\Vert \upsilon ^{0}\xi \bigr\Vert _{L^{p} (0,t; V_{\Gamma _{1}}^{p}) } \bigr)^{p -1} \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p} (0,t; V_{\Gamma _{1}}^{p})} } . \end{gathered} \end{aligned}$$

Since $f + \upsilon ^{0} \frac{\partial \xi }{\partial t} \in L^{p'} ( 0,T; {\mathbf{L}}^{2} (\Omega ) )$, we obtain

$$\begin{aligned} \begin{gathered} {\frac{1}{2} \bigl\Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} (t) \bigr\Vert _{\mathbf{L}^{2}(\Omega )}^{2}} \\ \quad\quad {} + 2(C_{\mathrm{Korn}, p})^{p} \mu _{0} \bigl\vert \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,t; V_{0.\operatorname{div}}^{p})} - \bigl\Vert \upsilon ^{0}\xi \bigr\Vert _{L^{p} (0,t; V^{p}_{ \Gamma _{1}})} \bigr\vert ^{p} \\ \quad \leq \widetilde{C} \biggl\Vert f + \upsilon ^{0} \frac{ \partial \xi }{\partial t} \biggr\Vert _{L^{p'} (0,t; { \mathbf{L}}^{2}( \Omega ))} \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p}(0,t; V_{0.\operatorname{div}}^{p})} + \int _{0}^{t} \int _{\Gamma _{0}} k \vert \tilde{s} \vert \,dx \,d\tilde{t} \\ \quad \quad {} + 2\mu _{1} \bigl( \Vert \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,t; V_{0.\operatorname{div}}^{p})} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p} (0,t; V_{\Gamma _{1}}^{p}) } \bigr)^{p -1} \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p} (0,t; V_{\Gamma _{1}}^{p})}, \end{gathered} \end{aligned}$$

(3.15)

where C̃ denotes the norm of the continuous injection of $V^{p}_{0}$ into ${\mathbf{L}}^{2} (\Omega )$.

Let us consider first $t=T$. If $\Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})} \neq0$, it follows that

$$\begin{aligned} \begin{gathered} 2(C_{\mathrm{Korn}, p})^{p} \mu _{0} \biggl\vert 1 - \frac{ \Vert \upsilon ^{0}\xi \Vert _{L^{p} (0,T; V^{p}_{\Gamma _{1}})} }{ \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})} } \biggr\vert ^{p} \\ \quad \leq \widetilde{C} \biggl\Vert f + \upsilon ^{0} \frac{\partial \xi }{\partial t} \biggr\Vert _{L^{p'} (0,T; {\mathbf{L}}^{2}( \Omega ))} \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p}(0,T; V_{0.\operatorname{div}}^{p})}^{1-p} + \frac{J(0)}{ \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})}^{p}} \\ \quad\quad{} + 2\mu _{1} \biggl( 1 + \frac{ \Vert \upsilon ^{0}\xi \Vert _{L^{p} (0,T; V_{\Gamma _{1}}^{p}) }}{ \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})} } \biggr)^{p -1} \frac{ \Vert \upsilon ^{0} \xi \Vert _{L^{p} (0,T; V_{\Gamma _{1}}^{p})} }{ \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{ p})} } . \end{gathered} \end{aligned}$$

However, the mapping

$$\begin{aligned} \begin{aligned} z \mapsto {}& {2(C_{\mathrm{Korn}, p})^{p} \mu _{0} \biggl\vert 1 - \frac{ \Vert \upsilon ^{0}\xi \Vert _{L^{p} (0,T; V_{\Gamma _{1}}^{p}) }}{z } \biggr\vert ^{p} } - {\widetilde{C} \biggl\Vert f + \upsilon ^{0} \frac{\partial \xi }{\partial t} \biggr\Vert _{L^{p'} (0,T; {\mathbf{L}}^{2}( \Omega ))} z^{1-p} } \\ &{} - \frac{J(0)}{z^{p}} - 2\mu _{1} \biggl( 1 + \frac{ \Vert \upsilon ^{0}\xi \Vert _{L^{p} (0,T; V_{\Gamma _{1}}^{p}) }}{z } \biggr)^{p -1} \frac{ \Vert \upsilon ^{0} \xi \Vert _{L^{p} (0,T; V_{\Gamma _{1}}^{p})} }{z } \end{aligned} \end{aligned}$$

admits $2(C_{\mathrm{Korn}, p})^{p} \mu _{0}>0$ as a limit when z tends to +∞. Thus, there exists a real number $C >0$, independent of $(\tilde{u}, \tilde{\theta })$, such that

$$\begin{aligned} \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})} \le C, \end{aligned}$$

(3.16)

which yields (3.11). Going back to (3.15), we obtain (3.12).

Let us choose now $\overline{\varphi }=\pm \tilde{\vartheta }\zeta$ with $\tilde{\vartheta }\in V^{p}_{0.\operatorname{div}}$ and $\zeta \in \mathcal{D}(0,T)$ in (3.14). We obtain

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \biggl\langle \frac{\partial \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} }{\partial t} , \pm \tilde{\vartheta }\zeta \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \\ \quad {} + \int _{0}^{T} \int _{\Omega } {\mathcal{F}} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D ( \pm \tilde{\vartheta }\zeta ) \,dx \,dt \\ \quad {} +J(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \pm \tilde{\vartheta }\zeta ) -J( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t} ,\pm \tilde{\vartheta }\zeta \biggr)_{{ \mathbf{L}}^{2}(\Omega )} \,dt. \end{gathered} \end{aligned}$$

(3.17)

But

$$\begin{aligned} \bigl\vert J(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \pm \tilde{\vartheta } \zeta ) -J( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \bigr\vert \le \int _{0}^{T} \int _{\Gamma _{0}} k \vert \tilde{\vartheta }\zeta \vert \,dx \,dt \end{aligned}$$

and, recalling that $k \in L^{p'} ( 0,T; L^{p'}_{+}(\Gamma _{0}) )$, we get

$$\begin{aligned} \begin{gathered} \bigl\vert J(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \pm \tilde{\vartheta }\zeta ) -J( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \bigr\vert \\ \quad \le \Vert \gamma _{p} \Vert _{{\mathcal{L}} ({\mathbf{W}}^{1,p} ( \Omega ) , {\mathbf{L}}^{p} (\partial \Omega ))} \Vert k \Vert _{L^{p'}(0,T; L^{p'}( \Gamma _{0}))} \Vert \tilde{\vartheta }\zeta \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})}, \end{gathered} \end{aligned}$$

where $\gamma _{p}$ denotes the trace operator from ${\mathbf{W}}^{1,p} (\Omega )$ into ${\mathbf{L}}^{p} (\partial \Omega )$.

On the other hand,

$$\begin{aligned} \begin{gathered} \biggl\vert \int _{0}^{T} \int _{\Omega } {\mathcal{F}} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D ( \tilde{\vartheta }\zeta ) \,dx \,dt \biggr\vert \\ \quad \le 2 \mu _{1} \bigl\Vert D \bigl( \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr\Vert _{L^{p}(0,T; ( L^{p} ( \Omega ))^{3 \times 3} )}^{p-1} \bigl\Vert D( \tilde{\vartheta } \zeta ) \bigr\Vert _{L^{p}(0,T; ( L^{p} (\Omega ))^{3 \times 3} )}, \end{gathered} \end{aligned}$$

and with (3.16), we obtain

$$\begin{aligned} \begin{gathered} \biggl\vert \int _{0}^{T} \int _{\Omega } {\mathcal{F}} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D ( \tilde{\vartheta }\zeta ) \,dx \,dt \biggr\vert \\ \quad \le \bigl( 2 \mu _{1} \bigl( C + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p-1} \bigr) \Vert \tilde{\vartheta }\zeta \Vert _{L^{p}(0,T; V_{0.\operatorname{div}}^{p})}. \end{gathered} \end{aligned}$$

Going back to (3.17), we obtain

$$\begin{aligned} \begin{gathered} { \biggl\vert \int _{0}^{T} \biggl\langle \frac{\partial \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} }{\partial t} , \tilde{\vartheta }\zeta \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \biggr\vert } \\ \quad \le \Vert \gamma _{p} \Vert _{{\mathcal{L}} ({\mathbf{W}}^{1,p} ( \Omega ) , {\mathbf{L}}^{p} (\partial \Omega ))} \Vert k \Vert _{L^{p'}(0,T; L^{p'}( \Gamma _{0}))} \Vert \tilde{\vartheta }\zeta \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})} \\ \quad\quad {} + \widetilde{C} \biggl\Vert f + \upsilon ^{0} \frac{\partial \xi }{\partial t} \biggr\Vert _{L^{p'} (0,T; {\mathbf{L}}^{2}( \Omega ))} \Vert \tilde{\vartheta } \zeta \Vert _{L^{p} (0,T; V_{0.\operatorname{div}}^{p})} \\ \quad\quad {} + \bigl( 2 \mu _{1} \bigl( C + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p-1} \bigr) \Vert \tilde{\vartheta } \zeta \Vert _{L^{p}(0,T; V_{0.\operatorname{div}}^{p})}, \end{gathered} \end{aligned}$$

which yields (3.13). □

Proposition 3.3

(Existence and uniqueness result for $(P^{\mathrm{heat}}_{\delta , (\tilde{u}, \tilde{\theta })})$)

Let $(\tilde{u}, \tilde{\theta })$ be given in $L^{p} ( 0,T; V^{p}_{0.\operatorname{div}} ) \times L^{\tilde{q}_{1}} ( 0,T; L^{\tilde{q}_{2}} (\Omega ) )$ with $\tilde{q}_{1} >1$ and $\tilde{q}_{2} >1$. Let c, K, r, and μ satisfy (2.3)–(2.10). Let $\xi \in W^{1,p'}(0,T)$ satisfying (1.6), $\theta ^{b}_{\delta } \in {\mathcal{D}} ( (0,T) \times \omega )$, $\theta ^{0}_{\delta } \in {\mathcal{D}} (\Omega )$ and $\upsilon ^{0}\in \mathbf{W}^{1,p}(\Omega )$ satisfying (2.12). Then problem $(P^{\mathrm{heat}}_{\delta , (\tilde{u}, \tilde{\theta })})$ admits a unique solution. Moreover, there exists a constant $C^{\mathrm{heat}}$, depending only on the data K and c, such that

$$\begin{aligned} \begin{gathered} \biggl\Vert \frac{ \partial \theta _{\delta , (\tilde{u}, \tilde{\theta })}}{\partial t} \biggr\Vert ^{2}_{L^{2} (0,T; L^{2}(\Omega ))} + \Vert \nabla \theta _{ \delta , (\tilde{u}, \tilde{\theta })} \Vert ^{2}_{L^{ \infty } (0,T; L^{2}( \Omega ) )} \\ \quad \le C^{\mathrm{heat}} \bigl( \bigl\Vert g_{\delta } ( \tilde{\theta }, \tilde{u} ) + r( \tilde{\theta }) \bigr\Vert ^{2}_{L^{2}(0,T; L^{2}(\Omega )) } + \bigl\Vert \theta ^{0}_{\delta } \bigr\Vert ^{2}_{H^{1}(\Omega )} + \bigl\Vert \theta ^{b}_{ \delta } \bigr\Vert ^{2}_{W^{1,2} (0,T; L^{2} (\omega ))} \bigr). \end{gathered} \end{aligned}$$

Proof

The result is straightforward with the Galerkin method. The details are left to the reader. □

Owing to the definition of $g_{\delta } (\tilde{\theta }, \tilde{u} )$ and the uniform boundedness of the mapping r, we obtain immediately

Corollary 3.4

Under the previous assumptions, there exists a constant $C^{\mathrm{heat}}_{\delta }$, depending only on δ, $\Vert \theta ^{0}_{\delta } \Vert _{H^{1}(\Omega )}$ and $\Vert \theta ^{b}_{\delta } \Vert _{W^{1,2} (0,T; L^{2} (\omega ))}$ such that

$$\begin{aligned} \biggl\Vert \frac{ \partial \theta _{\delta , (\tilde{u}, \tilde{\theta })}}{\partial t} \biggr\Vert _{L^{2} (0,T; L^{2}(\Omega ))} + \Vert \nabla \theta _{ \delta , (\tilde{u}, \tilde{\theta })} \Vert _{L^{\infty } (0,T; L^{2}( \Omega ) )} \le C^{\mathrm{heat}}_{\delta }. \end{aligned}$$

(3.18)

Proof

By using (2.7) and (3.5), we obtain

$$\begin{aligned} \begin{gathered} \bigl\Vert g_{\delta } (\tilde{\theta }, \tilde{u} ) + r( \tilde{\theta }) \bigr\Vert ^{2}_{L^{2}(0,T; L^{2}(\Omega )) } \\ \quad \le 2 \int _{0}^{T} \int _{\Omega } \bigl( g_{\delta } ( \tilde{\theta }, \tilde{u}) \bigr)^{2} \,dx \,dt + 2 \int _{0}^{T} \int _{ \Omega } \bigl( r ( \tilde{\theta }) \bigr)^{2} \,dx \,dt \\ \quad \le 2 \biggl( \frac{1}{\delta ^{2}} + r_{1}^{2} \biggr) \operatorname{meas} (\Omega ) T, \end{gathered} \end{aligned}$$

which allows us to conclude. □

It follows that, for any $(\tilde{u}, \tilde{\theta }) \in L^{p} ( 0,T; {\mathbf{L}}^{p} ( \Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$, and for any $\delta >0$, we may define $(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}, \pi _{ ( \tilde{u}, \tilde{\theta })}) \in L^{p} ( 0,T; V^{p}_{0.\operatorname{div}} ) \times L^{p'} ( 0,T; L^{p'}_{0} (\Omega ) )$ as the unique solution of $(P^{\mathrm{flow}}_{ (\tilde{u}, \tilde{\theta })})$ and $\theta _{\delta , (\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}, \tilde{\theta })} \in W^{1,2} ( (0,T) \times \Omega )$ as the unique solution of $(P^{\mathrm{heat}}_{\delta , (\overline{\upsilon }_{(\tilde{u}, \tilde{\theta })} , \tilde{\theta })})$. Then we define the mapping $T_{\delta }: L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) ) \to L^{p} ( 0,T; {\mathbf{L}}^{p} ( \Omega ) ) \times L^{2} ( 0,T; L^{2}( \Omega ) )$ by

$$\begin{aligned} T_{\delta } (\tilde{u}, \tilde{\theta }) = (\overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })}, \theta _{\delta , (\overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })}, \tilde{\theta })}), \end{aligned}$$

and we will prove that $T_{\delta }$ admits a fixed point.

Theorem 3.5

Let $\delta >0$. Let the mappings c, K, r, and μ satisfy (2.3)–(2.10), $f \in L^{p'} (0,T;{\mathbf{L}}^{2}(\Omega ) )$, $k \in L^{p'} (0,T;L_{+}^{p'}(\Gamma _{0}) )$, $s\in L^{ p} (0,T;{\mathbf{L}}^{p}(\Gamma _{0}) )$, $\xi \in W^{1,p'}(0,T)$ satisfying (1.6), $\theta _{\delta }^{0} \in \mathcal{D}(\Omega )$, $\theta _{\delta }^{b} \in \mathcal{D} ( (0,T) \times \omega )$ and $\upsilon ^{0}\in {\mathbf{W}}^{1,p}(\Omega )$ satisfying (2.12). Then the mapping $T_{\delta }$ admits a fixed point in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$.

