Global existence of solutions for the second grade fluid equations in a thin three-dimensional domain

Abstract

We consider the second grade fluid equations on a thin three-dimensional domain with periodic boundary conditions. We prove global existence and uniqueness of the solution for large initial data. We use an appropriate decomposition of solution u into a v part, which is solution of a $2 D$ second grade fluid equations and the remaining w part which has an initial data converging to 0 as the thickness of the thin domain goes to 0.

Keywords

Second grade fluid equations thin domains perturbation global existence

1. Introduction

A grade-two fluid is a fluid of differential type, a theoretical model introduced by Rivlin and Ericksen [19] for describing non-Newtonian behaviour. Its equations generalize the Navier-Stokes equations and it is believed to describe the motion of a water solution of polymers. Interestingly, some years ago, these equations were interpreted by Camassa, Holm, Marsden, Ratiu and Shkoller as a model of turbulence.

The existence and uniqueness of a solution of the second grade fluid equations for a bounded domain with Dirichlet boundary condition, is treated by Cioranescu and Ouazar [7]. In their work, they decompose the system into a Stokes problem and a transport equation satisfied by $ω - α Δ ω$ , where ω is the vorticity, and they introduced a special Galerkin basis in order to obtain global existence and uniqueness of a $H^{3}$ solution for the two-dimensional case and local existence for small data in the three-dimensional case. This last result was improved by Cioranescu and Girault in [6], where the authors showed that the solution in the three-dimensional case is global under some appropriate assumptions on the initial data and the forcing term. They also gave some propagation of the $H^{4}$ -regularity. We also mention that these results were also obtained by Busuioc and Ratiu [4] in the case of Navier boundary conditions.

The main difficulty in proving the global existence in the three-dimensional case for arbitrary initial data is due to the nonlinear term $rot (u - α Δ u) \times u$ . In fact, when estimating the solutions of the second grade fluid system in three dimensions, one cannot obtain the appropriate estimates in $H^{3}$ which could ensure the global existence of solutions, while in the two-dimensional case, due to some cancellations, the bad term $rot (u - α Δ u) \times u$ disappears.

The global existence and regularity of solutions of the second grade fluid in three-dimensional case for large initial data is still an open problem.

Thus a natural question arises, if we consider a thin three-dimensional domain, can we use the global existence on the two-dimensional domain to ameliorate the assumption on the three-dimensional space.

This problem has been previously studied for Navier-Stokes equations by Raugel and Sell [18], Iftimie and Raugel [12] and Iftimie Raugel and Sell [13], they proved the global existence of strong solutions for large initial data in the case of thin three dimensional domains.

The study of the second grade fluid in a thin three-dimensional domain is the subject of this paper. We will prove that if we exploit the fact that $Q_{ε}$ is a thin domain, the global solutions exist for a larger size of initial data and forcing term. We note that the second grade fluid equations differ from the Navier-Stokes equations. Indeed, the Navier-Stokes equations contain a regularizing term, this is not the case for the second grade fluid equations where the dissipation is very weak.

The study of the dynamics of an evolutionary equation on a thin domain is a particular case of the study of the dynamics under small perturbations. In [8–11], and [17], Hale and Raugel have compared the flow, the equilibrium points and the attractors of an evolutionary equation in a thin domain with those of the reduced equation on the limit domain when the thickness ε of the thin domain goes to 0.

In [1] and [2], for the case of a system of two damped wave equations in thin two-dimensional domains, we proved the persistence of existence of a periodic solution and we compared the dynamics of a system of damped wave equations on a thin domain with the dynamics of the limit system when the thickness of the thin domain goes to 0.

Let us now introduce the second grade fluid equations in a thin domain. Let $Ω \subset R^{2}$ be a bounded domain; let $Q_{ε} \subset R^{3}$ , with $ε > 0$ , be the bounded thin domain $\begin{matrix} Q_{ε} = {(X, Y) \in R^{3}; X \in Ω, 0 < Y < ε} . \end{matrix}$ We consider the second grade fluid equations $\begin{array}{rcl} (1.1) & \{\begin{matrix} \partial_{t} (U - α Δ U) - ν Δ U + U \cdot \nabla (U - α Δ U) + \sum_{j} (U_{j} - α Δ U_{j}) \nabla U_{j} \\ = - \nabla P + F, in Q_{ε} \\ U (X, Y, 0) = U_{0} (X, Y), in Q_{ε}, \end{matrix} \end{array}$ where U denotes the velocity of the fluid and P is the pressure. We suppose that the flow is incompressible: $\begin{matrix} div U = 0 . \end{matrix}$

In this paper we are interested in the case where $Ω = (0, l_{1}) \times (0, l_{2})$ , $l_{1}$ and $l_{2}$ are positive. We will assume that $ε ⩽ l_{2} ⩽ l_{1}$ and that the solution U of (1.1) satisfy the periodic boundary conditions $\begin{array}{rcl} \{\begin{matrix} U (X + l_{i} e_{i}, Y, t) = U (X, Y, t), i = 1, 2, \\ U (X, Y + ε, t) = U (X, Y, t), \end{matrix} \end{array}$ where ${e_{1}, e_{2}}$ is the natural basis in $R^{2}$ . In addition, we will assume that F and the initial data $U_{0}$ satisfy $\begin{matrix} \int_{Q_{ε}} F d X d Y = \int_{Q_{ε}} U_{0} d X d Y = 0 . \end{matrix}$ We can show that any solution U of (1.1), with $U (0) = U_{0}$ will also satisfy $\int_{Q_{ε}} U d X d Y = 0$ .

In the study of (1.1) on the thin domain $Q_{ε}$ , it is convenient to perform a change of variables, $X = x$ and $Y = ε y$ , which transforms $Q_{ε}$ into the fixed domain $Q = Ω \times (0, 1)$ , and the equation (1.1) into the equation $\begin{array}{rcl} (1.2) & \{\begin{matrix} \partial_{t} (u - α Δ_{ε} u) - ν Δ_{ε} u + u \cdot \nabla_{ε} (u - α Δ_{ε} u) + \sum_{j} (u_{j} - α Δ_{ε} u_{j}) \nabla_{ε} u_{j} \\ = - \nabla_{ε} p + f, in Q, \\ {div}_{ε} u = 0, \\ u (x, y, 0) = u_{0} (x, y), \end{matrix} \end{array}$ where $u (x, y) = U (X, Y)$ , $\nabla_{ε} = (\partial_{1}, \partial_{2}, \partial_{3})$ , $Δ_{ε} = \partial_{1}^{2} + \partial_{2}^{2} + \partial_{3}^{2}$ , $\partial_{i} = \partial_{x_{i}}$ , $i = 1, 2$ and $\partial_{3} = ε^{- 1} \partial_{y}$ . By this change of variables the parameters describing the domain are contained in the new differential operator.

Let the Sobolev spaces $H_{ε}^{m} {(Q)}^{3}$ , $m \in N$ , be the closure in $H^{m} {(Q)}^{3}$ of the space $\begin{matrix} {u \in C^{\infty} {(Q)}^{3} | u is periodic, {div}_{ε} u = 0, \int_{Q} u = 0} . \end{matrix}$ By classical interpolation theory, we also define the spaces $H_{ε}^{s} {(Q)}^{3}$ , for $s \in R_{+}$ . We denote by $H_{per}^{m} {(Q)}^{3} = {u \in H^{m} {(Q)}^{3} ∣ u is periodic, \int_{Q} u = 0}$ .

