# Vorticity Leray-$\alpha$ model for Navier-Stokes equations with viscosity depending on the distance to the wall

Guillaume Leloup

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Guillaume Leloup. Vorticity Leray-$\alpha$ model for Navier-Stokes equations with viscosity depending on the distance to the wall. 2024. hal-04424189

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# Vorticity Leray-$\alpha$ model for Navier-Stokes equations with viscosity depending on the distance to the wall.

Guillaume LELOUP

Univ Rennes, IRMAR, UMR CNRS 6625, and Odyssey Team, INRIA Rennes, France

E-mail: guillaume.leloup@univ-rennes.fr

## Abstract

We introduce a vorticity Leray-$\alpha$ model with eddy viscosity depending on $d(x, \partial\Omega)^\eta$ where $\partial\Omega$ is the boundary of the domain and $\eta \in ]0; 1[$. We prove that this system admits fairly regular weak solutions converging when $\alpha$ goes to 0 to the solution of a reference system.

**Key words** : Fluid mechanics, Navier-Stokes equations, Leray-$\alpha$ models, Vorticity.

**2010 MSC** : 35Q30, 76D03, 76D05, 35A35, 35B33.

## 1 Introduction

Let $\Omega$ be a $\mathcal{C}^2$ open bounded subset of $\mathbb{R}^3$. According to turbulence modelling, we introduce in this paper the following vorticity Leray-$\alpha$ model

$$ (1.1) \qquad \bar{\omega} \times \mathbf{u} - \operatorname{div}(\varrho^{\eta} \tilde{\nu} \nabla \mathbf{u}) + \lambda \mathbf{u} + \nabla \tilde{p} = \mathbf{f} \quad \text{in } \Omega, $$

$$ (1.2) \qquad \operatorname{div}(\mathbf{u}) = 0 \quad \text{in } \Omega, $$

$$ (1.3) \qquad \omega = \operatorname{curl}(\mathbf{u}) \quad \text{in } \Omega, $$

$$ (1.4) \qquad -\alpha^2 \Delta \bar{\omega} + \bar{\omega} = P(\omega) \quad \text{in } \mathbb{R}^3, $$

$$ (1.5) \qquad \mathbf{u} = 0 \quad \text{on } \partial\Omega, $$

where $\mathbf{u}$ models the long time average of the flow, $\tilde{p} = p - \frac{|\mathbf{u}|^2}{2}$ is the modified mean pressure, and $\mathbf{f} \in L^2(\Omega)^3$ is a given source term. The linear term $\lambda\mathbf{u}$ with $\lambda > 0$ is a damping term.

The eddy viscosity is of the form $\nu_{\text{turb}} = \varrho^{\eta} \tilde{\nu}$, where $0 < \eta < 1$ and $\tilde{\nu}$ is a continuous function satisfying

$$ (1.6) \qquad \forall t \in \mathbb{R}, \quad 0 < \nu_m \le \tilde{\nu}(t) \le \nu_M, $$

where $\nu_m$ and $\nu_M$ are constants.
The function $\varrho$ refers to the Prandtl mixing length (see [20]), which appears like a regular function behaving like the distance to the boundary $d(\cdot, \partial\Omega)$. More precisely, we consider $\varrho : \Omega \to \mathbb{R}_+$ such that

$$ (1.7) \qquad \lim_{\substack{d(x,\partial\Omega)\to0 \\ x\in\Omega}} \frac{\varrho(x)}{d(x,\partial\Omega)} = 1 \quad \text{and} \quad \forall n>0, \quad \varrho_n = \inf_{\substack{d(x,\partial\Omega)\ge\frac{1}{n} \\ x\in\Omega}} \varrho(x) > 0. $$

The operator $P$ in (1.4) is the extension by 0, which means

$$ (1.8) \qquad P(\omega) : x \mapsto \begin{cases} \omega(x) & \text{if } x \in \Omega, \\ 0 & \text{if } x \notin \Omega. \end{cases} $$

Equation (1.4) defining $\bar{\omega}$ is the Helmholtz equation (studied in detail for instance in [7, 13]), so that (1.1)-(1.5) is a kind of Leray-$\alpha$ model, which is a LES model.

In this paper we show that (1.1)-(1.5) with $\eta < 1$ admits sufficiently regular weak solutions satisfying the energy balance (3.2) (Theorem 3.1) that converge when $\alpha$ goes to zero to a solution of the following system

$$ (1.9) \qquad \omega \times \mathbf{u} - \operatorname{div}(\varrho^{\eta} \tilde{\nu} \nabla \mathbf{u}) + \lambda \mathbf{u} + \nabla p = \mathbf{f} \quad \text{in } \Omega, $$

$$ (1.10) \qquad \operatorname{div}(\mathbf{u}) = 0 \quad \text{in } \Omega, $$

$$ (1.11) \qquad \omega = \operatorname{curl}(\mathbf{u}) \quad \text{in } \Omega, $$

$$ (1.12) \qquad \mathbf{u} = 0 \quad \text{on } \partial\Omega. $$

As previously shown in [1], (1.9)-(1.12) admits weak solutions for $0 \le \eta < 1$, but the energy balance (3.2) can be satisfied if and only if $0 \le \eta < 1/5$. Consequently, the convergence stated in Theorem 3.2 is weak when $1/5 \le \eta < 1$ and strong if $0 \le \eta < 1/5$.

Leray-$\alpha$ models were first introduced by J. Leray in [15, 16], and have been studied in detail in [3, 4, 6, 8, 9, 18, 19]. In this kind of models, the convection term is given by $(\bar{\mathbf{u}} \cdot \nabla)\mathbf{u}$.

By the way, it is possible to apply the filter to both velocities, so that the convection term becomes $(\mathbf{u} \cdot \nabla)\mathbf{u}$. This is called Bardina-type models. These have been studied for instance in [3, 5, 8, 11, 17]. In particular, it is shown that solutions of the $\alpha$-models are regular and converge to the Leray solution of NSE, most of time with periodic boundary conditions. A comparison between these different models is made for instance in [3, 6, 8, 18].