Proof

Let $(\tilde{u}, \tilde{\theta }) \in L^{p} ( 0,T; {\mathbf{L}}^{p} ( \Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$. With the previous estimates, we have

$$\begin{aligned} \Vert \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert _{L^{p}(0,T; V^{p}_{0.\operatorname{div}} )} \le C^{\mathrm{flow}}, \qquad \Vert \theta _{\delta , ( \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })}, \tilde{\theta })} \Vert _{L^{\infty } (0,T; W^{1,2}_{ \Gamma _{1} \cup \Gamma _{L}} (\Omega ))} \le C^{\mathrm{heat}}_{\delta }. \end{aligned}$$

Thus, $T_{\delta } ( {\mathcal{C}}) \subset {\mathcal{C}}$ with ${\mathcal{C}}= {\overline{B}}_{L^{p}(0,T: {\mathbf{L}}^{p}(\Omega ))} (0, C_{p} C^{\mathrm{flow}}) \times {\overline{B}}_{L^{2}(0,T; L^{2}( \Omega )} (0, \sqrt{T} C_{2} C^{\mathrm{heat}}_{\delta })$, where $C_{p}$ and $C_{2}$ denote Poincaré’s constant on $V^{p}_{0.\operatorname{div}}$ and on $W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$, respectively. Moreover, the estimates (3.13) and (3.18) imply that $T_{\delta } ({\mathcal{C}})$ is bounded in the space $W^{1,p'} ( 0,T; (V^{p}_{0.\operatorname{div}})' ) \times W^{1,2} ( 0,T; L^{2}(\Omega ) )$, and we may conclude that $T_{\delta } ({\mathcal{C}})$ is relatively compact in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$.

Let us now prove that $T_{\delta }$ is continuous. Let $(\tilde{u}_{n}, \tilde{\theta }_{n})_{n \ge 1}$ be a sequence, which converges strongly in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$ to $(\tilde{u}, \tilde{\theta })$, and let us define

$$\begin{aligned} T_{\delta } (\tilde{u}_{n}, \tilde{\theta }_{n} ) = ( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \theta _{\delta , ( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n})}, \tilde{\theta }_{n})}) \quad \text{for all $n \ge 1$} \end{aligned}$$

and

$$\begin{aligned} T_{\delta } (\tilde{u}, \tilde{\theta }) = (\overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })}, \theta _{\delta , (\overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })}, \tilde{\theta })}). \end{aligned}$$

We have to prove that the sequence $(\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \theta _{ \delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n})}, \tilde{\theta }_{n}) })_{n \ge 1}$ converges strongly to $(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}, \theta _{ ( \overline{\upsilon }_{\delta , (\tilde{u}, \tilde{\theta })}, \tilde{\theta })})$ in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$.

The sequence $(\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \theta _{ \delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n})}, \tilde{\theta }_{n}) })_{n \ge 1}$ satisfies the estimates (3.11)–(3.13) and (3.18), so it admits strongly converging subsequences in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$.

Let us consider such a subsequence still denoted $(\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \theta _{ \delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n})}, \tilde{\theta }_{n})})_{n \ge 1}$.

With (3.14), we have

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \biggl\langle \frac{ \partial \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } }{\partial t} , {\overline{\varphi }} - \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \\ \qquad {} + \int _{0}^{T} \int _{\Omega } {\mathcal{F}} \bigl(\tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) : D({\overline{\varphi }} - \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } ) \,dx \,dt \\ \qquad {} + J( {\overline{\varphi }}) - J( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }) \\ \quad \ge \int _{0}^{T} \biggl( f + {\upsilon }^{0} \frac{ \partial \xi }{\partial t} , {\overline{\varphi }} - \overline{ \upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } \biggr)_{{ \mathbf{L}}^{2}(\Omega )} \,dt \quad \forall { \overline{\varphi }} \in L^{p} \bigl( 0,T; V^{p}_{0.\operatorname{div}} \bigr) \end{gathered} \end{aligned}$$

(3.19)

for all $n \ge 1$, we also have

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \biggl\langle \frac{ \partial \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} }{\partial t} , {\overline{\varphi }} - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \\ \qquad {} + \int _{0}^{T} \int _{\Omega } {\mathcal{F}} \bigl(\tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) : D({\overline{\varphi }} - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })} ) \,dx \,dt \\ \qquad {} + J( {\overline{\varphi }}) - J( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}) \\ \quad \ge \int _{0}^{T} \biggl( f + {\upsilon }^{0} \frac{ \partial \xi }{\partial t} , {\overline{\varphi }} - \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta })} \biggr)_{{ \mathbf{L}}^{2}( \Omega )} \,dt \quad \forall { \overline{\varphi }} \in L^{p} \bigl( 0,T; V^{p}_{0.\operatorname{div}} \bigr). \end{gathered} \end{aligned}$$

(3.20)

We choose ${\overline{\varphi }} = \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })}$ in (3.19) then ${\overline{\varphi }} = \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }$ in (3.20) and add the two inequalities. We obtain:

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \biggl\langle \frac{ \partial (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) }{\partial t} , \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta })} \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \\ \quad {} + \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}} \bigl(\tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad {} - {\mathcal{F}} \bigl(\tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \,dx \,dt \le 0 \end{gathered} \end{aligned}$$

and thus,

$$\begin{aligned} \begin{gathered} \frac{1}{2} \Vert \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert ^{2}_{{\mathbf{L}}^{2} (\Omega )} (T) \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}} \bigl(\tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad \quad {} - {\mathcal{F}} \bigl(\tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl(\overline{ \upsilon }_{ ( \tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \,dx \,dt \\ \quad \le \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}} \bigl(\tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - {\mathcal{F}} \bigl(\tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl(\overline{ \upsilon }_{ ( \tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \,dx \,dt . \end{gathered} \end{aligned}$$

We decompose ${\mathcal{F}}$ as ${\mathcal{F}} = {{\mathcal{F}}_{1} } + {{\mathcal{F}}_{2} }$ with

$$\begin{aligned} {{\mathcal{F}}_{1} } (\lambda _{2}) = \mu _{0} \Vert \lambda _{2} \Vert ^{p-2} \lambda _{2} \quad \text{if } \lambda _{2} \neq 0_{ \mathbb{R}^{3\times 3}}, \quad\quad {{\mathcal{F}}_{1} } ( \lambda _{2})= 0_{ \mathbb{R}^{3\times 3}} \quad \text{otherwise} \end{aligned}$$

and

$$\begin{aligned} \textstyle\begin{cases} {{\mathcal{F}}_{2} } (\lambda _{0}, \lambda _{1}, \lambda _{2}) = 2 {\overline{\mu }} (\lambda _{0} ,\lambda _{1}, \Vert \lambda _{2} \Vert ) \Vert \lambda _{2} \Vert ^{p-2}\lambda _{2} \quad \text{if } \lambda _{2} \neq 0_{\mathbb{R}^{3\times 3}}, \\ {{\mathcal{F}}_{2} } (\lambda _{0}, \lambda _{1}, \lambda _{2})= 0_{\mathbb{R}^{3\times 3}} \quad \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

where ${\overline{\mu }} = \mu - \frac{\mu _{0}}{2}$. By observing that μ̅ satisfies

$$\begin{aligned}& d\mapsto {\overline{\mu }} (\cdot , \cdot ,d) \text{ is monotone increasing on } \mathbb{R}_{+},\\& 0< \frac{\mu _{0}}{2} \leq {\overline{\mu }} (o,e,d) \leq \mu _{1} - \frac{\mu _{0}}{2} \quad \text{for all } (o,e,d)\in \mathbb{R}\times \mathbb{R}^{3}\times \mathbb{R}_{+}, \end{aligned}$$

we infer from Lemma 1 in [9] that $\lambda _{2} \mapsto {{\mathcal{F}}}_{2} (\cdot , \cdot , \lambda _{2})$ is monotone in $\mathbb{R}^{3\times 3}$. Hence,

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \int _{\Omega } \bigl( { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n} , \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n} ) } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad {} - { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n} , \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n} ) } - \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } ) \,dx \,dt \ge 0 \end{gathered} \end{aligned}$$

and thus,

$$\begin{aligned} \begin{gathered} \frac{1}{2} \Vert \overline{ \upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} \Vert ^{2}_{{\mathbf{L}}^{2} (\Omega )} (T) \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}}_{1} \bigl( D \bigl(\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) - {\mathcal{F}}_{1} \bigl( D \bigl(\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) \bigr) \\ \quad \quad : D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta })} ) \,dx \,dt \\ \quad \le \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}}_{2} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - {\mathcal{F}}_{2} \bigl( \tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta })} + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta })} ) \,dx \,dt . \end{gathered} \end{aligned}$$

(3.21)

Now we distinguish two cases.

• Case 1: $p \in [6/5, 2)$

With some algebraic computations, we can prove that

$$\begin{aligned} \bigl( \Vert \lambda \Vert + \bigl\Vert \lambda ' \bigr\Vert \bigr)^{2-p} \bigl( { { \mathcal{F}}_{1} } (\lambda ) - { { \mathcal{F}}_{1} } \bigl( \lambda ' \bigr) \bigr) : \bigl( \lambda - \lambda ' \bigr) \ge \mu _{0} (p-1) \bigl\Vert \lambda - \lambda ' \bigr\Vert ^{2}, \end{aligned}$$

which yields

$$\begin{aligned} \bigl( \Vert \lambda \Vert ^{p} + \bigl\Vert \lambda ' \bigr\Vert ^{p} \bigr)^{ \frac{2-p}{2}} \bigl( \bigl( { {\mathcal{F}}_{1} } (\lambda ) - { { \mathcal{F}}_{1} } \bigl(\lambda ' \bigr) \bigr) : \bigl( \lambda - \lambda ' \bigr) \bigr)^{\frac{p}{2}} \ge \frac{ ( \mu _{0} (p-1) )^{\frac{p}{2}}}{ 2^{ \frac{(p-1)(2-p)}{2} } } \bigl\Vert \lambda - \lambda ' \bigr\Vert ^{p} \end{aligned}$$

for all $(\lambda , \lambda ') \in {\mathbb{R}}^{3 \times 3} \times { \mathbb{R}}^{3 \times 3}$ (see, for instance, Theorem 4.1 in [10]).

It follows that

$$\begin{aligned}& \frac{ ( \mu _{0} (p-1) )^{\frac{p}{2}}}{ 2^{ \frac{(p-1)(2-p)}{2} } } \int _{0}^{T} \int _{\Omega } \bigl\Vert D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) }) \bigr\Vert ^{p} \,dx \,dt \\& \quad \le \biggl( \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) - {{\mathcal{F}}_{1} } \bigl( D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) \\& \quad\quad : D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) } ) \,dx \,dt \biggr)^{\frac{p}{2}} \\& \qquad {} \times \biggl( \int _{0}^{T} \int _{\Omega } \bigl( \bigl\Vert D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr\Vert ^{p} + \bigl\Vert D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr\Vert ^{p} \bigr) \,dx \,dt \biggr)^{\frac{2-p}{2}} \\& \quad \le \biggl( \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) - {{\mathcal{F}}_{1} } \bigl( D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) \\& \quad \quad : D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) } ) \,dx \,dt \biggr)^{\frac{p}{2}} \\& \qquad {} \times \bigl( \bigl\Vert \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})}^{p} + \bigl\Vert \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})}^{p} \bigr)^{\frac{2-p}{2}}. \end{aligned}$$

Thus, with (3.11), we get

$$\begin{aligned}& \frac{ ( \mu _{0} (p-1) )^{\frac{p}{2}}}{ 2^{ \frac{(p-1)(2-p)}{2} } } \bigl\Vert D( \overline{ \upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } ) \bigr\Vert _{L^{p}(0,T; (L^{p}(\Omega ))^{3\times 3})}^{p} \\& \quad \le \biggl( \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) - {{\mathcal{F}}_{1} } \bigl( D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) \\& \quad\quad : D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) }) \,dx \,dt \biggr)^{\frac{p}{2}} \\& \qquad {} \times \bigl( 2 \bigl(C^{\mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p} \bigr)^{ \frac{2-p}{2}}, \end{aligned}$$

and finally,

$$\begin{aligned} \begin{gathered} \frac{ \mu _{0} (p-1) }{ 2^{ (2-p) } ( C^{\mathrm{flow}} + \Vert \upsilon ^{0} \xi \Vert _{ L^{p} (0,T; V^{p}_{\Gamma _{1}} ) } )^{ (2-p) } } \bigl\Vert D( \overline{ \upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } ) \bigr\Vert _{ L^{p} (0,T; (L^{p} (\Omega ) )^{3\times 3} ) }^{2} \\ \quad \leq \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } + \upsilon ^{0} \xi \bigr) \bigr) - {{\mathcal{F}}_{1} } \bigl( D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) \\ \quad\quad : D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } ) \,dx \,dt . \end{gathered} \end{aligned}$$

Going back to (3.21), we obtain

$$\begin{aligned}& \frac{ \mu _{0} (p-1) }{ 2^{ (2-p) } ( C^{\mathrm{flow}} + \Vert \upsilon ^{0} \xi \Vert _{ L^{p} (0,T; V^{p}_{\Gamma _{1}} ) } )^{ (2-p) } } \bigl\Vert D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) }) \bigr\Vert _{ L^{p} (0,T; (L^{p} (\Omega ) )^{3\times 3} ) }^{2} \\& \quad \leq \int _{0}^{T} \int _{\Omega } \bigl( { { \mathcal{F}}}_{2} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \\& \quad\quad {} - {{\mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } ) \,dx \,dt \\& \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \\& \quad\quad {} - { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}(\Omega ))^{3 \times 3})} \\& \qquad {} \times \bigl\Vert D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{ \upsilon }_{ ( \tilde{u}, \tilde{\theta }) } ) \bigr\Vert _{L^{p}(0,T; (L^{p}(\Omega ))^{3 \times 3})}, \end{aligned}$$

which yields

$$\begin{aligned} \begin{gathered} \frac{ \mu _{0} (p-1) }{ 2^{ (2-p) } ( C^{\mathrm{flow}} + \Vert \upsilon ^{0} \xi \Vert _{ L^{p} (0,T; V^{p}_{\Gamma _{1}} ) } )^{ (2-p) } } \bigl\Vert D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) }) \bigr\Vert _{ L^{p} (0,T; (L^{p} (\Omega ) )^{3\times 3} ) } \\ \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}(\Omega ))^{3 \times 3})} . \end{gathered} \end{aligned}$$

(3.22)