We introduce the space $V_{ε}^{1} = H_{ε}^{1}$ equipped with the scalar product $\begin{matrix} {(u, v)}_{V_{ε}^{1}} = \int_{Q} u v + α \int_{Q} \nabla_{ε} u \nabla_{ε} v; \forall u, v \in V_{ε}^{1} . \end{matrix}$ We define the space $V_{ε}^{3} = {u \in H_{ε}^{1} ∣ {curl}_{ε} (u - α Δ_{ε} u) \in L^{2} {(Q)}^{3}}$ equipped with the scalar product $\begin{matrix} {(u, v)}_{V_{ε}^{3}} = \int_{Q} u v + \int_{Q} {curl}_{ε} (u - α Δ_{ε} u) {curl}_{ε} (v - α Δ_{ε} v) . \end{matrix}$ Also we define the space $V_{ε}^{4} = {u \in H_{ε}^{2} ∣ \nabla_{ε} {curl}_{ε} (u - α Δ_{ε} u) \in L^{2} {(Q)}^{3}}$ equipped with the scalar product $\begin{matrix} {(u, v)}_{V_{ε}^{4}} = \int_{Q} u v + \int_{Q} \nabla_{ε} {curl}_{ε} (u - α Δ_{ε} u) \nabla_{ε} {curl}_{ε} (v - α Δ_{ε} v) . \end{matrix}$ Let M be the projection of $L^{2} {(Q)}^{3}$ onto $L^{2} {(Ω)}^{3}$ , given by $\begin{array}{rcl} (M u) (x) = \int_{0}^{1} u (x, y) d y, \forall u \in L^{2} (Q) . \end{array}$ The aim of this paper is to prove the global existence of a solution for equation (1.2) for large initial data. For this purpose we decompose the solution in two vectors $u = v + w$ , where $v (0) = M u_{0}$ independent of the vertical variable, and $w (0) = (I - M) u_{0}$ with vertical vanishing mean, which allows to have v a solution of $2 D$ second grade fluid system with three components and w a solution of the remaining equation, where the lower Sobolev norms of the initial condition $w_{0}$ are small (of order $ε^{k}$ , $k > 0$ ) compared to the higher ones.

The global existence of strong solutions of $2 D$ second grade fluid system with three components is treated by B. Jaffal ([14,15]). In this work, if f belongs to $L_{t}^{2} (H_{per}^{1})$ , the author proved two different results of global existence of solutions. First, she proved the global existence under some restriction of the size of the third components of the initial data for an arbitrary coefficient α. Second, she considered a large initial data and she proved global existence if α small enough.

In the first part of this paper, in Section 3, we extend the results of B. Jaffal in the case of forcing term f belongs to $L_{t}^{\infty} (H_{per}^{1})$ . Interestingly enough, exploiting the fact that $Q_{ε}$ is a thin domain, we prove in the second part of this paper, the global existence of a solution w for large initial data $w_{0}$ , when the thickness of the thin domain ε goes to 0.

Before stating our main theorems, we introduce some notations. Let $u_{h} = (u_{1}, u_{2})$ be the horizontal component of the vector field $u = (u_{1}, u_{2}, u_{3})$ and $u_{3}$ its third component. We set $\begin{array}{rcl} K_{0} (M u_{0}, M f) & = & α ‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{h} Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + C_{0} α (1 + α) ‖ \nabla_{h} M f_{3} ‖_{L^{\infty} (L^{2})}^{2}, \\ K_{1} (M u_{0}, M f) & = & {‖ rot (M u_{0, h} - α Δ_{h} M u_{0, h}) ‖}_{L^{2}}^{2} + ‖ \nabla_{h} M u_{0, 3} ‖_{L^{2}}^{2} \\ + C_{0} (1 + α) (‖ M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2} + (1 + α) ‖ rot M f_{h} ‖_{L_{t}^{\infty} (L^{2})}^{2}), \\ K_{2} (M u_{0}, M f) & = & ‖ Δ_{h} rot M u_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot M u_{0} ‖_{L^{2}}^{2} + C_{0} (1 + α) ‖ Δ_{h} M f ‖_{L_{t}^{\infty} (L^{2})}^{2}, \end{array}$ and $\begin{matrix} K_{3} (M u_{0}, M f) = C_{1} (K_{0} + K_{1}) e^{C_{1} (1 + α) K_{1}}, \end{matrix}$ where $C_{0}$ and $C_{1}$ are positive constants which do not depend on the data, or on α.

In order to simplify the notation, we just write $K_{0}$ , $K_{1}$ , $K_{2}$ and $K_{3}$ .

For the case of arbitrary coefficient α, exploiting the fact that $Q_{ε}$ is a thin domain, we can replace the small data condition on $(u_{0}, f)$ by large data conditions.

Theorem 1.1.
For any positive number $δ_{0}$ , $0 ⩽ δ_{0} < \frac{1}{2}$ , there exist positive constants $ε_{0}$ , $r_{0}$ and $R_{0}$ such that, for all $0 < ε ⩽ ε_{0}$ , if the initial data $(M u_{0}, (I - M) u_{0}) \in V_{ε}^{4} \times V_{ε}^{3}$ and the forcing term $(M f, (I - M) f) \in L_{t}^{\infty} (H_{per}^{2}) \times L_{t}^{\infty} (H_{per}^{1})$ satisfy the assumptions $\begin{array}{l} K_{0} e^{K_{3}} ⩽ r_{0}^{2}, \\ (ε^{2 - 2 δ_{0}} + ε^{2} ln (1 + c ε^{- 2})) K_{2}^{2} e^{K_{3}} ⩽ R_{0}^{2}, \\ (ε^{2 - 2 δ_{0}} + ε^{2} ln (1 + c ε^{- 2})) ({‖ \nabla_{ε} (w_{0} - α Δ_{ε} w_{0}) ‖}_{L^{2}}^{2} + {‖ (I - M) f ‖}_{L^{\infty} (H^{1})}^{2}) e^{K_{3}} ⩽ R_{0}^{2}, \end{array}$ then system ( 1.2 ) admits a unique global solution u in the space $L^{\infty} (R_{+}, V_{ε}^{3}) \cap L^{2} (R_{+}, V_{ε}^{3})$ .

We now consider initial data and a forcing term of arbitrary size. In this case we remove every smallness assumption on the data $u_{0}$ and f, provided α and ε small enough.
Theorem 1.2.
For any positive number $δ_{0}$ , $0 ⩽ δ_{0} < \frac{1}{2}$ , there exist positive constants $ε_{0}$ , $α_{0}$ , $R_{0}$ such that, for all $0 < ε ⩽ ε_{0}$ , $0 < α ⩽ α_{0}$ if the initial data $(M u_{0}, (I - M) u_{0}) \in V_{ε}^{4} \times V_{ε}^{3}$ and the forcing term $(M f, (I - M) f) \in L_{t}^{\infty} (H_{per}^{2}) \times L_{t}^{\infty} (H_{per}^{1})$ satisfy $\begin{array}{l} α (‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} \nabla_{h} M u_{0, 3} ‖_{L^{2}}^{2} + ‖ \nabla_{h} M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2}) e^{K_{3}} ⩽ R_{0}^{2}, \\ α (ε^{2 - 2 δ_{0}} + ε^{2} ln (1 + c ε^{- 2})) K_{2}^{2} e^{K_{3}} ⩽ R_{0}^{2}, \\ (ε^{4} + α (ε^{2 - 2 δ_{0}} + ε^{2} ln (1 + c ε^{- 2}))) (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w_{0} ‖_{L^{2}}^{2} + {‖ (I - M) f ‖}_{L^{\infty} (H^{1})}^{2}) e^{K_{3}} ⩽ R_{0}^{2}, \end{array}$ then system ( 1.2 ) admits a unique global solution u in the space $L^{\infty} (R_{+}, V_{ε}^{3}) \cap L^{2} (R_{+}, V_{ε}^{3})$ .

The paper is organized as follows. In Section 2, we introduce the notations, definitions and some preliminary results. In Section 3, we study the solution of the reduced problem. In Section 4, we prove the global existence of the solution with initial data $(I - M) u_{0}$ . Thus, we give the proof of Theorems 1.1 and 1.2.
2. Notation and preliminary results

Let $λ_{1}$ be the smallest eigenvalue of the Stokes operator. Since $0 ⩽ ε ⩽ l_{2} ⩽ l_{1}$ , one has $λ_{1} = 4 π^{2} l_{1}^{- 2}$ and, in particular, for any $u \in V_{ε}^{1}$ $\begin{array}{rcl} (2.1) & ‖ u ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} u ‖_{L^{2}}^{2} ⩽ (λ_{1}^{- 1} + α) ‖ \nabla_{ε} u ‖_{L^{2}}^{2} . \end{array}$ As already remarked in [6], if u belongs to $V_{ε}^{3}$ , then u is in $H_{ε}^{3}$ and there exists a positive constant $c_{0}$ independent of α and ε such that, for any $u \in V_{ε}^{3}$ $\begin{array}{rcl} ‖ \nabla_{ε} u ‖_{L^{2}}^{2} + α ‖ Δ_{ε} u ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{ε} Δ_{ε} u ‖_{L^{2}}^{2} ⩽ c_{0} {‖ {curl}_{ε} (u - α Δ_{ε} u) ‖}_{L^{2}}^{2} . \end{array}$ Let $B_{ε}$ be the bilinear form on $H_{ε}^{0}$ defined, for $u, v \in H_{ε}^{0}$ , by $\begin{matrix} B_{ε} (u, v) = u \cdot \nabla_{ε} (v - α Δ_{ε} v) + \sum_{j = 1}^{3} (u_{j} - α Δ_{ε} u_{j}) \nabla_{ε} v_{j} . \end{matrix}$ Thus, we can write the equation (1.2) as $\begin{array}{rcl} \{\begin{matrix} \partial_{t} (u - α Δ_{ε} u) - ν Δ_{ε} u + B_{ε} (u, u) = - \nabla_{ε} p + f, in Q, \\ {div}_{ε} u = 0, \\ u (x, y, 0) = u_{0} (x, y) . \end{matrix} \end{array}$ The following properties have been proved by Hale and Raugel ([8,9,17]):

