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

Laetitia Paoli

► To cite this version:

Laetitia Paoli. Unsteady non-Newtonian fluid flow with heat transfer and Tresca's friction boundary conditions. Fixed Point Theory and Algorithms for Sciences and Engineering, 2022, 10.1186/s13663-022-00714-x. http://hal-04155324

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Submitted on 14 Mar 2025

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

Laetitia Paoli¹*

## 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 $\sigma = 2\mu(\theta, \nu, ||D(\nu)||) ||D(\nu)||^{p-2}D(\nu) - \pi \text{Id}$ where $\theta$ is the temperature, $\pi$ is the pressure, $\nu$ is the velocity, and $D(\nu)$ 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 $\delta$ tends to zero.

**MSC:** Primary 76A05; 35Q35; secondary 35Q79; 35K87; 76M30

**Keywords:** Unsteady non-Newtonian fluid flow; Shear thickening and shear thinning fluids; Heat transfer; Tresca's friction boundary conditions; Non-linear parabolic variational inequality; Fixed point technique

## 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

$$ \sigma = 2\mu(\theta, \nu, ||D(\nu)||) ||D(\nu)||^{p-2}D(\nu) - \pi \text{Id}_{\mathbb{R}^3}, \quad (1.1) $$

where $\mu$ is a given mapping, $\theta$ is the temperature, $\pi$ is the pressure, $\nu$ is the velocity, $D(\nu)$ is the strain rate tensor, and $p \in (1, +\infty)$ is a real parameter.

<footnote>*Correspondence: laetitia.paoli@univ-st-etienne.fr
¹Universite Jean Monnet, Saint-Etienne, France</footnote>
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 $\mu$ depends on $\theta$, $v$, and $|D(v)|$, 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

$$ \Omega = \{(x', x_3) \in \mathbb{R}^2 \times \mathbb{R} : x' \in \omega, 0 < x_3 < h(x')\}, $$

where $\omega$ 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:

$$ \left\{ \begin{array}{l} \frac{\partial v}{\partial t} - 2 \operatorname{div}(\mu(\theta, v, \|D(v)\|) \|D(v)\|^{p-2} D(v)) + \nabla \pi = f \quad \text{in } (0, T) \times \Omega, \\ \operatorname{div}(v) = 0 \quad \text{in } (0, T) \times \Omega, \end{array} \right. \quad (1.2) $$

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

$$ c \frac{\partial \theta}{\partial t} - \operatorname{div}(K \nabla \theta) = 2\mu(\theta, v, \|D(v)\|) \|D(v)\|^p + r(\theta) \quad \text{in } (0, T) \times \Omega, \quad (1.3) $$

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 $\mu$ depends on the temperature, and in (1.3), the right-hand side contains the heat source term $2\mu(\theta, v, \|D(v)\|) \|D(v)\|^p$, 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 $\Omega$ as $\partial\Omega = \Gamma_0 \cup \Gamma_L \cup \Gamma_1$ with

$$ \Gamma_0 = \{(x', x_3) \in \bar{\Omega} : x_3 = 0\}, \quad \Gamma_1 = \{(x', x_3) \in \bar{\Omega} : x_3 = h(x')\}, $$

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

$$ \int_{\Gamma_L} g \cdot n dY = 0, \quad g = 0 \text{ on } \Gamma_1, \quad g \cdot n = 0 \text{ on } \Gamma_0, \quad g \neq 0 \text{ on } \Gamma_L, \quad (1.4) $$

where $n = (n_1, n_2, n_3)$ is the unit outward normal vector to $\partial\Omega$, and $g \cdot n$ denotes the Euclidean inner product of the vectors $g$ and $n$ in $\mathbb{R}^3$. We define by $v_n = v \cdot n$ and $v_\tau = v - v_n n$ the normal and the tangential velocities on $\partial\Omega$. The normal and tangential components of the stress vector on $\partial\Omega$ are given by $\sigma_n$ and $\sigma_\tau$ with

$$ \sigma_n = \sum_{i,j=1}^{3} \sigma_{ij} n_j n_i, \quad \sigma_\tau = \left( \sum_{j=1}^{3} \sigma_{ij} n_j - \sigma_n n_i \right)_{1 \le i \le 3}. $$

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.,

$$ v = 0 \quad \text{on } (0, T) \times \Gamma_1, \quad v = g\xi \quad \text{on } (0, T) \times \Gamma_L, \quad (1.5) $$

where $\xi$ is a function depending only on the time variable such that

$$ \xi(0) = 1 \quad (1.6) $$

and

$$ v_n = 0 \quad \text{on } (0, T) \times \Gamma_0 \quad \text{(slip condition)}, \quad (1.7) $$

$$ \left. \begin{array}{l} \|\sigma_{\tau}\| = k \quad \Rightarrow \quad \exists \lambda \ge 0 \quad v_{\tau} = s - \lambda\sigma_{\tau}, \\ \|\sigma_{\tau}\| < k \quad \Rightarrow \quad v_{\tau} = s \end{array} \right\} \quad \text{on } (0, T) \times \Gamma_0 \quad \text{(Tresca's law)}, \quad (1.8) $$

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.,

$$ \theta = 0 \quad \text{on } (0, T) \times (\Gamma_1 \cup \Gamma_L), \quad (K\nabla\theta) \cdot n = \theta^b \quad \text{on } (0, T) \times \Gamma_0, \quad (1.9) $$

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 ($v_\delta, \pi_\delta, \theta_\delta$) converge to a solution to our original fluid flow/heat transfer problem as $\delta$ 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 $W^{1,p}(\Omega)$:

$$V_{\Gamma_1}^p = \{\varphi \in W^{1,p}(\Omega); \varphi = 0 \text{ on } \Gamma_1\},$$

$$V_0^p = \{\varphi \in W^{1,p}(\Omega); \varphi = 0 \text{ on } \Gamma_1 \cup \Gamma_L \text{ and } \varphi \cdot n = 0 \text{ on } \Gamma_0\}$$

and

$$V_{0, \text{div}}^p = \{\varphi \in V_0^p; \text{div}(\varphi) = 0 \text{ in } \Omega\}$$

for all $p > 1$ endowed with the norm

$$\|v\|_{1,p} = \left( \int_{\Omega} \|\nabla v\|^p dx \right)^{1/p}.$$

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

$$\left( \int_{\Omega} \|D(u)\|^p dx \right)^{1/p} = \|D(u)\|_{(L^p(\Omega))^{3 \times 3}} \le \|u\|_{1,p} \quad \forall u \in W^{1,p}(\Omega), \quad (2.1)$$

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

$$\left( \int_{\Omega} \|D(u)\|^p dx \right)^{1/p} = \|D(u)\|_{(L^p(\Omega))^{3 \times 3}} \ge C_{\text{Korn},p} \|u\|_{1,p} \quad \forall u \in V_{\Gamma_1}^p, \quad (2.2)$$

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

$$\|\psi\|_{\mathcal{Y}} = (\|\psi\|_{L^2(\Omega)}^2 + \|\text{div}(\psi)\|_{L^2(\Omega)}^2)^{1/2} \quad \forall \psi \in \mathcal{Y},$$

and let $H$ be its subspace given by $H = \{\psi \in L^2(\Omega); \text{div}(\psi) = 0 \text{ in } \Omega, \psi \cdot n = 0 \text{ on } \partial\Omega\}$. Owing to that $W^{1,p}(\Omega)$ is continuously embedded into $L^2(\Omega)$ if and only if $p \ge \frac{6}{5}$ in a 3D setting and that the space $\mathcal{V} = \{\Phi \in (\mathcal{D}(\Omega))^3; \text{div}(\Phi) = 0 \text{ in } \Omega\}$ is dense in $H$ (see Chap. 1, Theorem 2.8 in [20]), we obtain that the embedding of $V_{0, \text{div}}^p$ into $H$ is continuous and
dense if and only if $p \ge \frac{6}{5}$, and $(V_{0, \text{div}}^p, H, (V_{0, \text{div}}^p)^t)$ is a Gelfand triplet. Let us also recall that the trace operator is compact from $W^{1,p}(\Omega)$ into $L^p(\partial\Omega)$ 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

$$W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega) = \{\varphi \in W^{1,q}(\Omega) : \varphi = 0 \text{ on } \Gamma_1 \cup \Gamma_L\}$$

for all $q > 1$ endowed with the norm

$$\|\varphi\|_{W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega)} = \left( \int_{\Omega} \|\nabla \varphi\|^q dx \right)^{1/q}.$$

Let us now introduce the assumptions on the data.

The heat capacity and the thermal conductivity tensor satisfy

$$c \in W^{1,q'}(\Omega), \quad K \in (L^\infty(\Omega))^{3 \times 3} \qquad (2.3)$$

there exists $(c_0, c_1) \in \mathbb{R}^2$ such that

$$0 < c_0 \le c(x) \le c_1 \quad \text{for a.e. } x \in \Omega, \qquad (2.4)$$

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

$$\text{there exists } k_0 > 0 \text{ such that } \sum_{i,j=1}^{3} K_{ij}(x) \gamma_i \gamma_j \ge k_0 \sum_{i=1}^{3} |\gamma_i|^2 \qquad (2.5)$$

for all $\gamma = (\gamma_i)_{1 \le i \le 3} \in \mathbb{R}^3$ for a.e. $x \in \Omega$.

We also assume that

$$\text{the mapping } r: \mathbb{R} \to \mathbb{R} \text{ is continuous,} \qquad (2.6)$$

and

$$\text{there exists } r_1 \in \mathbb{R} \text{ such that } |r(z)| \le r_1 \quad \text{for all } z \in \mathbb{R}. \qquad (2.7)$$

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

$$(o, e, d) \mapsto \mu(o, e, d) \text{ is continuous on } \mathbb{R} \times \mathbb{R}^3 \times \mathbb{R}_+, \qquad (2.8)$$

$$d \mapsto \mu(\cdot, \cdot, d) \text{ is monotone increasing on } \mathbb{R}_+, \qquad (2.9)$$

$$\text{there exists } (\mu_0, \mu_1) \in \mathbb{R}^2 \text{ such that} \\ 0 < \mu_0 \le \mu(o, e, d) \le \mu_1 \quad \text{for all } (o, e, d) \in \mathbb{R} \times \mathbb{R}^3 \times \mathbb{R}_+, \qquad (2.10)$$

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

$$\begin{cases} \mathcal{F}(\lambda_0, \lambda_1, \lambda_2) = 2\mu(\lambda_0, \lambda_1, \|\lambda_2\|)\|\lambda_2\|^{p-2}\lambda_2 & \text{if } \lambda_2 \ne 0_{\mathbb{R}^{3 \times 3}}, \\ \mathcal{F}(\lambda_0, \lambda_1, \lambda_2) = 0_{\mathbb{R}^{3 \times 3}} & \text{otherwise.} \end{cases} \qquad (2.11)$$
With (2.10), we obtain immediately

$$|\mathcal{F}(\lambda_0, \lambda_1, \lambda_2)| \le 2\mu_1 \|\lambda_2\|^{p-1} \quad \forall (\lambda_0, \lambda_1, \lambda_2) \in \mathbb{R} \times \mathbb{R}^3 \times \mathbb{R}^{3 \times 3}.$$

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 $u \in L^{\tilde{p}}(0, T; W^{1,p}(\Omega))$, we have

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

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

$$\mathcal{F}(\theta, u, D(u)) : D(\bar{\varphi}) \in L^1(0, T; L^1(\Omega))$$

for all $\bar{\varphi} \in L^{\frac{\tilde{p}}{p-1}}(0, T; W^{1,p}(\Omega))$, and the right-hand side of the heat equation is well defined in $L^1(0, T; L^1(\Omega))$ if and only if $\tilde{p} \ge \frac{\tilde{p}}{p-1}$, 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 $\mathcal{A} : L^{\tilde{p}}(0, T; V_{\Gamma_1}^p) \to (L^{\tilde{p}}(0, T; V_{\Gamma_1}^p))'$ defined by

$$[\mathcal{A}(u), \bar{\varphi}] = \int_0^T \int_{\Omega} \mathcal{F}(\theta, u, D(u)) : D(\bar{\varphi}) \,dx\,dt \quad \forall (u, \bar{\varphi}) \in (L^{\tilde{p}}(0, T; V_{\Gamma_1}^p))^2,$$

where $[\cdot, \cdot]$ denotes the duality product between the space $L^{\tilde{p}}(0, T; V_{\Gamma_1}^p)$ and its dual $(L^{\tilde{p}}(0, T; V_{\Gamma_1}^p))'$. With (2.10), we have

$$\begin{aligned} |[\mathcal{A}(u), \bar{\varphi}]| & \le 2\mu_1 \|D(u)\|_{L^{\tilde{p}}(0,T;L^p(\Omega))}^{p-1} \|D(\bar{\varphi})\|_{L^{\frac{\tilde{p}}{p-1}}(0,T;L^p(\Omega))} \\ & \le 2\mu_1 T^{\frac{\tilde{p}-p}{\tilde{p}}} \|u\|_{L^{\tilde{p}}(0,T;V_{\Gamma_1}^p)}^{p-1} \|\bar{\varphi}\|_{L^{\tilde{p}}(0,T;V_{\Gamma_1}^p)} \end{aligned}$$

and

$$[\mathcal{A}(u), u - \bar{\varphi}] \ge 2(C_{\text{Korn},p})^p \mu_0 \|u\|_{L^p(0,T;V_{\Gamma_1}^p)}^p - 2\mu_1 \|u\|_{L^{\tilde{p}}(0,T;V_{\Gamma_1}^p)}^{p-1} \|\bar{\varphi}\|_{L^{\frac{\tilde{p}}{\tilde{p}-1}}(0,T;V_{\Gamma_1}^p)}$$

for any $(u, \bar{\varphi}) \in (L^{\tilde{p}}(0, T; V_{\Gamma_1}^p))^2$. 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 $v^0 \in W^{1,p}(\Omega)$ be a given initial velocity such that

$$\begin{aligned} \operatorname{div}(v^0) &= 0 \quad \text{in } \Omega, & v^0 &= 0 \quad \text{on } \Gamma_1, & v^0 &= g \quad \text{on } \Gamma_L, \\ v^0 \cdot n &= 0 \quad \text{on } \Gamma_0. \end{aligned} \qquad (2.12)$$

In order to deal with homogeneous boundary conditions on $(0, T) \times (\Gamma_1 \cup \Gamma_L)$, we set $\bar{v} = v - v^0 \xi$. We obtain the following weak formulation of the problem:

**Problem (P)** Let $c, K, r$, and $\mu$ satisfy (2.3)–(2.10). Let $f \in L^{p'}(0, T; L^2(\Omega))$, $k \in L^{p'}(0, T; L_{+}^{p'}(\Gamma_0))$, $s \in L^p(0, T; L^p(\Gamma_0))$, $\xi \in W^{1,p'}(0, T)$ satisfying (1.6), $\theta^b \in L^1((0, T) \times \omega)$, $\theta^0 \in L^1(\Omega)$ and $v^0 \in W^{1,p}(\Omega)$ satisfying (2.12).
Find $\theta \in L^q(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega))$, $\bar{u} \in C([0, T]; L^2(\Omega)) \cap L^p(0, T; V_{0, \text{div}}^p)$ with $\frac{\partial \bar{u}}{\partial t} \in L^p(0, T; V_{0, \text{div}}^p)'$ and $\pi \in H^{-1}(0, T; L_0^{p'}(\Omega))$ satisfying the following parabolic variational coupled problem:

$$ 
\begin{aligned}
& \left\langle \frac{\partial}{\partial t} (\bar{u}, \tilde{\vartheta})_{L^2(\Omega), \zeta} \right\rangle_{D'(0,T), D(0,T)} \\
& \qquad + \int_0^T \int_{\Omega} \mathcal{F}(\theta, \bar{u} + \nu^0 \xi, D(\bar{u} + \nu^0 \xi)) : D(\tilde{\vartheta}) \zeta \, dx \, dt \\
& \qquad - \left\langle \int_{\Omega} \pi \operatorname{div}(\tilde{\vartheta}) \, dx, \zeta \right\rangle_{D'(0,T), D(0,T)} + J(\bar{u} + \tilde{\vartheta} \zeta) - J(\bar{u}) \\
& \geq \int_0^T \left( f + \nu^0 \frac{\partial \xi}{\partial t}, \tilde{\vartheta} \right)_{L^2(\Omega)} \zeta \, dt \quad \forall \tilde{\vartheta} \in V_0^p, \forall \zeta \in D(0, T)
\end{aligned} 
\quad (2.13) $$

with the initial condition

$$ \bar{u}(0, \cdot) = \nu^0 - \nu^0 \xi(0) = 0 \quad \text{in } \Omega \quad (2.14) $$

and

$$ 
\begin{aligned}
& - \int_0^T \int_{\Omega} c \theta w \tilde{\xi}' dx dt + \int_0^T \int_{\Omega} (K \nabla \theta) \cdot \nabla w \tilde{\xi} \, dx dt \\
& = \int_0^T \int_{\Omega} (\mathcal{F}(\theta, \bar{u} + \nu^0 \xi, D(\bar{u} + \nu^0 \xi)) : D(\bar{u} + \nu^0 \xi)) w \tilde{\xi} \, dx dt \\
& \quad + \int_0^T \int_{\Omega} r(\theta) w \tilde{\xi} \, dx dt + \int_0^T \int_{\omega} \theta^b w \tilde{\xi} \, dx' dt + \int_{\Omega} c \theta^0 w \tilde{\xi}(0) \, dx \\
& \forall w \in W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega), \forall \tilde{\xi} \in C^{\infty}([0, T]) \text{ s.t. } \tilde{\xi}(T) = 0,
\end{aligned} 
\quad (2.15) $$

where

$$ J: \quad \begin{cases} L^p(0, T; V_0^p) \to \mathbb{R}, \\ \bar{\varphi} \mapsto \int_0^T \int_{\Gamma_0} k |\bar{\varphi} - \tilde{s}| \, dx \, dt, \quad \tilde{s} = s - (\nu^0)_\tau \xi \end{cases} $$