• Case 2: $p \in [2, + \infty )$

Once again, with some algebraic computations, we get

$$\begin{aligned} \bigl( {\mathcal{F}}_{1} (\lambda ) - { \mathcal{F}}_{1} \bigl(\lambda ' \bigr) \bigr) : \bigl( \lambda - \lambda ' \bigr) \ge \frac{\mu _{0}}{2^{p-1}} \bigl\Vert \lambda - \lambda ' \bigr\Vert ^{p} \end{aligned}$$

for all $(\lambda , \lambda ') \in {\mathbb{R}}^{3 \times 3} \times { \mathbb{R}}^{3 \times 3}$ (see, for instance, Theorem 4.1 in [11]). Going back to (3.21),

$$\begin{aligned} \begin{gathered} \frac{\mu _{0}}{2^{p-1} } \bigl\Vert D( \overline{ \upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) } ) \bigr\Vert _{ L^{p}(0,T; (L^{p}(\Omega ))^{3 \times 3})}^{p} \\ \quad \leq \int _{0}^{T} \int _{\Omega } \bigl( { { \mathcal{F}}}_{2} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) - {{ \mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) \\ \quad\quad : D( \overline{\upsilon }_{ ( \tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) } ) \,dx \,dt \\ \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }, \tilde{u}+ \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}(\Omega ))^{3 \times 3})} \\ \qquad {} \times \bigl\Vert D(\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) }) \bigr\Vert _{L^{p}(0,T; (L^{p}(\Omega ))^{3 \times 3})}. \end{gathered} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{gathered} \frac{\mu _{0}}{2^{p-1} } \bigl\Vert D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) } ) \bigr\Vert _{ L^{p}(0,T; (L^{p}(\Omega ))^{3 \times 3})}^{p-1} \\ \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }, \tilde{u}+ \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - { {\mathcal{F}}}_{2} \bigl( \tilde{\theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}(\Omega ))^{3 \times 3})} . \end{gathered} \end{aligned}$$

(3.23)

By possibly extracting a subsequence still denoted $( \tilde{u}_{n}, \tilde{\theta }_{n})_{n \ge 1}$, we have

$$\begin{aligned} \tilde{u}_{n} \longrightarrow \tilde{u} , \quad\qquad \tilde{\theta }_{n} \longrightarrow \tilde{\theta }\quad \text{a.e. in $(0,T) \times \Omega $}. \end{aligned}$$

By using the continuity and boundedness assumptions (2.8) and (2.10) for the mapping μ, we infer with Lebesgue’s theorem that

$$\begin{aligned} \begin{gathered} { {\mathcal{F}}}_{2} \bigl( \tilde{ \theta }_{n}, \tilde{u}_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \longrightarrow { { \mathcal{F}}}_{2} \bigl( \tilde{\theta }, \tilde{u} + \upsilon ^{0} \xi , D \bigl( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad \text{strongly in $L^{p'} (0,T; (L^{p'}(\Omega ) )^{3 \times 3})$}. \end{gathered} \end{aligned}$$

With (3.22) when $p \in [6/5, 2)$ or (3.23) when $p \in [2, + \infty )$, we obtain

$$\begin{aligned} \bigl\Vert D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } - \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } ) \bigr\Vert _{L^{p} (0,T; ( L^{p}(\Omega ) )^{3 \times 3} )} \longrightarrow 0, \end{aligned}$$

and with Korn’s inequality, we infer that

$$\begin{aligned} \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } \longrightarrow \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } \quad \text{strongly in $L^{p} (0,T; V^{p}_{0.\operatorname{div}} )$ and in $L^{p} ( 0,T; {\mathbf{L}}^{p}(\Omega ) )$}. \end{aligned}$$

By possibly extracting another subsequence still denoted $( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } )_{n \ge 1}$, we have

$$\begin{aligned} D( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } ) \longrightarrow D( \overline{ \upsilon }_{ (\tilde{u}, \tilde{\theta }) } ), \qquad \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } \longrightarrow \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } \quad \text{a.e. in $(0,T) \times \Omega $}. \end{aligned}$$

By using the continuity and boundedness properties of the mappings μ and r, we obtain

$$\begin{aligned} g_{\delta } (\tilde{\theta }_{n}, \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } ) \longrightarrow g_{\delta } (\tilde{\theta }, \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } ) \quad \text{strongly in $L^{2} ( 0,T; L^{2}(\Omega ) )$} \end{aligned}$$

and

$$\begin{aligned} r(\tilde{\theta }_{n}) \longrightarrow r(\tilde{\theta }) \quad \text{strongly in $L^{2} ( 0,T; L^{2}(\Omega ) )$}. \end{aligned}$$

Moreover, $( \theta _{\delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } , \tilde{\theta }_{n}) })_{n \ge 1}$ satisfies the estimate (3.18). It follows that there exists $\theta _{\delta }^{*} \in W^{1,2} ( (0,T) \times \Omega ) \cap L^{2} ( 0, T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} ( \Omega ) )$ such that, by possibly extracting a subsequence still denoted $( \theta _{\delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } , \tilde{\theta }_{n}) })_{n \ge 1}$, we have

$$\begin{aligned} \begin{aligned} \theta _{\delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } , \tilde{\theta }_{n}) } \longrightarrow \theta _{ \delta }^{*} \quad &\text{weakly in $W^{1,2} ( (0,T) \times \Omega )$} \\ &\text{and weakly* in $L^{\infty } ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} ( \Omega ) )$} \end{aligned} \end{aligned}$$

and with Simon’s lemma [34]

$$\begin{aligned} \theta _{\delta , (\overline{\upsilon }_{\delta , (\tilde{u}_{n}, \tilde{\theta }_{n}) } , \tilde{\theta }_{n}) } \longrightarrow \theta _{ \delta }^{*} \quad \text{strongly in $C^{0} ([0,T]; L^{2}(\Omega ) )$ and in $L^{2} ( 0,T; L^{2}(\Omega ) )$.} \end{aligned}$$

Hence, we may pass to the limit in all the terms of $(P^{\mathrm{heat}}_{\delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) } , \tilde{\theta }_{n}) })$ and obtain that $\theta _{\delta }^{*}$ is the solution of $(P^{\mathrm{heat}}_{\delta , (\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } , \tilde{\theta }) })$. By uniqueness of the solution of $(P^{\mathrm{heat}}_{\delta , (\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) } , \tilde{\theta }) })$, we infer that $\theta _{\delta }^{*} = \theta _{\delta , (\overline{\upsilon }_{ ( \tilde{u}, \tilde{\theta }) }, \tilde{\theta })}$.

Finally, recalling that $( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \theta _{ \delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \tilde{\theta }_{n})} )_{n \ge 1}$ satisfies the estimates (3.11)–(3.13) and (3.18), we infer that the whole sequence $( \overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \theta _{ \delta , (\overline{\upsilon }_{ (\tilde{u}_{n}, \tilde{\theta }_{n}) }, \tilde{\theta }_{n}) })_{n \ge 1}$ converges strongly to $( \overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) }, \theta _{ \delta , (\overline{\upsilon }_{ (\tilde{u}, \tilde{\theta }) }, \tilde{\theta }) } )$ in the space $L^{p} (0,T; {\mathbf{L}}^{p}(\Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$, which allows us to conclude. □

As a corollary, we obtain an existence result for the approximate problem $(P_{\delta })$.

Corollary 3.6

Let $\delta >0$. Let the mappings c, K, r, and μ satisfy (2.3)–(2.10), $f \in L^{p'} (0,T;{\mathbf{L}}^{2}(\Omega ) )$, $k \in L^{p'} (0,T;L_{+}^{p'}(\Gamma _{0}) )$, $s\in L^{ p} (0,T; {\mathbf{L}}^{p}(\Gamma _{0}) )$, $\xi \in W^{1,p'}(0,T)$ satisfying (1.6), $\theta _{\delta }^{0} \in \mathcal{D}(\Omega )$, $\theta _{\delta }^{b} \in \mathcal{D} ( (0,T) \times \omega )$ and $\upsilon ^{0}\in {\mathbf{W}}^{1,p}(\Omega )$ satisfying (2.12). Then problem $(P_{\delta })$ admits a solution.

Proof

With the previous theorem, we know that the mapping $T_{\delta }$ admits a fixed point, i.e., there exists $(\tilde{u}, \tilde{\theta }) \in L^{p} ( 0,T; {\mathbf{L}}^{p} ( \Omega ) ) \times L^{2} ( 0,T; L^{2}(\Omega ) )$ such that

$$\begin{aligned} T_{\delta } (\tilde{u}, \tilde{\theta }) = (\overline{\upsilon }_{( \tilde{u}, \tilde{\theta })}, \theta _{\delta , (\overline{\upsilon }_{( \tilde{u}, \tilde{\theta })}, \tilde{\theta })} ) = (\tilde{u}, \tilde{\theta }) . \end{aligned}$$

By the definition of $T_{\delta }$, we infer that $\tilde{u} = {\overline{\upsilon }}_{(\tilde{u}, \tilde{\theta })} \in C ( [0,T]; {\mathbf{L}}^{2}(\Omega ) ) \cap L^{p} ( 0,T; V^{p}_{0.\operatorname{div}} )$ with $\frac{\partial \tilde{u} }{\partial t} = \frac{\partial {\overline{\upsilon }}_{(\tilde{u}, \tilde{\theta })}}{\partial t} \in L^{p'} ( 0,T; (V^{p}_{0.\operatorname{div}})' )$, and there exists $\pi _{(\tilde{u}, \tilde{\theta })} \in H^{-1} ( 0,T; L^{p'}_{0} ( \Omega ) )$ such that $(\tilde{u}, \pi _{(\tilde{u}, \tilde{\theta }) }) = ( { \overline{\upsilon }}_{(\tilde{u}, \tilde{\theta })}, \pi _{( \tilde{u}, \tilde{\theta }) })$ is solution of $(P^{\mathrm{flow}}_{ (\tilde{u}, \tilde{\theta })})$, while $\tilde{\theta }= \theta _{\delta , (\overline{\upsilon }_{(\tilde{u}, \tilde{\theta })}, \tilde{\theta })} \in W^{1,2} ( (0,T) \times \Omega ) \cap L^{\infty } ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) ) \cap C ( [0,T] ; L^{2}( \Omega ) )$ is a solution of the problem $(P^{\mathrm{heat}}_{\delta , ({\overline{\upsilon }}_{(\tilde{u}, \tilde{\theta })}, \tilde{\theta })}) = (P^{\mathrm{heat}}_{\delta , ({ \overline{\upsilon }}_{( \tilde{u}, \tilde{\theta })}, \theta _{ \delta , (\overline{\upsilon }_{( \tilde{u}, \tilde{\theta })}, \tilde{\theta })} )} )$. Hence, the triplet $( {\overline{\upsilon }}_{(\tilde{u}, \tilde{\theta })}, \pi _{( \tilde{u}, \tilde{\theta })}, \theta _{\delta , (\overline{\upsilon }_{( \tilde{u}, \tilde{\theta })}, \tilde{\theta })})$ is a solution of $(P_{\delta })$. Aubin and Simon □

4 Existence result for the coupled fluid flow / heat transfer problem $(P)$

In order to prove the existence of a solution to the coupled fluid flow/heat transfer problem $(P)$, we consider a sequence of approximate solutions and prove its convergence to a triplet $({\overline{\upsilon }}, \pi , \theta )$, satisfying (2.13)–(2.15).

More precisely, let $\delta _{n}= \frac{1}{n}$ for all $n \ge 1$ and $({\overline{\upsilon }}_{n}, \pi _{n}, \theta _{n})$ be a solution of the approximate problem $(P_{\delta _{n}})$. For all $n\ge 1$, we have $\theta _{n} \in W^{1,2} ( (0,T) \times \Omega ) \cap L^{ \infty } ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) ) \cap C^{0} ( [0,T]; L^{2}(\Omega ) )$, $\overline{\upsilon }_{n}\in C ([0,T];\mathbf{L}^{2}(\Omega ) )\cap L^{ p} (0,T;V^{p}_{0.\operatorname{div}} )$ with $\frac{\partial \overline{\upsilon }_{n}}{\partial t} \in L^{p'} (0,T; (V^{p}_{0.\operatorname{div}})' )$ and $\pi _{n} \in H^{-1} (0,T;L_{0}^{p'}(\Omega ) )$ such that

$$\begin{aligned} \begin{gathered} \biggl\langle \frac{\partial }{\partial t}( \overline{\upsilon }_{n} , \tilde{\vartheta })_{{\mathbf{L}}^{2}( \Omega )}, \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) : D( \tilde{\vartheta }) \zeta \,dx \,dt \\ \quad\quad {} - \biggl\langle \int _{\Omega } \pi _{n} \operatorname{div}( \tilde{ \vartheta }) \,dx , \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} + J( \overline{ \upsilon }_{n} +\tilde{\vartheta } \zeta )-J( \overline{\upsilon }_{n}) \\ \quad \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t} , \tilde{\vartheta } \biggr)_{{ \mathbf{L}}^{2}( \Omega )}\zeta \,dt \quad \forall \tilde{\vartheta }\in V^{p}_{0}, \forall \zeta \in \mathcal{D}(0,T) \end{gathered} \end{aligned}$$

(4.1)

and

$$\begin{aligned} \begin{gathered} { \int _{0}^{T} \int _{\Omega } c \frac{\partial \theta _{n}}{\partial t} w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\Omega } (K \nabla \theta _{n}) \cdot \nabla w \widetilde{\zeta }\,dx \,dt} \\ \quad = \int _{0}^{T} \int _{\Omega } g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\Omega } r( \theta _{n}) w \widetilde{\zeta } \,dx \,dt \\ \quad\quad {} + \int _{0}^{T} \int _{\omega } \theta ^{b}_{\delta _{n}} w \widetilde{\zeta }\,dx' \,dt \quad \forall w \in W^{1, 2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ), \forall \widetilde{\zeta }\in \mathcal{D}(0,T) \end{gathered} \end{aligned}$$

(4.2)

with the initial conditions

$$\begin{aligned} \overline{\upsilon }_{n}(0, \cdot ) =0 \quad \text{in } \Omega \end{aligned}$$

(4.3)

and

$$\begin{aligned} \theta _{n} (0, \cdot ) = \theta ^{0}_{\delta _{n}} \quad \text{in } \Omega . \end{aligned}$$

(4.4)

Owing to that $\theta _{n} \in W^{1,2} ( (0,T) \times \Omega ) \cap L^{ \infty } ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) ) \cap C^{0} ( [0,T]; L^{2}(\Omega ) )$ the variational equality (4.2) remains true for all $\widetilde{\zeta }\in L^{2}(0,T; \mathbb{R})$. Let us choose $\widetilde{\zeta }\in C^{\infty } ( [0,T] )$ such that $\widetilde{\zeta }(T) =0$: with integration by part with respect to the time-variable, we obtain

$$\begin{aligned} \begin{gathered} - \int _{0}^{T} \int _{\Omega } c \theta _{n} w \widetilde{\zeta }' \,dx \,dt + \int _{0}^{T} \int _{\Omega } (K \nabla \theta _{n}) \cdot \nabla w \widetilde{\zeta }\,dx \,dt \\ \quad = \int _{0}^{T} \int _{\Omega } g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) w \widetilde{\zeta }\,dx \,dt + \int _{0}^{T} \int _{\Omega } r( \theta _{n}) w \widetilde{\zeta } \,dx \,dt \\ \quad\quad {} + \int _{0}^{T} \int _{\omega } \theta ^{b}_{\delta _{n}} w \widetilde{\zeta }\,dx' \,dt + \int _{\Omega } c \theta ^{0}_{ \delta _{n}} w \widetilde{\zeta }(0) \,dx \\ \quad \forall w \in W^{1, 2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ), \quad \forall \widetilde{\zeta }\in C^{ \infty } \bigl( [0,T] \bigr) \text{ s.t. } \widetilde{\zeta }(T) =0 . \end{gathered} \end{aligned}$$

(4.5)

Let us establish first some a priori estimates for $( \overline{\upsilon }_{n}, \pi _{n})_{n \ge 1}$.