In order to ensure coercivity results, a free-divergence assumption is required for $\bar{\mathbf{u}}$. Consequently, the standard Leray-$\alpha$ model adapted to (1.1) should be like

$$ (1.13) \qquad (\bar{\mathbf{u}} \cdot \nabla)\mathbf{u} - \operatorname{div}(\varrho^{\eta} \tilde{\nu} \nabla \mathbf{u}) + \lambda \mathbf{u} + \nabla p = \mathbf{f} \quad \text{in } \Omega, $$

$$ (1.14) \qquad -\alpha^2 \Delta \bar{\mathbf{u}} + \bar{\mathbf{u}} + \nabla q = \mathbf{u} \quad \text{in } \Omega, $$

$$ (1.15) \qquad \operatorname{div}(\mathbf{u}) = \operatorname{div}(\bar{\mathbf{u}}) = 0 \quad \text{in } \Omega, $$

$$ (1.16) \qquad \mathbf{u} = 0 \quad \text{on } \Gamma. $$

Unfortunately, this model does not provide sufficient results to ensure the existence of suitable solutions. Indeed, we work in weighted function spaces, which only provides a $W^{1,p}$ regularity for $\mathbf{u}$, with $1 \le p < 2$. This regularity is not enough, because of the Lagrange multiplier in (1.14).
This is why the vorticity has been introduced. This quantity appears in [8, 19] to introduce an α-model called NS-α. In this model, the vorticity term curl( **u** ) × **u** is replaced by curl( **u** ) × **u**.

Here we consider another version of this model, similar to the NS − ¯ ω model introduced by W.J. Layton and al. in [12, 14], where the vorticity term of (1.9)-(1.12) becomes curl( ¯ **u** ) × **u**. A theoretical study is made in [12], for a system of NSE with constant vis- cosity. This idea is relevant given that there is no free-divergence assumption required to prove coercivity in the calculations.

The paper is organized as follows. In section 2, we focus on the vorticity Helmholz equa- tion with a L p source term. The goal is proving some convergence results that are applied later.

In section 3, we start by showing the existence of a distributional solution of (1.1)-(1.5) using a fixed-point strategy, and then we prove the convergence to a solution of (1.9)-(1.12) when α goes to zero. The value of η has an influence on the sense of the convergence.

I would like to express my gratitude to my PhD advisor Professor Roger LEWANDOWSKI to have suggested this problem and guided me during this work.

2 Vorticity Leray- α model

This section starts by defining the function spaces in which solutions of (1.9)-(1.12) are searched. As already done in [1], some Sobolev regularity is given on these spaces. Then, the solution of the Helmholz vorticity equation is written with a convolution kernel, to ease the calculations in order to find results on ¯ ω starting from equivalent statements on ω .

2.1 Functions spaces and estimates

We first define a space of smooth functions

$$
(2.1) \qquad \mathcal{V} = \{ \mathbf{v} \in \mathcal{D}(\Omega)^3 / \operatorname{div}(\mathbf{v}) = 0 \},
$$

and V$_{η}$ is the closure of this space for the norm

$$
(2.2) \qquad \| \mathbf{u} \|_{V_{\eta}} = \left( \| \mathbf{u} \|_{0,2,\Omega}^2 + \| \varrho^{\eta/2} \nabla \mathbf{u} \|_{0,2,\Omega}^2 \right)^{1/2} .
$$

We start by stating the regularity of V$_{η}$ functions. Indeed, looking at the norm (2.2), we understand that such functions are not likely to be H 1 if η > 0.

We recall two embedding theorems. The first is from [1, 2] brings a H s estimate, with 1 / 2 < s ≤ 1.

Theorem 2.1. Let η ∈ [0; 1[. The continuous embedding

$$
(2.3) \qquad V_{\eta} \hookrightarrow H^{1-\frac{\eta}{2}}(\Omega)^{3}
$$

stands, which implies the Sobolev compact embedding

$$
(2.4) \qquad V_{\eta} \hookrightarrow L^r(\Omega)^3, \quad \text{where} \quad 1 \le r < \frac{6}{1+\eta}.
$$
The second result from [10] brings a $W^{1,p}$ estimate with $1 \le p < 2$.

**Theorem 2.2.** Let $\eta \in [0; 1[$. The continuous embedding

$$ (2.5) \qquad V_{\eta} \hookrightarrow W^{1,p}(\Omega)^3 $$

stands for $p \in [1; \frac{2}{1+\eta}]$.

**Remark 2.1.** The compact Sobolev embedding given by Theorem 2.2 is weaker than this given by Theorem 2.1. Indeed, $\frac{6}{1+3\eta}$ is lower than $\frac{6}{1+\eta}$.

**Remark 2.2.** These two results also give an estimate on the vorticity.

* If $u \in H^{1-\frac{\eta}{2}}(\Omega)^3$, then $\omega = \mathrm{curl}(u) \in H^{-\frac{\eta}{2}}(\Omega)^3$.

* If $u \in W^{1,p}(\Omega)^3$, then $\omega = \mathrm{curl}(u) \in L^p(\Omega)^3$.

## 2.2 Green kernel for the Helmholtz problem

Until the end of this section, we simply write $\omega$ instead of $P(\omega)$. This means

$$ (2.6) \qquad \omega \in L^p(\mathbb{R}^3)^3 \quad \text{with} \quad 1 \le p < \frac{2}{1+\eta} \quad \text{and} \quad 0 \le \eta \le 1. $$

We focus on the following Helmholtz equation, where $\alpha > 0$ is a constant parameter, supposed to be near 0.

$$ (2.7) \qquad -\alpha^2 \Delta \bar{\omega} + \bar{\omega} = \omega. $$

**Theorem 2.3 (Green function).** Let $\omega \in L^p(\mathbb{R}^3)^3$ defined in (2.6), and $\bar{\omega}$ satisfying (2.7). Then

$$ (2.8) \qquad \bar{\omega}(x) = K_{\alpha} * \omega(x) $$

for almost every $x \in \mathbb{R}^3$, where $K_\alpha$ is the scalar function given by

$$ (2.9) \qquad K_{\alpha}: \mathbb{R}^3 \to \mathbb{R} \\ x \mapsto \frac{e^{-|x|/\alpha}}{4\pi\alpha^2|x|}. $$

This result is already proved in [17].