Lemma 2.1.
If u belongs to $H^{j} {(Q)}^{3}, j ⩾ 0$ , then $M u$ belongs to $H^{j} {(Ω)}^{3}$ and $\begin{array}{rcl} (2.2) & ‖ M u ‖_{H^{j} {(Ω)}^{3}} ⩽ ‖ u ‖_{H^{j} {(Q)}^{3}} . \end{array}$ Moreover, there exists a positive constant c such that, for $0 ⩽ ε ⩽ ε_{0}$ ,
for any $u \in H^{1} {(Q)}^{3}$ , $\begin{array}{rcl} ‖ u - M u ‖_{L^{2} {(Q)}^{3}} ⩽ c ε {‖ \frac{1}{ε} \frac{\partial u}{\partial y} ‖}_{L^{2} {(Q)}^{3}} ⩽ c ε ‖ u ‖_{H^{1} {(Q)}^{3}} (Poincar é inequality), \end{array}$

for any $u \in H^{1} {(Q)}^{3}$ , we have the following Sobolev inequality, for $2 ⩽ p ⩽ 6$ , $\begin{matrix} ‖ u - M u ‖_{L^{p} {(Q)}^{3}} ⩽ c ε^{\frac{2}{p}} ‖ u ‖_{H^{1} {(Q)}^{3}} . \end{matrix}$

We now apply the projections M and $(I - M)$ to the initial condition $u_{0}$ . We decompose the solution $u = v + w$ , where $v (0) = M u_{0}$ and $w (0) = (I - M) u_{0}$ .

Then v satisfies the following equation $\begin{matrix} (2.3) & \{\begin{matrix} \partial_{t} (v - α Δ_{h} v) - ν Δ_{h} v + B_{ε} (v, v) = - \nabla_{h} M p + M f, in Ω \\ {div}_{h} v = 0, \\ v (x, 0) = M u_{0} (x), \end{matrix} \end{matrix}$ where $\nabla_{h} = (\partial_{1}, \partial_{2})$ , ${div}_{h} = \partial_{1} + \partial_{2}$ and $Δ_{h} = \partial_{1}^{2} + \partial_{2}^{2}$ .

The w-part of u is a solution of the equation $\begin{array}{rcl} (2.4) & \{\begin{matrix} \partial_{t} (w - α Δ_{ε} w) - ν Δ_{ε} w + B_{ε} (w, w) + B_{ε} (v, w) + B_{ε} (w, v) \\ = - \nabla_{ε} (I - M) p + (I - M) f, in Q \\ {div}_{ε} w = 0, \\ w (x, 0) = (I - M) u_{0} . \end{matrix} \end{array}$
3. Reduced equation

System (2.3) essentially differs from the second grade fluid in three-dimensional case since the unknown v of (2.3) is a three-dimensional vector but only depending on the horizontal variables.

If the forcing term f belongs to $L_{t}^{2} (H_{per}^{1})$ , B. Jaffal [15] proved two different results of the global existence of strong solution of (2.3). The first result, she considered an arbitrary coefficient α and she supposed that the third components of the vertical average of the initial data and the forcing term are small compared to the horizontal components. In the second result, she considered a forcing term and large initial data of arbitrary size but she restricted the size of α.

The purpose of this section is to extend the methods of [15], in order to show the global existence of strong solution of (2.3) in the case of the forcing term f belonging to $L_{t}^{\infty} (H_{per}^{1})$ . We distinguish the case of arbitrary coefficient α and the case of small α.

Case of arbitrary α

We obtain the following result

Proposition 3.1.
There exists a positive constant $r_{0}$ , such that for $M u_{0} \in V_{ε}^{3}$ and the forcing term $f \in L_{t}^{\infty} (H_{per}^{1})$ satisfying the assumption $\begin{matrix} K_{0} (M u_{0}, M f) e^{K_{3} (M u_{0}, M f)} ⩽ r_{0}^{2}, \end{matrix}$ system ( 2.3 ) admits a unique solution v in the space $L^{\infty} (R_{+}, V_{ε}^{3}) \cap L^{2} (R_{+}, V_{ε}^{3})$ .

Furthermore, there exists $β_{1} = \frac{ν}{2 (λ_{1}^{- 1} + 2 α)}$ such that, solution v of ( 2.3 ) satisfies the priori estimates for all $t ⩾ 0$ $\begin{array}{l} {‖ rot (v - α Δ_{h} v) ‖}_{L^{2}}^{2} + \frac{ν}{4} \int_{0}^{t} e^{β_{1} (s - t)} (‖ Δ_{h} v ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v ‖_{L^{2}}^{2}) d s \\ ⩽ ({‖ rot (M u_{0} - α Δ_{h} M u_{0}) ‖}_{L^{2}}^{2} \\ + C_{0} (1 + α) (‖ M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2} + α ‖ \nabla_{h} M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2} + (1 + α) ‖ rot M f_{h} ‖_{L_{t}^{\infty} (L^{2})}^{2})) \\ \times exp (C (‖ M u_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M u_{0} ‖_{L^{2}}^{2} + C_{0} (1 + α) ‖ M f ‖_{L^{\infty} (L^{2})}^{2})) \\ (3.1) & ⩽ C_{1} (K_{0} + K_{1}) e^{C_{1} (1 + α) K_{1}} = K_{3} (M u_{0}, M f) . \end{array}$ If moreover, $M u_{0} \in V_{ε}^{4}$ and $M f \in L^{\infty} (R^{+}, H_{per}^{2})$ , solution v of system ( 2.3 ) is in $L^{\infty} (R_{+}, V_{ε}^{4}) \cap L^{2} (R_{+}, V_{ε}^{4})$ and for all $0 < β ⩽ β_{0} = \frac{ν}{2 (λ_{1}^{- 1} + α)}$ , $\begin{array}{l} ‖ Δ_{h} rot v ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β (s - t)} ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} \\ ⩽ (‖ Δ_{h} rot M u_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot M u_{0} ‖_{L^{2}}^{2} + C (1 + α) ‖ Δ_{h} M f ‖_{L_{t}^{\infty} (L^{2})}^{2}) \\ (3.2) & \times exp (C_{1} (K_{0} + K_{1}) e^{C_{1} (1 + α) K_{1}}) = K_{2} (M u_{0}, M f) e^{K_{3} (M u_{0}, M f)}, \forall t ⩾ 0 . \end{array}$
Proof.
Using a Galerkin method with the same basis as the one used to prove global existence of the second grade fluid in the three-dimensional case (see [6]), we can prove the existence of solution for (2.3).