Proposition 4.1

(A priori estimates of $( \overline{\upsilon }_{n}, \pi _{n})_{n \ge 1}$)

Let μ satisfy (2.8)–(2.10), $f \in L^{p'} (0,T; {\mathbf{L}}^{2}(\Omega ) )$, $k \in L^{p'} (0,T;L_{+}^{p'}(\Gamma _{0}) )$, $s\in L^{ p} (0,T; {\mathbf{L}}^{p}(\Gamma _{0}) )$, $\xi \in W^{1,p'}(0,T)$ satisfying (1.6), and $\upsilon ^{0}\in {\mathbf{W}}^{1,p}(\Omega )$ satisfying (2.12). Then there exists a constant $\widetilde{C}^{\mathrm{flow}}$, independent of n, such that for all $n \ge 1$:

$$\begin{aligned}& \Vert \overline{\upsilon }_{n} \Vert _{L^{p}(0,T; V^{p}_{0.\operatorname{div}})} \le \widetilde{C}^{\mathrm{flow}}, \end{aligned}$$

(4.6)

$$\begin{aligned}& \Vert \overline{\upsilon }_{n} \Vert _{L^{\infty } (0,T; {\mathbf{L}}^{2}( \Omega ) )} \le \widetilde{C}^{\mathrm{flow}}, \end{aligned}$$

(4.7)

$$\begin{aligned}& \biggl\Vert \frac{\partial \overline{\upsilon }_{n} }{\partial t} \biggr\Vert _{L^{p'} (0,T; (V^{p}_{0.\operatorname{div}})')} \le \widetilde{C}^{ \mathrm{flow}} \end{aligned}$$

(4.8)

and

$$\begin{aligned} \Vert \pi _{n} \Vert _{H^{-1} (0,T; L^{p'}_{0} (\Omega ))} \le \widetilde{C}^{ \mathrm{flow}}. \end{aligned}$$

(4.9)

Proof

By using the same computations as in Proposition 3.2, we obtain immediately (4.6)–(4.8).

Now let us prove (4.9). We choose $\overline{\varphi } = \pm \tilde{\vartheta }\zeta$ with $\tilde{\vartheta }\in {\mathbf{W}}^{1, p}_{0}(\Omega )$ and $\zeta \in \mathcal{D}(0,T)$ in (4.1). We obtain

$$\begin{aligned} \begin{gathered} \biggl\langle \int _{\Omega } \pi _{n} \operatorname{div}( \tilde{ \vartheta }) \,dx, \zeta \biggr\rangle _{{ \mathcal{D}}'(0,T), { \mathcal{D}} (0,T)} \\ \quad = - \int _{0}^{T} ( \overline{\upsilon }_{n}, \tilde{\vartheta })_{{{\mathbf{L}}^{2}( \Omega )}} \zeta ' \,dt \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) : D( \tilde{\vartheta }) \zeta \,dx \,dt \\ \quad\quad {} - \int _{0}^{T} \biggl(f + \upsilon ^{0} \frac{\partial \xi }{\partial t} ,\tilde{\vartheta } \biggr)_{{ \mathbf{L}}^{2}( \Omega )} \zeta \,dt . \end{gathered} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{gathered} \biggl\vert \biggl\langle \int _{\Omega } \pi _{n} \operatorname{div}( \tilde{ \vartheta }) \,dx, \zeta \biggr\rangle _{{ \mathcal{D}}'(0,T), { \mathcal{D}} (0,T)} \biggr\vert \\ \quad \le \sqrt{T} \Vert \overline{\upsilon }_{n} \Vert _{L^{\infty } (0,T; {\mathbf{L}}^{2}( \Omega ))} \bigl\Vert \tilde{\vartheta }\zeta ' \bigr\Vert _{L^{2} (0,T; { \mathbf{L}}^{2}(\Omega ))} \\ \quad\quad {} + 2 \mu _{1} \bigl( \Vert \overline{\upsilon }_{n} \Vert _{L^{p}(0,T; V^{p}_{0.\operatorname{div}})} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p-1} \Vert \tilde{\vartheta }\zeta \Vert _{L^{p}(0,T; V_{0}^{p})} \\ \quad\quad {} + \biggl\Vert f + \upsilon ^{0} \frac{\partial \xi }{\partial t} \biggr\Vert _{L^{p'} (0,T; {\mathbf{L}}^{2}( \Omega ))} \Vert \tilde{\vartheta }\zeta \Vert _{L^{p} (0,T; {\mathbf{L}}^{2}( \Omega ))} \end{gathered} \end{aligned}$$

and with (4.6)–(4.8), we get

$$\begin{aligned} \begin{gathered} \biggl\vert \biggl\langle \int _{\Omega } \pi _{n} \operatorname{div}( \tilde{ \vartheta }) \,dx, \zeta \biggr\rangle _{{ \mathcal{D}}'(0,T), { \mathcal{D}} (0,T)} \biggr\vert \\ \quad \le \biggl( \sqrt{T} \widetilde{C} \widetilde{C}^{ \mathrm{flow}} + 2 \mu _{1} C_{\infty } T^{1/p} \bigl( \widetilde{C}^{ \mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{ \Gamma _{1}})} \bigr)^{p-1} \\ \qquad {} + \widetilde{C} C_{\infty } T^{1/p} \biggl\Vert f + \upsilon ^{0} \frac{\partial \xi }{\partial t} \biggr\Vert _{L^{p'} (0,T; {\mathbf{L}}^{2}( \Omega ))} \biggr) \Vert \tilde{\vartheta }\zeta \Vert _{H^{1}_{0} (0,T; V_{0}^{p})} , \end{gathered} \end{aligned}$$

where C̃ denotes the norm of the continuous injection of $V^{p}_{0}$ into ${\mathbf{L}}^{2}(\Omega )$, and $C_{\infty }$ is the norm of the continuous injection of $H^{1}(0,T; \mathbb{R})$ into $L^{\infty }(0,T; \mathbb{R})$.

Moreover, for any $p>1$, there exists a linear and continuous operator $P_{p}: L^{p}_{0}(\Omega ) \to {\mathbf{W}}^{1,p}_{0}(\Omega )$ such that

$$\begin{aligned} \operatorname{div} \bigl( P_{p} (\varpi ) \bigr) = \varpi \quad \forall \varpi \in L^{p}_{0}( \Omega ) \end{aligned}$$

(see Corollary 3.1 in [2]). It follows that for any $\varpi \in L^{p}_{0}(\Omega )$ and $\zeta \in {\mathcal{D}}(0,T)$, we have

$$\begin{aligned} \begin{gathered} \biggl\vert \biggl\langle \int _{\Omega } \pi _{n} \varpi \,dx, \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}} (0,T)} \biggr\vert \\ \quad \le \Vert P_{p} \Vert _{{\mathcal{L}}(L^{p}_{0}(\Omega ), {\mathbf{W}}^{1,p}_{0}(\Omega ))} \biggl( \sqrt{T} \widetilde{C} \widetilde{C}^{\mathrm{flow}} + 2 \mu _{1} C_{\infty } T^{1/p} \bigl( \widetilde{C}^{\mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{ \Gamma _{1}})} \bigr)^{p-1} \\ \quad\quad {} + \widetilde{C} C_{\infty } T^{1/p} \biggl\Vert f + \upsilon ^{0} \frac{\partial \xi }{\partial t} \biggr\Vert _{L^{p'} (0,T; {\mathbf{L}}^{2}( \Omega ))} \biggr) \Vert \varpi \zeta \Vert _{H^{1}_{0} (0,T; L^{p} (\Omega ))}. \end{gathered} \end{aligned}$$

Hence, there exists a real number $C'>0$, independent of n, such that for all $n \ge 1$, we have

$$\begin{aligned} \biggl\vert \biggl\langle \int _{\Omega } \pi _{n} \varpi \,dx, \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}} (0,T)} \biggr\vert \le C' \Vert \varpi \zeta \Vert _{H^{1}_{0} (0,T; L^{p} (\Omega ))} \quad \forall \varpi \in L^{p}_{0}( \Omega ), \forall \zeta \in { \mathcal{D}}(0,T). \end{aligned}$$

Furthermore, for any $\varpi ^{*} \in L^{p}(\Omega )$, we may define $\varpi \in L^{p}_{0}(\Omega )$ by

$$\begin{aligned} \varpi =\varpi ^{*} -\frac{1}{\operatorname{meas}( \Omega ) } \int _{ \Omega } \varpi ^{*} \,dx. \end{aligned}$$

We have $\Vert \varpi \Vert _{L^{p}(\Omega )} \le 2 \Vert \varpi ^{*} \Vert _{L^{p}( \Omega )}$, and since $\pi _{n} \in H^{-1} (0,T; L^{p'}_{0}(\Omega ) )$, we have

$$\begin{aligned}& \biggl\langle \int _{\Omega } \pi _{n} \biggl( \varpi ^{*}- \frac{1}{\operatorname{meas}( \Omega )} \int _{\Omega } \varpi ^{*} \,dx \biggr) \,dx , \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}}(0,T)} \\& \quad = \biggl\langle \int _{\Omega } \pi _{n} \varpi ^{*} \,dx , \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}}(0,T)} \\& \qquad {} - \frac{1}{\operatorname{meas}( \Omega )} \biggl( \int _{\Omega }\varpi ^{*} \,dx \biggr) ) \biggl\langle \int _{ \Omega } \pi _{n} \,dx , \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), { \mathcal{D}}(0,T)} \\& \quad = \biggl\langle \int _{\Omega } \pi _{n} \varpi ^{*} \,dx , \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}}(0,T)}. \end{aligned}$$

It follows that

$$\begin{aligned} \begin{gathered} \biggl\vert \biggl\langle \int _{\Omega } \pi _{n} \varpi ^{*} \,dx, \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}} (0,T)} \biggr\vert \\ \quad = \biggl\vert \biggl\langle \int _{\Omega } \pi _{n} \varpi \,dx, \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), { \mathcal{D}} (0,T)} \biggr\vert \\ \quad \le 2 C' \bigl\Vert \varpi ^{*} \zeta \bigr\Vert _{H^{1}_{0} (0,T; L^{p} ( \Omega ))} \quad \forall \varpi ^{*} \in L^{p} ( \Omega ), \forall \zeta \in {\mathcal{D}}(0,T). \end{gathered} \end{aligned}$$

Hence, by possibly modifying $\widetilde{C}^{\mathrm{flow}}$, we have

$$\begin{aligned} \begin{gathered} \biggl\vert \biggl\langle \int _{\Omega } \pi _{n} \varpi ^{*} \,dx, \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}} (0,T)} \biggr\vert \le \widetilde{C}^{\mathrm{flow}} \bigl\Vert \varpi ^{*} \zeta \bigr\Vert _{H^{1}_{0} (0,T; L^{p} (\Omega ))} \\ \quad \forall \varpi ^{*} \in L^{p} (\Omega ), \forall \zeta \in {\mathcal{D}}(0,T), \end{gathered} \end{aligned}$$

and we may conclude by using the density of $\mathcal{D}(0,T)\otimes L^{p}(\Omega )$ into $H_{0}^{1} (0,T;L^{p}(\Omega ) )$. □

Let us now obtain some a priori estimates for $(\theta _{n})_{n \ge 1}$.

Proposition 4.2

(Further a priori estimates for $\theta _{n}$)

Let $q \in (1, \frac{5}{4} )$. Under the previous assumptions, there exists a constant $C^{\mathrm{heat}}_{q}>0$, independent of n, such that

$$\begin{aligned} \Vert \theta _{n} \Vert _{L^{q}(0,T; W^{1,q}_{ \Gamma _{1} \cup \Gamma _{L}} (\Omega ))} \le C^{\mathrm{heat}}_{q} \end{aligned}$$

(4.10)

for all $n \ge 1$.

Proof

Let $\widetilde{w} = w \widetilde{\zeta }$ with $w \in W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}}(\Omega )$ and $\zeta \in {\mathcal{D}}(0,T)$. With (4.2), we have

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \int _{\Omega } c \frac{\partial \theta _{n }}{\partial t} \widetilde{w} \,dx \,dt + \int _{0}^{T} \int _{\Omega } (K \nabla \theta _{n}) \cdot \nabla \widetilde{w} \,dx \,dt \\ \quad = \int _{0}^{T} \int _{\Omega } g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \widetilde{w} \,dx \,dt + \int _{0}^{T} \int _{\Omega } r( \theta _{n}) \widetilde{w} \,dx \,dt + \int _{0}^{T} \int _{\omega } \theta ^{b}_{\delta _{n}} \widetilde{w} \,dx' \,dt \end{gathered} \end{aligned}$$

(4.11)

and by density of ${\mathcal{D}}(0,T) \otimes W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}}( \Omega )$ into $L^{2} (0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}}(\Omega ) )$ the same equality holds for all $\widetilde{w} \in L^{2} ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}}( \Omega ) )$.

In order to obtain (4.10), we perform the same kind of computations as in [6]. More precisely, for all $m \ge 1$ let $\Psi _{0}^{m} : \mathbb{R} \to \mathbb{R}$ and $\Psi _{0} : \mathbb{R} \to \mathbb{R}$ be given by

$$\begin{aligned} \Psi _{0}^{m} (s) = \textstyle\begin{cases} 0 &\text{if } s \ge 1 + \frac{1}{m}, \\ 1 - m(s-1) &\text{if } 1< s< 1 + \frac{1}{m}, \\ 1 &\text{if } -1 \le s \le 1 , \\ 1 + m(s+1) &\text{if } - 1 - \frac{1}{m} < s < -1, \\ 0 &\text{if } s \le -1 - \frac{1}{m}, \end{cases}\displaystyle \quad \quad \Psi _{0} (s) = \textstyle\begin{cases} 0 &\text{if } s>1, \\ 1 &\text{if } -1 \le s \le 1, \\ 0 &\text{if } s< -1 \end{cases}\displaystyle \end{aligned}$$

and let

$$\begin{aligned} \begin{gathered} \psi _{0}^{m} (s) = \int _{0}^{s} \Psi _{0}^{m} (z) \,dz, \quad\quad \psi _{0} (s) = \int _{0}^{s} \Psi _{0} (z) \,dz \quad \text{for all $s \in \mathbb{R}$}, \\ \Phi _{0}^{m} (s) = \int _{0}^{s} \psi _{0}^{m} (z) \,dz, \quad\quad \Phi _{0} (s) = \int _{0}^{s} \psi _{0} (z) \,dz \quad \text{for all $s \in \mathbb{R}$}. \end{gathered} \end{aligned}$$

We choose $\widetilde{w} = \psi _{0}^{m} (\theta _{n}) {\mathbf{1}}_{[0,t]}$, where ${\mathbf{1}}_{[0,t]}$ is the indicatrix function of the time-interval $[0,t]$ with $t \in (0,T]$, in (4.11). We obtain

$$\begin{aligned} \begin{gathered} \int _{\Omega } c(x) \Phi _{0}^{m} \bigl( \theta _{n } (t,x) \bigr) \,dx \\ \quad\quad {} + \int _{0}^{t} \int _{\Omega } \Psi ^{m}_{0} \bigl( \theta _{n } (s,x) \bigr) \bigl( K(x) \nabla \theta _{n } (s,x) \bigr) \cdot \nabla \theta _{n} (s,x) \,dx \,ds \\ \quad = \int _{0}^{t} \int _{\Omega } g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n} ) \psi _{0}^{m} ( \theta _{n }) \,dx \,ds + \int _{0}^{t} \int _{\Omega } r( \theta _{n}) \psi _{0}^{m}( \theta _{n }) \,dx \,ds \\ \quad\quad {} + \int _{0}^{t} \int _{\omega } \theta ^{b}_{\delta _{n}} \psi _{0}^{m} (\theta _{n }) \,dx \,ds + \int _{\Omega } c(x) \Phi _{0}^{m} \bigl( \theta ^{0}_{\delta _{n}} \bigr) \,dx . \end{gathered} \end{aligned}$$