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

$$ 
\left\{ \begin{array}{ll} 
\displaystyle \frac{\partial \theta}{\partial t} - \operatorname{div}(a(x, \nabla \theta)) = g & \text{in } \Omega \times (0, T), \\ 
\theta = 0 & \text{in } \partial \Omega \times (0, T) 
\end{array} \right. 
\quad (2.16) $$

with $g \in L^1(0, T; L^1(\Omega))$ and the coercivity property $a(x, \gamma) \cdot \gamma \ge \alpha_a \|\gamma\|^2 (\alpha_a > 0)$ for all $\gamma \in \mathbb{R}^3$ and for almost every $x \in \Omega$ admit a solution $\theta \in L^q(0, T; W_0^{1,q}(\Omega))$ with $1 < q < \frac{5}{4}$
(see [6]). For a given fluid velocity $v = \bar{v} + v^0 \xi \in L^p(0, T; V_{\Gamma_1}^p)$, 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 $\theta \in L^q(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega))$. 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 $(\bar{v}_\delta, \pi_\delta, \theta_\delta)$ to ($P_\delta$) by using a fixed point argument. Then in Sect. 4, we will prove that the sequence $(\bar{v}_\delta, \pi_\delta, \theta_\delta)_{\delta>0}$ 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 $\mu$ satisfy (2.3)–(2.10). Let $f \in L^p'(0, T; L^2(\Omega))$, $k \in L^p'(0, T; L_+^p(\Gamma_0))$, $s \in L^p(0, T; L^p(\Gamma_0))$, $\xi \in W^{1,p'}(0, T)$ satisfying (1.6), $\theta^b \in L^1((0, T) \times \omega)$, $\theta^0 \in L^1(\Omega)$ and $v^0 \in W^{1,p}(\Omega)$ satisfying (2.12).

Find $\theta_\delta \in W^{1,2}((0, T) \times \Omega) \cap L^\infty(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)) \cap C^0([0, T]; L^2(\Omega))$, $\bar{v}_\delta \in C([0, T]; L^2(\Omega)) \cap L^p(0, T; V_{0, \text{div}}^p)$ with $\frac{\partial \bar{v}_\delta}{\partial t} \in L^p'(0, T; (V_{0, \text{div}}^p)'')$ and $\pi_\delta \in H^{-1}(0, T; L_0^{p'}(\Omega))$, satisfying the following parabolic variational coupled problem:

$$ \left\langle \frac{\partial}{\partial t} (\bar{v}_\delta, \tilde{v})_{L^2(\Omega)}, \zeta \right\rangle_{D'(0,T), D(0,T)} \\ + \int_0^T \int_{\Omega} \mathcal{F}(\theta_\delta, \bar{v}_\delta + v^0 \xi, D(\bar{v}_\delta + v^0 \xi)) : D(\tilde{v}) \zeta \,dx\,dt \\ - \left\langle \int_{\Omega} \pi_\delta \operatorname{div}(\tilde{v}) \,dx\,\zeta \right\rangle_{D'(0,T), D(0,T)} + J(\bar{v}_\delta + \tilde{v})\zeta - J(\bar{v}_\delta) \\ \geq \int_0^T \left( f + v^0 \frac{\partial \xi}{\partial t}, \tilde{v} \right)_{L^2(\Omega)} \zeta \,dt \quad \forall \tilde{v} \in V_0^p, \forall \zeta \in D(0, T) \quad (3.1) $$

and

$$ \begin{aligned} & \int_0^T \int_{\Omega} c \frac{\partial \theta_\delta}{\partial t} \tilde{w} \zeta \,dx\,dt + \int_0^T \int_{\Omega} (K \nabla \theta_\delta) \cdot \nabla \tilde{w} \zeta \,dx\,dt \\ = & \int_0^T \int_{\Omega} g_\delta(\theta_\delta, \bar{v}_\delta) \tilde{w} \zeta \,dx\,dt + \int_0^T \int_{\Omega} r(\theta_\delta) \tilde{w} \zeta \,dx\,dt \\ & + \int_0^T \int_{\omega} \theta_\delta^b \tilde{w} \zeta \,dx'\,dt \quad \forall w \in W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega), \forall \tilde{\zeta} \in D(0, T) \end{aligned} \quad (3.2) $$

with the initial conditions

$$ \bar{v}_\delta(0, \cdot) = 0 \quad \text{in } \Omega \quad (3.3) $$
and

$$ \theta_{\delta}(0, \cdot) = \theta_{\delta}^{0} \quad \text{in } \Omega, \qquad (3.4) $$

where

$$ g_{\delta}(\theta_{\delta}, \bar{v}_{\delta}) = \frac{2\mu(\theta_{\delta}, \bar{v}_{\delta} + v^{0}\xi, |D(\bar{v}_{\delta} + v^{0}\xi)|)|D(\bar{v}_{\delta} + v^{0}\xi)|^{p}}{1 + 2\delta\mu(\theta_{\delta}, \bar{v}_{\delta} + v^{0}\xi, |D(\bar{v}_{\delta} + v^{0}\xi)|)|D(\bar{v}_{\delta} + v^{0}\xi)|^{p}} \qquad (3.5) $$

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

$$ \|\theta_{\delta}^{b} - \theta^{b}\|_{L^{1}((0,T) \times \omega)} \leq \delta, \quad \|\theta_{\delta}^{0} - \theta^{0}\|_{L^{1}(\Omega)} \leq \delta. \qquad (3.6) $$

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 ($P^{\text{flow}}_{(\tilde{u}, \tilde{\theta})}$ and ($P^{\text{heat}}_{\delta, (\tilde{u}, \tilde{\theta})}$):

**Problem (P^{flow}_{(\tilde{u}, \tilde{\theta})})** Let $(\tilde{u}, \tilde{\theta})$ be given in $L^{p}(0, T; L^{p}(\Omega)) \times L^{\tilde{q}_{1}}(0, T; L^{\tilde{q}_{2}}(\Omega))$ with $\tilde{q}_{1} > 1$ and $\tilde{q}_{2} > 1$. Find $\bar{v}_{(\tilde{u}, \tilde{\theta})} \in C([0, T]; L^{2}(\Omega)) \cap L^{p}(0, T; V_{0, \text{div}}^{p})$ with $\frac{\partial \bar{v}_{(\tilde{u}, \tilde{\theta})}}{\partial t} \in L^{p'}(0, T; (V_{0, \text{div}}^{p})')$ and $\pi_{(\tilde{u}, \tilde{\theta})} \in H^{-1}(0, T; L_{0}^{p'}(\Omega))$, satisfying the following parabolic variational inequality

$$ \left\langle \frac{\partial}{\partial t} (\bar{v}_{(\tilde{u}, \tilde{\theta})}, \tilde{\theta})_{L^2(\Omega), \zeta} \right\rangle_{D'(0,T), D(0,T)} \\ + \int_0^T \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) : D(\tilde{\theta}) \zeta \, dx \, dt \\ - \left\langle \int_{\Omega} \pi_{(\tilde{u}, \tilde{\theta})} \operatorname{div}(\tilde{\theta}) \, dx, \zeta \right\rangle_{D'(0,T), D(0,T)} + J(\bar{v}_{(\tilde{u}, \tilde{\theta})} + \tilde{\theta} \zeta) - J(\bar{v}_{(\tilde{u}, \tilde{\theta})}) \\ \geq \int_0^T \left( f + v^0 \frac{\partial \xi}{\partial t}, \tilde{\theta} \right)_{L^2(\Omega)} \zeta \, dt \quad \forall \tilde{\theta} \in V_0^p, \forall \zeta \in \mathcal{D}(0, T) $$

with the initial condition

$$ \bar{v}_{(\tilde{u}, \tilde{\theta})}(0, \cdot) = 0 \quad \text{in } \Omega \qquad (3.8) $$

and

**Problem (P^{heat}_{\delta, (\tilde{u}, \tilde{\theta})})** Let $(\tilde{u}, \tilde{\theta})$ be given in $L^{p}(0, T; V_{0, \text{div}}^{p}) \times L^{\tilde{q}_{1}}(0, T; L^{\tilde{q}_{2}}(\Omega))$ with $\tilde{q}_{1} > 1$ and $\tilde{q}_{2} > 1$. Find $\theta_{\delta, (\tilde{u}, \tilde{\theta})} \in W^{1,2}((0, T) \times \Omega) \cap L^{\infty}(0, T; W_{\Gamma_{1} \cup \Gamma_{L}}^{1,2}(\Omega)) \cap C^{0}([0, T]; L^{2}(\Omega))$, satisfying the following parabolic variational equality:

$$ \int_0^T \int_{\Omega} c \frac{\partial \theta_{\delta, (\tilde{u}, \tilde{\theta})}}{\partial t} w \tilde{\zeta} \, dx \, dt + \int_0^T \int_{\Omega} (K \nabla \theta_{\delta, (\tilde{u}, \tilde{\theta})}) \cdot \nabla w \tilde{\zeta} \, dx \, dt \\ = \int_0^T \int_{\Omega} g_{\delta}(\tilde{\theta}, \tilde{u}) w \tilde{\zeta} \, dx \, dt + \int_0^T \int_{\Omega} r(\tilde{\theta}) w \tilde{\zeta} \, dx \, dt \qquad (3.9) \\ + \int_0^T \int_{\omega} \theta_{\delta}^b w \tilde{\zeta} \, dx' \, dt \quad \forall w \in W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega), \forall \tilde{\zeta} \in \mathcal{D}(0, T) $$
with the initial condition

$$ \theta_{\delta, (\tilde{u}, \tilde{\theta})}(0, \cdot) = \theta_{\delta}^{0} \quad \text{in } \Omega. \qquad (3.10) $$

Remark 3.1 Let us emphasize that problem ($P^{\text{flow}}_{(\tilde{u}, \tilde{\theta})}$) does not depend on $\delta$.

Then we have

**Proposition 3.2** (Existence and uniqueness result for ($P^{\text{flow}}_{(\tilde{u}, \tilde{\theta})}$)) Let $(\tilde{u}, \tilde{\theta})$ be given in $L^p(0, T; L^p(\Omega)) \times L^{\tilde{q}_1}(0, T; L^{\tilde{q}_2}(\Omega))$ with $\tilde{q}_1 > 1$ and $\tilde{q}_2 > 1$. Let $\mu$ satisfy (2.8)–(2.10), $f \in L^p(0, T; L^2(\Omega))$, $k \in L^p(0, T; L_+^p(\Gamma_0))$, $s \in L^p(0, T; L^p(\Gamma_0))$, $\xi \in W^{1,p'}(0, T)$ satisfying (1.6), and $v^0 \in W^{1,p}(\Omega)$ satisfying (2.12). Then problem ($P^{\text{flow}}_{(\tilde{u}, \tilde{\theta})}$) admits a unique solution. Moreover, there exists a constant $C^{\text{flow}}$, independent of $(\tilde{\theta}, \tilde{u})$, such that

$$ \|\bar{v}_{(\tilde{u}, \tilde{\theta})}\|_{L^p(0, T; V_{0, \text{div}}^p)} \le C^{\text{flow}}, \qquad (3.11) $$

$$ \|\bar{v}_{(\tilde{u}, \tilde{\theta})}\|_{L^{\infty}(0, T; L^2(\Omega))} \le C^{\text{flow}} \qquad (3.12) $$

and

$$ \left\| \frac{\partial \bar{v}_{(\tilde{u}, \tilde{\theta})}}{\partial t} \right\|_{L^{p'}(0, T; V_{0, \text{div}}^p)} \le C^{\text{flow}}. \qquad (3.13) $$

**Proof** The existence and uniqueness of a solution to the problem ($P^{\text{flow}}_{(\tilde{u}, \tilde{\theta})}$) 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 $\bar{\varphi} = \tilde{\vartheta}\zeta$ with $\tilde{\vartheta} \in V_{0, \text{div}}^p$ and $\zeta \in \mathcal{D}(0, T)$, we have

$$ \left\langle \frac{\partial}{\partial t} (\bar{v}_{(\tilde{u}, \tilde{\theta})}, \tilde{\vartheta})_{L^2(\Omega)}, \zeta \right\rangle_{\mathcal{D}'(0, T), \mathcal{D}(0, T)} \\ = \int_0^T \left\langle \frac{\partial \bar{v}_{(\tilde{u}, \tilde{\theta})}}{\partial t}, \bar{\varphi} \right\rangle_{(V_{0, \text{div}}^p)^{\prime}, V_{0, \text{div}}^p} dt \\ + \int_0^T \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) : D(\bar{\varphi}) dx dt \\ + J(\bar{v}_{(\tilde{u}, \tilde{\theta})} + \bar{\varphi}) - J(\bar{v}_{(\tilde{u}, \tilde{\theta})}) \ge \int_0^T \left( f + v^0 \frac{\partial \xi}{\partial t}, \bar{\varphi} \right)_{L^2(\Omega)} dt. \qquad (3.14) $$

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

Let us choose $\bar{\varphi} = -\bar{v}_{(\tilde{u}, \tilde{\theta})} \mathbf{1}_{[0,t]}$, 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

$$ \int_0^t \left\langle \frac{\partial \bar{v}_{(\tilde{u}, \tilde{\theta})}}{\partial t}, \bar{v}_{(\tilde{u}, \tilde{\theta})} \right\rangle_{(V_{0, \text{div}}^p)^{\prime}, V_{0, \text{div}}^p} d\tilde{t} \\ + \int_0^t \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) : D(\bar{v}_{(\tilde{u}, \tilde{\theta})}) dx d\tilde{t} \\ \le \int_0^t \left( f + v^0 \frac{\partial \xi}{\partial t}, \bar{v}_{(\tilde{u}, \tilde{\theta})} \right)_{L^2(\Omega)} d\tilde{t} + \int_0^t \int_{\Gamma_0} k|\tilde{s}| dx d\tilde{t}. $$
With (2.10) and (2.1)–(2.2), we get

$$
\begin{aligned}
& \int_0^t \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + \nu^0 \xi, D(\overline{\nu}(\tilde{u}, \tilde{\theta}) + \nu^0 \xi)) : D(\overline{\nu}(\tilde{u}, \tilde{\theta})) dx dt \\
& \geq 2(C_{\text{Korn},p})^p \mu_0 \int_0^t \|\overline{\nu}(\tilde{u}, \tilde{\theta}) + \nu^0 \xi\|_{1,p}^p dt \\
& \quad - 2\mu_1 \int_0^t \|\nu^0 \xi\|_{1,p} \|\overline{\nu}(\tilde{u}, \tilde{\theta}) + \nu^0 \xi\|_{1,p}^{p-1} dt \\
& \geq 2(C_{\text{Korn},p})^p \mu_0 \left\|\overline{\nu}(\tilde{u}, \tilde{\theta})\right\|_{L^p(0,t;V_{0,\text{div}}^p)} - \left\|\nu^0 \xi\right\|_{L^p(0,t;V_{1,1}^p)}^p \\
& \quad - 2\mu_1 \left(\left\|\overline{\nu}(\tilde{u}, \tilde{\theta})\right\|_{L^p(0,t;V_{0,\text{div}}^p)} + \left\|\nu^0 \xi\right\|_{L^p(0,t;V_{1,1}^p)}^{p-1}\right) \left\|\nu^0 \xi\right\|_{L^p(0,t;V_{1,1}^p)}.
\end{aligned}
$$

Since $f + \nu^0 \frac{\partial \xi}{\partial t} \in L^p(0, T; L^2(\Omega))$, we obtain

$$
\begin{aligned}
& \frac{1}{2} \| \overline{\nu}(\tilde{u}, \tilde{\theta}) (t) \|^2_{L^2(\Omega)} \\
& \quad + 2(C_{\text{Korn},p})^p \mu_0 \left\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \right\|_{L^p(0,t;V_{0,\text{div}}^p)} - \left\| \nu^0 \xi \right\|_{L^p(0,t;V_{1,1}^p)}^p \\
& \le \tilde{C} \left\| f + \nu^0 \frac{\partial \xi}{\partial t} \right\|_{L^{p'}(0,t;L^2(\Omega))} \left\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \right\|_{L^p(0,t;V_{0,\text{div}}^p)} + \int_0^t \int_{\Gamma_0} k|\tilde{s}| dx dt \\
& \quad + 2\mu_1 \left( \left\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \right\|_{L^p(0,t;V_{0,\text{div}}^p)} + \left\| \nu^0 \xi \right\|_{L^p(0,t;V_{1,1}^p)}^{p-1} \right) \left\| \nu^0 \xi \right\|_{L^p(0,t;V_{1,1}^p)},
\end{aligned}
\quad (3.15)
$$

where $\tilde{C}$ denotes the norm of the continuous injection of $V_0^p$ into $L^2(\Omega)$.