**Lemma 2.1.** The function $K_\alpha$ given by (2.9) is in $L^q(\mathbb{R}^3, \mathbb{R}_+)$ for $q \in [1; 3[$ and

$$ (2.10) \qquad \int_{\mathbb{R}^3} K_{\alpha}(x)^q dx = \Gamma(3-q) \frac{(4\pi\alpha^3)^{1-q}}{q^{3-q}}, $$

where $\Gamma$ refers to the Euler function: $\Gamma(t) = \int_0^{+\infty} s^{t-1}e^{-s} ds$, for any $t > 0$.

**Remark 2.3.** When $q = 1$, we obtain in particular

$$ (2.11) \qquad \int_{\mathbb{R}^3} K_{\alpha}(x) dx = 1. $$
Proof. We use a change of variables which transforms the classical cartesian coordinates of $\mathbb{R}^3$ into spherical coordinates $(r, \theta, \varphi) \in \mathbb{R}_+ \times [0; \pi] \times [0; 2\pi]$, so that the 3D Lebesgue measure is written

$$ (2.12) \qquad d\lambda(y) = r^2 \sin(\theta) dr d\theta d\varphi. $$

The Fubini-Tonelli theorem then yields

$$ (2.13) \qquad \iiint_{\mathbb{R}^3} K_\alpha(y)^q dy = \int_{r=0}^{+\infty} \int_{\theta=0}^{\pi} \int_{\varphi=0}^{2\pi} \frac{e^{-qr/\alpha}}{(4\pi)^q \alpha^{2q} r^{2q}} r^2 \sin(\theta) dr d\theta d\varphi \\ = \frac{(4\pi)^{1-q}}{\alpha^{2q}} \int_0^{+\infty} r^{2-q} e^{-qr/\alpha} dr. $$

The change of variables $s = \frac{qr}{\alpha}$ yields

$$ (2.14) \qquad \int_0^{+\infty} r^{2-q} e^{-qr/\alpha} dr = \left(\frac{\alpha}{q}\right)^{3-q} \int_0^{+\infty} s^{2-q} e^{-s} ds. $$

This integral is defined if and only if $2-q > -1$, which means $q < 3$. When this condition is satisfied, (2.13) yields

$$ (2.15) \qquad \int_{\mathbb{R}^3} K_\alpha(y)^q dy = \frac{(4\pi)^{1-q}}{\alpha^{2q}} \left(\frac{\alpha}{q}\right)^{3-q} \int_0^{+\infty} s^{2-q} e^{-s} ds, $$

which corresponds to (2.10). $\square$

Since the vector function $\bar{\omega}$ is defined as a convolution, the Young inequality gives some regularity results for $\bar{\omega}$. Indeed, the extension by 0 preserves the $L^p$ norms.

**Lemma 2.2** ($L^\gamma$-regularity for $\bar{\omega}$). Let $\omega \in L^p(\mathbb{R}^3)^3$ with $p$ defined by (2.6), and $K_\alpha \in L^q(\mathbb{R}^3)$ with $q \in [1; 3[$.

* If $0 \le \eta < 1/3$, $\bar{\omega} \in L^\gamma(\mathbb{R}^3)^3$ with $1 \le \gamma \le +\infty$.

* If $\eta = 1/3$, $\bar{\omega} \in L^\gamma(\mathbb{R}^3)^3$ with $1 \le \gamma < +\infty$.

* If $1/3 < \eta \le 1$, $\bar{\omega} \in L^\gamma(\mathbb{R}^3)^3$ with $1 \le \gamma < \frac{6}{3\eta - 1}$.

**Proof.** Since $K_\alpha \in L^q(\mathbb{R}^3)$ and $\omega \in L^p(\mathbb{R}^3)^3$, the Young inequality yields

$$ (2.16) \qquad \bar{\omega} \in L^\gamma(\mathbb{R}^3)^3 \text{ with } 1 + \frac{1}{\gamma} = \frac{1}{p} + \frac{1}{q}. $$

The assumptions on $p$ and $q$ yield

$$ (2.17) \qquad \frac{1+\eta}{2} + \frac{1}{3} < \frac{1}{p} + \frac{1}{q} \le 2, $$

which means

$$ (2.18) \qquad \frac{\eta}{2} - \frac{1}{6} < \frac{1}{\gamma} \le 1. $$

We see that $\frac{\eta}{2} - \frac{1}{6}$ is always strictly lower than 1, and it is positive if and only if $\eta > 1/3$. Consequently, when $\eta \le 1/3$ we directly obtain the result. And if $\eta > 1/3$, (2.18) yields

$$ (2.19) \qquad 1 \le \gamma < \frac{6}{3\eta - 1}. $$

Hence the result.

$\square$
## 2.3 Main convergence results

The two next results are the main statements of this paper. First, we consider the case where $\alpha$ is fixed. The following lemma will be used to find solutions of the approximated system (1.1)-(1.5), by a fixed-point method.

**Lemma 2.3 (Convergence when $\alpha$ is fixed).** Let $(\omega_n)_{n \in \mathbb{N}}$ be a sequence of $L^p(\mathbb{R}^3)^3$ functions $(1 \le p < +\infty)$ that weakly converges to $\omega \in L^p(\mathbb{R}^3)^3$. Let $(\bar{\omega}_n)_{n \in \mathbb{N}}$ be the sequence of the solutions of (2.7) with source term $\omega_n$, and $\bar{\omega}$ the solution of (2.7) with source term $\omega$.

Then $(\bar{\omega}_n)_{n \in \mathbb{N}}$ weakly converges in $L^p(\mathbb{R}^3)^3$ to $\bar{\omega}$.

*Proof.* Let $\phi \in L^p(\mathbb{R}^3)$, the Fubini theorem yields

$$ (2.20) \qquad \int_{\mathbb{R}^3} \bar{\omega}_n(x)\phi(x) \,dx = \iint_{x,y \in \mathbb{R}^3} \omega_n(y) K_\alpha(x-y)\phi(x) \,dx \,dy \\ = \int_{y \in \mathbb{R}^3} \omega_n(y) \left( \int_{x \in \mathbb{R}^3} K_\alpha(x-y)\phi(x) \,dx \right) \,dy. $$

Since $\phi \in L^p(\mathbb{R}^3)$ and $K_\alpha \in L^1(\mathbb{R}^3)$, the function $[y \mapsto \int_{\mathbb{R}^3} K_\alpha(x-y)\phi(x) \,dx]$ is in $L^p(\mathbb{R}^3)$. Hence the weak convergence of $(\omega_n)_{n \in \mathbb{N}}$ yields the result.