By observing that $\vert \Psi _{0}^{m} (s) \vert \le 1$, $\vert \psi ^{m}_{0} (s) \vert \le 2$ and $\vert \Phi _{0}^{m} (s) \vert \le 2 \vert s\vert$ for all $s \in \mathbb{R}$ and for all $m \ge 1$, we may pass to the limit as m tends to +∞ by using Lebesgue’s theorem, and we get

$$\begin{aligned} \begin{gathered} \int _{\Omega } c(x) \Phi _{0} \bigl( \theta _{n } (t,x) \bigr) \,dx \\ \quad\quad {} + \int _{0}^{t} \int _{\Omega } \Psi _{0} \bigl( \theta _{n } (s,x) \bigr) \bigl( K(x) \nabla \theta _{n } (s,x) \bigr) \cdot \nabla \theta _{n } (s,x) \,dx \,ds \\ \quad = \int _{0}^{t} \int _{\Omega } g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \psi _{0} (\theta _{n }) \,dx \,ds + \int _{0}^{t} \int _{\Omega } r( \theta _{n}) \psi _{0}(\theta _{n }) \,dx \,ds \\ \quad\quad {} + \int _{0}^{t} \int _{\omega } \theta ^{b}_{\delta _{n}} \psi _{0} (\theta _{n }) \,dx' \,ds + \int _{\Omega } c(x) \Phi _{0} \bigl( \theta ^{0}_{\delta _{n}} \bigr) \,dx . \end{gathered} \end{aligned}$$

Furthermore, $\vert s \vert - \frac{1}{2} \le \Phi _{0} (s) \le \vert s\vert$ and $\vert \psi _{0} (s) \vert \le 1$ for all $s \in \mathbb{R}$. Thus,

$$\begin{aligned} \begin{gathered} c_{0} \int _{\Omega } \bigl\vert \theta _{n} (t,x) \bigr\vert \,dx + k_{0} \int _{0}^{t} \int _{\Omega } \underbrace{ \Psi _{0} \bigl(\theta _{n } (s,x) \bigr) \bigl\Vert \nabla \theta _{n } (s,x) \bigr\Vert ^{2}}_{ \ge 0} \,dx \,ds \\ \quad \le \bigl\Vert g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \bigr\Vert _{L^{1} (0,T; L^{1} (\Omega ))} + \bigl\Vert r( \theta _{n}) \bigr\Vert _{L^{1}(0,T; L^{1}(\Omega ))} + \bigl\Vert \theta ^{b}_{ \delta _{n}} \bigr\Vert _{L^{1}(( 0,T) \times \omega )} \\ \quad\quad {} + \frac{1}{2} \Vert c \Vert _{L^{1}(\Omega )} + \bigl\Vert c \theta ^{0}_{\delta _{n}} \bigr\Vert _{L^{1}(\Omega )} \end{gathered} \end{aligned}$$

(4.12)

for all $t \in [0,T]$. Next, for all $k \ge 1$ and for all $m \ge 1$, we define $\Psi _{k}^{m} : \mathbb{R} \to \mathbb{R}$ and $\Psi _{k} : \mathbb{R} \to \mathbb{R}$ by

$$\begin{aligned}& \Psi _{k}^{m} (s) = \textstyle\begin{cases} 0 &\text{if } s \ge k+ 1 + \frac{1}{m}, \\ 1 - m (s- (k+1) ) &\text{if } k+1< s< k+ 1 + \frac{1}{m}, \\ 1 &\text{if } k + \frac{1}{m} \le s \le k+ 1 , \\ m ( s - k ) &\text{if } k < s < k + \frac{1}{m} , \\ 0 &\text{if } -k \le s \le k , \\ -m ( s + k ) &\text{if } -k - \frac{1}{m}< s < -k , \\ 1 &\text{if } -k-1 \le s \le -k - \frac{1}{m}, \\ 1 + m (s - (-k-1) ) &\text{if } -k - 1 - \frac{1}{m} < s < -k -1, \\ 0 &\text{if } s \le -k -1 - \frac{1}{m}, \end{cases}\displaystyle \\& \Psi _{k} (s) = \textstyle\begin{cases} 0 &\text{if } s>k+ 1, \\ 1 &\text{if } k < s \le k + 1, \\ 0 &\text{if } -k \le s \le k, \\ 1 &\text{if } -k -1 \le s < -k , \\ 0 &\text{if } s< -k -1 \end{cases}\displaystyle \end{aligned}$$

and let

$$\begin{aligned} \begin{gathered} \psi _{k}^{m} (s) = \int _{0}^{s} \Psi _{k}^{m} (z) \,dz, \quad\quad \psi _{k} (s) = \int _{0}^{s} \Psi _{k} (z) \,dz \quad \text{for all $s \in \mathbb{R}$}, \\ \Phi _{k}^{m} (s) = \int _{0}^{s} \psi _{k}^{m} (z) \,dz, \quad\quad \Phi _{k} (s) = \int _{0}^{s} \psi _{k} (z) \,dz \quad \text{for all $s \in \mathbb{R}$}. \end{gathered} \end{aligned}$$

Now we choose $\widetilde{w} = \psi _{k}^{m} (\theta _{n })$ in (4.11) with $k \ge 0$ and $m \ge 1$. We get

$$\begin{aligned} \begin{gathered} \int _{\Omega } c(x) \Phi _{k}^{m} \bigl( \theta _{n } (t,x) \bigr) \,dx \\ \quad\quad {} + \int _{0}^{t} \int _{\Omega } \Psi _{k}^{m} \bigl( \theta _{n } (s,x) \bigr) \bigl( K(x) \nabla \theta _{n} (s,x) \bigr) \cdot \nabla \theta _{n } (s,x) \,dx \,ds \\ \quad = \int _{0}^{t} \int _{\Omega } g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \psi _{k}^{m} ( \theta _{n}) \,dx \,ds + \int _{0}^{t} \int _{\Omega } r( \theta _{n}) \psi _{k}^{m} ( \theta _{n}) \,dx \,ds \\ \quad\quad {} + \int _{0}^{t} \int _{\omega } \theta ^{b}_{\delta _{n}} \psi _{k}^{m} (\theta _{n}) \,dx' \,ds + \int _{\Omega } c(x) \Phi _{k}^{m} \bigl( \theta ^{0}_{\delta _{n}} \bigr) \,dx . \end{gathered} \end{aligned}$$

We may observe that $\vert \Psi _{k}^{m} (s) \vert \le 1$, $\vert \psi ^{m}_{k} (s) \vert \le 2$, and $\vert \Phi _{k}^{m} (s) \vert \le 2 \vert s\vert$ for all $s \in \mathbb{R}$, for all $k \ge 0$ and for all $m \ge 1$. By passing to the limit as m tends to +∞, we obtain

$$\begin{aligned} \begin{gathered} \int _{\Omega } c(x) \Phi _{k} \bigl( \theta _{n} (t,x) \bigr) \,dx \\ \quad\quad {} + \int _{0}^{t} \int _{\Omega } \Psi _{k} \bigl( \theta _{n} (s,x) \bigr) \bigl( K(x) \nabla \theta _{n} (s,x) \bigr) \cdot \nabla \theta _{n} (s,x) \,dx \,ds \\ \quad = \int _{0}^{t} \int _{\Omega } g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \psi _{k} (\theta _{n}) \,dx \,ds + \int _{0}^{t} \int _{\Omega } r( \theta _{n}) \psi _{k}(\theta _{n}) \,dx \,ds \\ \quad\quad {} + \int _{0}^{t} \int _{\omega } \theta ^{b}_{\delta _{n}} \psi _{k} (\theta _{n}) \,dx' \,ds + \int _{\Omega } c(x) \Phi _{k} \bigl( \theta ^{0}_{\delta _{n}} \bigr) \,dx . \end{gathered} \end{aligned}$$

Obviously $0 \le \Phi _{k} (s) \le \vert s\vert$ and $\vert \psi _{k} (s) \vert \le 1$ for all $s \in \mathbb{R}$ and for all $k \ge 0$. Hence,

$$\begin{aligned} \begin{gathered} k_{0} \int _{0}^{T} \int _{\Omega } \Psi _{k} (\theta _{n}) \Vert \nabla \theta _{n} \Vert ^{2} \,dx \,dt \\ \quad \le \bigl\Vert g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \bigr\Vert _{L^{1} (0,T; L^{1} ( \Omega ))} \\ \quad\quad {} + \bigl\Vert r( \theta _{n}) \bigr\Vert _{L^{1}(0,T; L^{1}( \Omega ))} + \bigl\Vert \theta ^{b}_{\delta _{n}} \bigr\Vert _{L^{1}(( 0,T) \times \omega )} + \bigl\Vert c \theta ^{0}_{\delta _{n}} \bigr\Vert _{L^{1}(\Omega )}. \end{gathered} \end{aligned}$$

(4.13)

Let us define $C_{\delta _{n}}$ as

$$\begin{aligned} \begin{aligned} C_{\delta _{n}} &= \bigl\Vert g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \bigr\Vert _{L^{1} (0,T; L^{1} (\Omega ))} + \bigl\Vert r( \theta _{n}) \bigr\Vert _{L^{1}(0,T; L^{1}(\Omega ))} \\ &\quad {} + \bigl\Vert \theta ^{b}_{\delta _{n}} \bigr\Vert _{L^{1}((0,T) \times \omega )} + \frac{1}{2} \Vert c \Vert _{L^{1}(\Omega )} + \bigl\Vert c \theta ^{0}_{ \delta _{n}} \bigr\Vert _{L^{1}(\Omega )} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{gathered} Q_{0} = \bigl\{ (t,x) \in (0,T) \times \Omega ; \bigl\vert \theta _{n } (t,x) \bigr\vert \le 1 \bigr\} , \\ Q_{k} = \bigl\{ (t,x) \in (0,T) \times \Omega ; k < \bigl\vert \theta _{n } (t,x) \bigr\vert \le k+1 \bigr\} \quad \text{for all } k \in \mathbb{N}^{*} . \end{gathered} \end{aligned}$$

With (4.13), we have

$$\begin{aligned} \int _{Q_{k}} \Vert \nabla \theta _{n} \Vert ^{2} \,dx \,dt \le \frac{C_{\delta _{n}}}{k_{0}} \quad \forall k \in \mathbb{N}. \end{aligned}$$

Since $q \in (1, \frac{5}{4} )$, we may use Hölder’s inequality, which yields

$$\begin{aligned} \begin{aligned} \int _{Q_{k}} \Vert \nabla \theta _{n} \Vert ^{q} \,dx \,dt &\le \biggl( \int _{Q_{k}} \Vert \nabla \theta _{n } \Vert ^{2} \,dx \,dt \biggr)^{q/2} \bigl( \operatorname{meas}( Q_{k}) \bigr)^{1 - \frac{q}{2}} \\ &\le \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \bigl( \operatorname{meas}( Q_{k}) \bigr)^{ \frac{2-q}{2}} \quad \forall k \in \mathbb{N}. \end{aligned} \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} \int _{0}^{T} \int _{\Omega } \Vert \nabla \theta _{n} \Vert ^{q} \,dx \,dt &= \int _{\bigcup _{k \ge 0} Q_{k}} \Vert \nabla \theta _{n} \Vert ^{q} \,dx \,dt \le \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \bigl( T \operatorname{meas}( \Omega ) \bigr)^{ \frac{2-q}{2}} \\ &\quad {}+ \sum_{k \ge 1} \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \frac{1}{k^{r \frac{2-q}{2}}} \biggl( \int _{Q_{k}} \vert \theta _{n} \vert ^{r} \,dx \,dt \biggr)^{\frac{2-q}{2}} \end{aligned} \end{aligned}$$

and using again Hölder’s inequality

$$\begin{aligned} \begin{aligned} \int _{0}^{T} \int _{\Omega } \Vert \nabla \theta _{n} \Vert ^{q} \,dx \,dt &\le \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \bigl( T \operatorname{meas}( \Omega ) \bigr)^{ \frac{2-q}{2}} \\ &\quad {} + \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \biggl( \sum _{k \ge 1} \frac{1}{k^{r \frac{2-q}{q}}} \biggr)^{q/2} \biggl( \int _{0}^{T} \int _{\Omega } \vert \theta _{n } \vert ^{r} \,dx \,dt \biggr)^{\frac{2-q}{2}} \end{aligned} \end{aligned}$$

(4.14)

for all $r \ge 1$ such that $r \frac{2-q}{q} >1$. Next, we choose $r= \frac{4}{3} q$. It follows that $\frac{q^{*} (1-r)}{1-q^{*}} = q$, where $q^{*}= \frac{3q}{3-q}$ and

$$\begin{aligned} \int _{0}^{T} \int _{\Omega } \vert \theta _{n} \vert ^{r} \,dx \,dt \le \int _{0}^{T} \biggl( \int _{\Omega } \vert \theta _{n} \vert \,dx \biggr)^{\frac{r}{4} } \biggl( \int _{\Omega } \vert \theta _{n} \vert ^{q^{*}} \,dx \biggr)^{ \frac{3r}{4q^{*}}} \,dt. \end{aligned}$$

Then (4.12) implies that

$$\begin{aligned} \begin{aligned} \Vert \theta _{n} \Vert ^{r}_{L^{r} (0,T; L^{r}(\Omega ))} &= \int _{0}^{T} \int _{\Omega } \vert \theta _{n} \vert ^{r} \,dx \,dt \\ &\le \biggl( \frac{C_{\delta _{n}} }{c_{0}} \biggr)^{ \frac{r}{4}} \int _{0}^{T} \biggl( \int _{\Omega } \vert \theta _{n} \vert ^{q^{*}} \,dx \biggr)^{\frac{q}{q^{*}} } \,dt = \biggl( \frac{C_{\delta _{n} }}{c_{0}} \biggr)^{\frac{r}{4}} \Vert \theta _{n} \Vert _{L^{q} (0,T; L^{q^{*}}(\Omega ))}^{q}. \end{aligned} \end{aligned}$$

Going back to (4.14),

$$\begin{aligned} \begin{aligned} \int _{0}^{T} \Vert \theta _{n} \Vert _{W^{1,q}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )}^{q} \,dt &= \int _{0}^{T} \int _{\Omega } \Vert \nabla \theta _{n} \Vert ^{q} \,dx \,dt \\ &\le \biggl( \frac{C_{\delta _{n} }}{k_{0}} \biggr)^{q/2} \bigl(T \operatorname{meas}( \Omega ) \bigr)^{ \frac{2-q}{2}} \\ &\quad {} + \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \biggl( \sum _{k \ge 1} \frac{1}{k^{r \frac{2-q}{q}}} \biggr)^{q/2} \biggl( \frac{C_{\delta _{n}}}{c_{0}} \biggr)^{\frac{r}{4} \frac{2-q}{2} } \Vert \theta _{n} \Vert _{L^{q} (0,T; L^{q^{*}}(\Omega ))}^{q \frac{2-q}{2}} . \end{aligned} \end{aligned}$$