The $V_{ε}^{1}$ -estimate is the same as in the two-dimensional second grade fluid with two components (see for example [16]). Thus, we have that for any $0 < β ⩽ β_{0} = \frac{ν}{2 (λ_{1}^{- 1} + α)}$ $\begin{array}{rcl} {‖ v (t) ‖}_{L^{2}}^{2} + α ‖ \nabla_{h} v ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β (s - t)} ‖ \nabla_{h} v ‖_{L^{2}}^{2} d s & ⩽ & ‖ M u_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M u_{0} ‖_{L^{2}}^{2} \\ + \frac{1}{λ_{1} ν β} ‖ M f ‖_{L^{\infty} (L^{2})}^{2} . \end{array}$ In order to obtain the $V_{ε}^{3}$ estimates, we first prove estimate (3.4), stated below, which is contained in [15]. For the reader’s convenience, we quickly recall this proof: we decompose system (2.3) into two systems (2.3) h and (2.3) 3 satisfied by $v_{h} = (v_{1}, v_{2})$ and $v_{3}$ respectively. $\begin{array}{l} (2.3)_{h} & \{\begin{matrix} \partial_{t} (v_{h} - α Δ_{h} v_{h}) - ν Δ_{h} v_{h} + {(B_{ε} (v, v))}_{h} = - \nabla_{h} M p_{h} + M f_{h}, in Ω \\ v_{h} (x, 0) = M u_{0, h} (x) . \end{matrix} \\ (2.3)_{3} & \{\begin{matrix} \partial_{t} (v_{3} - α Δ_{h} v_{3}) - ν Δ_{h} v_{3} + {(B_{ε} (v, v))}_{3} = - \nabla_{h} M p_{3} + M f_{3}, in Ω \\ v_{3} (x, 0) = M u_{0, 3} (x) . \end{matrix} \end{array}$ We take the vorticity of equation (2.3) h and taking the $L^{2}$ -inner product of this equation with $rot (v_{h} - α Δ_{h} v_{h})$ and the $L^{2}$ inner product of (2.3) 3 with $- Δ_{h} (v_{3} - α Δ_{h} v_{3})$ we add the resulting equations. Using the Hölder inequality, the Cauchy-Schwartz inequality and the Sobolev injections we obtain: $\begin{array}{l} \partial_{t} ({‖ rot (v_{h} - α Δ_{h} v_{h}) ‖}_{L^{2}}^{2} + {‖ \nabla_{h} (v_{3} - α Δ_{h} v_{3}) ‖}_{L^{2}}^{2}) \\ + \frac{3 ν}{2} (‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α (‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2})) \\ ⩽ C_{0} ‖ \nabla_{h} v_{h} ‖_{L^{2}}^{2} (‖ \nabla_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2}) + α C_{0} ‖ rot v_{h} ‖_{L^{2}}^{2} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ + α C_{0} ‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} ‖ \nabla_{h} v_{3} ‖_{L^{2}}^{2} + α^{3} C_{0} ‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ (3.3) & + C_{0} (‖ M f_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M f_{3} ‖_{L^{2}}^{2} + (1 + α) ‖ rot M f_{h} ‖_{L^{2}}^{2}) . \end{array}$ We can remark that the term $‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}$ creates a difficulty in the proof of the global existence of solution since it contains the higher derivatives of v.

In order to overcome this difficulty, we suppose that $\begin{matrix} α ‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{h} Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} ⩽ r_{0}^{2}, \end{matrix}$ where $r_{0}$ is a positive constant.

Then there exists a time $T > 0$ such that $\begin{matrix} α ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} ⩽ r_{0}^{2}, \forall t ⩽ T . \end{matrix}$ If we assume $r_{0}$ small enough, such that $C_{0} r_{0}^{2} ⩽ \frac{ν}{2}$ , then we obtain $\begin{array}{l} \partial_{t} ({‖ rot (v_{h} - α Δ_{h} v_{h}) ‖}_{L^{2}}^{2} + {‖ \nabla_{h} (v_{3} - α Δ_{h} v_{3}) ‖}_{L^{2}}^{2}) \\ + ν (‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α (‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2})) \\ ⩽ C_{0} ‖ \nabla_{h} v_{h} ‖_{L^{2}}^{2} (‖ \nabla_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2}) + α C_{0} ‖ rot v_{h} ‖_{L^{2}}^{2} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ + α C_{0} ‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} ‖ \nabla_{h} v_{3} ‖_{L^{2}}^{2} \\ (3.4) & + C_{0} (‖ M f_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M f_{3} ‖_{L^{2}}^{2} + (1 + α) ‖ rot M f_{h} ‖_{L^{2}}^{2}) . \end{array}$ We now remark that by inequality (2.1), we have $\begin{matrix} \begin{matrix} {‖ rot (v - α Δ_{h} v) ‖}_{L^{2}}^{2} \\ = {‖ rot (v_{h} - α Δ_{h} v_{h}) ‖}_{L^{2}}^{2} + {‖ \nabla_{h} (v_{3} - α Δ_{h} v_{3}) ‖}_{L^{2}}^{2} \\ = ‖ rot v_{h} ‖_{L^{2}}^{2} + 2 α ‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} + α^{2} ‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} \\ + ‖ \nabla_{h} v_{3} ‖_{L^{2}}^{2} + 2 α ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α^{2} ‖ Δ_{h} \nabla_{h} v_{3} ‖_{L^{2}}^{2} \\ ⩽ (λ_{1}^{- 1} + 2 α) (‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α (‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ Δ_{h} \nabla_{h} v_{3} ‖_{L^{2}}^{2})) . \end{matrix} \end{matrix}$ Thus, we obtain $\begin{array}{l} \partial_{t} {‖ rot (v - α Δ_{h} v) ‖}_{L^{2}}^{2} + \frac{ν}{2 (λ_{1}^{- 1} + 2 α)} {‖ rot (v - α Δ_{h} v) ‖}_{L^{2}}^{2} \\ + \frac{ν}{2} (‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α (‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2})) \\ ⩽ C_{0} ‖ \nabla_{h} v_{h} ‖_{L^{2}}^{2} (‖ \nabla_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2}) + α C_{0} ‖ rot v_{h} ‖_{L^{2}}^{2} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ + α C_{0} ‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} ‖ \nabla_{h} v_{3} ‖_{L^{2}}^{2} + C_{0} (‖ M f_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M f_{3} ‖_{L^{2}}^{2} + (1 + α) ‖ rot M f_{h} ‖_{L^{2}}^{2}) . \end{array}$ Integrating in time and applying the Gronwall lemma, we get $\begin{array}{l} {‖ rot (v - α Δ_{h} v) ‖}_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β_{1} (s - t)} (‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ + α (‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2})) \\ ⩽ ({‖ rot (M u_{0} - α Δ_{h} M u_{0}) ‖}_{L^{2}}^{2} + \frac{C_{0}}{β_{1}} (‖ M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2} \\ + α ‖ \nabla_{h} M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2} + (1 + α) ‖ rot M f_{h} ‖_{L_{t}^{\infty} (L^{2})}^{2})) exp (\frac{3}{2} C_{0} \int_{0}^{t} e^{β_{1} (s - t)} ‖ \nabla_{h} v ‖_{L^{2}}^{2}), \end{array}$ where $β_{1} = \frac{ν}{2 (λ_{1}^{- 1} + 2 α)}$ .

Assuming that $β ⩽ β_{1} ⩽ β_{0}$ , the $V_{ε}^{1}$ -estimate imply $\begin{matrix} \int_{0}^{t} e^{β_{1} (s - t)} ‖ \nabla_{h} v ‖_{L^{2}}^{2} ⩽ \frac{2}{ν} (‖ M u_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M u_{0} ‖_{L^{2}}^{2}) + \frac{2}{λ_{1} ν^{2} β} ‖ M f ‖_{L^{\infty} (L^{2})}^{2} . \end{matrix}$ Thus $\begin{array}{l} {‖ rot (v - α Δ_{h} v) ‖}_{L^{2}}^{2} \\ + \frac{ν}{4} \int_{0}^{t} e^{β_{1} (s - t)} (‖ \nabla_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α (‖ Δ_{h} rot v_{h} ‖_{L^{2}}^{2} + ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2})) \\ ⩽ ({‖ rot (M u_{0} - α Δ_{h} M u_{0}) ‖}_{L^{2}}^{2} \\ + \frac{C_{0}}{β_{1}} (‖ M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2} + α ‖ \nabla_{h} M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2} + (1 + α) ‖ rot M f_{h} ‖_{L_{t}^{\infty} (L^{2})}^{2})) \\ \times exp (C (‖ M u_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M u_{0} ‖_{L^{2}}^{2} + (1 + α) ‖ M f ‖_{L^{\infty} (L^{2})}^{2})) \\ ⩽ C (K_{0} + K_{1}) e^{C_{1} (1 + α) K_{1}} . \end{array}$ It remains to show that $α ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} ⩽ r_{0}^{2}$ , $\forall t ⩽ T$ .