□

The following theorem gives the convergence of the solutions $\bar{\omega}$ of (2.7) when $\alpha$ goes to zero. This will be used later to show the convergence of a sequence of solutions of (1.1)-(1.5) to a solution of (1.9)-(1.12).

**Theorem 2.4.** Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of positive numbers such that $\alpha_n \underset{n \to +\infty}{\longrightarrow} 0$, and $p$ be defined by (2.6). Let $(\omega_{\alpha_n})_{n \in \mathbb{N}}$ be a sequence of $L^p(\mathbb{R}^3)^3$ functions that weakly converges to $\omega \in L^p(\mathbb{R}^3)^3$.

Let $(\bar{\omega}_{\alpha_n})_{n \in \mathbb{N}}$ satisfy for all $n \in \mathbb{N}$

$$ (2.21) \qquad -\alpha_n^2 \Delta \bar{\omega}_{\alpha_n} + \bar{\omega}_{\alpha_n} = \omega_{\alpha_n}. $$

Then $(\bar{\omega}_{\alpha_n})_{n \in \mathbb{N}}$ weakly converges to $\omega$ in $L^p(\mathbb{R}^3)^3$.

*Proof.* Let $\phi \in \mathcal{D}(\mathbb{R}^3)$, by the Fubini theorem

$$ (2.22) \qquad \int_{\mathbb{R}^3} \bar{\omega}_{\alpha_n}(x)\phi(x) \,dx = \iint_{x,y \in \mathbb{R}^3} \omega_{\alpha_n}(y) K_{\alpha_n}(x-y)\phi(x) \,dx \,dy \\ = \int_{y \in \mathbb{R}^3} \omega_{\alpha_n}(y) \left( \int_{x \in \mathbb{R}^3} K_{\alpha_n}(x-y)\phi(x) \,dx \right) \,dy. $$

The change of variables $z = \frac{x-y}{\alpha_n}$ in the middle integral yields

$$ (2.23) \qquad \int_{\mathbb{R}^3} K_{\alpha_n}(x-y)\phi(x) \,dx = \int_{\mathbb{R}^3} \frac{e^{-|x-y|/\alpha_n}}{4\pi\alpha_n^2|x-y|}\phi(x) \,dx = \int_{\mathbb{R}^3} \frac{e^{-|z|}}{4\pi|z|}\phi(y+\alpha_n z) \,dz. $$

Then (2.22) becomes

$$ (2.24) \qquad \int_{\mathbb{R}^3} \bar{\omega}_{\alpha_n}(x)\phi(x) \,dx = \int_{z \in \mathbb{R}^3} K_1(z) \left( \int_{y \in \mathbb{R}^3} \omega_{\alpha_n}(y)\phi(y+\alpha_n z) \,dy \right) \,dz. $$
We now aim at showing the following convergence for any fixed $z \in \mathbb{R}^3$

$$ (2.25) \qquad \int_{\mathbb{R}^3} \omega_{\alpha_n}(y)\phi(y + \alpha_n z) dy \xrightarrow{n \to +\infty} \int_{\mathbb{R}^3} \omega(y)\phi(y) dy. $$

To do so, we decompose the difference

$$ (2.26) \qquad \int_{\mathbb{R}^3} \omega_{\alpha_n}(y)\phi(y + \alpha_n z) dy - \int_{\mathbb{R}^3} \omega(y)\phi(y) dy = \int_{\mathbb{R}^3} (\omega_{\alpha_n}(y) - \omega(y))\phi(y) dy \\ + \int_{\mathbb{R}^3} \omega_{\alpha_n}(y) (\phi(y + \alpha_n z) - \phi(y)) dy. $$

On one hand, the weak convergence of $(\omega_{\alpha_n})_{n \in \mathbb{N}}$ yields

$$ (2.27) \qquad \int_{\mathbb{R}^3} (\omega_{\alpha_n}(y) - \omega(y)) \phi(y) dy \xrightarrow{n \to +\infty} 0. $$

On the other hand, the Hölder inequality yields

$$ (2.28) \qquad \int_{\mathbb{R}^3} \omega_{\alpha_n}(y) (\phi(y + \alpha_n z) - \phi(y)) dy \le \|\omega_{\alpha_n}\|_p \left( \int_{\mathbb{R}^3} |\phi(y + \alpha_n z) - \phi(y)|^{p'} dy \right)^{1/p'} \\ \qquad \qquad \xrightarrow{n \to +\infty} 0, $$

by the Lebesgue theorem. Hence (2.25).

Then the following domination stands:

$$ (2.29) \qquad \left| K_1(z) \left( \int_{y \in \mathbb{R}^3} \omega_{\alpha_n}(y) \phi(y + \alpha_n z) dy \right) \right| \le K_1(z) \|\omega_{\alpha_n}\|_p \|\phi\|_{p'}, $$

which is $L^1(\mathbb{R}^3)$.

Therefore by the Lebesgue theorem, (2.25) and (2.29)

$$ (2.30) \qquad \int_{\mathbb{R}^3} \bar{\omega}_{\alpha_n}(x) \phi(x) dx \xrightarrow{n \to +\infty} \int_{\mathbb{R}^3} \omega(y) \phi(y) dy. $$

The density of $\mathcal{D}(\mathbb{R}^3)$ in $L^p(\mathbb{R}^3)$ yields the result.

□

# 3 Solutions of the Leray-$\alpha$ approximated system

This section aims at proving the existence of solutions of (1.9)-(1.12), in three steps. First, we solve a linearized system by the Lax-Milgram theorem. Such a solution is unique, and can be written $Tu$. Then the existence of solutions of (1.1)-(1.5) is shown by using the Leray-Schauder fixed-point theorem on the operator $T$.

We then obtain a sequence $(u_{\alpha_n})_{n \in \mathbb{N}}$ of solutions, corresponding to a sequence of parameters $(\alpha_n)_{n \in \mathbb{N}}$ converging to 0. Finally, Theorem 2.4 provides the convergence to a solution of (1.9)-(1.12).