(4.15)

Owing to that $W^{1, q}( \Omega )$ is continuously embedded into $L^{q^{*}}( \Omega )$, we obtain

$$\begin{aligned} \begin{aligned} \Vert \theta _{n} \Vert _{L^{q}(0,T; L^{q^{*}}(\Omega ))}^{q}&= \int _{0}^{T} \Vert \theta _{n} \Vert _{L^{q^{*}}(\Omega )}^{q} \,dt \\ &\le \bigl(C'_{q} \bigr)^{q} \biggl( \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \bigl( T \operatorname{meas}( \Omega ) \bigr)^{ \frac{2-q}{2}} \\ &\quad{} + \biggl( \frac{C_{\delta _{n} }}{k_{0}} \biggr)^{q/2} \biggl( \sum _{k \ge 1} \frac{1}{k^{r \frac{2-q}{q}}} \biggr)^{q/2} \biggl( \frac{C_{\delta _{n} }}{c_{0}} \biggr)^{ \frac{r}{4} \frac{2-q}{2} } \Vert \theta _{n} \Vert _{L^{q} (0,T; L^{q^{*}}( \Omega ))}^{q \frac{2-q}{2}} \biggr), \end{aligned} \end{aligned}$$

where $C'_{q}$ is the norm of the identity mapping from $W^{1,q}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$ into $L^{q^{*}} (\Omega )$. By observing that $\frac{2-q}{2} <1$ and recalling that $r \frac{2-q}{q} = \frac{4}{3} (2-q) > 1$, we obtain

$$\begin{aligned} \begin{aligned} \Vert \theta _{n} \Vert ^{q}_{L^{q}(0,T; L^{q^{*}}(\Omega ))} &\le \max \biggl( 1, \bigl(C'_{q} \bigr)^{2} \biggl( \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \bigl( T \operatorname{meas}( \Omega ) \bigr)^{ \frac{2-q}{2}} \\ &\quad {} + \biggl( \frac{C_{\delta _{n}}}{k_{0}} \biggr)^{q/2} \biggl( \sum _{k \ge 1} \frac{1}{k^{ \frac{4}{3} (2-q) }} \biggr)^{q/2} \biggl( \frac{C_{\delta _{n}}}{c_{0}} \biggr)^{\frac{q}{3} \frac{2-q}{2} } \biggr)^{2/q} \biggr). \end{aligned} \end{aligned}$$

(4.16)

Finally, recalling the definition of $g_{\delta _{n}}$ and using (2.4) and (2.7), we obtain

$$\begin{aligned} C_{\delta _{n}} &= \biggl\Vert \frac{2\mu (\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , \vert D( \overline{\upsilon }_{n} + \upsilon ^{0} \xi ) \vert ) \vert D(\overline{\upsilon }_{n} + \upsilon ^{0} \xi ) \vert ^{p} }{ 1 + 2\delta _{n} \mu (\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , \vert D( \overline{\upsilon }_{n} + \upsilon ^{0} \xi ) \vert ) \vert D( \overline{\upsilon }_{n} + \upsilon ^{0} \xi ) \vert ^{p}} \biggr\Vert _{L^{1} (0,T; L^{1} (\Omega ))} \\ &\quad {} + \bigl\Vert r( \theta _{n}) \bigr\Vert _{L^{1}(0,T; L^{1}(\Omega ))} + \bigl\Vert \theta ^{b}_{ \delta _{n}} \bigr\Vert _{L^{1}((0,T) \times \omega )} + \frac{1}{2} \Vert c \Vert _{L^{1}( \Omega )} + \bigl\Vert c \theta ^{0}_{\delta _{n}} \bigr\Vert _{L^{1}(\Omega )} \\ &\le 2 \mu _{1} \bigl\Vert D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr\Vert ^{p}_{L^{p}(0,T; (L^{p}(\Omega ))^{3 \times 3} )} + \biggl( r_{1} + \frac{c_{1}}{2} \biggr) \operatorname{meas}( \Omega ) T \\ &\quad {} + \bigl\Vert \theta ^{b} \bigr\Vert _{L^{1}((0,T) \times \omega )} + c_{1} \bigl\Vert \theta ^{0} \bigr\Vert _{L^{1}(\Omega )} + \delta _{n} (1+c_{1}) \\ &\le 2 \mu _{1} \bigl( \Vert \overline{\upsilon }_{n} \Vert _{L^{p}(0,T; V^{p}_{0.\operatorname{div}})} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p} + \biggl( r_{1} + \frac{c_{1}}{2} \biggr) \operatorname{meas}( \Omega ) T \\ &\quad {} + \bigl\Vert \theta ^{b} \bigr\Vert _{L^{1}((0,T) \times \omega )} + c_{1} \bigl\Vert \theta ^{0} \bigr\Vert _{L^{1}(\Omega )} + \frac{1}{n} (1+c_{1}) \end{aligned}$$

and with (4.6)

$$\begin{aligned} \begin{aligned} C_{\delta _{n}}& \le 2 \mu _{1} \bigl(\widetilde{C}^{ \mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p} + \biggl( r_{1} + \frac{c_{1}}{2} \biggr) \operatorname{meas}(\Omega ) T \\ &\quad {} + \bigl\Vert \theta ^{b} \bigr\Vert _{L^{1}((0,T) \times \omega )} + c_{1} \bigl\Vert \theta ^{0} \bigr\Vert _{L^{1}(\Omega )} + (1+c_{1}) \end{aligned} \end{aligned}$$

for all $n \ge 1$. Then (4.15)–(4.16) allows us to conclude. □

As a corollary, we also obtain an estimate for $\frac{\partial \theta _{n }}{\partial t}$.

Proposition 4.3

(A priori estimate for $\frac{\partial \theta _{n }}{\partial t}$)

Let $q \in ( 1, \frac{5}{4} )$. Under the previous assumptions, we have the following estimate:

$$\begin{aligned} \begin{gathered} \biggl\Vert \frac{\partial \theta _{n}}{\partial t} \biggr\Vert _{L^{1}(0,T; (W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} ( \Omega ))')} \\ \quad \le \frac{1}{c_{0}} C_{q'} \bigl( 2 \mu _{1} \bigl( \widetilde{C}^{\mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{ \Gamma _{1}})} \bigr)^{p} + r_{1} \operatorname{meas}(\Omega ) T + \bigl\Vert \theta ^{b} \bigr\Vert _{L^{1}((0,T) \times \omega )} + 1 \bigr) \\ \quad\quad {} + \frac{T^{1/q'}}{c_{0}^{2}} \Vert K \Vert _{(L^{\infty }( \Omega ))^{3 \times 3} } C^{\mathrm{heat}}_{q} \bigl( c_{0} + C_{q'} \Vert \nabla c \Vert _{{\mathbf{L}}^{q'}( \Omega ) } \bigr), \end{gathered} \end{aligned}$$

where $C_{q'}$ is the norm of the continuous injection of $W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$ into $C^{0} ({\overline{\Omega }})$.

Proof

Since $q \in (1, \frac{5}{4} )$, we have $q' = \frac{q}{q-1} >5$, and $W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$ is continuously embedded into $W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$ and into $C^{0} ({\overline{\Omega }})$. Moreover, owing to that $c \in W^{1, q'} (\Omega )$, we also have $\frac{ w}{c} \in W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) \subset W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$ for all $w \in W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$. Let $\psi = w \widetilde{\zeta }$ with $w \in W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega )$ and $\widetilde{\zeta }\in {\mathcal{D}}(0,T)$. With $\widetilde{w} = \frac{\psi }{c} = \frac{w}{c} \zeta$ in (4.11), we get

$$\begin{aligned} \begin{gathered} \biggl\vert \int _{0}^{T} \int _{\Omega } \frac{\partial \theta _{n } }{\partial t} \psi \,dx \,dt \biggr\vert \\ \quad \le \biggl\vert \int _{0}^{T} \int _{\Omega } \frac{1}{c} (K \nabla \theta _{n}) \cdot \nabla \psi \,dx \,dt \biggr\vert \\ \quad \quad {} + \biggl\vert \int _{0}^{T} \int _{\Omega } \frac{\psi }{c^{2}} (K \nabla \theta _{n}) \cdot \nabla c \,dx \,dt \biggr\vert + \biggl\vert \int _{0}^{T} \int _{\Omega } \frac{1}{c} \bigl( g_{ \delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) + r( \theta _{n} ) \bigr) \psi \,dx \,dt \biggr\vert \\ \quad \quad {} + \biggl\vert \int _{0}^{T} \int _{\omega } \frac{1}{c} \theta ^{b}_{ \delta _{n}} \psi \,dx' \,dt \biggr\vert . \end{gathered} \end{aligned}$$

Recalling that $\theta _{n} \in W^{1,2} ( (0,T) \times \Omega )$, the same equality remains true for all $\psi \in L^{2} ( 0,T; W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} ( \Omega ) )$. Let us now consider $\psi \in L^{\infty } ( 0,T; W^{1,q'}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) )$. We obtain

$$\begin{aligned}& \biggl\vert \int _{0}^{T} \int _{\Omega } \frac{\partial \theta _{n}}{\partial t} \psi \,dx \,dt \biggr\vert \\& \quad \le \frac{T^{1/q'}}{c_{0}} \Vert K \Vert _{(L^{\infty }(\Omega ))^{3 \times 3} } \Vert \nabla \theta _{n } \Vert _{L^{q} (0,T; {\mathbf{L}}^{q} ( \Omega ))} \Vert \nabla \psi \Vert _{L^{\infty }(0,T; {\mathbf{L}}^{q'}( \Omega ))} \\& \quad\quad {} + \frac{T^{1/q'}}{c_{0}^{2}} \Vert K \Vert _{(L^{\infty }( \Omega ))^{3 \times 3} } \Vert \nabla \theta _{n} \Vert _{L^{q} (0,T; { \mathbf{L}}^{q} (\Omega ))} \Vert \psi \Vert _{L^{\infty } (0,T; C^{0}({ \overline{\Omega }}))} \Vert \nabla c \Vert _{{\mathbf{L}}^{q'}( \Omega ) } \\& \quad\quad {} + \frac{1}{c_{0}} \bigl( \bigl\Vert g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n} ) + r ( \theta _{n} ) \bigr\Vert _{L^{1}(0,T; L^{1} (\Omega ))} + \bigl\Vert \theta ^{b}_{\delta _{n}} \bigr\Vert _{L^{1}((0,T)\times \omega ) } \bigr) \Vert \psi \Vert _{L^{\infty } (0,T; C^{0}({ \overline{\Omega }}))} \\& \quad \le \frac{T^{1/q'}}{c_{0}} \Vert K \Vert _{(L^{\infty }(\Omega ))^{3 \times 3} } \Vert \nabla \theta _{n } \Vert _{L^{q} (0,T; {\mathbf{L}}^{q} ( \Omega ))} \Vert \nabla \psi \Vert _{L^{\infty } (0,T; {\mathbf{L}}^{q'}( \Omega ))} \\& \quad\quad {} + \frac{T^{1/q'}}{c_{0}^{2}} \Vert K \Vert _{(L^{\infty }( \Omega ))^{3 \times 3} } \Vert \nabla \theta _{n} \Vert _{L^{q} (0,T; { \mathbf{L}}^{q} (\Omega ))} \Vert \psi \Vert _{L^{\infty } (0,T; C^{0}({ \overline{\Omega }}))} \Vert \nabla c \Vert _{{\mathbf{L}}^{q'}( \Omega ) } \\& \quad\quad {} + \frac{1}{c_{0}} \bigl( 2 \mu _{1} \bigl( \widetilde{C}^{\mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{ \Gamma _{1}})} \bigr)^{p} + r_{1} \operatorname{meas}(\Omega ) T + \bigl\Vert \theta ^{b} \bigr\Vert _{L^{1}((0,T) \times \omega )} + 1 \bigr) \\& \quad \quad {} \times \Vert \psi \Vert _{L^{\infty } (0,T; C^{0}({\overline{\Omega }}))} \end{aligned}$$

which allows us to conclude. □

It follows that, by possibly extracting a subsequence still denoted $(\overline{\upsilon }_{n}, \pi _{n}, \theta _{n})_{n \ge 1}$, there exists a triplet $(\overline{\upsilon }, \pi , \theta )$ such that

$$\begin{aligned}& \begin{aligned} \overline{\upsilon }_{n} \longrightarrow \overline{\upsilon } \quad &\text{weakly in $L^{p} ( 0,T; V^{p}_{0.\operatorname{div}} )$ and } \\ \quad &\text{weakly* in $L^{\infty } ( 0,T; {\mathbf{L}}^{2} (\Omega ) )$}, \end{aligned} \end{aligned}$$

(4.17)

$$\begin{aligned}& \frac{ \partial \overline{\upsilon }_{n}}{\partial t} \longrightarrow \frac{ \partial \overline{\upsilon }}{\partial t} \quad \text{weakly in $L^{p'} ( 0,T; (V^{p}_{0.\operatorname{div}})' )$}, \end{aligned}$$

(4.18)

$$\begin{aligned}& \pi _{n} \longrightarrow \pi \quad \text{weakly* in $H^{-1} (0,T; L^{p'}_{0} (\Omega ) )$}, \end{aligned}$$

(4.19)

and

$$\begin{aligned} \theta _{n} \longrightarrow \theta \quad \text{weakly in $L^{q} ( 0,T; W^{1,q}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) )$.} \end{aligned}$$

(4.20)

Moreover, with the Aubin and Simon lemmas [29, 34], by possibly extracting another subsequence still denoted $(\overline{\upsilon }_{n}, \pi _{n}, \theta _{n})_{n \ge 1}$, we have

$$\begin{aligned} \begin{aligned} \overline{\upsilon }_{n} \longrightarrow \overline{\upsilon } \quad &\text{strongly in $C^{0} ( [0,T]; H )$} \\ \quad &\text{and in $L^{p} ( 0,T; {\mathbf{L}}^{p} (\Omega ) )$}, \end{aligned} \end{aligned}$$

(4.21)

where we recall that $H = \{ \psi \in {\mathbf{L}}^{2}(\Omega ); \operatorname{div}(\psi ) =0 \text{ in } \Omega , \psi . n = 0 \text{ on } \partial \Omega \}$, and with Aubin’s generalized lemma [29], we also have

$$\begin{aligned} \theta _{n} \longrightarrow \theta \quad \text{strongly in $L^{q} ( 0,T; L^{q}(\Omega ) )$.} \end{aligned}$$

(4.22)

We infer that

$$\begin{aligned} \overline{\upsilon }_{n} (0, \cdot ) \longrightarrow \overline{\upsilon } (0, \cdot ) = 0 \quad \text{strongly in ${\mathbf{L}}^{2}(\Omega )$.} \end{aligned}$$

(4.23)

Furthermore, by possibly extracting again a subsequence still denoted $(\overline{\upsilon }_{n}, \pi _{n}, \theta _{n})_{n \ge 1}$, we have

$$\begin{aligned} \overline{\upsilon }_{n} \longrightarrow \overline{\upsilon }, \quad\quad \theta _{n} \longrightarrow \theta \quad \text{a.e. in $(0,T) \times \Omega $.} \end{aligned}$$

(4.24)

These convergence properties do not allow us to pass directly to the limit as n tends to +∞, and we need a better convergence result for $(\overline{\upsilon }_{n})_{n \ge 1}$ to overcome the difficulty due to the non-linearity of the mapping ${\mathcal{F}}$.