Let us consider first $t = T$. If $\|\overline{\nu}(\tilde{u}, \tilde{\theta})\|_{L^p(0,T;V_{0,\text{div}}^p)} \neq 0$, it follows that

$$
\begin{aligned}
& 2(C_{\text{Korn},p})^p \mu_0 \left| 1 - \frac{\| \nu^0 \xi \|_{L^p(0,T;V_{1,1}^p)}}{\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \|_{L^p(0,T;V_{0,\text{div}}^p)}} \right|^p \\
& \le \tilde{C} \left\| f + \nu^0 \frac{\partial \xi}{\partial t} \right\|_{L^{p'}(0,T;L^2(\Omega))} \left\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \right\|_{L^p(0,T;V_{0,\text{div}}^p)}^{1-p} + \frac{J(0)}{\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \|_{L^p(0,T;V_{0,\text{div}}^p)}^p} \\
& \quad + 2\mu_1 \left( 1 + \frac{\| \nu^0 \xi \|_{L^p(0,T;V_{1,1}^p)}}{\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \|_{L^p(0,T;V_{0,\text{div}}^p)}} \right)^{p-1} \frac{\| \nu^0 \xi \|_{L^p(0,T;V_{1,1}^p)}}{\| \overline{\nu}(\tilde{u}, \tilde{\theta}) \|_{L^p(0,T;V_{0,\text{div}}^p)}}.
\end{aligned}
$$

However, the mapping

$$
\begin{aligned}
z \mapsto & 2(C_{\text{Korn},p})^p \mu_0 \left| 1 - \frac{\| \nu^0 \xi \|_{L^p(0,T;V_{1,1}^p)}}{z} \right|^p - \tilde{C} \left\| f + \nu^0 \frac{\partial \xi}{\partial t} \right\|_{L^{p'}(0,T;L^2(\Omega))} z^{1-p} \\
& - \frac{J(0)}{z^p} - 2\mu_1 \left( 1 + \frac{\| \nu^0 \xi \|_{L^p(0,T;V_{1,1}^p)}}{z} \right)^{p-1} \frac{\| \nu^0 \xi \|_{L^p(0,T;V_{1,1}^p)}}{z}
\end{aligned}
$$

admits $2(C_{\text{Korn},p})^p \mu_0 > 0$ as a limit when $z$ tends to $+\infty$. Thus, there exists a real number $C > 0$, independent of $(\tilde{u}, \tilde{\theta})$, such that

$$ \| \overline{\nu}(\tilde{u}, \tilde{\theta}) \|_{L^p(0, T; V_{0, \text{div}}^p)} \le C, \quad (3.16) $$

which yields (3.11). Going back to (3.15), we obtain (3.12).
Let us choose now $\bar{\varphi} = \pm \tilde{\vartheta} \xi$ with $\tilde{\vartheta} \in V_{0,div}^p$ and $\xi \in D(0, T)$ in (3.14). We obtain

$$ \int_0^T \left\langle \frac{\partial \bar{u}(\bar{u}, \tilde{\theta})}{\partial t}, \pm \tilde{\vartheta} \xi \right\rangle_{(V_{0,div}^p)', V_{0,div}^p} dt \\ + \int_0^T \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + \nu^0 \xi, D(\bar{u}(\bar{u}, \tilde{\theta}) + \nu^0 \xi)) : D(\pm \tilde{\vartheta} \xi) dx dt \quad (3.17) \\ + J(\bar{u}(\bar{u}, \tilde{\theta}) \pm \tilde{\vartheta} \xi) - J(\bar{u}(\bar{u}, \tilde{\theta})) \geq \int_0^T \left( f + \nu^0 \frac{\partial \xi}{\partial t}, \pm \tilde{\vartheta} \xi \right)_{L^2(\Omega)} dt. $$

But

$$ |J(\bar{u}(\bar{u}, \tilde{\theta}) \pm \tilde{\vartheta} \xi) - J(\bar{u}(\bar{u}, \tilde{\theta}))| \leq \int_0^T \int_{\Gamma_0} k |\tilde{\vartheta} \xi| dx dt $$

and, recalling that $k \in L^p(0, T; L_+^p(\Gamma_0))$, we get

$$ \begin{aligned} & |J(\bar{u}(\bar{u}, \tilde{\theta}) \pm \tilde{\vartheta} \xi) - J(\bar{u}(\bar{u}, \tilde{\theta}))| \\ & \leq \|\gamma_p\|_{L^1(\Omega), L^p(\partial\Omega)} \|k\|_{L^{p'}(0, T; L^{p'}(\Gamma_0))} \|\tilde{\vartheta} \xi\|_{L^p(0, T; V_{0,div}^p)} \end{aligned} $$

where $\gamma_p$ denotes the trace operator from $W^{1,p}(\Omega)$ into $L^p(\partial\Omega)$.
On the other hand,

$$ \begin{aligned} & \left| \int_0^T \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + \nu^0 \xi, D(\bar{u}(\bar{u}, \tilde{\theta}) + \nu^0 \xi)) : D(\tilde{\vartheta} \xi) dx dt \right| \\ & \leq 2\mu_1 \left\| D(\bar{u}(\bar{u}, \tilde{\theta}) + \nu^0 \xi) \right\|_{L^{p'}(0, T; L^p(\Omega))^{3 \times 3}}^{p-1} \left\| D(\tilde{\vartheta} \xi) \right\|_{L^p(0, T; L^p(\Omega))^{3 \times 3}}, \end{aligned} $$

and with (3.16), we obtain

$$ \begin{aligned} & \left| \int_0^T \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + \nu^0 \xi, D(\bar{u}(\bar{u}, \tilde{\theta}) + \nu^0 \xi)) : D(\tilde{\vartheta} \xi) dx dt \right| \\ & \leq (2\mu_1(C + \|\nu^0 \xi\|_{L^p(0, T; V_{\Gamma_1}^p)})^{p-1}) \|\tilde{\vartheta} \xi\|_{L^p(0, T; V_{0,div}^p)}. \end{aligned} $$

Going back to (3.17), we obtain

$$ \begin{aligned} & \left| \int_0^T \left\langle \frac{\partial \bar{u}(\bar{u}, \tilde{\theta})}{\partial t}, \tilde{\vartheta} \xi \right\rangle_{(V_{0,div}^p)', V_{0,div}^p} dt \right| \\ & \leq \|\gamma_p\|_{L^1(\Omega), L^p(\partial\Omega)} \|k\|_{L^{p'}(0, T; L^{p'}(\Gamma_0))} \|\tilde{\vartheta} \xi\|_{L^p(0, T; V_{0,div}^p)} \\ & \qquad + \tilde{C} \left\| f + \nu^0 \frac{\partial \xi}{\partial t} \right\|_{L^{p'}(0, T; L^2(\Omega))} \|\tilde{\vartheta} \xi\|_{L^p(0, T; V_{0,div}^p)} \\ & \qquad + (2\mu_1(C + \|\nu^0 \xi\|_{L^p(0, T; V_{\Gamma_1}^p)})^{p-1}) \|\tilde{\vartheta} \xi\|_{L^p(0, T; V_{0,div}^p)}, \end{aligned} $$

which yields (3.13).

□
**Proposition 3.3** (Existence and uniqueness result for $(P_{\delta,(\tilde{u},\tilde{\theta})}^{\text{heat}})$) Let $(\tilde{u}, \tilde{\theta})$ be given in $L^p(0, T; V_{0,\text{div}}^p) \times L^{\tilde{q}_1}(0, T; L^{\tilde{q}_2}(\Omega))$ with $\tilde{q}_1 > 1$ and $\tilde{q}_2 > 1$. Let $c, K, r, \text{ and } \mu$ satisfy (2.3)-(2.10). Let $\xi \in W^{1,p'}(0, T)$ satisfying (1.6), $\theta_{\delta}^{b} \in \mathcal{D}((0, T) \times \omega)$, $\theta_{\delta}^{0} \in \mathcal{D}(\Omega)$ and $\upsilon^{0} \in W^{1,p}(\Omega)$ satisfying (2.12). Then problem $(P_{\delta,(\tilde{u},\tilde{\theta})}^{\text{heat}})$ admits a unique solution. Moreover, there exists a constant $C^{\text{heat}}$, depending only on the data $K$ and $c$, such that

$$ \left\| \frac{\partial \theta_{\delta,(\tilde{u},\tilde{\theta})}}{\partial t} \right\|_{L^2(0,T;L^2(\Omega))}^2 + \left\| \nabla \theta_{\delta,(\tilde{u},\tilde{\theta})} \right\|_{L^{\infty}(0,T;L^2(\Omega))}^2 \\ \le C^{\text{heat}} \left( \|g_{\delta}(\tilde{\theta}, \tilde{u}) + r(\tilde{\theta})\|_{L^2(0,T;L^2(\Omega))}^2 + \|\theta_{\delta}^{0}\|_{H^1(\Omega)}^2 + \|\theta_{\delta}^{b}\|_{W^{1,2}(0,T;L^2(\omega))}^2 \right). $$

*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 $C_{\delta}^{\text{heat}}$, depending only on $\delta$, $\|\theta_{\delta}^{0}\|_{H^1(\Omega)}$ and $\|\theta_{\delta}^{b}\|_{W^{1,2}(0,T;L^2(\omega))}$ such that

$$ \left\| \frac{\partial \theta_{\delta,(\tilde{u},\tilde{\theta})}}{\partial t} \right\|_{L^2(0,T;L^2(\Omega))} + \left\| \nabla \theta_{\delta,(\tilde{u},\tilde{\theta})} \right\|_{L^{\infty}(0,T;L^2(\Omega))} \le C_{\delta}^{\text{heat}}. \quad (3.18) $$

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

$$ \begin{aligned} & \|g_{\delta}(\tilde{\theta}, \tilde{u}) + r(\tilde{\theta})\|_{L^2(0,T;L^2(\Omega))}^2 \\ & \le 2 \int_0^T \int_{\Omega} (g_{\delta}(\tilde{\theta}, \tilde{u}))^2 dx dt + 2 \int_0^T \int_{\Omega} (r(\tilde{\theta}))^2 dx dt \\ & \le 2 \left( \frac{1}{\delta^2} + r_1^2 \right) \text{meas}(\Omega) T, \end{aligned} $$

which allows us to conclude. □

It follows that, for any $(\tilde{u}, \tilde{\theta}) \in L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$, and for any $\delta > 0$, we may define $(\overline{v}_{(\tilde{u},\tilde{\theta})}, \pi_{(\tilde{u},\tilde{\theta})}) \in L^p(0, T; V_{0,\text{div}}^p) \times L^p'(0, T; L_0^{p'}(\Omega))$ as the unique solution of $(P_{(\tilde{u},\tilde{\theta})}^{\text{flow}})$ and $\theta_{\delta,(\overline{v}_{(\tilde{u},\tilde{\theta})},\tilde{\theta})} \in W^{1,2}((0, T) \times \Omega)$ as the unique solution of $(P_{\delta,(\overline{v}_{(\tilde{u},\tilde{\theta})},\tilde{\theta})}^{\text{heat}})$. Then we define the mapping $T_{\delta}: L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega)) \to L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$ by

$$ T_{\delta}(\tilde{u}, \tilde{\theta}) = (\overline{v}_{(\tilde{u}, \tilde{\theta})}, \theta_{\delta,(\overline{v}_{(\tilde{u}, \tilde{\theta})}, \tilde{\theta})}), $$

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

**Theorem 3.5** Let $\delta > 0$. Let the mappings $c, K, r$, and $\mu$ satisfy (2.3)-(2.10), $f \in L^p'(0, T; L^2(\Omega))$, $k \in L^p'(0, T; L_0^{p'}(\Gamma_0))$, $s \in L^p(0, T; L^p(\Gamma_0))$, $\xi \in W^{1,p'}(0, T)$ satisfying (1.6), $\theta_{\delta}^{0} \in \mathcal{D}(\Omega)$, $\theta_{\delta}^{b} \in \mathcal{D}((0, T) \times \omega)$ and $\upsilon^{0} \in W^{1,p}(\Omega)$ satisfying (2.12). Then the mapping $T_{\delta}$ admits a fixed point in $L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$.
Proof Let $(\tilde{u}, \tilde{\theta}) \in L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$. With the previous estimates, we have

$$ \|\overline{\nu}(\tilde{u}, \tilde{\theta})\|_{L^p(0, T; V_{0, \text{div}}^p)} \le C^{\text{flow}}, \quad \|\theta_{\delta, (\overline{\nu}(\tilde{u}, \tilde{\theta}), \tilde{\theta})\|_{L^{\infty}(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1, 2}(\Omega))} \le C_{\delta}^{\text{heat}}. $$

Thus, $T_{\delta}(C) \subset C$ with $C = \overline{B}_{L^p(0, T; L^p(\Omega))}(0, C_p C^{\text{flow}}) \times \overline{B}_{L^2(0, T; L^2(\Omega)}(0, \sqrt{T}C_2 C_{\delta}^{\text{heat}})$, where $C_p$ and $C_2$ denote Poincaré's constant on $V_{0, \text{div}}^p$ and on $W_{\Gamma_1 \cup \Gamma_L}^{1, 2}(\Omega)$, respectively. Moreover, the estimates (3.13) and (3.18) imply that $T_{\delta}(C)$ is bounded in the space $W^{1, p'}(0, T; (V_{0, \text{div}}^p)^{\prime}) \times W^{1, 2}(0, T; L^2(\Omega))$, and we may conclude that $T_{\delta}(C)$ is relatively compact in $L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$.