## 3.1 Variational formulation

The function $\bar{\omega}$ in (1.1)-(1.5) is obtained this way.

* We consider the extension by 0 of $\omega \in L^p(\Omega)^3$, still named the same,

* $\bar{\omega}$ satisfies (2.7), so that $\bar{\omega} \in L^\gamma(\mathbb{R}^3)^3$, where $\gamma$ is given by Corollary 2.2.

* we take the restriction of $\bar{\omega}$ to $\Omega$, still named the same.
We write the variational formulation of system (1.1)-(1.5)

$$ (3.1) \qquad \begin{cases} \mathbf{u} \in V_{\eta}, \\ \forall \mathbf{v} \in V_{\eta}, \quad \int_{\Omega} (\bar{\omega}(\mathbf{u}) \times \mathbf{u}) \cdot \mathbf{v} + \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \mathbf{u} : \nabla \mathbf{v} + \lambda \int_{\Omega} \mathbf{u} \cdot \mathbf{v} = \int_{\Omega} \mathbf{f} \cdot \mathbf{v}. \end{cases} $$

Taking $\mathbf{v} = \mathbf{u}$ in (3.1) yields the following energy balance

$$ (3.2) \qquad \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla \mathbf{u}|^2 + \lambda \int_{\Omega} |\mathbf{u}|^2 = \int_{\Omega} \mathbf{f} \cdot \mathbf{u}. $$

These two equations have a sense if and only if the integrals are defined. The condition is given by the following lemma.

**Lemma 3.1** (Critical exponents). Let $\mathbf{f} \in L^2(\Omega)^3$ and $\lambda > 0$. For any $\eta \in ]0; 1[$, all the integrals in (3.1) are well defined.

*Proof.* The definition of $V_{\eta}$ directly yields the three last integrals in (3.1) are finite. In order to show that $\int_{\Omega} (\bar{\omega} \times \mathbf{u}) \cdot \mathbf{v}$ is finite for $\mathbf{u}, \mathbf{v} \in V_{\eta}$, we have to distinguish the two cases of Lemma 2.2.

If $\eta \le 1/3$, $\bar{\omega} \in L^{\infty}(\Omega)^3$ and $\mathbf{u}, \mathbf{v} \in L^r(\Omega)^3$ for $1 \le r < \frac{6}{1+\eta}$. The Hölder inequality then yields

$$ (3.3) \qquad \int_{\Omega} |(\bar{\omega} \times \mathbf{u}) \cdot \mathbf{v}| \le \|\bar{\omega}\|_{0,\infty,\Omega} \|\mathbf{u}\|_{0,r,\Omega} \|\mathbf{v}\|_{0,r,\Omega}, \quad \text{if } \frac{2}{r} = 1. $$

This condition can be fulfilled if and only if $\frac{6}{1+\eta} > 2$, which means $\eta < 2$. This is the case, since $\eta \le 1/3$.

If, $\eta > 1/3$, $\bar{\omega} \in L^s(\Omega)^3$ with $1 \le s < \frac{6}{3\eta - 1}$. The functions $\mathbf{u}$ and $\mathbf{v}$ have the same regularity as before. The Hölder inequality yields

$$ (3.4) \qquad \int_{\Omega} |(\bar{\omega} \times \mathbf{u}) \cdot \mathbf{v}| \le \|\bar{\omega}\|_{0,s,\Omega} \|\mathbf{u}\|_{0,r,\Omega} \|\mathbf{v}\|_{0,r,\Omega}, \quad \text{if } \frac{1}{s} + \frac{2}{r} = 1. $$

This equality of exponents can be fulfilled if and only if $\frac{3\eta - 1}{6} + \frac{1+\eta}{3} < 1$, which is equivalent to $\eta < 1$. $\square$

Now we can initiate the method explained at the beginning of this section.

## 3.2 Solutions for the vorticity Leray-$\alpha$ system

The variational problem (3.1) is linearized by considering a vorticity field $\mathbf{w}$ which does not depend on $\mathbf{u}$.

$$ (3.5) \qquad \begin{cases} \mathbf{u} \in V_{\eta}, \\ \forall \mathbf{v} \in V_{\eta}, \quad \int_{\Omega} (\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v} + \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \mathbf{u} : \nabla \mathbf{v} + \lambda \int_{\Omega} \mathbf{u} \cdot \mathbf{v} = \int_{\Omega} \mathbf{f} \cdot \mathbf{v}, \end{cases} $$
**Lemma 3.2 (Linearized problem).** Let $\eta \in [0; 1[, \mathbf{f} \in L^2(\Omega)$, $\lambda > 0$ and $\mathbf{w} \in L^s(\Omega)$ for

$$ (3.6) \qquad \left\{ \begin{array}{ll} 1 \le s \le +\infty & \text{if } \eta < 1/3, \\ 1 \le s < +\infty & \text{if } \eta = 1/3, \\ 1 \le s < \frac{6}{3\eta-1} & \text{if } \eta > 1/3. \end{array} \right. $$

The variational problem (3.5) admits a unique solution.

*Proof.* The Lax-Milgram Theorem is applied to the bilinear form $B(\mathbf{u}, \mathbf{v}) = A(\mathbf{u}, \mathbf{v}) + b_{\mathbf{w}}(\mathbf{u}, \mathbf{v})$, where $A$ and $b_{\mathbf{w}}$ are defined by

$$ (3.7) \qquad A(\mathbf{u}, \mathbf{v}) = \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \mathbf{u} : \nabla \mathbf{v} + \lambda \int_{\Omega} \mathbf{u} \cdot \mathbf{v}, $$

$$ (3.8) \qquad b_{\mathbf{w}}(\mathbf{u}, \mathbf{v}) = \int_{\Omega} (\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}. $$

First, we write

$$ (3.9) \qquad |A(\mathbf{u}, \mathbf{v})| \le (\lambda + \nu_M) \|\mathbf{u}\|_{V_{\eta}} \|\mathbf{v}\|_{V_{\eta}}. $$