Proposition 4.4

Under the previous assumptions, $( D(\overline{\upsilon }_{n}) )_{n \ge 1}$ converges strongly to $D(\overline{\upsilon })$ in $L^{p} (0,T; ({ L}^{p}(\Omega ))^{3 \times 3} )$.

Proof

Let $(\hat{\upsilon }, \hat{\pi })$ be the unique solution of the problem $(P^{\mathrm{flow}}_{( \overline{\upsilon }, \theta )})$. We have

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \biggl\langle \frac{ \partial \hat{\upsilon } }{\partial t} , {\overline{\varphi }} - \hat{\upsilon } \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \\ \quad {} + \int _{0}^{T} \int _{\Omega } { \mathcal{F}} \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) : D({ \overline{\varphi }} - \hat{\upsilon } ) \,dx \,dt \\ \quad {} + J( {\overline{\varphi }}) - J( \hat{\upsilon }) \ge \int _{0}^{T} \biggl( f + {\upsilon }^{0} \frac{ \partial \xi }{\partial t} , {\overline{\varphi }} - \hat{\upsilon } \biggr)_{{\mathbf{L}}^{2}(\Omega )} \,dt \quad \forall { \overline{\varphi }} \in L^{p} \bigl( 0,T; V^{p}_{0.\operatorname{div}} \bigr). \end{gathered} \end{aligned}$$

(4.25)

For all $n \ge 1$, we also have

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \biggl\langle \frac{ \partial \overline{\upsilon }_{n} }{\partial t} , { \overline{\varphi }} - \overline{\upsilon }_{n} \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \\ \quad {} + \int _{0}^{T} \int _{\Omega } { \mathcal{F}} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) : D({ \overline{\varphi }} - \overline{\upsilon }_{n} ) \,dx \,dt \\ \quad {} + J( {\overline{\varphi }}) - J( \overline{\upsilon }_{n}) \ge \int _{0}^{T} \biggl( f + {\upsilon }^{0} \frac{ \partial \xi }{\partial t} , {\overline{\varphi }} - \overline{ \upsilon }_{n} \biggr)_{{\mathbf{L}}^{2}(\Omega )} \,dt \quad \forall { \overline{\varphi }} \in L^{p} \bigl( 0,T; V^{p}_{0.\operatorname{div}} \bigr). \end{gathered} \end{aligned}$$

(4.26)

Let us choose ${\overline{\varphi }} = \hat{\upsilon }$ in (4.26) and ${\overline{\varphi }} = \overline{\upsilon }_{n}$ in (4.25). By adding the two inequalities, we obtain:

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \biggl\langle \frac{ \partial (\overline{\upsilon }_{n} - \hat{\upsilon }) }{\partial t} , \overline{\upsilon }_{n} - \hat{\upsilon } \biggr\rangle _{(V_{0.\operatorname{div}}^{p})', V_{0.\operatorname{div}}^{p}} \,dt \\ \quad {} + \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad {} - {\mathcal{F}} \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl(\hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt \le 0 \end{gathered} \end{aligned}$$

and thus,

$$\begin{aligned} \begin{gathered} \frac{1}{2} \Vert \overline{ \upsilon }_{n} - \hat{\upsilon } \Vert ^{2}_{{\mathbf{L}}^{2} (\Omega )} (T) \\ \quad \quad {} + \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - {\mathcal{F}} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl(\hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt \\ \quad \le \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}} \bigl(\theta , \overline{ \upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - {\mathcal{F}} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl(\hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt . \end{gathered} \end{aligned}$$

(4.27)

We perform the same kind of computations as in Theorem 3.5. More precisely, we split the second term of the left-hand side as follows:

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}} \bigl( \theta _{n} , \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad \quad {} - {\mathcal{F}} \bigl( \theta _{n}, \overline{ \upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{ \upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{ \upsilon }_{n}- \hat{\upsilon } ) \,dx \,dt \\ \quad = { \int _{0}^{T} \int _{\Omega } \bigl( {{\mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) - {{ \mathcal{F}}_{1} } \bigl( D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n}-\hat{\upsilon } ) \,dx \,dt } \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \bigl( {{\mathcal{F}}_{2} } \bigl( \theta _{n} , \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad \quad {} - {{\mathcal{F}}_{2} } \bigl( \theta _{n} , \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n}- \hat{\upsilon } ) \,dx \,dt, \end{gathered} \end{aligned}$$

where we recall that

$$\begin{aligned}& {{\mathcal{F}}_{1} } (\lambda _{2}) = \mu _{0} \Vert \lambda _{2} \Vert ^{p-2} \lambda _{2} \quad \text{if } \lambda _{2} \neq 0_{ \mathbb{R}^{3\times 3}}, \quad\quad {{\mathcal{F}}_{1} } ( \lambda _{2})= 0_{ \mathbb{R}^{3\times 3}} \quad \text{otherwise}, \\& \textstyle\begin{cases} {{\mathcal{F}}_{2} } (\lambda _{0}, \lambda _{1}, \lambda _{2}) = 2 {\overline{\mu }} (\lambda _{0} ,\lambda _{1}, \Vert \lambda _{2} \Vert ) \Vert \lambda _{2} \Vert ^{p-2}\lambda _{2} \quad \text{if } \lambda _{2} \neq 0_{\mathbb{R}^{3\times 3}}, \\ {{\mathcal{F}}_{2} } (\lambda _{0}, \lambda _{1}, \lambda _{2})= 0_{\mathbb{R}^{3\times 3}} \quad \text{otherwise} \end{cases}\displaystyle \end{aligned}$$

and ${\overline{\mu }} = \mu - \frac{\mu _{0}}{2}$. Since $\lambda _{2} \mapsto {{\mathcal{F}}}_{2} (\cdot , \cdot , \lambda _{2})$ is monotone in $\mathbb{R}^{3\times 3}$ (see Lemma 1 in [9]), we have

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \int _{\Omega } \bigl( { {\mathcal{F}}}_{2} \bigl( \theta _{n} , \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad {} - { {\mathcal{F}}}_{2} \bigl(\theta _{n} , \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n}- \hat{\upsilon } ) \,dx \,dt \ge 0, \end{gathered} \end{aligned}$$

and (4.27) yields

$$\begin{aligned} \begin{gathered} \frac{1}{2} \Vert \overline{ \upsilon }_{n} - \hat{\upsilon } \Vert ^{2}_{{\mathbf{L}}^{2} (\Omega )} (T) \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}}_{1} \bigl( D \bigl(\overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) - { \mathcal{F}}_{1} \bigl( D \bigl(\hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt \\ \quad \le \int _{0}^{T} \int _{\Omega } \bigl( {\mathcal{F}}_{2} \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - {\mathcal{F}}_{2} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt . \end{gathered} \end{aligned}$$

(4.28)

Then we distinguish two cases.

• Case 1: $p \in [6/5, 2)$

Recalling that

$$\begin{aligned} \bigl( \Vert \lambda \Vert ^{p} + \bigl\Vert \lambda ' \bigr\Vert ^{p} \bigr)^{ \frac{2-p}{2}} \bigl( \bigl( { {\mathcal{F}}_{1} } (\lambda ) - { { \mathcal{F}}_{1} } \bigl(\lambda ' \bigr) \bigr) : \bigl( \lambda - \lambda ' \bigr) \bigr)^{\frac{p}{2}} \ge \frac{ ( \mu _{0} (p-1) )^{\frac{p}{2}}}{ 2^{ \frac{(p-1)(2-p)}{2} } } \bigl\Vert \lambda - \lambda ' \bigr\Vert ^{p} \end{aligned}$$

for all $(\lambda , \lambda ') \in {\mathbb{R}}^{3 \times 3} \times { \mathbb{R}}^{3 \times 3}$, we obtain

$$\begin{aligned} \begin{gathered} \frac{ ( \mu _{0} (p-1) )^{\frac{p}{2}}}{ 2^{ \frac{(p-1)(2-p)}{2} } } \int _{0}^{T} \int _{\Omega } \bigl\Vert D( \overline{\upsilon }_{n} - \hat{\upsilon }) \bigr\Vert ^{p} \,dx \,dt \\ \quad \le \biggl( \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) - {{\mathcal{F}}_{1} } \bigl( D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt \biggr)^{\frac{p}{2}} \\ \qquad {} \times \biggl( \int _{0}^{T} \int _{\Omega } \bigl( \bigl\Vert D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr\Vert ^{p} + \bigl\Vert D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr\Vert ^{p} \bigr) \,dx \,dt \biggr)^{\frac{2-p}{2}} \\ \quad \le \biggl( \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) - {{\mathcal{F}}_{1} } \bigl( D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt \biggr)^{\frac{p}{2}} \\ \qquad {} \times \bigl( \bigl\Vert \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})}^{p} + \bigl\Vert \hat{\upsilon } + \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})}^{p} \bigr)^{\frac{2-p}{2}}. \end{gathered} \end{aligned}$$

Thus with (3.11) and (4.6), we get

$$\begin{aligned} \begin{gathered} \frac{ ( \mu _{0} (p-1) )^{\frac{p}{2}}}{ 2^{ \frac{(p-1)(2-p)}{2} } } \bigl\Vert D( \overline{ \upsilon }_{n} - \hat{\upsilon } ) \bigr\Vert _{L^{p}(0,T; (L^{p}(\Omega ))^{3\times 3})}^{p} \\ \quad \le \biggl( \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) - {{\mathcal{F}}_{1} } \bigl( D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt \biggr)^{\frac{p}{2}} \\ \qquad {} \times \bigl( \bigl( \widetilde{C}^{\mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p} + \bigl( C^{ \mathrm{flow}} + \bigl\Vert \upsilon ^{0} \xi \bigr\Vert _{L^{p}(0,T; V^{p}_{\Gamma _{1}})} \bigr)^{p} \bigr)^{\frac{2-p}{2}}. \end{gathered} \end{aligned}$$

By observing that $\widetilde{C}^{\mathrm{flow}} \ge C^{\mathrm{flow}}$, we have finally

$$\begin{aligned} \begin{gathered} \frac{ \mu _{0} (p-1) }{ 2^{ (2-p) } (\widetilde{C}^{\mathrm{flow}} + \Vert \upsilon ^{0} \xi \Vert _{ L^{p} (0,T; V^{p}_{\Gamma _{1}} ) } )^{ (2-p) } } \bigl\Vert D( \overline{\upsilon }_{n}-\hat{\upsilon }) \bigr\Vert _{ L^{p} (0,T; (L^{p} (\Omega ) )^{3\times 3} ) }^{2} \\ \quad \leq \int _{0}^{T} \int _{\Omega } \bigl( {{ \mathcal{F}}_{1} } \bigl( D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) - {{ \mathcal{F}}_{1} } \bigl( D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n} - \hat{\upsilon }) \,dx \,dt \\ \quad \leq \int _{0}^{T} \int _{\Omega } \bigl( { { \mathcal{F}}}_{2} \bigl( \theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{ \upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - {{\mathcal{F}}}_{2} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n}- \hat{\upsilon } ) \,dx \,dt \\ \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - { {\mathcal{F}}}_{2} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}( \Omega ))^{3 \times 3})} \\ \quad \quad {} \times \bigl\Vert D(\overline{\upsilon }_{n}- \hat{ \upsilon }) \bigr\Vert _{L^{p}(0,T; (L^{p}(\Omega ))^{3 \times 3})} \end{gathered} \end{aligned}$$

which yields

$$\begin{aligned} \begin{gathered} \frac{ \mu _{0} (p-1) }{ 2^{ (2-p) } (\widetilde{C}^{\mathrm{flow}} + \Vert \upsilon ^{0} \xi \Vert _{ L^{p} (0,T; V^{p}_{\Gamma _{1}} ) } )^{ (2-p) } } \bigl\Vert D( \overline{\upsilon }_{n}-\hat{\upsilon }) \bigr\Vert _{ L^{p} (0,T; (L^{p} (\Omega ) )^{3\times 3} ) } \\ \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl( \theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - { {\mathcal{F}}}_{2} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}( \Omega ))^{3 \times 3})} . \end{gathered} \end{aligned}$$

(4.29)

• Case 2: $p \in [2, + \infty )$

Recalling that

$$\begin{aligned} \bigl( {\mathcal{F}}_{1} (\lambda ) - {\mathcal{F}}_{1} \bigl(\lambda ' \bigr) \bigr) : \bigl(\lambda - \lambda ' \bigr) \ge \frac{\mu _{0} }{2^{p-1}} \bigl\Vert \lambda - \lambda ' \bigr\Vert ^{p} \end{aligned}$$

for all $(\lambda , \lambda ') \in {\mathbb{R}}^{3 \times 3} \times { \mathbb{R}}^{3 \times 3}$, we obtain

$$\begin{aligned} \begin{gathered} \frac{\mu _{0}}{2^{p-1} } \bigl\Vert D(\overline{ \upsilon }_{n}- \hat{\upsilon }) \bigr\Vert _{ L^{p}(0,T; (L^{p}(\Omega ))^{3\times 3})}^{p} \\ \quad \leq \int _{0}^{T} \int _{\Omega } \bigl( { { \mathcal{F}}}_{1} \bigl( D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) - {{ \mathcal{F}}}_{1} \bigl( D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n}- \hat{\upsilon } ) \,dx \,dt \\ \quad \leq \int _{0}^{T} \int _{\Omega } \bigl( { { \mathcal{F}}}_{2} \bigl( \theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{ \upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - {{\mathcal{F}}}_{2} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr) : D( \overline{\upsilon }_{n}- \hat{\upsilon } ) \,dx \,dt \\ \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad \quad {} - { {\mathcal{F}}}_{2} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}( \Omega ))^{3 \times 3})} \\ \quad \quad {} \times \bigl\Vert D(\overline{\upsilon }_{n}- \hat{ \upsilon }) \bigr\Vert _{L^{p}(0,T; (L^{p}(\Omega ))^{3 \times 3})} \end{gathered} \end{aligned}$$

which yields

$$\begin{aligned} \begin{gathered} \frac{\mu _{0}}{2^{p-1} } \bigl\Vert D( \overline{\upsilon }_{n}- \hat{\upsilon }) \bigr\Vert _{ L^{p}(0,T; (L^{p}(\Omega ))^{3\times 3})}^{p-1} \\ \quad \le \bigl\Vert { {\mathcal{F}}}_{2} \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad\quad {} - { {\mathcal{F}}}_{2} \bigl( \theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \bigr\Vert _{L^{p'}(0,T; (L^{p'}( \Omega ))^{3 \times 3})} .\end{gathered} \end{aligned}$$

(4.30)

By using the convergence properties (4.24) for the sequences $(\theta _{n})_{n \ge 1}$ and $(\overline{\upsilon }_{n})_{n \ge 1}$ and the continuity and boundedness assumptions (2.8) and (2.10) for the mapping μ, we obtain

$$\begin{aligned} \begin{gathered} {\mathcal{F}}_{2} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \longrightarrow {\mathcal{F}}_{2} \bigl( \theta , \hat{ \upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad \text{strongly in $L^{p'} (0,T; (L^{p'}(\Omega ))^{3 \times 3} )$.} \end{gathered} \end{aligned}$$