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 $L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$ to $(\tilde{u}, \tilde{\theta})$, and let us define

$$ T_{\delta}(\tilde{u}_n, \tilde{\theta}_n) = (\overline{\nu}(\tilde{u}_n, \tilde{\theta}_n), \theta_{\delta, (\overline{\nu}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)}) \quad \text{for all } n \ge 1 $$

and

$$ T_{\delta}(\tilde{u}, \tilde{\theta}) = (\overline{\nu}(\tilde{u}, \tilde{\theta}), \theta_{\delta, (\overline{\nu}(\tilde{u}, \tilde{\theta}), \tilde{\theta}))). $$

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

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

Let us consider such a subsequence still denoted $(\overline{\nu}(\tilde{u}_n, \tilde{\theta}_n), \theta_{\delta, (\overline{\nu}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)})_{n \ge 1}$. With (3.14), we have

$$ \begin{aligned} & \int_0^T \left\langle \frac{\partial \overline{\nu}(\tilde{u}_n, \tilde{\theta}_n)}{\partial t}, \overline{\varphi} - \overline{\nu}(\tilde{u}_n, \tilde{\theta}_n) \right\rangle_{(V_{0, \text{div}}^p)^{\prime}, V_{0, \text{div}}^p} dt \\ & + \int_0^T \int_{\Omega} \mathcal{F}(\tilde{\theta}_n, \tilde{u}_n + \nu^0 \xi, D(\overline{\nu}(\tilde{u}_n, \tilde{\theta}_n) + \nu^0 \xi)) : D(\overline{\varphi} - \overline{\nu}(\tilde{u}_n, \tilde{\theta}_n)) dx dt \\ & + J(\overline{\varphi}) - J(\overline{\nu}(\tilde{u}_n, \tilde{\theta}_n)) \\ & \ge \int_0^T \left( f + \nu^0 \frac{\partial \xi}{\partial t}, \overline{\varphi} - \overline{\nu}(\tilde{u}_n, \tilde{\theta}_n) \right)_{L^2(\Omega)} dt \quad \forall \overline{\varphi} \in L^p(0, T; V_{0, \text{div}}^p) \end{aligned} \quad (3.19) $$

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

$$ \begin{aligned} & \int_0^T \left\langle \frac{\partial \overline{\nu}(\tilde{u}, \tilde{\theta})}{\partial t}, \overline{\varphi} - \overline{\nu}(\tilde{u}, \tilde{\theta}) \right\rangle_{(V_{0, \text{div}}^p)^{\prime}, V_{0, \text{div}}^p} dt \\ & + \int_0^T \int_{\Omega} \mathcal{F}(\tilde{\theta}, \tilde{u} + \nu^0 \xi, D(\overline{\nu}(\tilde{u}, \tilde{\theta}) + \nu^0 \xi)) : D(\overline{\varphi} - \overline{\nu}(\tilde{u}, \tilde{\theta})) dx dt \\ & + J(\overline{\varphi}) - J(\overline{\nu}(\tilde{u}, \tilde{\theta})) \\ & \ge \int_0^T \left( f + \nu^0 \frac{\partial \xi}{\partial t}, \overline{\varphi} - \overline{\nu}(\tilde{u}, \tilde{\theta}) \right)_{L^2(\Omega)} dt \quad \forall \overline{\varphi} \in L^p(0, T; V_{0, \text{div}}^p). \end{aligned} \quad (3.20) $$
We choose $\bar{\varphi} = \bar{v}_{(\tilde{u}, \tilde{\theta})}$ in (3.19) then $\bar{\varphi} = \bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)}$ in (3.20) and add the two inequalities. We obtain:

$$ \int_0^T \left\langle \frac{\partial(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \bar{v}_{(\tilde{u}, \tilde{\theta})})}{\partial t}, \bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \bar{v}_{(\tilde{u}, \tilde{\theta})} \right\rangle_{(V_{0.\mathrm{div}}^p)^T, V_{0.\mathrm{div}}^p} dt \\ + \int_0^T \int_{\Omega} (\mathcal{F}(\tilde{\theta}_n, \tilde{u}_n + v^0\xi, D(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} + v^0\xi)) \\ - \mathcal{F}(\tilde{\theta}, \tilde{u} + v^0\xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0\xi))) : D(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \bar{v}_{(\tilde{u}, \tilde{\theta})}) dxdt \le 0 $$

and thus,

$$ \frac{1}{2} \|\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \bar{v}_{(\tilde{u}, \tilde{\theta})}\|_{L^2(\Omega)}^2(T) \\ + \int_0^T \int_{\Omega} (\mathcal{F}(\tilde{\theta}_n, \tilde{u}_n + v^0\xi, D(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} + v^0\xi)) \\ - \mathcal{F}(\tilde{\theta}_n, \tilde{u}_n + v^0\xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0\xi))) : D(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \bar{v}_{(\tilde{u}, \tilde{\theta})}) dxdt \\ \le \int_0^T \int_{\Omega} (\mathcal{F}(\tilde{\theta}, \tilde{u} + v^0\xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0\xi)) \\ - \mathcal{F}(\tilde{\theta}_n, \tilde{u}_n + v^0\xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0\xi))) : D(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \bar{v}_{(\tilde{u}, \tilde{\theta})}) dxdt. $$

We decompose $\mathcal{F}$ as $\mathcal{F} = \mathcal{F}_1 + \mathcal{F}_2$ with

$$ \mathcal{F}_1(\lambda_2) = \mu_0 \|\lambda_2\|^{p-2}\lambda_2 \quad \text{if } \lambda_2 \neq 0_{\mathbb{R}^{3 \times 3}}, \quad \mathcal{F}_1(\lambda_2) = 0_{\mathbb{R}^{3 \times 3}} \quad \text{otherwise} $$

and

$$ \begin{cases} \mathcal{F}_2(\lambda_0, \lambda_1, \lambda_2) = 2\bar{\mu}(\lambda_0, \lambda_1, \|\lambda_2\|)\|\lambda_2\|^{p-2}\lambda_2 & \text{if } \lambda_2 \neq 0_{\mathbb{R}^{3 \times 3}}, \\ \mathcal{F}_2(\lambda_0, \lambda_1, \lambda_2) = 0_{\mathbb{R}^{3 \times 3}} & \text{otherwise,} \end{cases} $$

where $\bar{\mu} = \mu - \frac{\mu_0}{2}$. By observing that $\bar{\mu}$ satisfies

$d \mapsto \bar{\mu}(\cdot, \cdot, d)$ is monotone increasing on $\mathbb{R}_+$,

$$ 0 < \frac{\mu_0}{2} \le \bar{\mu}(o, e, d) \le \mu_1 - \frac{\mu_0}{2} \quad \text{for all } (o, e, d) \in \mathbb{R} \times \mathbb{R}^3 \times \mathbb{R}_+ $$

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,

$$ \int_0^T \int_{\Omega} (\mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0\xi, D(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} + v^0\xi)) \\ - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0\xi, D(\bar{v}_{(\tilde{u}, \tilde{\theta})} + v^0\xi))) : D(\bar{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \bar{v}_{(\tilde{u}, \tilde{\theta})}) dxdt \ge 0 $$
and thus,

$$
\begin{equation}
\begin{aligned}
& \frac{1}{2} \| \overline{v}(\tilde{u}_n, \tilde{\theta}_n) - \overline{v}(\tilde{u}, \tilde{\theta}) \|_{L^2(\Omega)}^2 (T) \\
& \quad + \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\overline{v}(\tilde{u}_n, \tilde{\theta}_n) + v^0 \xi)) - \mathcal{F}_1(D(\overline{v}(\tilde{u}, \tilde{\theta}) + v^0 \xi))) \\
& \qquad : D(\overline{v}(\tilde{u}_n, \tilde{\theta}_n) - \overline{v}(\tilde{u}, \tilde{\theta})) dx dt \\
& \le \int_0^T \int_{\Omega} (\mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}(\tilde{u}, \tilde{\theta}) + v^0 \xi)) \\
& \quad - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}(\tilde{u}, \tilde{\theta}) + v^0 \xi))) : D(\overline{v}(\tilde{u}_n, \tilde{\theta}_n) - \overline{v}(\tilde{u}, \tilde{\theta})) dx dt.
\end{aligned}
\tag{3.21}
\end{equation}
$$

Now we distinguish two cases.

Case 1: $p \in [6/5, 2)$

With some algebraic computations, we can prove that

$$
(\|\lambda\| + \|\lambda'\|)^{2-p} (\mathcal{F}_1(\lambda) - \mathcal{F}_1(\lambda')) : (\lambda - \lambda') \geq \mu_0(p-1) \|\lambda - \lambda'\|^2,
$$

which yields

$$
(\|\lambda\|^p + \|\lambda'\|^p)^{\frac{2-p}{2}} ((\mathcal{F}_1(\lambda) - \mathcal{F}_1(\lambda')) : (\lambda - \lambda'))^{\frac{p}{2}} \geq \frac{(\mu_0(p-1))^{\frac{p}{2}}}{2^{\frac{(p-1)(2-p)}{2}}} \|\lambda - \lambda'\|^p
$$

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

$$
\begin{align*}
& \frac{(\mu_0(p-1))^\frac{p}{2}}{2^{\frac{(p-1)(2-p)}{2}}} \int_0^T \int_\Omega \|D(\bar{v}(\tilde{u}_n, \tilde{\theta}_n) - \bar{v}(\tilde{u}, \tilde{\theta}))\|^p dx dt \\
& \le \left( \int_0^T \int_\Omega (\mathcal{F}_1(D(\bar{v}(\tilde{u}_n, \tilde{\theta}_n) + v^0 \xi)) - \mathcal{F}_1(D(\bar{v}(\tilde{u}, \tilde{\theta}) + v^0 \xi))) \right. \\
& \qquad \left. : D(\bar{v}(\tilde{u}_n, \tilde{\theta}_n) - \bar{v}(\tilde{u}, \tilde{\theta})) dx dt \right)^{\frac{p}{2}} \\
& \quad \times \left( \int_0^T \int_\Omega (\|D(\bar{v}(\tilde{u}_n, \tilde{\theta}_n) + v^0 \xi)\|^p + \|D(\bar{v}(\tilde{u}, \tilde{\theta}) + v^0 \xi)\|^p) dx dt \right)^{\frac{2-p}{2}} \\
& \le \left( \int_0^T \int_\Omega (\mathcal{F}_1(D(\bar{v}(\tilde{u}_n, \tilde{\theta}_n) + v^0 \xi)) - \mathcal{F}_1(D(\bar{v}(\tilde{u}, \tilde{\theta}) + v^0 \xi))) \right. \\
& \qquad \left. : D(\bar{v}(\tilde{u}_n, \tilde{\theta}_n) - \bar{v}(\tilde{u}, \tilde{\theta})) dx dt \right)^{\frac{p}{2}} \\
& \quad \times \left( \|\bar{v}(\tilde{u}_n, \tilde{\theta}_n) + v^0 \xi\|_{L^p(0,T; V_{\Gamma_1}^p)}^p + \|\bar{v}(\tilde{u}, \tilde{\theta}) + v^0 \xi\|_{L^p(0,T; V_{\Gamma_1}^p)}^p \right)^{\frac{2-p}{2}}.
\end{align*}
$$
Thus, with (3.11), we get

$$ \begin{aligned} & \frac{(\mu_0(p-1))^p}{2^{\frac{(p-1)(2-p)}{2}}} \|D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^p \\ & \le \left( \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} + v^0 \xi)) - \mathcal{F}_1(D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi))) \right)^p \\ & : D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) dx dt \right)^{\frac{p}{2}} \\ & \times (2(C^{\text{flow}} + \|v^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^p)^{\frac{2-p}{2}}, \end{aligned} $$

and finally,

$$ \begin{aligned} & \frac{\mu_0(p-1)}{2^{(2-p)}(C^{\text{flow}} + \|v^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^{(2-p)}} \|D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^2 \\ & \le \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} + v^0 \xi)) - \mathcal{F}_1(D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi))) \\ & : D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) dx dt. \end{aligned} $$

Going back to (3.21), we obtain

$$ \begin{aligned} & \frac{\mu_0(p-1)}{2^{(2-p)}(C^{\text{flow}} + \|v^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^{(2-p)}} \|D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^2 \\ & \le \int_0^T \int_{\Omega} (\mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \\ & - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi))) : D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) dx dt \\ & \le \|\mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \\ & - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi))\|_{L^{p'}(0,T;(L^{p'}(\Omega)))^{3 \times 3}} \\ & \times \|D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}} \end{aligned} $$

which yields

$$ \begin{aligned} & \frac{\mu_0(p-1)}{2^{(2-p)}(C^{\text{flow}} + \|v^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^{(2-p)}} \|D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^2 \\ & \le \mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \\ & - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \|_{L^{p'}(0,T;(L^{p'}(\Omega)))^{3 \times 3}}. \end{aligned} \quad (3.22) $$

Case 2: $p \in [2, +\infty)$

Once again, with some algebraic computations, we get

$$ (\mathcal{F}_1(\lambda) - \mathcal{F}_1(\lambda')) : (\lambda - \lambda') \ge \frac{\mu_0}{2^{p-1}} \|\lambda - \lambda'\|^p $$
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),

$$ 
\begin{aligned}
& \frac{\mu_0}{2^{p-1}} \| D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) \|_{L^p(0, T; L^p(\Omega))^{3 \times 3}}^p \\
& \le \int_0^T \int_{\Omega} (\mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi))) \\
& : D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) dx dt \\
& \le \| \mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \\
& \quad - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \|_{L^{p'}(0, T; L^{p'}(\Omega))^{3 \times 3}} \\
& \quad \times \| D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) \|_{L^p(0, T; L^p(\Omega))^{3 \times 3}}.
\end{aligned}
$$

Thus,

$$ 
\begin{aligned}
& \frac{\mu_0}{2^{p-1}} \| D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) \|_{L^p(0, T; L^p(\Omega))^{3 \times 3}}^{p-1} \\
& \le \| \mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \\
& \quad - \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \|_{L^{p'}(0, T; L^{p'}(\Omega))^{3 \times 3}}.
\end{aligned}
\quad (3.23) $$

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

$$ \tilde{u}_n \to \tilde{u}, \quad \tilde{\theta}_n \to \tilde{\theta} \quad \text{a.e. in } (0, T) \times \Omega. $$

By using the continuity and boundedness assumptions (2.8) and (2.10) for the mapping $\mu$, we infer with Lebesgue's theorem that

$$ 
\begin{aligned}
& \mathcal{F}_2(\tilde{\theta}_n, \tilde{u}_n + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \to \mathcal{F}_2(\tilde{\theta}, \tilde{u} + v^0 \xi, D(\overline{v}_{(\tilde{u}, \tilde{\theta})} + v^0 \xi)) \\
& \text{strongly in } L^{p'}(0, T; L^{p'}(\Omega))^{3 \times 3}.
\end{aligned}
$$

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

$$ \| D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} - \overline{v}_{(\tilde{u}, \tilde{\theta})}) \|_{L^p(0, T; L^p(\Omega))^{3 \times 3}} \to 0, $$

and with Korn's inequality, we infer that

$$ \overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} \to \overline{v}_{(\tilde{u}, \tilde{\theta})} \quad \text{strongly in } L^p(0, T; V_{0, \text{div}}^p) \text{ and in } L^p(0, T; L^p(\Omega)). $$

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

$$ D(\overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)}) \to D(\overline{v}_{(\tilde{u}, \tilde{\theta})}), \quad \overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)} \to \overline{v}_{(\tilde{u}, \tilde{\theta})} \quad \text{a.e. in } (0, T) \times \Omega. $$

By using the continuity and boundedness properties of the mappings $\mu$ and $r$, we obtain

$$ g_{\delta}(\tilde{\theta}_n, \overline{v}_{(\tilde{u}_n, \tilde{\theta}_n)}) \to g_{\delta}(\tilde{\theta}, \overline{v}_{(\tilde{u}, \tilde{\theta})}) \quad \text{strongly in } L^2(0, T; L^2(\Omega)) $$
and

$$r(\tilde{\theta}_n) \rightarrow r(\tilde{\theta}) \quad \text{strongly in } L^2(0, T; L^2(\Omega)).$$

Moreover, $(\theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)})_{n \ge 1}$ satisfies the estimate (3.18). It follows that there exists $\theta_{\delta}^* \in W^{1,2}((0, T) \times \Omega) \cap L^2(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega))$ such that, by possibly extracting a subsequence still denoted $(\theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)})_{n \ge 1}$, we have

$$\theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)} \rightarrow \theta_{\delta}^* \quad \text{weakly in } W^{1,2}((0, T) \times \Omega) \\ \text{and weakly* in } L^{\infty}(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega))$$

and with Simon's lemma [34]

$$\theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)} \rightarrow \theta_{\delta}^* \quad \text{strongly in } C^0([0, T]; L^2(\Omega)) \text{ and in } L^2(0, T; L^2(\Omega)).$$

Hence, we may pass to the limit in all the terms of $(P_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)}^{\text{heat}})$ and obtain that $\theta_{\delta}^*$ is the solution of $(P_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)}^{\text{heat}})$. By uniqueness of the solution of $(P_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)}^{\text{heat}})$, we infer that $\theta_{\delta}^* = \theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)}$.