Let $r \in [1; \frac{6}{1+\eta}]$, we have

$$ (3.10) \qquad |b_{\mathbf{w}}(\mathbf{u}, \mathbf{v})| \le c \|\mathbf{w}\|_{0,s,\Omega} \|\mathbf{u}\|_{0,r,\Omega} \|\mathbf{v}\|_{0,r,\Omega} \le C \|\mathbf{u}\|_{V_{\eta}} \|\mathbf{v}\|_{V_{\eta}}. $$

Inequalities (3.9) and (3.10) yield the continuity of the bilinear form $B$ on $V_{\eta}$. Let us show it is coercive, by considering any $\mathbf{v} \in V_{\eta}$.

$$ (3.11) \qquad B(\mathbf{v}, \mathbf{v}) = \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla \mathbf{v}|^2 + \lambda \int_{\Omega} |\mathbf{v}|^2 \ge \min(\nu_m, \lambda) \|\mathbf{v}\|_{V_{\eta}}^2. $$

Finally, the linear form $\ell: \mathbf{v} \mapsto \int_{\Omega} \mathbf{f} \cdot \mathbf{v}$ is continuous on $V_{\eta}$, since $\mathbf{f} \in L^2(\Omega)^3$. This concludes the proof. $\square$

Lemma 3.2 yields the existence of an operator $T$ associating to any function $\mathbf{u} \in V_{\eta}$ the solution of the linearized problem where $\mathbf{w} = \bar{\omega}(\mathbf{u})$, which means

$$ (3.12) \qquad \forall \mathbf{v} \in V_{\eta}, \quad \int_{\Omega} (\bar{\omega}(\mathbf{u}) \times T\mathbf{u}) \cdot \mathbf{v} + \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla T\mathbf{u} : \nabla \mathbf{v} + \lambda \int_{\Omega} T\mathbf{u} \cdot \mathbf{v} = \int_{\Omega} \mathbf{f} \cdot \mathbf{v}. $$

Replacing $\mathbf{v}$ by $T\mathbf{u}$ in (3.12) yields

$$ (3.13) \qquad \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla T\mathbf{u}|^2 + \lambda \int_{\Omega} |T\mathbf{u}|^2 = \int_{\Omega} \mathbf{f} \cdot T\mathbf{u}, $$

that yields the bound

$$ (3.14) \qquad \|T\mathbf{u}\|_{V_{\eta}} \le \frac{\|\mathbf{f}\|_{0,2,\Omega}}{\min(\nu_m, \lambda)} =: M. $$

Then we use the Leray-Schauder fixed point theorem to prove the following result.

**Theorem 3.1.** Let $\mathbf{f} \in L^2(\Omega)^3$ and $\lambda > 0$. The operator $T$ defined by (3.12) has a fixed point in the closed ball $\bar{B}_{V_{\eta}}(0, M)$ simply named $B$. This means that (3.1) admits a solution in $B$.
*Proof.* The fixed point theorem will be applied on $B$, which is a nonempty closed bounded convex subset of $V_\eta$. The continuity of $T$ and compactness of $T(B)$ need to be proved. To do so, we consider a sequence $(\mathbf{u}_n)_{n\in\mathbb{N}}$ of functions in $B$ that weakly converges to a certain $\mathbf{u}$ in $V_\eta$. The goal is showing that $(T\mathbf{u}_n)_{n\in\mathbb{N}}$ strongly converges to $Tu$ in $V_\eta$. We recall that $Tu_n$ is defined by

$$ (3.15) \quad \forall \mathbf{v} \in V_\eta, \quad \int_{\Omega} (\bar{\omega}(\mathbf{u}_n) \times T\mathbf{u}_n) \cdot \mathbf{v} + \int_{\Omega} \varrho^\eta \tilde{\nu} \nabla T\mathbf{u}_n : \nabla \mathbf{v} + \lambda \int_{\Omega} T\mathbf{u}_n \cdot \mathbf{v} = \int_{\Omega} \mathbf{f} \cdot \mathbf{v}. $$

Since $(T\mathbf{u}_n)_{n\in\mathbb{N}}$ is bounded in $V_\eta$, a subsequence still named the same way weakly converges to a certain $\varphi$ in $V_\eta$. The first step of the proof consists in showing that $\varphi = Tu$. The weak convergence and the compact Sobolev embedding (2.4) directly yield for any $\mathbf{v} \in V_\eta$

$$ (3.16) \quad \int_{\Omega} \varrho^\eta \tilde{\nu} \nabla T \mathbf{u}_n : \nabla \mathbf{v} \underset{n \to +\infty}{\longrightarrow} \int_{\Omega} \varrho^\eta \tilde{\nu} \nabla \varphi : \nabla \mathbf{v}, $$

$$ (3.17) \quad \lambda \int_{\Omega} T \mathbf{u}_n \cdot \mathbf{v} \underset{n \to +\infty}{\longrightarrow} \int_{\Omega} \varphi \cdot \mathbf{v}. $$

For the vorticity term, we write the decomposition

$$ (3.18) \quad \int_{\Omega} (\bar{\omega}(\mathbf{u}_n) \times T\mathbf{u}_n) \cdot \mathbf{v} - \int_{\Omega} (\bar{\omega}(\mathbf{u}) \times \varphi) \cdot \mathbf{v} = \int_{\Omega} ((\bar{\omega}(\mathbf{u}_n) - \bar{\omega}(\mathbf{u})) \times \varphi) \cdot \mathbf{v} \\ \qquad + \int_{\Omega} (\bar{\omega}(\mathbf{u}_n) \times (T\mathbf{u}_n - \varphi)) \cdot \mathbf{v}. $$

On one hand, the assumption and Theorem 2.2 imply $\mathbf{u}_n \underset{n \to +\infty}{\longrightarrow} \mathbf{u}$ in $W^{1,p}(\Omega)^3$ for

$$ (3.19) \quad 1 \le p < \frac{2}{1+\eta}, \quad \text{with } 0 \le \eta < 1. $$

This yields $\mathrm{curl}(\mathbf{u}_n) \underset{n \to +\infty}{\longrightarrow} \mathrm{curl}(\mathbf{u})$ in $L^p(\Omega)^3$, and the extensions by 0 satisfy

$$ (3.20) \quad \omega(\mathbf{u}_n) \underset{n \to +\infty}{\longrightarrow} \omega(\mathbf{u}) \quad \text{in } L^p(\mathbb{R}^3)^3, $$