For this purpose, we take the $L^{2}$ scalar product of system (2.3) 3 with $α Δ_{h}^{2} v_{3}$ , we get, after some integrations by parts and using the Sobolev injections $\begin{array}{l} α \partial_{t} (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}) + ν α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ ⩽ α ‖ Δ_{h} v_{h} ‖_{L^{2}}^{2} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α^{3} ‖ \nabla_{h} Δ_{h} v_{h} ‖_{L^{2}}^{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M f_{3} ‖_{L^{2}}^{2} . \end{array}$ We remark that, by inequality (2.1), we have $\begin{matrix} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} ⩽ (λ_{1}^{- 1} + α) ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} . \end{matrix}$ Then $\begin{array}{l} α \partial_{t} (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}) + α β_{0} (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}) + \frac{ν α}{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ ⩽ α ‖ Δ_{h} v_{h} ‖_{L^{2}}^{2} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α^{3} ‖ \nabla_{h} Δ_{h} v_{h} ‖_{L^{2}}^{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} M f_{3} ‖_{L^{2}}^{2} . \end{array}$ Integrating and applying the Gronwall lemma, we get $\begin{array}{l} α (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}) + \frac{ν α}{2} \int_{0}^{t} e^{β_{0} (s - t)} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ ⩽ α (‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + \frac{1}{β_{0}} ‖ \nabla_{h} M f_{3} ‖_{L^{2}}^{2}) \\ (3.5) & \times exp (\int_{0}^{t} e^{β_{0} (s - t)} (‖ Δ_{h} v_{h} ‖_{L^{2}}^{2} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{h} ‖_{L^{2}}^{2})) . \end{array}$ Since $β_{1} ⩽ β_{0}$ $\begin{array}{rcl} \int_{0}^{t} e^{β_{0} (s - t)} (‖ Δ_{h} v_{h} ‖_{L^{2}}^{2} ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{h} ‖_{L^{2}}^{2}) ⩽ \frac{4}{ν} C (K_{0} + K_{1}) e^{C_{1} (1 + α) K_{1}} . \end{array}$ Thus $\begin{array}{l} α (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}) + \frac{ν α}{2} \int_{0}^{t} e^{β_{0} (s - t)} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ (3.6) & ⩽ K_{0} exp (C_{1} (K_{0} + K_{1}) e^{C_{1} (1 + α) K_{1}}), \end{array}$ where $C_{1}$ is a positive constant that does not depend on α.

Suppose now that $\begin{matrix} K_{0} exp (K_{3}) = K_{0} exp (C_{1} (K_{0} + K_{1}) e^{C_{1} (1 + α) K_{1}}) ⩽ r_{0}^{2} . \end{matrix}$ Thus, the above condition and inequality (3.6) imply that, for all $t ⩽ T$ $\begin{matrix} α ‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} ⩽ r_{0}^{2} . \end{matrix}$ Therefore, the global existence of a solution for (2.3) is proved.

In what follows, we show that if $M u_{0}$ belongs to $V_{ε}^{4}$ and $M f$ is in $L^{\infty} (R^{+}, H_{per}^{2})$ , the solution of system (2.3) is in $L^{\infty} (R^{+}, V_{ε}^{4}) \cap L^{2} (R^{+}, V_{ε}^{4})$ .

In [6], Cioranescu and Girault have proved that, the solution belongs to $L_{loc}^{\infty} (R^{+}, V_{ε}^{4})$ . In [15], Jaffal proves the global existence of a solution, if $M u_{0}$ belongs to $V_{ε}^{4}$ and $M f$ is in $L^{2} (R^{+}, H_{per}^{2})$ .

Following the approach of [15], we can extend this result for the case when $M f$ is in $L^{\infty} (R^{+}, H_{per}^{2})$ .

We take the $L^{2}$ inner product of the vorticity of the first equation of system (2.3) with $rot Δ_{h}^{2} v$ . Using several integrations by parts, the Hölder inequality, the Sobolev embeddings and the Young inequality, we obtain the following inequality $\begin{array}{l} \partial_{t} (‖ Δ_{h} rot v ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2}) + ν ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} \\ ⩽ C (‖ Δ_{h} v ‖_{L^{2}}^{2} ‖ Δ_{h} rot v ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{h} Δ_{h} v ‖_{L^{2}}^{2} ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} + ‖ Δ_{h} M f ‖_{L^{2}}^{2}) . \end{array}$ Applying inequality (2.1), we yield $\begin{array}{l} \partial_{t} (‖ Δ_{h} rot v ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2}) \\ + \frac{ν}{2 (λ_{1}^{- 1} + α)} (‖ Δ_{h} rot v ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2}) + \frac{ν}{2} ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} \\ ⩽ C (‖ Δ_{h} v ‖_{L^{2}}^{2} ‖ Δ_{h} rot v ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{h} Δ_{h} v ‖_{L^{2}}^{2} ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} + ‖ Δ_{h} M f ‖_{L^{2}}^{2}) . \end{array}$ Integrating between 0 and t and applying the Gronwall lemma, we obtain, for all $β ⩽ β_{0}$ $\begin{array}{l} ‖ Δ_{h} rot v ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β (s - t)} ‖ \nabla_{h} Δ_{h} rot v ‖_{L^{2}}^{2} \\ ⩽ (‖ Δ_{h} rot M u_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} rot M u_{0} ‖_{L^{2}}^{2} + C β^{- 1} ‖ \nabla_{h} M f ‖_{L_{t}^{\infty} (L^{2})}^{2}) \\ \times exp (c \int_{0}^{t} e^{β (s - t)} (‖ Δ_{h} v ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v ‖_{L^{2}}^{2}) d s) . \end{array}$ Then, choosing $β = β_{1} ⩽ β_{0}$ and using inequality (3.1), inequality (3.2) is proved. □

Case of small α

If we restrict the size of α, we can prove the global existence of a solution of system (2.3) for large initial data. In fact, in the previous proof the condition that we imposed on the initial data is equivalent to $\begin{matrix} ‖ \nabla_{h} rot M u_{0} ‖_{L^{2}}^{2} + α ‖ Δ_{h} rot M u_{0} ‖_{L^{2}}^{2} ⩽ \frac{r_{0}^{2}}{α}, \end{matrix}$ this implies that when α is very small, we can choose large initial data. In particular when α vanishes (in the Navier-Stokes case) we obtain the global existence of solution for $2 D$ Navier-Stokes equations with three components for arbitrary initial data. Proposition 3.2.
There exist positive constants $α_{0}$ , $R_{0}$ , such that for all $0 < α ⩽ α_{0}$ for all $M u_{0} \in V_{ε}^{3}$ and for all $M f \in L_{t}^{\infty} (R^{+}, H_{per}^{1})$ satisfying $\begin{matrix} α (‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} \nabla_{h} M u_{0, 3} ‖_{L^{2}}^{2} + ‖ \nabla_{h} M f_{3} ‖_{L_{t}^{\infty} (L^{2})}^{2}) e^{K_{3} (M u_{0}, M f)} ⩽ R_{0}^{2}, \end{matrix}$ system ( 2.3 ) has a unique solution v in the space $L^{\infty} (R_{+}, V_{ε}^{3}) \cap L^{2} (R_{+}, V_{ε}^{3})$ .

If moreover, $M u_{0} \in V_{ε}^{4}$ and $M f \in L_{t}^{\infty} (R^{+}, H_{per}^{2})$ , then solution v of ( 2.3 ) belongs to $L^{\infty} (R_{+}, V_{ε}^{4}) \cap L^{2} (R_{+}, V_{ε}^{4})$ .
Proof.
Estimate (3.3) still holds in the case of small α. We assume for $R > 0$ that $\begin{matrix} ‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} \nabla_{h} M u_{0, 3} ‖_{L^{2}}^{2} ⩽ \frac{R^{2}}{4} . \end{matrix}$ Let α be small enough such that $\begin{matrix} α (‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} \nabla_{h} M u_{0, 3} ‖_{L^{2}}^{2}) ⩽ α \frac{R^{2}}{4} ⩽ \frac{ν}{2 C_{0}} . \end{matrix}$ Then, using the continuity of the solution, there exists a time T such that $\begin{matrix} α (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} \nabla_{h} v_{3} ‖_{L^{2}}^{2}) ⩽ \frac{ν}{2 C_{0}} . \end{matrix}$ Thus, we obtain inequality (3.4). Applying the same strategy as in the case of arbitrary α, we can find estimate (3.1).