Finally, recalling that $(\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)})_{n \ge 1}$ satisfies the estimates (3.11)–(3.13) and (3.18), we infer that the whole sequence $(\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)})_{n \ge 1}$ converges strongly to $(\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \theta_{\delta, (\bar{v}(\tilde{u}_n, \tilde{\theta}_n), \tilde{\theta}_n)})$ in the space $L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$, 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 $\mu$ satisfy (2.3)–(2.10), $f \in L^p(0, T; L^2(\Omega))$, $k \in L^p(0, T; L_+^p(\Gamma_0))$, $s \in L^p(0, T; L^p(\Gamma_0))$, $\xi \in W^{1,p'}(0, T)$ satisfying (1.6), $\theta_{\delta}^0 \in \mathcal{D}(\Omega)$, $\theta_{\delta}^b \in \mathcal{D}((0, T) \times \omega)$ and $\nu^0 \in W^{1,p}(\Omega)$ 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 $(\tilde{u}, \tilde{\theta}) \in L^p(0, T; L^p(\Omega)) \times L^2(0, T; L^2(\Omega))$ such that

$$T_{\delta}(\tilde{u}, \tilde{\theta}) = (\bar{v}(\tilde{u}, \tilde{\theta}), \theta_{\delta, (\bar{v}(\tilde{u}, \tilde{\theta}), \tilde{\theta})}) = (\tilde{u}, \tilde{\theta}).$$

By the definition of $T_{\delta}$, we infer that $\tilde{u} = \bar{v}(\tilde{u}, \tilde{\theta}) \in C([0, T]; L^2(\Omega)) \cap L^p(0, T; V_{0, \text{div}}^p)$ with $\frac{\partial \tilde{u}}{\partial t} = \frac{\partial \bar{v}(\tilde{u}, \tilde{\theta})}{\partial t} \in L^p(0, T; (V_{0, \text{div}}^p)')$, and there exists $\pi(\tilde{u}, \tilde{\theta}) \in H^{-1}(0, T; L_0^p(\Omega))$ such that $(\tilde{u}, \pi(\tilde{u}, \tilde{\theta})) = (\bar{v}(\tilde{u}, \tilde{\theta}), \pi(\tilde{u}, \tilde{\theta}))$ is solution of $(P_{\delta, (\bar{v}(\tilde{u}, \tilde{\theta}), \tilde{\theta})}^{\text{flow}})$, while $\tilde{\theta} = \theta_{\delta, (\bar{v}(\tilde{u}, \tilde{\theta}), \tilde{\theta})} \in W^{1,2}((0, T) \times \Omega) \cap L^{\infty}(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)) \cap C([0, T]; L^2(\Omega))$ is a solution of the problem $(P_{\delta, (\bar{v}(\tilde{u}, \tilde{\theta}), \tilde{\theta})}^{\text{heat}}) = (P_{\delta, (\bar{v}(\tilde{u}, \tilde{\theta}), \tilde{\theta})}^{\text{heat}}, \theta_{\delta, (\bar{v}(\tilde{u}, \tilde{\theta}), \tilde{\theta})}^{\text{heat}})$). Hence, the triplet $(\bar{v}(\tilde{u}, \tilde{\theta}), \pi(\tilde{u}, \tilde{\theta}), \theta_{\delta, (\bar{v}(\tilde{u}, \tilde{\theta}), \tilde{\theta})})$ 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 $(\bar{v}, \pi, \theta)$, satisfying (2.13)–(2.15).
More precisely, let $\delta_n = \frac{1}{n}$ for all $n \ge 1$ and $(\bar{v}_n, \pi_n, \theta_n)$ be a solution of the approximate problem $(P_{\delta_n})$. For all $n \ge 1$, we have $\theta_n \in W^{1,2}((0, T) \times \Omega) \cap L^\infty(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)) \cap C^0([0, T]; L^2(\Omega))$, $\bar{v}_n \in C([0, T]; L^2(\Omega)) \cap L^p(0, T; V_{0, \text{div}}^p)$ with $\frac{\partial \bar{v}_n}{\partial t} \in L^{p'}(0, T; (V_{0, \text{div}}^p)^{\prime})$ and $\pi_n \in H^{-1}(0, T; L_0^{p'}(\Omega))$ such that

$$ 
\begin{aligned}
& \left\langle \frac{\partial}{\partial t} (\bar{v}_n, \tilde{\vartheta})_{L^2(\Omega), \zeta} \right\rangle_{D'(0,T), D(0,T)} \\
& \qquad + \int_0^T \int_{\Omega} \mathcal{F}(\theta_n, \bar{v}_n + \nu^0 \xi, D(\bar{v}_n + \nu^0 \xi)) : D(\tilde{\vartheta}) \zeta \,dx\,dt \\
& \qquad - \left\langle \int_{\Omega} \pi_n \operatorname{div}(\tilde{\vartheta}) \,dx, \zeta \right\rangle_{D'(0,T), D(0,T)} + J(\bar{v}_n + \tilde{\vartheta}) \zeta - J(\bar{v}_n) \\
& \qquad \ge \int_0^T \left( f + \nu^0 \frac{\partial \xi}{\partial t}, \tilde{\vartheta} \right)_{L^2(\Omega)} \zeta \,dt \quad \forall \tilde{\vartheta} \in V_0^p, \forall \zeta \in \mathcal{D}(0,T)
\end{aligned} 
\quad (4.1) $$

and

$$ 
\begin{aligned}
& \int_0^T \int_{\Omega} c \frac{\partial \theta_n}{\partial t} w \tilde{\zeta} \,dx\,dt + \int_0^T \int_{\Omega} (K \nabla \theta_n) \cdot \nabla w \tilde{\zeta} \,dx\,dt \\
& = \int_0^T \int_{\Omega} g_{\delta_n}(\theta_n, \bar{v}_n) w \tilde{\zeta} \,dx\,dt + \int_0^T \int_{\Omega} r(\theta_n) w \tilde{\zeta} \,dx\,dt \\
& \qquad + \int_0^T \int_{\Omega} \theta_{\delta_n}^b w \tilde{\zeta} \,dx'\,dt \quad \forall w \in W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega), \forall \tilde{\zeta} \in \mathcal{D}(0,T)
\end{aligned} 
\quad (4.2) $$

with the initial conditions

$$ \bar{v}_n(0, \cdot) = 0 \quad \text{in } \Omega \quad (4.3) $$

and

$$ \theta_n(0, \cdot) = \theta_{\delta_n}^0 \quad \text{in } \Omega. \quad (4.4) $$

Owing to that $\theta_n \in W^{1,2}((0, T) \times \Omega) \cap L^\infty(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)) \cap C^0([0, T]; L^2(\Omega))$ the variational equality (4.2) remains true for all $\tilde{\zeta} \in L^2(0, T; \mathbb{R})$. Let us choose $\tilde{\zeta} \in C^\infty([0, T])$ such that $\tilde{\zeta}(T) = 0$: with integration by part with respect to the time-variable, we obtain

$$ 
\begin{aligned}
& - \int_0^T \int_{\Omega} c \theta_n w \tilde{\zeta}' \,dx\,dt + \int_0^T \int_{\Omega} (K \nabla \theta_n) \cdot \nabla w \tilde{\zeta} \,dx\,dt \\
& = \int_0^T \int_{\Omega} g_{\delta_n}(\theta_n, \bar{v}_n) w \tilde{\zeta} \,dx\,dt + \int_0^T \int_{\Omega} r(\theta_n) w \tilde{\zeta} \,dx\,dt \\
& \qquad + \int_0^T \int_{\Omega} \theta_{\delta_n}^b w \tilde{\zeta} \,dx'\,dt + \int_{\Omega} c \theta_{\delta_n}^0 w \tilde{\zeta}(0) \,dx \\
& \qquad \forall w \in W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega), \quad \forall \tilde{\zeta} \in C^\infty([0, T]) \text{ s.t. } \tilde{\zeta}(T) = 0.
\end{aligned} 
\quad (4.5) $$

Let us establish first some a priori estimates for $(\bar{v}_n, \pi_n)_{n \ge 1}$.
**Proposition 4.1** (A priori estimates of $(\bar{v}_n, \pi_n)_{n \ge 1}$) Let $\mu$ satisfy (2.8)–(2.10), $f \in L^p(0, T; L^2(\Omega))$, $k \in L^{p'}(0, T; L_+^{p'}(\Gamma_0))$, $s \in L^p(0, T; L^p(\Gamma_0))$, $\xi \in W^{1,p'}(0, T)$ satisfying (1.6), and $v^0 \in W^{1,p}(\Omega)$ satisfying (2.12). Then there exists a constant $\tilde{C}^{\text{flow}}$, independent of $n$, such that for all $n \ge 1$:

$$ \|\bar{v}_n\|_{L^p(0,T;V_{0,\text{div}}^p)} \le \tilde{C}^{\text{flow}}, \quad (4.6) $$

$$ \|\bar{v}_n\|_{L^{\infty}(0,T;L^2(\Omega))} \le \tilde{C}^{\text{flow}}, \quad (4.7) $$

$$ \left\| \frac{\partial \bar{v}_n}{\partial t} \right\|_{L^{p'}(0,T;V_{0,\text{div}}^p)^{\prime}} \le \tilde{C}^{\text{flow}} \quad (4.8) $$

and

$$ \|\pi_n\|_{H^{-1}(0,T;L_0^{p'}(\Omega))} \le \tilde{C}^{\text{flow}}. \quad (4.9) $$

*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 $\bar{\varphi} = \pm \tilde{\vartheta}\zeta$ with $\tilde{\vartheta} \in W_0^{1,p}(\Omega)$ and $\zeta \in D(0, T)$ in (4.1). We obtain

$$ \left\langle \int_{\Omega} \pi_n \operatorname{div}(\tilde{\vartheta}) dx, \zeta \right\rangle_{D'(0,T), D(0,T)} \\ = - \int_{0}^{T} (\bar{v}_n, \tilde{\vartheta})_{L^2(\Omega)} \zeta' dt \\ + \int_{0}^{T} \int_{\Omega} \mathcal{F}(\theta_n, \bar{v}_n + v^0 \xi, D(\bar{v}_n + v^0 \xi)) : D(\tilde{\vartheta}) \zeta dx dt \\ - \int_{0}^{T} \left( f + v^0 \frac{\partial \xi}{\partial t}, \tilde{\vartheta} \right)_{L^2(\Omega)} \zeta dt. $$

Thus,

$$ \left| \left\langle \int_{\Omega} \pi_n \operatorname{div}(\tilde{\vartheta}) dx, \zeta \right\rangle_{D'(0,T), D(0,T)} \right| \\ \le \sqrt{T} \|\bar{v}_n\|_{L^{\infty}(0,T;L^2(\Omega))} \|\tilde{\vartheta}\zeta'\|_{L^2(0,T;L^2(\Omega))} \\ + 2\mu_1 (\|\bar{v}_n\|_{L^p(0,T;V_{0,\text{div}}^p)} + \|v^0\xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^{p-1} \|\tilde{\vartheta}\zeta\|_{L^p(0,T;V_0^p)} \\ + \left\| f + v^0 \frac{\partial \xi}{\partial t} \right\|_{L^{p'}(0,T;L^2(\Omega))} \|\tilde{\vartheta}\zeta\|_{L^p(0,T;L^2(\Omega))} $$

and with (4.6)–(4.8), we get

$$ \left| \left\langle \int_{\Omega} \pi_n \operatorname{div}(\tilde{\vartheta}) dx, \zeta \right\rangle_{D'(0,T), D(0,T)} \right| \\ \le \left( \sqrt{T} \tilde{C} \tilde{C}^{\text{flow}} + 2\mu_1 C_{\infty} T^{1/p} (\tilde{C}^{\text{flow}} + \|v^0\xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^{p-1} \right. \\ \left. + \tilde{C} C_{\infty} T^{1/p} \left\| f + v^0 \frac{\partial \xi}{\partial t} \right\|_{L^{p'}(0,T;L^2(\Omega))} \right) \|\tilde{\vartheta}\zeta\|_{H_0^1(0,T;V_0^p)} $$
where $\tilde{C}$ denotes the norm of the continuous injection of $V_0^p$ into $L^2(\Omega)$, 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 $P_p : L_0^p(\Omega) \to W_0^{1,p}(\Omega)$ such that

$$ \mathrm{div}(P_p(\omega)) = \omega \quad \forall \omega \in L_0^p(\Omega) $$

(see Corollary 3.1 in [2]). It follows that for any $\omega \in L_0^p(\Omega)$ and $\zeta \in \mathcal{D}(0, T)$, we have

$$ \left| \left\langle \int_{\Omega} \pi_n \omega \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| \\ \le \|P_p\|_{L^p(\Omega), W_0^{1,p}(\Omega)} \left( \sqrt{T} \tilde{C} \tilde{C}^{\text{flow}} + 2\mu_1 C_\infty T^{1/p} (\tilde{C}^{\text{flow}} + \|\nu^0 \xi\|_{L^p(0,T;V_{T_1}^p)})^{p-1} \right. \\ \left. + \tilde{C} C_\infty T^{1/p} \left\| f + \nu^0 \frac{\partial \xi}{\partial t} \right\|_{L^{p'}(0,T;L^2(\Omega))} \right) \|\omega \zeta\|_{H_0^1(0,T;L^p(\Omega))}. $$

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

$$ \left| \left\langle \int_{\Omega} \pi_n \omega \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| \le C' \|\omega \zeta\|_{H_0^1(0,T;L^p(\Omega))} \quad \forall \omega \in L_0^p(\Omega), \forall \zeta \in \mathcal{D}(0, T). $$

Furthermore, for any $\omega^* \in L^p(\Omega)$, we may define $\omega \in L_0^p(\Omega)$ by

$$ \omega = \omega^* - \frac{1}{\mathrm{meas}(\Omega)} \int_{\Omega} \omega^* \, dx. $$

We have $\|\omega\|_{L^p(\Omega)} \le 2\|\omega^*\|_{L^p(\Omega)}$, and since $\pi_n \in H^{-1}(0, T; L_0^{p'}(\Omega))$, we have

$$ \left| \left\langle \int_{\Omega} \pi_n \left( \omega^* - \frac{1}{\mathrm{meas}(\Omega)} \int_{\Omega} \omega^* \, dx \right) \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| \\ = \left| \left\langle \int_{\Omega} \pi_n \omega^* \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right. \\ \left. - \frac{1}{\mathrm{meas}(\Omega)} \left( \int_{\Omega} \omega^* \, dx \right) \left\langle \int_{\Omega} \pi_n \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| \\ = \left| \left\langle \int_{\Omega} \pi_n \omega^* \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| . $$

It follows that

$$ \left| \left\langle \int_{\Omega} \pi_n \omega^* \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| \\ = \left| \left\langle \int_{\Omega} \pi_n \omega \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| \\ \le 2C' \|\omega^* \zeta\|_{H_0^1(0,T;L^p(\Omega))} \quad \forall \omega^* \in L^p(\Omega), \forall \zeta \in \mathcal{D}(0, T). $$
Hence, by possibly modifying $\tilde{C}^{\text{flow}}$, we have

$$ \left| \left\langle \int_{\Omega} \pi_n \varpi^* dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \right| \le \tilde{C}^{\text{flow}} \|\varpi^* \zeta\|_{H_0^1(0,T; L^p(\Omega))} \\ \forall \varpi^* \in L^p(\Omega), \forall \zeta \in \mathcal{D}(0, T), $$

and we may conclude by using the density of $\mathcal{D}(0, T) \otimes L^p(\Omega)$ into $H_0^1(0, T; L^p(\Omega))$. $\square$

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 $C_q^{\text{heat}} > 0$, independent of $n$, such that

$$ \|\theta_n\|_{L^q(0,T; W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega))} \le C_q^{\text{heat}} \quad (4.10) $$

for all $n \ge 1$.

*Proof* Let $\tilde{w} = w\tilde{\zeta}$ with $w \in W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)$ and $\zeta \in \mathcal{D}(0, T)$. With (4.2), we have

$$ \int_0^T \int_{\Omega} c \frac{\partial \theta_n}{\partial t} \tilde{w} dx dt + \int_0^T \int_{\Omega} (K \nabla \theta_n) \cdot \nabla \tilde{w} dx dt \\ = \int_0^T \int_{\Omega} g_{\delta_n}(\theta_n, \bar{\nu}_n) \tilde{w} dx dt + \int_0^T \int_{\Omega} r(\theta_n) \tilde{w} dx dt + \int_0^T \int_{\Omega} \theta_{\delta_n}^b \tilde{w} dx' dt \quad (4.11) $$

and by density of $\mathcal{D}(0, T) \otimes W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)$ into $L^2(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega))$ the same equality holds for all $\tilde{w} \in L^2(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega))$.