In particular, $(\omega(\mathbf{u}_n))_{n\in\mathbb{N}}$ is bounded in $L^p(\Omega)$, with the same values of $p$. Therefore, Lemma 2.2 yields

$$ (3.21) \quad \|\bar{\omega}(\mathbf{u}_n)\|_{0,s,\Omega} \le \|K_\alpha\|_{0,q,\Omega} \|\omega(\mathbf{u}_n)\|_{0,p,\Omega}, $$

with $s$ given by (3.6) and $1 \le q < 3$. Thus, $(\bar{\omega}(\mathbf{u}_n))_{n\in\mathbb{N}}$ is bounded in $L^s(\Omega)$. We obtain

$$ (3.22) \quad \left| \int_{\Omega} (\bar{\omega}(\mathbf{u}_n) \times (T\mathbf{u}_n - \varphi)) \cdot \mathbf{v} \right| \le c_1 \|\bar{\omega}(\mathbf{u}_n)\|_{0,s,\Omega} \|T\mathbf{u}_n - \varphi\|_{0,r,\Omega} \|\mathbf{v}\|_{0,r,\Omega} \underset{n \to +\infty}{\longrightarrow} 0, $$

where $r$ is given by (2.4).

On the other hand, convergence (3.20) and Lemma 2.3 yield

$$ (3.23) \quad \bar{\omega}(\mathbf{u}_n) \underset{n \to +\infty}{\longrightarrow} \bar{\omega}(\mathbf{u}) \quad \text{in } L^p(\mathbb{R}^3)^3. $$

And the restrictions weakly converge in $L^p(\Omega)^3$. This yields

$$ (3.24) \quad \int_{\Omega} ((\bar{\omega}(\mathbf{u}_n) - \bar{\omega}(\mathbf{u})) \times \varphi) \cdot \mathbf{v} \underset{n \to +\infty}{\longrightarrow} 0. $$
To sum up, convergences (3.16), (3.17), (3.22) and (3.24) yield

$$ (3.25) \quad \forall \mathbf{v} \in V_{\eta}, \quad \int_{\Omega} (\bar{\omega}(\mathbf{u}) \times \varphi) \cdot \mathbf{v} + \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \varphi : \nabla \mathbf{v} + \lambda \int_{\Omega} \varphi \cdot \mathbf{v} = \int_{\Omega} \mathbf{f} \cdot \mathbf{v}. $$

That means $\varphi = T\mathbf{u}$.

Now the strong convergence has to be proved. Taking $\mathbf{v} = T\mathbf{u}_n$ in (3.15) and $\mathbf{v} = T\mathbf{u}$ in (3.12) yields

$$ (3.26) \quad \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla T \mathbf{u}_n|^2 + \lambda \int_{\Omega} |T \mathbf{u}_n|^2 = \int_{\Omega} \mathbf{f} \cdot T \mathbf{u}_n, $$

$$ (3.27) \quad \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla T \mathbf{u}|^2 + \lambda \int_{\Omega} |T \mathbf{u}|^2 = \int_{\Omega} \mathbf{f} \cdot T \mathbf{u}. $$

What was done before directly yields

$$ (3.28) \quad \lambda \int_{\Omega} |T \mathbf{u}_n|^2 \xrightarrow{n \to +\infty} \lambda \int_{\Omega} |T \mathbf{u}|^2, $$

$$ (3.29) \quad \int_{\Omega} \mathbf{f} \cdot T \mathbf{u}_n \xrightarrow{n \to +\infty} \int_{\Omega} \mathbf{f} \cdot T \mathbf{u}, $$

so that we deduce

$$ (3.30) \quad \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla T \mathbf{u}_n|^2 \xrightarrow{n \to +\infty} \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla T \mathbf{u}|^2. $$

The weak convergence and (3.30) yield the strong convergence $T\mathbf{u}_n \xrightarrow{n \to +\infty} T\mathbf{u}$ in $V_{\eta}$ for the norm

$$ (3.31) \quad N_{\nu}(\mathbf{u}) := (\|\mathbf{u}\|_{0,2,\Omega}^2 + \|\varrho^{\eta/2} \tilde{\nu}^{1/2} \nabla \mathbf{u}\|_{0,2,\Omega}^2)^{1/2}. $$

Given that this norm and $\|\cdot\|_{V_{\eta}}$ are equivalent, we obtain the strong convergence $T\mathbf{u}_n \xrightarrow{n \to +\infty} T\mathbf{u}$ in $V_{\eta}$, which concludes the proof. $\square$

### 3.3 Convergence to the solutions of the original problem

In this subsection, we consider a sequence of parameters $(\alpha_n)_{n \in \mathbb{N}}$ converging to 0, and a sequence $(\mathbf{u}_{\alpha_n})_{n \in \mathbb{N}}$ of solutions of (1.1)-(1.5) with the parameter $\alpha_n$ in (1.4). We prove that this sequence converges to a solution $\mathbf{u}$ of (1.9)-(1.12).

**Theorem 3.2.** Let $\eta \in [0; 1[$, $\lambda > 0$ and $\mathbf{f} \in L^2(\Omega)^3$. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of positive numbers such that $\alpha_n \xrightarrow{n \to +\infty} 0$, and $(\mathbf{u}_{\alpha_n})_{n \in \mathbb{N}}$ a sequence of functions satisfying for any $\mathbf{v} \in V_{\eta}$

$$ (3.32) \quad \int_{\Omega} (\bar{\omega}(\mathbf{u}_{\alpha_n}) \times \mathbf{u}_{\alpha_n}) \cdot \mathbf{v} + \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \mathbf{u}_{\alpha_n} : \nabla \mathbf{v} + \lambda \int_{\Omega} \mathbf{u}_{\alpha_n} \cdot \mathbf{v} = \int_{\Omega} \mathbf{f} \cdot \mathbf{v}, $$

where $\bar{\omega}(\mathbf{u}_{\alpha_n})$ is obtained with the method detailed in section 3, with equation

$$ (3.33) \quad -\alpha_n^2 \Delta \bar{\omega}(\mathbf{u}_{\alpha_n}) + \bar{\omega}(\mathbf{u}_{\alpha_n}) = \omega(\mathbf{u}_{\alpha_n}). $$
• If $0 \le \eta < 1/5$, a subsequence of $(\mathbf{u}_{\alpha_n})_{n \in \mathbb{N}}$ strongly converges in $V_\eta$ to a function $\mathbf{u}$ satisfying for any $\mathbf{v} \in V_\eta$

$$ (3.34) \qquad \int_{\Omega} (\omega(\mathbf{u}) \times \mathbf{u}) \cdot \mathbf{v} + \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \mathbf{u} : \nabla \mathbf{v} + \lambda \int_{\Omega} \mathbf{u} \cdot \mathbf{v} = \int_{\Omega} \mathbf{f} \cdot \mathbf{v}. $$

• If $1/5 \le \eta < 1$, a subsequence of $(\mathbf{u}_{\alpha_n})_{n \in \mathbb{N}}$ converges at the distribution sense to a function $\mathbf{u} \in V_\eta$ satisfying (3.34) for any $\mathbf{v} \in \mathcal{V}$.