Finally, it remains to show that $α (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ Δ_{h} \nabla_{h} v_{3} ‖_{L^{2}}^{2}) ⩽ \frac{ν}{2 C_{0}}$ for all t. To do this, we return to inequality (3.5) $\begin{array}{l} α (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}) + \frac{ν α}{2} \int_{0}^{t} e^{β_{0} (s - t)} ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2} \\ ⩽ α (‖ Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} M u_{0, 3} ‖_{L^{2}}^{2} + 2 \frac{λ_{1}^{- 1} + α}{ν} ‖ \nabla_{h} M f_{3} ‖_{L^{2}}^{2}) e^{K_{3}} . \end{array}$ Choosing α small enough such that $\begin{matrix} α \frac{R^{2}}{2} e^{K_{3}} ⩽ \frac{ν}{4 C_{0}}, \end{matrix}$ we obtain that $α (‖ Δ_{h} v_{3} ‖_{L^{2}}^{2} + α ‖ \nabla_{h} Δ_{h} v_{3} ‖_{L^{2}}^{2}) ⩽ \frac{ν}{2 C_{0}}$ and the global existence of solution is proved.

If $M u_{0} \in V_{ε}^{4}$ and $M f \in L^{\infty} (H_{per}^{2})$ then one can perform the same estimate as in the case of arbitrary α. □

4. Global existence of the solution w

The aim of this section is to prove Theorem 1.1 and Theorem 1.2. For this end, we prove the global existence of w. We point out that, since we get good Poincaré estimates (constants depending on ε) of $w_{0}$ , and exploiting the fact that $Q_{ε}$ is a thin domain as $ε \to 0$ , we can prove the existence of global solution for larger data conditions.

In the first time we want to obtain a priori estimate on the $V_{ε}^{1}$ norms of the solution of the equation (2.4).

Lemma 4.1.

Assume that hypotheses of Proposition 3.1 or Proposition 3.2 hold. Then there exists a unique solution w of ( 2.4 ) belonging to $L^{\infty} (R^{+}, V_{ε}^{1}) \cap L^{2} (R^{+}, V_{ε}^{1})$ .

Furthermore, w satisfies $\begin{array}{l} ‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β (s - t)} {‖ \nabla_{ε} w (s) ‖}_{L^{2}}^{2} d s \\ ⩽ (‖ w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + C β^{- 1} {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{- 1})}^{2}) e^{K_{3}}, \\ (4.1) & \forall t ⩾ 0 and \forall β ⩽ β_{0} = \frac{ν}{2 (λ_{1}^{- 1} + α)} . \end{array}$

Proof.

Taking the inner product in $L^{2} {(Q)}^{3}$ of equation (2.4) with w, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) + ν ‖ \nabla_{ε} w ‖_{L^{2}}^{2} \\ = \int_{Q} B_{ε} (w, w) w + \int_{Q} (B_{ε} (v, w) + B_{ε} (w, v)) w + \int_{Q} (I - M) f w \\ (4.2) & = I_{1} + I_{2} + I_{3} . \end{array}$ Since ${div}_{ε} w = 0$ , $I_{1} = 0$ .

Integrating by parts and using the Hölder inequality, we obtain $\begin{array}{rcl} I_{2} & = & \sum_{i, j = 1}^{3} \int_{Q} v_{i} \partial_{i} (w_{j} - α Δ_{ε} w_{j}) w_{j} + (v_{i} - α Δ_{ε} v_{i}) \partial_{j} w_{i} w_{j} \\ + \int_{Q} w_{i} \partial_{i} (v_{j} - α Δ_{ε} v_{j}) w_{j} + (w_{i} - α Δ_{ε} w_{i}) \partial_{j} v_{i} w_{j} \\ = & \sum_{i, j = 1}^{3} - α \int_{Q} \nabla_{ε} v_{i} \nabla_{ε} w_{j} \partial_{i} w_{j} + \int_{Q} \partial_{j} v_{i} w_{i} w_{j} \\ + α \int_{Q} \nabla_{ε} w_{i} \nabla_{ε} w_{j} \partial_{j} v_{i} + α \int_{Q} \nabla_{ε} w_{i} w_{j} \partial_{j} \nabla_{ε} v_{i} \\ ⩽ & 2 α ‖ \nabla_{ε} v ‖_{L^{\infty}} ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + ‖ \nabla_{ε} v ‖_{L^{2}} ‖ w ‖_{L^{4}}^{2} + α ‖ Δ_{ε} v ‖_{L^{4}} ‖ w ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{2}} . \end{array}$ By the Sobolev embeddings, the Nirenberg inequality and the Young inequality, we obtain $\begin{array}{rcl} I_{2} & ⩽ & C α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}} ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + ‖ \nabla_{ε} v ‖_{L^{2}} ‖ w ‖_{L^{2}} ‖ \nabla_{ε} w ‖_{L^{2}} \\ ⩽ & \frac{ν}{4} ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + C (α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + ‖ \nabla_{ε} v ‖_{L^{2}}^{2}) (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) . \end{array}$ Finally, by the Young inequality we have $\begin{matrix} I_{3} ⩽ \frac{ν}{4} ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + C {‖ (I - M) f ‖}_{H^{- 1}}^{2} . \end{matrix}$ Collecting all these bounds, we obtain $\begin{array}{l} \frac{d}{d t} (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) + ν ‖ \nabla_{ε} w ‖_{L^{2}}^{2} \\ ⩽ C (α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + ‖ \nabla_{ε} v ‖_{L^{2}}^{2}) (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) + C {‖ (I - M) f ‖}_{H^{- 1}}^{2} . \end{array}$ Applying (2.1), we get the following inequalities for any $t ⩾ 0$ $\begin{array}{l} \frac{d}{d t} (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) + \frac{ν}{2 (λ_{1}^{- 1} + α)} (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) + \frac{ν}{2} ‖ \nabla_{ε} w ‖_{L^{2}}^{2} \\ ⩽ C (α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + ‖ \nabla_{ε} v ‖_{L^{2}}^{2}) (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) + C {‖ (I - M) f ‖}_{H^{- 1}}^{2} . \end{array}$ Integrating in time the previous inequality, we obtain $\begin{array}{l} ‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β (s - t)} {‖ \nabla_{ε} w (s) ‖}_{L^{2}}^{2} d s \\ ⩽ ‖ w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + \frac{C}{β} {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{- 1})}^{2} \\ + C \int_{0}^{t} e^{β (s - t)} (α {‖ \nabla_{ε} Δ_{ε} v (s) ‖}_{L^{2}}^{2} + {‖ \nabla_{ε} v (s) ‖}_{L^{2}}^{2}) ({‖ w (s) ‖}_{L^{2}}^{2} + α {‖ \nabla_{ε} w (s) ‖}_{L^{2}}^{2}) d s, \\ for all t ⩾ 0, for any β ⩽ β_{0} = \frac{ν}{2 (λ_{1}^{- 1} + α)} . \end{array}$ By the Gronwall lemma, we obtain for $t ⩾ 0$ $\begin{array}{l} ‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β (s - t)} {‖ \nabla_{ε} w (s) ‖}_{L^{2}}^{2} d s \\ ⩽ (‖ w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + \frac{C}{β} {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{- 1})}^{2}) \\ \times exp (\int_{0}^{t} C e^{β (s - t)} (α {‖ \nabla_{ε} Δ_{ε} v (s) ‖}_{L^{2}}^{2} + {‖ \nabla_{ε} v (s) ‖}_{L^{2}}^{2}) d s) . \end{array}$ We remark that, by inequality (3.1), we have $\begin{matrix} C \int_{0}^{+ \infty} e^{β (s - t)} ({‖ \nabla_{ε} v (s) ‖}_{L^{2}}^{2} + α {‖ \nabla_{ε} Δ_{ε} v (s) ‖}_{L^{2}}^{2}) d s ⩽ K_{3} . \end{matrix}$ Thus, inequality (4.1) is proved. □

We next want to prove the global existence of the solution w in $V_{ε}^{3}$ . We distinguish the case of arbitrary coefficient α and the case of small α.

Proof of Theorem 1.1: Case of arbitrary coefficient α .

Taking the inner product in $L^{2} {(Q)}^{3}$ of equation (2.4) with $- Δ_{ε} (w - α Δ_{ε} w)$ , we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} + ν (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ = \int_{Q} B_{ε} (w, w) Δ_{ε} (w - α Δ_{ε} w) + \int_{Q} B_{ε} (v, w) Δ_{ε} (w - α Δ_{ε} w) + \int_{Q} B_{ε} (w, v) Δ_{ε} (w - α Δ_{ε} w) \\ - \int_{Q} (I - M) f Δ_{ε} (w - α Δ_{ε} w) \\ = B_{1} + B_{2} + B_{3} + B_{4} . \end{array}$ Now we want to estimate separately the terms $B_{k}$ , $1 ⩽ k ⩽ 4$ .