Then with (4.29) if $p \in [6/5, 2)$ and (4.30) if $p \in [2, + \infty )$, we obtain

$$\begin{aligned} \lim_{n\rightarrow +\infty } \bigl\Vert D(\overline{\upsilon }_{n}- \hat{\upsilon }) \bigr\Vert _{L^{p}(0,T; ({ L}^{p}(\Omega ))^{3 \times 3} )} = 0. \end{aligned}$$

Finally, Korn’s inequality implies that the sequence $(\overline{\upsilon }_{n})_{n \ge 1}$ converges strongly to υ̂ in $L^{p} ( 0,T; V^{p}_{0.\operatorname{div}})$, which yields $\hat{\upsilon } = \overline{\upsilon }$. □

By possibly extracting another subsequence, still denoted $( \overline{\upsilon }_{n}, \pi _{n} , \theta _{n})_{n \ge 1}$, we obtain that

$$\begin{aligned} D( \overline{\upsilon }_{n}) \longrightarrow D(\overline{\upsilon }) \quad \text{a.e. in $(0,T) \times \Omega $}. \end{aligned}$$

By using (2.8) and (2.10), we infer from Lebesgue’s theorem that

$$\begin{aligned} \begin{gathered} {\mathcal{F}} \bigl(\theta _{n}, \overline{\upsilon }_{n} + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon }_{n} + \upsilon ^{0} \xi \bigr) \bigr) \longrightarrow {\mathcal{F}} \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) \\ \quad \text{strongly in $L^{p'} (0,T; (L^{p'}(\Omega ))^{3 \times 3} )$} \end{gathered} \end{aligned}$$

and

$$\begin{aligned} \begin{gathered} g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) \longrightarrow 2 \mu \bigl(\theta , \overline{\upsilon } + \upsilon ^{0} \xi , \bigl\Vert D \bigl( \overline{ \upsilon } + \upsilon ^{0} \xi \bigr) \bigr\Vert \bigr) \bigl\Vert D(\overline{ \upsilon } + \upsilon _{0} \xi ) \bigr\Vert ^{p} \\ \quad = {\mathcal{F}} \bigl( \theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) : D \bigl( \overline{\upsilon } + \upsilon ^{0} \xi \bigr) \\ \quad \text{strongly in $L^{1} (0,T; L^{1}(\Omega ) )$.} \end{gathered} \end{aligned}$$

Moreover,

$$\begin{aligned} \overline{\upsilon }_{n} \longrightarrow \hat{\upsilon }= \overline{\upsilon } \quad \text{strongly in $L^{p} ( 0,T; V^{p}_{0.\operatorname{div}})$.} \end{aligned}$$

Reminding that the mapping J is continuous on $L^{p} ( 0,T; V^{p}_{0})$, we get

$$\begin{aligned} J( \overline{\upsilon }_{n} + \widetilde{\vartheta }\zeta ) \longrightarrow J( \overline{\upsilon } + \widetilde{\vartheta } \zeta ) \quad \forall \widetilde{\vartheta }\in V^{p}_{0}, \forall \zeta \in {\mathcal{D}}(0,T). \end{aligned}$$

Finally, by using the continuity and boundedness assumptions (2.6)–(2.7) for the mapping r, we also have

$$\begin{aligned} r(\theta _{n}) \longrightarrow r(\theta ) \quad \text{strongly in $L^{1} ( 0,T; L^{1}(\Omega ) )$.} \end{aligned}$$

It follows that we can pass to the limit in (4.1) and (4.5), which allows us to conclude that the triplet $(\overline{\upsilon }, \pi , \theta )$ is a solution to the coupled fluid flow/heat transfer problem $(P)$.

Remark 4.5

We may observe that $\pi = \hat{\pi }$. Indeed,

$$\begin{aligned} \begin{gathered} \biggl\langle \frac{\partial }{\partial t}( \overline{ \upsilon } , \tilde{\vartheta })_{{\mathbf{L}}^{2}(\Omega )}, \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \overline{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) : D( \tilde{\vartheta }) \zeta \,dx \,dt \\ \quad\quad {} - \biggl\langle \int _{\Omega } \pi \operatorname{div}( \tilde{\vartheta }) \,dx, \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} + J( \overline{\upsilon }+\tilde{ \vartheta }\zeta )-J( \overline{\upsilon }) \\ \quad \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t}, \tilde{\vartheta } \biggr)_{{ \mathbf{L}}^{2}( \Omega )} \zeta \,dt \quad \forall \tilde{\vartheta }\in V^{p}_{0}, \forall \zeta \in \mathcal{D}(0,T) \end{gathered} \end{aligned}$$

and

$$\begin{aligned} \begin{gathered} \biggl\langle \frac{\partial }{\partial t}(\hat{\upsilon }, \tilde{\vartheta })_{{\mathbf{L}}^{2}(\Omega )},\zeta \biggr\rangle _{ \mathcal{D}'(0,T), \mathcal{D}(0,T)} \\ \quad\quad {} + \int _{0}^{T} \int _{\Omega } \mathcal{F} \bigl( \theta , \overline{\upsilon } + \upsilon ^{0} \xi , D \bigl( \hat{\upsilon } + \upsilon ^{0} \xi \bigr) \bigr) : D( \tilde{\vartheta }) \zeta \,dx \,dt \\ \quad\quad {} - \biggl\langle \int _{\Omega } \hat{\pi } \operatorname{div}( \tilde{\vartheta }) \,dx , \zeta \biggr\rangle _{\mathcal{D}'(0,T), \mathcal{D}(0,T)} + J( \hat{\upsilon } +\tilde{ \vartheta }\zeta )-J( \hat{\upsilon }) \\ \quad \geq \int _{0}^{T} \biggl( f + \upsilon ^{0} \frac{\partial \xi }{\partial t}, \tilde{\vartheta } \biggr)_{{ \mathbf{L}}^{2}( \Omega )} \zeta \,dt \quad \forall \tilde{\vartheta }\in V^{p}_{0}, \forall \zeta \in \mathcal{D}(0,T) . \end{gathered} \end{aligned}$$

Owing to that $\overline{\upsilon } = \hat{\upsilon }$, we obtain that

$$\begin{aligned} \biggl\langle \int _{\Omega } (\pi - \hat{\pi }) \operatorname{div}( \tilde{ \vartheta }) \,dx , \zeta \biggr\rangle _{{ \mathcal{D}}'(0,T), { \mathcal{D}}(0,T)} =0 \quad \forall \tilde{\vartheta }\in {\mathbf{W}}^{1,p}_{0} (\Omega ), \forall \zeta \in {\mathcal{D}}(0,T). \end{aligned}$$

Hence, for any $\varpi \in L^{p}_{0}(\Omega )$, we may choose $\widetilde{\vartheta }= P_{p} (\varpi )$, where $P_{p}$ is the operator introduced in Proposition 4.1, and we get

$$\begin{aligned} \biggl\langle \int _{\Omega } (\pi - \hat{\pi }) \varpi \,dx, \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), {\mathcal{D}} (0,T)} = 0. \end{aligned}$$

Furthermore, for any $\varpi ^{*} \in L^{p}(\Omega )$, we may define $\varpi \in L^{p}_{0}(\Omega )$ by

$$\begin{aligned} \varpi =\varpi ^{*} -\frac{1}{\operatorname{meas}( \Omega ) } \int _{ \Omega } \varpi ^{*} \,dx, \end{aligned}$$

and since $\pi - \hat{\pi }\in H^{-1} (0,T; L^{p}_{0}(\Omega ) )$, we have

$$\begin{aligned}& \biggl\langle \int _{\Omega } ( \pi - \hat{\pi }) \varpi \,dx , \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), { \mathcal{D}}(0,T)} \\& \quad = \biggl\langle \int _{\Omega }( \pi - \hat{\pi }) \varpi ^{*} \,dx , \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), { \mathcal{D}}(0,T)} \\& \quad\quad {} - \frac{1}{\operatorname{meas}( \Omega )} \biggl( \int _{\Omega }\varpi ^{*} \,dx \biggr) ) \biggl\langle \int _{ \Omega } (\pi - \hat{\pi }) \,dx , \zeta \biggr\rangle _{{ \mathcal{D}}'(0,T), {\mathcal{D}}(0,T)} \\& \quad = \biggl\langle \int _{\Omega } ( \pi - \hat{\pi }) \varpi ^{*} \,dx , \zeta \biggr\rangle _{{\mathcal{D}}'(0,T), { \mathcal{D}}(0,T)} =0 \quad \forall \varpi ^{*} \in L^{p} (\Omega ), \forall \zeta \in { \mathcal{D}}(0,T), \end{aligned}$$

which implies $\pi = \hat{\pi }$.

Remark 4.6

The strong convergence of $( g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) )_{n \ge 1}$ and $( r (\theta _{n}) )_{n \ge 1}$ in the space $L^{1} ( 0,T; L^{1}(\Omega ) )$ implies that $\theta \in C^{0} ( [0,T]; L^{1}(\Omega ) )$ and $\theta ( 0, \cdot ) = \theta ^{0}$. Indeed, starting from (4.11), we have also

$$\begin{aligned} \begin{gathered} \int _{0}^{T} \int _{\Omega } c \frac{\partial (\theta _{n } - \theta _{n'}) }{\partial t} \widetilde{w} \,dx \,dt + \int _{0}^{T} \int _{\Omega } \bigl(K \nabla ( \theta _{n} - \theta _{n'}) \bigr) \cdot \nabla \widetilde{w} \,dx \,dt \\ \quad = \int _{0}^{T} \int _{\Omega } \bigl( g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) - g_{\delta _{n'}} ( \theta _{n'}, \overline{\upsilon }_{n'}) \bigr) \widetilde{w} \,dx \,dt + \int _{0}^{T} \int _{\Omega } \bigl( r( \theta _{n}) - r( \theta _{n'}) \bigr) \widetilde{w} \,dx \,dt \\ \quad\quad {} + \int _{0}^{T} \int _{\omega } \bigl( \theta ^{b}_{ \delta _{n}} - \theta ^{b}_{\delta _{n'}} \bigr) \widetilde{w} \,dx' \,dt \end{gathered} \end{aligned}$$

for all $\widetilde{w} \in L^{2} ( 0,T; W^{1,2}_{\Gamma _{1} \cup \Gamma _{L}} (\Omega ) )$, for all $n \ge 1$, and $n' \ge 1$. By choosing $\widetilde{w} = \psi _{0}^{m} (\theta _{n} - \theta _{n'}) { \mathbf{1}}_{[0,t]}$, with $t \in (0,T]$ and by passing to the limit, as m tends to +∞, we get

$$\begin{aligned}& \int _{\Omega } c(x) \Phi _{0} \bigl( \theta _{n} (t,x) - \theta _{n'} (t,x) \bigr) \,dx \\& \quad\quad {} + \int _{0}^{t} \int _{\Omega } \Psi _{0} \bigl( \theta _{n} (s,x) - \theta _{n'} (s,x) \bigr) \\& \quad\quad {} \times \bigl( K(x) \nabla \bigl( \theta _{n} (s,x) - \theta _{n'} (s,x) \bigr) \bigr) \cdot \nabla \bigl(\theta _{n} (s,x) - \theta _{n'} (s,x) \bigr) \,dx \,ds \\& \quad = \int _{0}^{t} \int _{\Omega } \bigl( g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) - g_{\delta _{n'}} ( \theta _{n'}, \overline{\upsilon }_{n'}) \bigr) \psi _{0} ( \theta _{n} - \theta _{n'}) \,dx \,ds \\& \quad\quad {} + \int _{0}^{t} \int _{\Omega } \bigl( r( \theta _{n}) - r( \theta _{n'}) \bigr) \psi _{0} (\theta _{n} - \theta _{n'}) \,dx \,ds \\& \quad\quad {} + \int _{0}^{t} \int _{\omega } \bigl( \theta ^{b}_{ \delta _{n}} - \theta ^{b}_{\delta _{n'}} \bigr) \psi _{0} (\theta _{n} - \theta _{n'}) \,dx' \,ds + \int _{\Omega } c(x) \Phi _{0} \bigl( \theta ^{0}_{ \delta _{n}} - \theta ^{0}_{\delta _{n'}} \bigr) \,dx, \end{aligned}$$

which yields

$$\begin{aligned} \begin{gathered} c_{0} \int _{\Omega } \Phi _{0} \bigl( \theta _{n} (t,x) - \theta _{n'} \bigr) \,dx \\ \quad \le \bigl\Vert g_{\delta _{n}} ( \theta _{n}, \overline{\upsilon }_{n}) - g_{\delta _{n'}} ( \theta _{n'}, \overline{ \upsilon }_{n'}) \bigr\Vert _{L^{1}(0,T; L^{1}(\Omega ))} \\ \quad\quad {} + \bigl\Vert r( \theta _{n}) - r( \theta _{n'}) \bigr\Vert _{L^{1}(0,T; L^{1}(\Omega ))} + \bigl\Vert \theta ^{b}_{\delta _{n}} - \theta ^{b}_{ \delta _{n'}} \bigr\Vert _{L^{1}( (0,T) \times \omega )} + \bigl\Vert c \bigl( \theta ^{0}_{\delta _{n}} - \theta ^{0}_{\delta _{n'}} \bigr) \bigr\Vert _{L^{1}( \Omega )}. \end{gathered} \end{aligned}$$

Next, we observe that

$$\begin{aligned} \Phi _{0} (s) = \textstyle\begin{cases} \frac{s^{2}}{2} &\text{if } \vert s \vert \le 1, \\ s - \frac{1}{2} &\text{if } s >1, \\ -s - \frac{1}{2} &\text{if } s < -1 \end{cases}\displaystyle \end{aligned}$$

which implies $\Phi _{0}(s) \ge \frac{\vert s\vert }{2}$ if $\vert s\vert \ge 1$. It follows that

$$\begin{aligned} \begin{gathered} c_{0} \int _{\Omega } \bigl\vert \theta _{n} (t , x) - \theta _{n'} (t , x) \,dx \bigr\vert \\ \quad \le c_{0} \biggl\{ \biggl( \int _{\Omega } \Phi _{0} \bigl(\theta _{n} (t,x) - \theta _{n'} (t,x) \bigr) \biggr)^{1/2} \operatorname{meas} (\Omega )^{1/2} + 2 \int _{\Omega } \Phi _{0} \bigl(\theta _{n} (t,x) - \theta _{n'} (t,x) \bigr) \biggr\} , \end{gathered} \end{aligned}$$

which allows us to conclude that $(\theta _{n})_{n \ge 1}$ is a Cauchy sequence in $C^{0} ([0,T]; L^{1}(\Omega ) )$. As a consequence, we may relax the regularity of the test-function ζ̃ from $C^{\infty } ( [0,T] )$ in (2.15).

Moreover, following [28], we can prove that θ is an entropy solution of (1.3)–(1.9). For more details on the definition of entropy (or renormalized) solutions, the reader is referred to [5, 12, 28] and the references therein.

Finally, let us emphasize that the strong convergence of $(\theta _{n})_{n \ge 1}$ to θ in the space $C^{0} ( [0,T]; L^{1}( \Omega ) )$, which is the key point to show that θ is an entropy solution of (1.3)–(1.9), will not be true anymore if convective effects are taken into account in the heat equation.