*Proof.* Let $\mathbf{u}_{\alpha_n} \in V_\eta$ satisfying (3.32). Taking $\mathbf{v} = \mathbf{u}_{\alpha_n}$ directly yields $\|\mathbf{u}_{\alpha_n}\|_{V_\eta} \le M$, where $M > 0$ is defined in (3.14).

Therefore $(\mathbf{u}_{\alpha_n})_{n \in \mathbb{N}}$ is bounded, so there is a subsequence still named the same way which weakly converges in $V_\eta$ to a certain function $\mathbf{u} \in V_\eta$. As in the proof of Theorem 3.1, we directly obtain for any $\mathbf{v} \in V_\eta$

$$ (3.35) \qquad \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \mathbf{u}_{\alpha_n} : \nabla \mathbf{v} \underset{n \to +\infty}{\longrightarrow} \int_{\Omega} \varrho^{\eta} \tilde{\nu} \nabla \mathbf{u} : \nabla \mathbf{v}, $$

$$ (3.36) \qquad \lambda \int_{\Omega} \mathbf{u}_{\alpha_n} \cdot \mathbf{v} \underset{n \to +\infty}{\longrightarrow} \lambda \int_{\Omega} \mathbf{u} \cdot \mathbf{v}. $$

Thus, we only need to focus on the transport term, and we write a decomposition

$$ (3.37) \qquad \int_{\Omega} (\bar{\omega}(\mathbf{u}_{\alpha_n}) \times \mathbf{u}_{\alpha_n}) \cdot \mathbf{v} - \int_{\Omega} (\omega(\mathbf{u}) \times \mathbf{u}) \cdot \mathbf{v} = \int_{\Omega} ((\bar{\omega}(\mathbf{u}_{\alpha_n}) - \omega(\mathbf{u})) \times \mathbf{u}) \cdot \mathbf{v} \\ + \int_{\Omega} (\bar{\omega}(\mathbf{u}_{\alpha_n}) \times (\mathbf{u}_{\alpha_n} - \mathbf{u})) \cdot \mathbf{v}. $$

**Case 1** ($0 \le \eta < 1/5$): we consider any test function $\mathbf{v} \in V_\eta$.

On one hand, $\mathbf{u}_{\alpha_n} \underset{n \to +\infty}{\stackrel{V_\eta}{\longrightarrow}} \mathbf{u}$ which implies $\omega(\mathbf{u}_{\alpha_n}) \underset{n \to +\infty}{\stackrel{L^p}{\longrightarrow}} \omega(\mathbf{u})$. Thus, Theorem 2.4 yields $\bar{\omega}(\mathbf{u}_{\alpha_n}) \underset{n \to +\infty}{\stackrel{L^p}{\longrightarrow}} \omega(\mathbf{u})$. We deduce

$$ (3.38) \qquad \int_{\Omega} ((\bar{\omega}(\mathbf{u}_{\alpha_n}) - \omega(\mathbf{u})) \times \mathbf{u}) \cdot \mathbf{v} \underset{n \to +\infty}{\longrightarrow} 0. $$

On the other hand, taking $s$ given by (3.6), we have

$$ (3.39) \qquad \left| \int_{\Omega} (\bar{\omega}(\mathbf{u}_{\alpha_n}) \times (\mathbf{u}_{\alpha_n} - \mathbf{u})) \cdot \mathbf{v} \right| \le \|\bar{\omega}(\mathbf{u}_{\alpha_n})\|_{0,s,\Omega} \|\mathbf{u}_{\alpha_n} - \mathbf{u}\|_{0,r,\Omega} \|\mathbf{v}\|_{0,r,\Omega} \underset{n \to +\infty}{\longrightarrow} 0, $$

with the same arguments as in Theorem 3.1.

Consequently, (3.37) yields

$$ (3.40) \qquad \int_{\Omega} (\bar{\omega}(\mathbf{u}_{\alpha_n}) \times \mathbf{u}_{\alpha_n}) \cdot \mathbf{v} \underset{n \to +\infty}{\longrightarrow} \int_{\Omega} (\omega(\mathbf{u}) \times \mathbf{u}) \cdot \mathbf{v}. $$

Passing to the limit in (3.32) shows that $\mathbf{u}$ satisfies (3.34).

Now the strong convergence needs to be proved. Taking $\mathbf{v} = \mathbf{u}_{\alpha_n}$ in (3.32) and $\mathbf{v} = \mathbf{u}$ in (3.34) yields

$$ (3.41) \qquad \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla \mathbf{u}_{\alpha_n}|^2 + \lambda \int_{\Omega} |\mathbf{u}_{\alpha_n}|^2 = \int_{\Omega} \mathbf{f} \cdot \mathbf{u}_{\alpha_n}, $$

$$ (3.42) \qquad \int_{\Omega} \varrho^{\eta} \tilde{\nu} |\nabla \mathbf{u}|^2 + \lambda \int_{\Omega} |\mathbf{u}|^2 = \int_{\Omega} \mathbf{f} \cdot \mathbf{u}. $$
We conclude with the same method as used in the proof of Theorem 3.1.

**Case 2** ($1/5 \le \eta < 1$) : we consider a test function $v \in \mathcal{V}$.

The proof is the same as before, but **u** and $\mathbf{u}_{\alpha_n}$ cannot be taken as test functions, so that the convergence is not strong. $\square$

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