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

$$ \Psi_0^m(s) = \begin{cases} 0 & \text{if } s \ge 1 + \frac{1}{m}, \\ 1 - m(s - 1) & \text{if } 1 < s < 1 + \frac{1}{m}, \\ 1 & \text{if } -1 \le s \le 1, \\ 1 + m(s + 1) & \text{if } -1 - \frac{1}{m} < s < -1, \\ 0 & \text{if } s \le -1 - \frac{1}{m}, \end{cases} \qquad \Psi_0(s) = \begin{cases} 0 & \text{if } s > 1, \\ 1 & \text{if } -1 \le s \le 1, \\ 0 & \text{if } s < -1 \end{cases} $$

and let

$$ \psi_0^m(s) = \int_0^s \psi_0^m(z) dz, \qquad \psi_0(s) = \int_0^s \psi_0(z) dz \quad \text{for all } s \in \mathbb{R}, $$

$$ \Phi_0^m(s) = \int_0^s \psi_0^m(z) dz, \qquad \Phi_0(s) = \int_0^s \psi_0(z) dz \quad \text{for all } s \in \mathbb{R}. $$
We choose $\tilde{w} = \psi_0^m(\theta_n) \mathbf{1}_{[0,t]}$, 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

$$
\begin{aligned}
& \int_{\Omega} c(x) \Phi_0^m(\theta_n(t, x)) dx \\
& + \int_0^t \int_{\Omega} \Psi_0^m(\theta_n(s, x)) (K(x) \nabla \theta_n(s, x)) \cdot \nabla \theta_n(s, x) dx ds \\
& = \int_0^t \int_{\Omega} g_{\delta_n}(\theta_n, \bar{\nu}_n) \psi_0^m(\theta_n) dx ds + \int_0^t \int_{\Omega} r(\theta_n) \psi_0^m(\theta_n) dx ds \\
& + \int_0^t \int_{\omega} \theta_{\delta_n}^b \psi_0^m(\theta_n) dx ds + \int_{\Omega} c(x) \Phi_0^m(\theta_{\delta_n}^0) dx.
\end{aligned}
$$

By observing that $|\Psi_0^m(s)| \le 1$, $|\psi_0^m(s)| \le 2$ and $|\Phi_0^m(s)| \le 2|s|$ for all $s \in \mathbb{R}$ and for all $m \ge 1$,
we may pass to the limit as $m$ tends to $+\infty$ by using Lebesgue's theorem, and we get

$$
\begin{aligned}
& \int_{\Omega} c(x) \Phi_0(\theta_n(t, x)) dx \\
& + \int_0^t \int_{\Omega} \Psi_0(\theta_n(s, x)) (K(x) \nabla \theta_n(s, x)) \cdot \nabla \theta_n(s, x) dx ds \\
& = \int_0^t \int_{\Omega} g_{\delta_n}(\theta_n, \bar{\nu}_n) \psi_0(\theta_n) dx ds + \int_0^t \int_{\Omega} r(\theta_n) \psi_0(\theta_n) dx ds \\
& + \int_0^t \int_{\omega} \theta_{\delta_n}^b \psi_0(\theta_n) dx' ds + \int_{\Omega} c(x) \Phi_0(\theta_{\delta_n}^0) dx.
\end{aligned}
$$

Furthermore, $|s| - \frac{1}{2} \le \Phi_0(s) \le |s|$ and $|\psi_0(s)| \le 1$ for all $s \in \mathbb{R}$. Thus,

$$
\begin{aligned}
& c_0 \int_{\Omega} |\theta_n(t, x)| dx + k_0 \int_0^t \int_{\Omega} \underbrace{\Psi_0(\theta_n(s, x)) \|\nabla \theta_n(s, x)\|^2}_{\ge 0} dx ds \\
& \le \|g_{\delta_n}(\theta_n, \bar{\nu}_n)\|_{L^1(0,T;L^1(\Omega))} + \|r(\theta_n)\|_{L^1(0,T;L^1(\Omega))} + \|\theta_{\delta_n}^b\|_{L^1((0,T)\times\omega)} \\
& \qquad + \frac{1}{2} \|c\|_{L^1(\Omega)} + \|c\theta_{\delta_n}^0\|_{L^1(\Omega)}
\end{aligned}
\qquad (4.12)
$$

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

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

and let

$$ \psi_k^m(s) = \int_0^s \Psi_k^m(z) dz, \quad \psi_k(s) = \int_0^s \Psi_k(z) dz \quad \text{for all } s \in \mathbb{R}, $$

$$ \Phi_k^m(s) = \int_0^s \psi_k^m(z) dz, \quad \Phi_k(s) = \int_0^s \psi_k(z) dz \quad \text{for all } s \in \mathbb{R}. $$

Now we choose $\tilde{w} = \psi_k^m(\theta_n)$ in (4.11) with $k \ge 0$ and $m \ge 1$. We get

$$ \begin{aligned} & \int_{\Omega} c(x) \Phi_k^m(\theta_n(t,x)) dx \\ & \qquad + \int_0^t \int_{\Omega} \Psi_k^m(\theta_n(s,x)) (K(x) \nabla \theta_n(s,x)) \cdot \nabla \theta_n(s,x) dx ds \\ & = \int_0^t \int_{\Omega} g_{\delta_n}(\theta_n, \bar{\nu}_n) \psi_k^m(\theta_n) dx ds + \int_0^t \int_{\Omega} r(\theta_n) \psi_k^m(\theta_n) dx ds \\ & \qquad + \int_0^t \int_{\omega} \theta_{\delta_n}^b \psi_k^m(\theta_n) dx' ds + \int_{\Omega} c(x) \Phi_k^m(\theta_{\delta_n}^0) dx. \end{aligned} $$

We may observe that $|\psi_k^m(s)| \le 1$, $|\psi_k^m(s)| \le 2$, and $|\Phi_k^m(s)| \le 2|s|$ 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 $+\infty$, we obtain

$$ \begin{aligned} & \int_{\Omega} c(x) \Phi_k(\theta_n(t,x)) dx \\ & \qquad + \int_0^t \int_{\Omega} \Psi_k(\theta_n(s,x)) (K(x) \nabla \theta_n(s,x)) \cdot \nabla \theta_n(s,x) dx ds \\ & = \int_0^t \int_{\Omega} g_{\delta_n}(\theta_n, \bar{\nu}_n) \psi_k(\theta_n) dx ds + \int_0^t \int_{\Omega} r(\theta_n) \psi_k(\theta_n) dx ds \\ & \qquad + \int_0^t \int_{\omega} \theta_{\delta_n}^b \psi_k(\theta_n) dx' ds + \int_{\Omega} c(x) \Phi_k(\theta_{\delta_n}^0) dx. \end{aligned} $$

Obviously $0 \le \Phi_k(s) \le |s|$ and $|\psi_k(s)| \le 1$ for all $s \in \mathbb{R}$ and for all $k \ge 0$. Hence,

$$ \begin{aligned} & k_0 \int_0^T \int_{\Omega} \Psi_k(\theta_n) \|\nabla \theta_n\|^2 dx dt \\ & \le \|g_{\delta_n}(\theta_n, \bar{\nu}_n)\|_{L^1(0,T;L^1(\Omega))} \\ & \quad + \|r(\theta_n)\|_{L^1(0,T;L^1(\Omega))} + \|\theta_{\delta_n}^b\|_{L^1((0,T)\times\omega)} + \|c\theta_{\delta_n}^0\|_{L^1(\Omega)}. \end{aligned} \quad (4.13) $$
Let us define $C_{\delta_n}$ as

$$C_{\delta_n} = \|g_{\delta_n}(\theta_n, \bar{\nu}_n)\|_{L^1(0,T;L^1(\Omega))} + \|r(\theta_n)\|_{L^1(0,T;L^1(\Omega))} \\ + \|\theta_{\delta_n}^b\|_{L^1((0,T)\times\omega)} + \frac{1}{2}\|c\|_{L^1(\Omega)} + \|c\theta_{\delta_n}^0\|_{L^1(\Omega)}$$

and

$$Q_0 = \{(t,x) \in (0, T) \times \Omega; |\theta_n(t,x)| \le 1\},$$

$$Q_k = \{(t,x) \in (0, T) \times \Omega; k < |\theta_n(t,x)| \le k+1\} \quad \text{for all } k \in \mathbb{N}^*.$$

With (4.13), we have

$$\int_{Q_k} \|\nabla \theta_n\|^2 dx dt \le \frac{C_{\delta_n}}{k_0} \quad \forall k \in \mathbb{N}.$$

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

$$\int_{Q_k} \|\nabla \theta_n\|^q dx dt \le \left( \int_{Q_k} \|\nabla \theta_n\|^2 dx dt \right)^{q/2} (\text{meas}(Q_k))^{1-q/2} \\ \le \left( \frac{C_{\delta_n}}{k_0} \right)^{q/2} (\text{meas}(Q_k))^{2-q/2} \quad \forall k \in \mathbb{N}.$$

Then

$$\int_0^T \int_{\Omega} \|\nabla \theta_n\|^q dx dt = \int_{\bigcup_{k \ge 0} Q_k} \|\nabla \theta_n\|^q dx dt \le \left(\frac{C_{\delta_n}}{k_0}\right)^{q/2} (T \operatorname{meas}(\Omega))^{2-q/2} \\ + \sum_{k \ge 1} \left(\frac{C_{\delta_n}}{k_0}\right)^{q/2} \frac{1}{k^{2-q/2}} \left(\int_{Q_k} |\theta_n|^r dx dt\right)^{2-q/2}$$

and using again Hölder's inequality

$$\int_0^T \int_{\Omega} \|\nabla \theta_n\|^q dx dt \le \left(\frac{C_{\delta_n}}{k_0}\right)^{q/2} (T \operatorname{meas}(\Omega))^{2-q/2} \\ + \left(\frac{C_{\delta_n}}{k_0}\right)^{q/2} \left(\sum_{k \ge 1} \frac{1}{k^{2-q/2}}\right)^{q/2} \left(\int_0^T \int_{\Omega} |\theta_n|^r dx dt\right)^{2-q/2} \quad (4.14)$$

for all $r \ge 1$ such that $r^{2-q/2} > 1$. 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

$$\int_0^T \int_{\Omega} |\theta_n|^r dx dt \le \int_0^T \left( \int_{\Omega} |\theta_n| dx \right)^{\frac{r}{4}} \left( \int_{\Omega} |\theta_n|^{q^*} dx \right)^{\frac{3r}{4q^*}} dt.$$
Then (4.12) implies that

$$ \begin{aligned} \|\theta_n\|_{L^r(0,T;L^r(\Omega))}^r &= \int_0^T \int_\Omega |\theta_n|^r dx dt \\ &\le \left(\frac{C_{\delta_n}}{c_0}\right)^{\frac{r}{4}} \int_0^T \left(\int_\Omega |\theta_n|^{q^*} dx\right)^{\frac{q}{q^*}} dt = \left(\frac{C_{\delta_n}}{c_0}\right)^{\frac{r}{4}} \|\theta_n\|_{L^{q^*}(0,T;L^{q^*}(\Omega))}^q. \end{aligned} $$

Going back to (4.14),

$$ \begin{aligned} \int_0^T \|\theta_n\|_{W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega)}^q dt &= \int_0^T \int_\Omega \|\nabla \theta_n\|^q dx dt \\ &\le \left(\frac{C_{\delta_n}}{k_0}\right)^{q/2} (T \operatorname{meas}(\Omega))^{\frac{2-q}{2}} \\ &\quad + \left(\frac{C_{\delta_n}}{k_0}\right)^{q/2} \left(\sum_{k \ge 1} \frac{1}{k^{\frac{2-q}{q}}}\right)^{q/2} \left(\frac{C_{\delta_n}}{c_0}\right)^{\frac{r}{4}\frac{2-q}{2}} \|\theta_n\|_{L^{q^*}(0,T;L^{q^*}(\Omega))}^q \end{aligned} \quad (4.15) $$

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

$$ \begin{aligned} \|\theta_n\|_{L^q(0,T;L^{q^*}(\Omega))}^q &= \int_0^T \|\theta_n\|_{L^{q^*}(\Omega)}^q dt \\ &\le (C'_q)^q \left( \left( \left( \frac{C_{\delta_n}}{k_0} \right)^{q/2} (T \operatorname{meas}(\Omega))^{\frac{2-q}{2}} \right. \right. \\ &\qquad \left. \left. + \left( \frac{C_{\delta_n}}{k_0} \right)^{q/2} \left( \sum_{k \ge 1} \frac{1}{k^{\frac{2-q}{q}}} \right)^{q/2} \left( \frac{C_{\delta_n}}{c_0} \right)^{\frac{r}{4}\frac{2-q}{2}} \|\theta_n\|_{L^{q^*}(0,T;L^{q^*}(\Omega))}^{q\frac{2-q}{2}} \right) \right), \end{aligned} $$

where $C'_q$ is the norm of the identity mapping from $W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega)$ 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

$$ \begin{aligned} \|\theta_n\|_{L^q(0,T;L^{q^*}(\Omega))}^q &\le \max \left( 1, (C'_q)^2 \left( \left( \left( \frac{C_{\delta_n}}{k_0} \right)^{q/2} (T \operatorname{meas}(\Omega))^{\frac{2-q}{2}} \right. \right. \right. \\ &\qquad \left. \left. + \left( \frac{C_{\delta_n}}{k_0} \right)^{q/2} \left( \sum_{k \ge 1} \frac{1}{k^{\frac{2-q}{q}}} \right)^{q/2} \left( \frac{C_{\delta_n}}{c_0} \right)^{\frac{q}{3}\frac{2-q}{2}} \right)^{2/q} \right). \end{aligned} \quad (4.16) $$

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

$$ \begin{aligned} C_{\delta_n} &= \left\| \frac{2\mu(\theta_n, \bar{\nu}_n + \nu^0\xi, |D(\bar{\nu}_n + \nu^0\xi)|)|D(\bar{\nu}_n + \nu^0\xi)|^p}{1 + 2\delta_n\mu(\theta_n, \bar{\nu}_n + \nu^0\xi, |D(\bar{\nu}_n + \nu^0\xi)|)|D(\bar{\nu}_n + \nu^0\xi)|^p} \right\|_{L^1(0,T;L^1(\Omega))} \\ &\quad + \|r(\theta_n)\|_{L^1(0,T;L^1(\Omega))} + \|\theta_{\delta_n}^b\|_{L^1((0,T)\times\omega)} + \frac{1}{2}\|c\|_{L^1(\Omega)} + \|c\theta_{\delta_n}^0\|_{L^1(\Omega)} \\ &\le 2\mu_1 \|D(\bar{\nu}_n + \nu^0\xi)\|_{L^p(0,T;L^p(\Omega))}^p + \left(r_1 + \frac{c_1}{2}\right) \operatorname{meas}(\Omega)T \\ &\quad + \|\theta^b\|_{L^1((0,T)\times\omega)} + c_1 \|\theta^0\|_{L^1(\Omega)} + \delta_n(1+c_1) \\ &\le 2\mu_1 (\|\bar{\nu}_n\|_{L^p(0,T;V_{0,\mathrm{div}}^p)} + \|\nu^0\xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^p + \left(r_1 + \frac{c_1}{2}\right) \operatorname{meas}(\Omega)T \end{aligned} $$
$$+ \|\theta^b\|_{L^1((0,T)\times\Omega)} + c_1 \|\theta^0\|_{L^1(\Omega)} + \frac{1}{n}(1+c_1)$$

and with (4.6)

$$C_{\delta_n} \le 2\mu_1(\tilde{C}^{\text{flow}} + \|\nu^0\xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^p + (r_1 + \frac{c_1}{2}) \text{meas}(\Omega)T \\ + \|\theta^b\|_{L^1((0,T)\times\omega)} + c_1 \|\theta^0\|_{L^1(\Omega)} + (1+c_1)$$

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

<figure><img src="image_3.png" /></figure>

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:

$$\begin{aligned} \left\| \frac{\partial \theta_n}{\partial t} \right\|_{L^1(0,T;W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega))'} & \le \frac{1}{c_0} C_{q'} (2\mu_1(\tilde{C}^{\text{flow}} + \|\nu^0\xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^p + r_1 \text{meas}(\Omega)T + \|\theta^b\|_{L^1((0,T)\times\omega)} + 1) \\ & \quad + \frac{T^{1/q'}}{c_0^2} \|K\|_{(L^\infty(\Omega))^{3\times3}} C_q^{\text{heat}}(c_0 + C_{q'} \|\nabla c\|_{L^{q'}(\Omega)}), \end{aligned}$$

where $C_{q'}$ is the norm of the continuous injection of $W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega)$ into $C^0(\bar{\Omega})$.

*Proof* Since $q \in (1, \frac{5}{4})$, we have $q' = \frac{q}{q-1} > 5$, and $W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega)$ is continuously embedded into $W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)$ and into $C^0(\bar{\Omega})$. Moreover, owing to that $c \in W^{1,q'}(\Omega)$, we also have $\frac{w}{c} \in W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega) \subset W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega)$ for all $w \in W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega)$. Let $\psi = w\tilde{\zeta}$ with $w \in W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega)$ and $\tilde{\zeta} \in D(0, T)$. With $\tilde{w} = \frac{\psi}{c} = \frac{w}{c}\zeta$ in (4.11), we get

$$\begin{aligned} & \left| \int_0^T \int_{\Omega} \frac{\partial \theta_n}{\partial t} \psi \,dx\,dt \right| \\ & \le \left| \int_0^T \int_{\Omega} \frac{1}{c} (K \nabla \theta_n) \cdot \nabla \psi \,dx\,dt \right| \\ & \quad + \left| \int_0^T \int_{\Omega} \frac{\psi}{c^2} (K \nabla \theta_n) \cdot \nabla c \,dx\,dt \right| \\ & \quad + \left| \int_0^T \int_{\Omega} \frac{1}{c} (g_{\delta_n}(\theta_n, \bar{\nu}_n) + r(\theta_n)) \psi \,dx\,dt \right| \\ & \quad + \left| \int_0^T \int_{\Omega} \frac{1}{c} \theta_{\delta_n}^b \psi \,dx'\,dt \right|. \end{aligned}$$

Recalling that $\theta_n \in W^{1,2}((0, T) \times \Omega)$, the same equality remains true for all $\psi \in L^2(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega))$. Let us now consider $\psi \in L^\infty(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,q'}(\Omega))$. We obtain

$$\left| \int_0^T \int_{\Omega} \frac{\partial \theta_n}{\partial t} \psi \,dx\,dt \right| \le \frac{T^{1/q'}}{c_0} \|K\|_{(L^\infty(\Omega))^{3\times3}} \|\nabla \theta_n\|_{L^q(0,T;L^q(\Omega))} \|\nabla \psi\|_{L^\infty(0,T;L^{q'}(\Omega))}$$
$$
\begin{align*}
&+ \frac{T^{1/q'}}{c_0^2} \|K\|_{L^\infty(\Omega)}^{3 \times 3} \|\nabla \theta_n\|_{L^q(0,T;L^q(\Omega))} \|\psi\|_{L^\infty(0,T;C^0(\bar{\Omega}))} \|\nabla c\|_{L^{q'}(\Omega)} \\
&+ \frac{1}{c_0} \left( \|g_{\delta_n}(\theta_n, \bar{v}_n) + r(\theta_n)\|_{L^1(0,T;L^1(\Omega))} + \|\theta_{\delta_n}^b\|_{L^1((0,T) \times \omega)} \right) \|\psi\|_{L^\infty(0,T;C^0(\bar{\Omega}))} \\
&\le \frac{T^{1/q'}}{c_0} \|K\|_{L^\infty(\Omega)}^{3 \times 3} \|\nabla \theta_n\|_{L^q(0,T;L^q(\Omega))} \|\nabla \psi\|_{L^\infty(0,T;L^{q'}(\Omega))} \\
&+ \frac{T^{1/q'}}{c_0^2} \|K\|_{L^\infty(\Omega)}^{3 \times 3} \|\nabla \theta_n\|_{L^q(0,T;L^q(\Omega))} \|\psi\|_{L^\infty(0,T;C^0(\bar{\Omega}))} \|\nabla c\|_{L^{q'}(\Omega)} \\
&+ \frac{1}{c_0} \left( 2\mu_1 (\tilde{C}^{\text{flow}} + \|v^0\xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^p + r_1 \text{meas}(\Omega)T + \|\theta^b\|_{L^1((0,T) \times \omega)} + 1 \right) \\
&\quad \times \|\psi\|_{L^\infty(0,T;C^0(\bar{\Omega}))}
\end{align*}
$$

which allows us to conclude.

<figure><img src="image_4.png" /></figure>

It follows that, by possibly extracting a subsequence still denoted $(\bar{v}_n, \pi_n, \theta_n)_{n \ge 1}$, there
exists a triplet $(\bar{v}, \pi, \theta)$ such that

$$
\bar{v}_n \to \bar{v} \quad \begin{array}{@{}l@{}}
\text{weakly in } L^p(0, T; V_{0.\text{div}}^p) \text{ and} \\
\text{weakly}^* \text{ in } L^\infty(0, T; \mathbf{L}^2(\Omega)),
\end{array}
\qquad (4.17)
$$

$$
\frac{\partial \bar{v}_n}{\partial t} \to \frac{\partial \bar{v}}{\partial t} \quad \text{weakly in } L^p(0, T; (V_{0.\text{div}}^p)',) \qquad (4.18)
$$

$$
\pi_n \to \pi \quad \text{weakly}^* \text{ in } H^{-1}(0, T; L_0^{p'}(\Omega)), \qquad (4.19)
$$

and

$$
\theta_n \to \theta \quad \text{weakly in } L^q(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,q}(\Omega)). \qquad (4.20)
$$

Moreover, with the Aubin and Simon lemmas [29, 34], by possibly extracting another sub-
sequence still denoted $(\bar{v}_n, \pi_n, \theta_n)_{n \ge 1}$, we have

$$
\begin{align}
\bar{v}_n &\to \bar{v} \quad \begin{array}{@{}l@{}}
\text{strongly in } C^0([0, T]; H) \\
\text{and in } L^p(0, T; \mathbf{L}^p(\Omega)),
\end{array}
\tag{4.21}
\end{align}
$$

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

$$
\theta_n \to \theta \quad \text{strongly in } L^q(0, T; L^q(\Omega)). \qquad (4.22)
$$

We infer that

$$
\bar{v}_n(0, \cdot) \to \bar{v}(0, \cdot) = 0 \quad \text{strongly in } \mathbf{L}^2(\Omega). \qquad (4.23)
$$

Furthermore, by possibly extracting again a subsequence still denoted $(\bar{v}_n, \pi_n, \theta_n)_{n \ge 1}$, we
have

$$
\bar{v}_n \to \bar{v}, \quad \theta_n \to \theta \quad \text{a.e. in } (0, T) \times \Omega. \tag{4.24}
$$
These convergence properties do not allow us to pass directly to the limit as $n$ tends to $+\infty$, and we need a better convergence result for $(\bar{v}_n)_{n\ge 1}$ to overcome the difficulty due to the non-linearity of the mapping $\mathcal{F}$.

**Proposition 4.4** Under the previous assumptions, $(D(\bar{v}_n))_{n\ge 1}$ converges strongly to $D(\bar{v})$ in $L^p(0, T; (L^p(\Omega))^{3\times3})$.

*Proof* Let $(\hat{v}, \hat{\pi})$ be the unique solution of the problem $(P^{\text{flow}}_{(\bar{v},\theta)})$. We have

$$ \int_0^T \left\langle \frac{\partial \hat{v}}{\partial t}, \bar{\varphi} - \hat{v} \right\rangle_{(V_{0.\text{div}}^p)^\prime, V_{0.\text{div}}^p} dt \\ + \int_0^T \int_\Omega \mathcal{F}(\theta, \bar{v} + v^0\xi, D(\hat{v} + v^0\xi)) : D(\bar{\varphi} - \hat{v}) dx dt \quad (4.25) \\ + J(\bar{\varphi}) - J(\hat{v}) \ge \int_0^T \left( f + v^0 \frac{\partial \xi}{\partial t}, \bar{\varphi} - \hat{v} \right)_{L^2(\Omega)} dt \quad \forall \bar{\varphi} \in L^p(0, T; V_{0.\text{div}}^p). $$

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

$$ \int_0^T \left\langle \frac{\partial \bar{v}_n}{\partial t}, \bar{\varphi} - \bar{v}_n \right\rangle_{(V_{0.\text{div}}^p)^\prime, V_{0.\text{div}}^p} dt \\ + \int_0^T \int_\Omega \mathcal{F}(\theta_n, \bar{v}_n + v^0\xi, D(\bar{v}_n + v^0\xi)) : D(\bar{\varphi} - \bar{v}_n) dx dt \quad (4.26) \\ + J(\bar{\varphi}) - J(\bar{v}_n) \ge \int_0^T \left( f + v^0 \frac{\partial \xi}{\partial t}, \bar{\varphi} - \bar{v}_n \right)_{L^2(\Omega)} dt \quad \forall \bar{\varphi} \in L^p(0, T; V_{0.\text{div}}^p). $$

Let us choose $\bar{\varphi} = \hat{v}$ in (4.26) and $\bar{\varphi} = \bar{v}_n$ in (4.25). By adding the two inequalities, we obtain:

$$ \int_0^T \left\langle \frac{\partial(\bar{v}_n - \hat{v})}{\partial t}, \bar{v}_n - \hat{v} \right\rangle_{(V_{0.\text{div}}^p)^\prime, V_{0.\text{div}}^p} dt \\ + \int_0^T \int_\Omega (\mathcal{F}(\theta_n, \bar{v}_n + v^0\xi, D(\bar{v}_n + v^0\xi)) \\ - \mathcal{F}(\theta, \bar{v} + v^0\xi, D(\hat{v} + v^0\xi))) : D(\bar{v}_n - \hat{v}) dx dt \le 0 $$

and thus,

$$ \frac{1}{2} \|\bar{v}_n - \hat{v}\|_{L^2(\Omega)}^2 (T) \\ + \int_0^T \int_\Omega (\mathcal{F}(\theta_n, \bar{v}_n + v^0\xi, D(\bar{v}_n + v^0\xi)) \\ - \mathcal{F}(\theta_n, \bar{v}_n + v^0\xi, D(\hat{v} + v^0\xi))) : D(\bar{v}_n - \hat{v}) dx dt \quad (4.27) \\ \le \int_0^T \int_\Omega (\mathcal{F}(\theta, \bar{v} + v^0\xi, D(\hat{v} + v^0\xi)) \\ - \mathcal{F}(\theta_n, \bar{v}_n + v^0\xi, D(\hat{v} + v^0\xi))) : D(\bar{v}_n - \hat{v}) dx dt. $$
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:

$$
\begin{aligned}
& \int_0^T \int_{\Omega} (\mathcal{F}(\theta_n, \bar{v}_n + v^0 \xi, D(\bar{v}_n + v^0 \xi)) \\
& \qquad - \mathcal{F}(\theta_n, \bar{v}_n + v^0 \xi, D(\hat{v} + v^0 \xi))) : D(\bar{v}_n - \hat{v}) dx dt \\
& = \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\bar{v}_n + v^0 \xi)) - \mathcal{F}_1(D(\hat{v} + v^0 \xi))) : D(\bar{v}_n - \hat{v}) dx dt \\
& \quad + \int_0^T \int_{\Omega} (\mathcal{F}_2(\theta_n, \bar{v}_n + v^0 \xi, D(\bar{v}_n + v^0 \xi)) \\
& \qquad - \mathcal{F}_2(\theta_n, \bar{v}_n + v^0 \xi, D(\hat{v} + v^0 \xi))) : D(\bar{v}_n - \hat{v}) dx dt,
\end{aligned}
$$

where we recall that

$$
\begin{array}{ll}
\mathcal{F}_1(\lambda_2) = \mu_0 \|\lambda_2\|^{p-2} \lambda_2 & \text{if } \lambda_2 \neq 0 \mathbb{R}^{3 \times 3}, \quad \mathcal{F}_1(\lambda_2) = 0 \mathbb{R}^{3 \times 3} \text{ otherwise,} \\
\left\{
\begin{array}{l}
\mathcal{F}_2(\lambda_0, \lambda_1, \lambda_2) = 2\bar{\mu}(\lambda_0, \lambda_1, \|\lambda_2\|)\|\lambda_2\|^{p-2}\lambda_2 \quad \text{if } \lambda_2 \neq 0 \mathbb{R}^{3 \times 3}, \\
\mathcal{F}_2(\lambda_0, \lambda_1, \lambda_2) = 0 \mathbb{R}^{3 \times 3} \quad \text{otherwise}
\end{array}
\right.
&
\end{array}
$$

and $\bar{\mu} = \mu - \frac{\mu_0}{2}$. 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

$$
\int_0^T \int_{\Omega} (\mathcal{F}_2(\theta_n, \bar{v}_n + v^0 \xi, D(\bar{v}_n + v^0 \xi)) - \mathcal{F}_2(\theta_n, \bar{v}_n + v^0 \xi, D(\hat{v} + v^0 \xi))) : D(\bar{v}_n - \hat{v}) dx dt \ge 0,
$$

and (4.27) yields

$$
\begin{align}
& \frac{1}{2} \|\bar{v}_n - \hat{v}\|_{L^2(\Omega)}^2 (T) \nonumber \\
& \quad + \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\bar{v}_n + v^0 \xi)) - \mathcal{F}_1(D(\hat{v} + v^0 \xi))) : D(\bar{v}_n - \hat{v}) dx dt \tag{4.28} \\
& \le \int_0^T \int_{\Omega} (\mathcal{F}_2(\theta_n, \bar{v} + v^0 \xi, D(\bar{v} + v^0 \xi)) \nonumber \\
& \quad - \mathcal{F}_2(\theta_n, \bar{v}_n + v^0 \xi, D(\hat{v} + v^0 \xi))) : D(\bar{v}_n - \hat{v}) dx dt. \nonumber
\end{align}
$$

Then we distinguish two cases.

Case 1: $p \in [6/5, 2)$

Recalling that

$$
(\|\lambda\|^p + \|\lambda'\|^p)^{\frac{2-p}{2}} ((\mathcal{F}_1(\lambda) - \mathcal{F}_1(\lambda')) : (\lambda - \lambda'))^{\frac{p}{2}} \geq \frac{(\mu_0(p-1))^2}{2^{\frac{p-1}{2}(2-p)}} \|\lambda - \lambda'\|^p
$$
for all $(\lambda, \lambda') \in \mathbb{R}^{3 \times 3} \times \mathbb{R}^{3 \times 3}$, we obtain

$$ 
\begin{aligned}
& \frac{(\mu_0(p-1))^2}{2^{\frac{(p-1)(2-p)}{2}}} \int_0^T \int_{\Omega} \|D(\bar{u}_n - \hat{v})\|^p dx dt \\
& \le \left( \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\bar{u}_n + \nu^0 \xi)) - \mathcal{F}_1(D(\hat{v} + \nu^0 \xi))) : D(\bar{u}_n - \hat{v}) dx dt \right)^{\frac{p}{2}} \\
& \quad \times \left( \int_0^T \int_{\Omega} (\|D(\bar{u}_n + \nu^0 \xi)\|^p + \|D(\hat{v} + \nu^0 \xi)\|^p) dx dt \right)^{\frac{2-p}{2}} \\
& \le \left( \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\bar{u}_n + \nu^0 \xi)) - \mathcal{F}_1(D(\hat{v} + \nu^0 \xi))) : D(\bar{u}_n - \hat{v}) dx dt \right)^{\frac{p}{2}} \\
& \quad \times \left( \|\bar{u}_n + \nu^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)}^p + \|\hat{v} + \nu^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)}^p \right)^{\frac{2-p}{2}}.
\end{aligned}
$$

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

$$ 
\begin{aligned}
& \frac{(\mu_0(p-1))^2}{2^{\frac{(p-1)(2-p)}{2}}} \|D(\bar{u}_n - \hat{v})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^p \\
& \le \left( \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\bar{u}_n + \nu^0 \xi)) - \mathcal{F}_1(D(\hat{v} + \nu^0 \xi))) : D(\bar{u}_n - \hat{v}) dx dt \right)^{\frac{p}{2}} \\
& \quad \times \left( (\tilde{C}^{\text{flow}} + \|\nu^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^p + (C^{\text{flow}} + \|\nu^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p)})^p \right)^{\frac{2-p}{2}}.
\end{aligned}
$$

By observing that $\tilde{C}^{\text{flow}} \ge C^{\text{flow}}$, we have finally

$$ 
\begin{aligned}
& \frac{\mu_0(p-1)}{2^{(2-p)(\tilde{C}^{\text{flow}} + \|\nu^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p}))^{(2-p)}} \|D(\bar{u}_n - \hat{v})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^2 \\
& \le \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\bar{u}_n + \nu^0 \xi)) - \mathcal{F}_1(D(\hat{v} + \nu^0 \xi))) : D(\bar{u}_n - \hat{v}) dx dt \\
& \le \int_0^T \int_{\Omega} (\mathcal{F}_2(\theta, \bar{u} + \nu^0 \xi, D(\hat{v} + \nu^0 \xi)) \\
& \quad - \mathcal{F}_2(\theta_n, \bar{u}_n + \nu^0 \xi, D(\hat{v} + \nu^0 \xi))) : D(\bar{u}_n - \hat{v}) dx dt \\
& \le \|\mathcal{F}_2(\theta, \bar{u} + \nu^0 \xi, D(\hat{v} + \nu^0 \xi)) \\
& \quad - \mathcal{F}_2(\theta_n, \bar{u}_n + \nu^0 \xi, D(\hat{v} + \nu^0 \xi))\|_{L^{p'}(0,T;(L^{p'}(\Omega)))^{3 \times 3}} \\
& \quad \times \|D(\bar{u}_n - \hat{v})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}
\end{aligned}
$$

which yields

$$ 
\begin{aligned}
& \frac{\mu_0(p-1)}{2^{(2-p)(\tilde{C}^{\text{flow}} + \|\nu^0 \xi\|_{L^p(0,T;V_{\Gamma_1}^p}))^{(2-p)}} \|D(\bar{u}_n - \hat{v})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^2 \\
& \le \|\mathcal{F}_2(\theta, \bar{u} + \nu^0 \xi, D(\hat{v} + \nu^0 \xi)) \\
& \quad - \mathcal{F}_2(\theta_n, \bar{u}_n + \nu^0 \xi, D(\hat{v} + \nu^0 \xi))\|_{L^{p'}(0,T;(L^{p'}(\Omega)))^{3 \times 3}}.
\end{aligned}
\qquad (4.29)
$$
Case 2: $p \in [2, +\infty)$

Recalling that

$$ (\mathcal{F}_1(\lambda) - \mathcal{F}_1(\lambda')) : (\lambda - \lambda') \geq \frac{\mu_0}{2^{p-1}} \|\lambda - \lambda'\|^p $$

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

$$ \begin{aligned} & \frac{\mu_0}{2^{p-1}} \|D(\bar{v}_n - \hat{v})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^p \\ & \leq \int_0^T \int_{\Omega} (\mathcal{F}_1(D(\bar{v}_n + v^0\xi)) - \mathcal{F}_1(D(\hat{v} + v^0\xi))) : D(\bar{v}_n - \hat{v}) dx dt \\ & \leq \int_0^T \int_{\Omega} (\mathcal{F}_2(\theta, \bar{v} + v^0\xi, D(\hat{v} + v^0\xi)) \\ & \quad - \mathcal{F}_2(\theta_n, \bar{v}_n + v^0\xi, D(\hat{v} + v^0\xi))) : D(\bar{v}_n - \hat{v}) dx dt \\ & \leq \|\mathcal{F}_2(\theta, \bar{v} + v^0\xi, D(\hat{v} + v^0\xi)) \\ & \quad - \mathcal{F}_2(\theta_n, \bar{v}_n + v^0\xi, D(\hat{v} + v^0\xi))\|_{L^{p'}(0,T;(L^{p'}(\Omega)))^{3 \times 3}} \\ & \quad \times \|D(\bar{v}_n - \hat{v})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}} \end{aligned} $$

which yields

$$ \begin{aligned} & \frac{\mu_0}{2^{p-1}} \|D(\bar{v}_n - \hat{v})\|_{L^p(0,T;(L^p(\Omega)))^{3 \times 3}}^{p-1} \\ & \leq \|\mathcal{F}_2(\theta, \bar{v} + v^0\xi, D(\hat{v} + v^0\xi)) \\ & \quad - \mathcal{F}_2(\theta_n, \bar{v}_n + v^0\xi, D(\hat{v} + v^0\xi))\|_{L^{p'}(0,T;(L^{p'}(\Omega)))^{3 \times 3}}. \end{aligned} \quad (4.30) $$

By using the convergence properties (4.24) for the sequences $(\theta_n)_{n \geq 1}$ and $(\bar{v}_n)_{n \geq 1}$ and the continuity and boundedness assumptions (2.8) and (2.10) for the mapping $\mu$, we obtain

$$ \begin{aligned} \mathcal{F}_2(\theta_n, \bar{v}_n + v^0\xi, D(\hat{v} + v^0\xi)) & \to \mathcal{F}_2(\theta, \bar{v} + v^0\xi, D(\hat{v} + v^0\xi)) \\ \text{strongly in } & L^p(0, T; (L^p(\Omega)))^{3 \times 3}. \end{aligned} $$

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

$$ \lim_{n \to +\infty} \|D(\bar{v}_n - \hat{v})\|_{L^p(0, T; (L^p(\Omega)))^{3 \times 3}} = 0. $$

Finally, Korn's inequality implies that the sequence $(\bar{v}_n)_{n \geq 1}$ converges strongly to $\hat{v}$ in $L^p(0, T; V_{0, \text{div}}^p)$, which yields $\hat{v} = \bar{v}$. $\square$

By possibly extracting another subsequence, still denoted $(\bar{v}_n, \pi_n, \theta_n)_{n \geq 1}$, we obtain that

$$ D(\bar{v}_n) \to D(\bar{v}) \quad \text{a.e. in } (0, T) \times \Omega. $$

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

$$ \begin{aligned} \mathcal{F}(\theta_n, \bar{v}_n + v^0\xi, D(\bar{v}_n + v^0\xi)) & \to \mathcal{F}(\theta, \bar{v} + v^0\xi, D(\bar{v} + v^0\xi)) \\ \text{strongly in } & L^{p'}(0, T; (L^{p'}(\Omega)))^{3 \times 3} \end{aligned} $$
and

$$
\begin{aligned}
g_{\delta_n}(\theta_n, \bar{u}_n) &\to 2\mu(\theta, \bar{u} + \nu^0\xi, \|D(\bar{u} + \nu^0\xi)\|) \|D(\bar{u} + \nu_0\xi)\|^p \\
&= \mathcal{F}(\theta, \bar{u} + \nu^0\xi, D(\bar{u} + \nu^0\xi)) : D(\bar{u} + \nu^0\xi) \\
&\text{strongly in } L^1(0, T; L^1(\Omega)).
\end{aligned}
$$

Moreover,

$$
\bar{u}_n \to \hat{u} = \bar{u} \quad \text{strongly in } L^p(0, T; V_{0, \text{div}}^p).
$$

Reminding that the mapping $J$ is continuous on $L^p(0, T; V_0^p)$, we get

$$
J(\bar{u}_n + \tilde{\vartheta}\zeta) \to J(\bar{u} + \tilde{\vartheta}\zeta) \quad \forall \tilde{\vartheta} \in V_0^p, \forall \zeta \in \mathcal{D}(0, T).
$$

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

$$
r(\theta_n) \to r(\theta) \quad \text{strongly in } L^1(0, T; L^1(\Omega)).
$$

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

Remark 4.5 We may observe that $\pi = \hat{\pi}$. Indeed,

$$
\left\langle \frac{\partial}{\partial t} (\bar{u}, \tilde{\vartheta})_{L^2(\Omega)}, \zeta \right\rangle_{D'(0,T), D(0,T)} \\
+ \int_0^T \int_{\Omega} \mathcal{F}(\theta, \bar{u} + \nu^0 \xi, D(\bar{u} + \nu^0 \xi)) : D(\tilde{\vartheta}) \zeta \,dx\,dt \\
- \left\langle \int_{\Omega} \pi \operatorname{div}(\tilde{\vartheta}) \,dx\,\zeta \right\rangle_{D'(0,T), D(0,T)} + J(\bar{u} + \tilde{\vartheta})\zeta - J(\bar{u}) \\
\geq \int_0^T \left( f + \nu^0 \frac{\partial \zeta}{\partial t}, \tilde{\vartheta} \right)_{L^2(\Omega)} \zeta \,dt \quad \forall \tilde{\vartheta} \in V_0^p, \forall \zeta \in \mathcal{D}(0,T)
$$

and

$$
\left\langle \frac{\partial}{\partial t} (\hat{v}, \tilde{v})_{L^2(\Omega)}, \zeta \right\rangle_{D'(0,T), D(0,T)} \\
+ \int_0^T \int_{\Omega} \mathcal{F}(\theta, \bar{v} + \nu^0 \xi, D(\bar{v} + \nu^0 \xi)) : D(\tilde{v}) \zeta \,dx\,dt \\
- \left\langle \int_{\Omega} \hat{\pi} \operatorname{div}(\tilde{v}) \,dx\,\zeta \right\rangle_{D'(0,T), D(0,T)} + J(\hat{v} + \tilde{v})\zeta - J(\hat{v}) \\
\geq \int_0^T \left( f + \nu^0 \frac{\partial \zeta}{\partial t}, \tilde{v} \right)_{L^2(\Omega)} \zeta \,dt \quad \forall \tilde{v} \in V_0^p, \forall \zeta \in \mathcal{D}(0,T).
$$
Owing to that $\bar{v} = \hat{v}$, we obtain that

$$ \left\langle \int_{\Omega} (\pi - \hat{\pi}) \operatorname{div}(\tilde{\vartheta}) dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} = 0 \quad \forall \tilde{\vartheta} \in \mathbf{W}_0^{1,p}(\Omega), \forall \zeta \in \mathcal{D}(0,T). $$

Hence, for any $\omega \in L_0^p(\Omega)$, we may choose $\tilde{\vartheta} = P_p(\omega)$, where $P_p$ is the operator introduced in Proposition 4.1, and we get

$$ \left\langle \int_{\Omega} (\pi - \hat{\pi}) \omega \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} = 0. $$

Furthermore, for any $\omega^* \in L^p(\Omega)$, we may define $\omega \in L_0^p(\Omega)$ by

$$ \omega = \omega^* - \frac{1}{\operatorname{meas}(\Omega)} \int_{\Omega} \omega^* dx, $$

and since $\pi - \hat{\pi} \in H^{-1}(0, T; L_0^p(\Omega))$, we have

$$ \begin{aligned} & \left\langle \int_{\Omega} (\pi - \hat{\pi}) \omega \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \\ &= \left\langle \int_{\Omega} (\pi - \hat{\pi}) \omega^* \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \\ & \quad - \frac{1}{\operatorname{meas}(\Omega)} \left( \int_{\Omega} \omega^* \, dx \right) \left\langle \int_{\Omega} (\pi - \hat{\pi}) \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} \\ &= \left\langle \int_{\Omega} (\pi - \hat{\pi}) \omega^* \, dx, \zeta \right\rangle_{\mathcal{D}'(0,T), \mathcal{D}(0,T)} = 0 \quad \forall \omega^* \in L^p(\Omega), \forall \zeta \in \mathcal{D}(0,T), \end{aligned} $$

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

**Remark 4.6** The strong convergence of $(g_{\delta_n}(\theta_n, \bar{v}_n))_{n \ge 1}$ 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

$$ \begin{aligned} & \int_0^T \int_{\Omega} c \frac{\partial(\theta_n - \theta_{n'})}{\partial t} \tilde{w} \, dx \, dt + \int_0^T \int_{\Omega} (K \nabla(\theta_n - \theta_{n'})) \cdot \nabla \tilde{w} \, dx \, dt \\ &= \int_0^T \int_{\Omega} (g_{\delta_n}(\theta_n, \bar{v}_n) - g_{\delta_{n'}}(\theta_{n'}, \bar{v}_{n'})) \tilde{w} \, dx \, dt + \int_0^T \int_{\Omega} (r(\theta_n) - r(\theta_{n'})) \tilde{w} \, dx \, dt \\ & \quad + \int_0^T \int_{\Omega} (\theta_{\delta_n}^b - \theta_{\delta_{n'}}^b) \tilde{w} \, dx' \, dt \end{aligned} $$

for all $\tilde{w} \in L^2(0, T; W_{\Gamma_1 \cup \Gamma_L}^{1,2}(\Omega))$, for all $n \ge 1$, and $n' \ge 1$. By choosing $\tilde{w} = \psi_0^m(\theta_n - \theta_{n'}) \mathbf{1}_{[0,t]}$, with $t \in (0, T]$ and by passing to the limit, as $m$ tends to $+\infty$, we get

$$ \int_{\Omega} c(x) \Phi_0(\theta_n(t, x) - \theta_{n'}(t, x)) \, dx \\ + \int_0^t \int_{\Omega} \Psi_0(\theta_n(s, x) - \theta_{n'}(s, x)) \, dx \, dt $$
$$ 
\begin{aligned}
& \times (K(x)\nabla(\theta_n(s,x) - \theta_{n'}(s,x))) \cdot \nabla(\theta_n(s,x) - \theta_{n'}(s,x)) dx ds \\
= & \int_0^t \int_{\Omega} (g_{\delta_n}(\theta_n, \bar{\nu}_n) - g_{\delta_{n'}}(\theta_{n'}, \bar{\nu}_{n'})) \psi_0(\theta_n - \theta_{n'}) dx ds \\
& + \int_0^t \int_{\Omega} (r(\theta_n) - r(\theta_{n'})) \psi_0(\theta_n - \theta_{n'}) dx ds \\
& + \int_0^t \int_{\Omega} (\theta_{\delta_n}^b - \theta_{\delta_{n'}}^b) \psi_0(\theta_n - \theta_{n'}) dx' ds + \int_{\Omega} c(x) \Phi_0(\theta_{\delta_n}^0 - \theta_{\delta_{n'}}^0) dx,
\end{aligned}
$$

which yields

$$ 
\begin{aligned}
& c_0 \int_{\Omega} \Phi_0(\theta_n(t,x) - \theta_{n'}) dx \\
\leq & \|g_{\delta_n}(\theta_n, \bar{\nu}_n) - g_{\delta_{n'}}(\theta_{n'}, \bar{\nu}_{n'})\|_{L^1(0,T;L^1(\Omega))} \\
& + \|r(\theta_n) - r(\theta_{n'})\|_{L^1(0,T;L^1(\Omega))} + \|\theta_{\delta_n}^b - \theta_{\delta_{n'}}^b\|_{L^1((0,T)\times\omega)} + \|c(\theta_{\delta_n}^0 - \theta_{\delta_{n'}}^0)\|_{L^1(\Omega)}.
\end{aligned}
$$

Next, we observe that

$$ \Phi_0(s) = \begin{cases} \frac{s^2}{2} & \text{if } |s| \le 1, \\ s - \frac{1}{2} & \text{if } s > 1, \\ -s - \frac{1}{2} & \text{if } s < -1 \end{cases} $$

which implies $\Phi_0(s) \ge \frac{|s|}{2}$ if $|s| \ge 1$. It follows that

$$ c_0 \int_{\Omega} |\theta_n(t,x) - \theta_{n'}(t,x) dx| \le c_0 \left\{ \left( \int_{\Omega} \Phi_0(\theta_n(t,x) - \theta_{n'}(t,x)) \right)^{1/2} \mathrm{meas}(\Omega)^{1/2} + 2 \int_{\Omega} \Phi_0(\theta_n(t,x) - \theta_{n'}(t,x)) \right\}, $$

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 $\tilde{\zeta}$ from $C^\infty([0, T])$ in (2.15).

Moreover, following [28], we can prove that $\theta$ 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 $\theta$ in the space $C^0([0, T]; L^1(\Omega))$, which is the key point to show that $\theta$ 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.

### Acknowledgements

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# Declarations

## Competing interests

The author declares that she has no competing interests.

## Authors' contributions

I declare that I am the sole author of this work. The author read and approved the final manuscript.

## Authors' information

Lyon University, F-42023 Saint-Etienne, Institut Camille Jordan CNRS UMR 5208, 23 Rue Michelon, 42023 Saint-Etienne Cedex 2, France.

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Received: 22 September 2021 Accepted: 12 January 2022 Published online: 24 January 2022

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