Integrating the first term $B_{1}$ by parts, using the fact that ${div}_{ε} w = 0$ , we obtain that $\begin{array}{rcl} B_{1} & = & \sum_{i, j = 1}^{3} \int_{Q} (w_{i} \partial_{i} (w_{j} - α Δ_{ε} w_{j}) + (w_{i} - α Δ_{ε} w_{i}) \partial_{j} w_{i}) Δ_{ε} (w_{j} - α Δ_{ε} w_{j}) \\ = & - \sum_{i, j, m = 1}^{3} \int_{Q} \partial_{m} w_{i} \partial_{i} (w_{j} - α Δ_{ε} w_{j}) \partial_{m} (w_{j} - α Δ_{ε} w_{j}) \\ - \sum_{i, j, m = 1}^{3} [\int_{Q} (\partial_{m} (w_{i} - α Δ_{ε} w_{i}) \partial_{j} w_{i} - \partial_{j} (w_{i} - α Δ_{ε} w_{i}) \partial_{m} w_{i}) \partial_{m} (w_{j} - α Δ_{ε} w_{j})] . \end{array}$ Therefore, by the Hölder inequality we infer that $\begin{array}{rcl} B_{1} & ⩽ & ‖ \nabla_{ε} w ‖_{L^{4}}^{2} ‖ \nabla_{ε} w ‖_{L^{2}} + α ‖ \nabla_{ε} w ‖_{L^{\infty}} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) . \end{array}$ We use the Nirenberg inequality and interpolation between $L^{2}$ and $H^{2}$ , we obtain $\begin{array}{rcl} ‖ \nabla_{ε} w ‖_{L^{2}} ‖ \nabla_{ε} w ‖_{L^{4}}^{2} ⩽ c ‖ \nabla_{ε} w ‖_{L^{2}}^{2} ‖ Δ_{ε} w ‖_{L^{2}} ⩽ c ‖ w ‖_{L^{2}} ‖ Δ_{ε} w ‖_{L^{2}}^{2} . \end{array}$ We deduce from the above inequality and the Young inequality that $\begin{array}{rcl} (4.3) & B_{1} ⩽ (\frac{ν}{16} + C (‖ w ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{ε} w ‖_{L^{\infty}}^{2})) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) . \end{array}$ We next estimate $B_{2}$ . Using an integration by parts and the fact that ${div}_{ε} v = 0$ , we deduce that $\begin{array}{rcl} B_{2} & = & - \sum_{i, j, m} \int \partial_{m} v_{i} \partial_{i} (w_{j} - α Δ_{ε} w_{j}) \partial_{m} (w_{j} - α Δ_{ε} w_{j}) \\ - \sum_{i, j, m} [\int (\partial_{m} (v_{i} - α Δ_{ε} v_{i}) \partial_{j} w_{i} - \partial_{j} (v_{i} - α Δ_{ε} v_{i}) \partial_{m} w_{i}) \partial_{m} (w_{j} - α Δ_{ε} w_{j})] . \end{array}$ We use the Hölder inequalities to get $\begin{array}{rcl} B_{2} & ⩽ & ‖ \nabla_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} (‖ \nabla_{ε} w ‖_{L^{2}} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}) \\ + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}} ‖ \nabla_{ε} w ‖_{L^{4}}^{2} + α^{2} ‖ \nabla_{ε} v ‖_{L^{\infty}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ + α^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} . \end{array}$ By the Sobolev and the Nirenberg inequalities, we obtain $\begin{array}{rcl} B_{2} & ⩽ & C [‖ Δ_{ε} v ‖_{L^{2}} ‖ Δ_{ε} w ‖_{L^{2}} (‖ \nabla_{ε} w ‖_{L^{2}} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}) \\ + α ‖ Δ_{ε} \nabla_{ε} v ‖_{L^{2}} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ + α^{2} ‖ Δ_{ε} \nabla_{ε} v ‖_{L^{2}}^{\frac{1}{2}} ‖ Δ_{ε}^{2} v ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla_{ε} w ‖_{L^{2}}^{\frac{1}{2}} ‖ Δ_{ε} w ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}] . \end{array}$ Applying the Young inequality, we obtain the following estimate $\begin{array}{rcl} B_{2} & ⩽ & (\frac{ν}{16} + C α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ + C (‖ Δ_{ε} v ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} \\ (4.4) & + C α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} ‖ Δ_{ε}^{2} v ‖_{L^{2}}^{2} . \end{array}$ Likewise, the estimate of the term $B_{3}$ also follows from integration by parts and the Hölder inequalities $\begin{array}{rcl} B_{3} & = & - \sum_{i, j, m = 1}^{3} \int_{Q} [\partial_{m} w_{i} \partial_{i} (v_{j} - α Δ_{ε} v_{j}) + w_{i} \partial_{m} \partial_{i} (v_{j} - α Δ_{ε} v_{j})] \partial_{m} (w_{j} - α Δ_{ε} w_{j}) \\ - \sum_{i, j, m = 1}^{3} \int_{Q} [\partial_{m} (w_{i} - α Δ_{ε} w_{i}) \partial_{j} v_{i} - \partial_{j} (w_{i} - α Δ_{ε} w_{i}) \partial_{m} v_{i}] \partial_{m} (w_{j} - α Δ_{ε} w_{j}) \\ ⩽ & [‖ \nabla_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} (‖ \nabla_{ε} w ‖_{L^{2}} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}) + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}} ‖ \nabla_{ε} w ‖_{L^{4}}^{2} \\ + α^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} + α^{2} ‖ \nabla_{ε} v ‖_{L^{\infty}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}] \\ + [(‖ Δ_{ε} v ‖_{L^{2}} + {‖ Δ_{ε}^{2} v ‖}_{L^{2}}) ‖ w ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} + α ‖ Δ_{ε} v ‖_{L^{4}} ‖ w ‖_{L^{4}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}] \\ + [α^{2} {‖ Δ_{ε}^{2} v ‖}_{L^{2}} ‖ w ‖_{L^{\infty}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}] = J_{1} + J_{2} + J_{3} . \end{array}$ The term $J_{1}$ is entirely similar to the term $B_{2}$ $\begin{array}{rcl} J_{1} & ⩽ & (\frac{ν}{48} + C α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ + C (‖ Δ_{ε} v ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} + C α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} . \end{array}$ By using the classical Sobolev embeddings, the Nirenberg inequality and the Young inequality, we obtain $\begin{array}{rcl} J_{2} & ⩽ & (\frac{ν}{48} + C ‖ w ‖_{L^{2}}^{2}) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ + C (‖ Δ_{ε} v ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} + C {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} . \end{array}$ It remains to estimate $J_{3}$ , we recall that by the Sobolev embeddings and by interpolation, we have $\begin{matrix} ‖ w ‖_{L^{\infty}} ⩽ C ‖ Δ_{ε} w ‖_{L^{2}} ⩽ C ‖ \nabla_{ε} w ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{\frac{1}{2}} . \end{matrix}$ Finally, by the Young inequality, we obtain $\begin{array}{rcl} J_{3} ⩽ (\frac{ν}{48} + C α ‖ \nabla_{ε} w ‖_{L^{2}}^{2}) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) + C α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{4} . \end{array}$ Collecting all these bounds, we obtain the following inequality $\begin{array}{rcl} B_{3} & ⩽ & (\frac{ν}{16} + C (‖ w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w ‖_{L^{2}}^{2})) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ + C (‖ Δ_{ε} v ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} \\ (4.5) & + C ({‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} + α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{4}) . \end{array}$ Applying the Hölder inequality and the Young inequality, we yield $\begin{array}{rcl} B_{4} & = & \int_{Q} (I - M) f Δ_{ε} (w - α Δ_{ε} w) \\ ⩽ & {‖ (I - M) f ‖}_{L^{2}} ‖ Δ_{ε} w ‖_{L^{2}} + α {‖ \nabla_{ε} (I - M) f ‖}_{L^{2}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} \\ (4.6) & ⩽ & \frac{ν}{16} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) + C (1 + α) {‖ (I - M) f ‖}_{H^{1}}^{2} . \end{array}$ Estimates (4.3)-(4.6) imply that $\begin{array}{l} \frac{d}{d t} {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} + (\frac{3 ν}{2} - C (‖ w ‖_{V_{ε}^{1}}^{2} + α^{2} ‖ \nabla_{ε} w ‖_{L^{\infty}}^{2})) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ ⩽ C (‖ Δ_{ε} v ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} \\ + C (1 + α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2}) {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} + C (1 + α) {‖ (I - M) f ‖}_{H^{1}}^{2} . \end{array}$ Its remains to show that $\frac{3 ν}{2} - C (‖ w ‖_{V_{ε}^{1}}^{2} + α^{2} ‖ \nabla_{ε} w ‖_{L^{\infty}}^{2}) > 0$ . Fot this purpose, we remark that since $w_{0} = (I - M) u_{0}$ , inequality (4.1) and Lemma 2.1 imply that $\begin{array}{rcl} ‖ w ‖_{V_{ε}^{1}}^{2} & ⩽ & (‖ w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + (1 + α) {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{- 1})}^{2}) e^{K_{3}} \\ ⩽ & c ε^{2} (‖ \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + (1 + α) {‖ (I - M) f ‖}_{L_{t}^{\infty} (L^{2})}^{2}) e^{K_{3}} . \end{array}$ Let R be a positive constant such that $\begin{array}{rcl} (4.7) & c ({‖ \nabla_{ε} (w_{0} - α Δ_{ε} w_{0}) ‖}_{L^{2}}^{2} + {(1 + α)}^{2} {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2} + (1 + α^{3} + α^{5}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} ⩽ R^{2}, \end{array}$ and we assume ε small enough such that $\begin{array}{rcl} ε^{2} R^{2} ⩽ \frac{ν}{4 C} . \end{array}$ With these assumptions, we deduce that $\begin{array}{rcl} (4.8) & C ‖ w ‖_{V_{ε}^{1}}^{2} & ⩽ & \frac{ν}{4} . \end{array}$ On the other hand, the Brezis-Gallouet inequality (see [3]) implies that $\begin{array}{rcl} ‖ \nabla_{ε} w_{0} ‖_{L^{\infty}}^{2} & ⩽ & c ‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} (1 + ln (1 + \frac{‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2}}{‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2}})) . \end{array}$ We notice that, since $w_{0} = (I - M) u_{0}$ , we have $\begin{matrix} ‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} ⩽ c ε^{2} ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} . \end{matrix}$ Using the fact that $x \mapsto x ln (1 + \frac{y}{x})$ is an increasing function if $0 < x < y$ , and the preceding inequality, we obtain $\begin{array}{rcl} (4.9) & ‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} ln (1 + \frac{‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2}}{‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2}}) ⩽ c ε^{2} ln (1 + c ε^{- 2}) ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} . \end{array}$ The above inequality and inequality (4.7) imply that $\begin{array}{rcl} α^{2} ‖ \nabla_{ε} w_{0} ‖_{L^{\infty}}^{2} & ⩽ & c ε^{2} (1 + ln (1 + c ε^{- 2})) α^{2} ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} ⩽ ε^{2} (1 + ln (1 + c ε^{- 2})) R^{2} . \end{array}$ We assume ε small enough such that $\begin{array}{rcl} ε^{2} (1 + ln (1 + c ε^{- 2})) R^{2} ⩽ \frac{ν}{4 C} . \end{array}$ As w is a continuous solution, then there exists $T > 0$ , such that for $t \in [0, T]$ $\begin{array}{rcl} (4.10) & C α^{2} {‖ \nabla_{ε} w (t) ‖}_{L^{\infty}}^{2} ⩽ \frac{ν}{4} . \end{array}$ Relations (4.8) and (4.10) imply that for $t \in [0, T]$ $\begin{matrix} C (‖ w ‖_{V_{ε}^{1}}^{2} + α^{2} ‖ \nabla_{ε} w ‖_{L^{\infty}}^{2}) ⩽ \frac{ν}{2} . \end{matrix}$ In particular, we obtain for $t \in [0, T]$ $\begin{array}{l} \frac{d}{d t} {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} + ν (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ ⩽ C (‖ Δ_{ε} v ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} \\ + C (1 + α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2}) {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} + C (1 + α) {‖ (I - M) f ‖}_{H^{1}}^{2} . \end{array}$ We remark that, $\begin{array}{rcl} {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} & = & ‖ \nabla_{ε} w ‖_{L^{2}}^{2} + 2 α ‖ Δ_{ε} w ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ ⩽ & (λ_{1}^{- 1} + 2 α) (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) . \end{array}$ Thus, $\begin{array}{l} \frac{d}{d t} {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} + \frac{ν}{2 (λ_{1}^{- 1} + 2 α)} {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} \\ + \frac{ν}{2} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2}) \\ ⩽ C (‖ Δ_{ε} v ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) ‖}_{L^{2}}^{2} \\ + C (1 + α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2}) {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} + C (1 + α) {‖ (I - M) f ‖}_{H^{1}}^{2} . \end{array}$ Let $β_{1} = \frac{ν}{2 (λ_{1}^{- 1} + 2 α)}$ , integrating this equation in time between 0 and t $\begin{array}{l} {‖ \nabla_{ε} (w - α Δ_{ε} w) (t) ‖}_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β_{1} (s - t)} ({‖ Δ_{ε} w (s) ‖}_{L^{2}}^{2} + α {‖ \nabla_{ε} Δ_{ε} w (s) ‖}_{L^{2}}^{2}) d s \\ ⩽ {‖ \nabla_{ε} (w_{0} - α Δ_{ε} w_{0}) ‖}_{L^{2}}^{2} + C β_{1}^{- 1} (1 + α) {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2} \\ + C \int_{0}^{t} e^{β_{1} (s - t)} (1 + α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2}) {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} d s \\ + \int_{0}^{t} e^{β_{1} (s - t)} ({‖ Δ_{ε} v (s) ‖}_{L^{2}}^{2} + α {‖ \nabla_{ε} Δ_{ε} v (s) ‖}_{L^{2}}^{2}) {‖ \nabla_{ε} (w - α Δ_{ε} w) (s) ‖}_{L^{2}}^{2} d s . \end{array}$ According to Proposition 3.1, we have $\begin{matrix} C \int_{0}^{t} e^{β_{1} (s - t)} ({‖ Δ_{ε} v (s) ‖}_{L^{2}}^{2} + α {‖ \nabla_{ε} Δ_{ε} v (s) ‖}_{L^{2}}^{2}) d s ⩽ K_{3}, \end{matrix}$ and $\begin{array}{l} C \int_{0}^{t} e^{β_{1} (s - t)} (1 + α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2}) {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} \\ ⩽ C (1 + α^{5} ‖ \nabla_{ε} Δ_{ε} v ‖_{L_{t}^{\infty} (L^{2})}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L_{t}^{\infty} (L^{2})}^{2}) \int_{0}^{t} e^{β_{1} (s - t)} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} d s \\ ⩽ C (1 + α^{3} + α^{5}) K_{2}^{2} e^{2 K_{3}} . \end{array}$ By the Gronwall lemma, we obtain for all $t \in [0, T]$ $\begin{array}{l} {‖ \nabla_{ε} (w - α Δ_{ε} w) (t) ‖}_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β_{1} (s - t)} ({‖ Δ_{ε} w (s) ‖}_{L^{2}}^{2} + α {‖ \nabla_{ε} Δ_{ε} w (s) ‖}_{L^{2}}^{2}) \\ ⩽ ({‖ \nabla_{ε} (w_{0} - α Δ_{ε} w_{0}) ‖}_{L^{2}}^{2} + c β_{1}^{- 1} (1 + α) {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2} \\ (4.11) & + C (1 + α^{3} + α^{5}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} . \end{array}$

Let us prove by contradiction that inequality (4.10) is valid for all $t \in R^{+}$ . We let $[0, T_{0})$ denote the maximal time interval for which the above inequality is valid, then $\forall 0 ⩽ t ⩽ T_{0}$ $\begin{matrix} C α^{2} {‖ \nabla_{ε} w (t) ‖}_{L^{\infty}}^{2} ⩽ \frac{ν}{4}, \end{matrix}$ and $\begin{matrix} C α^{2} {‖ \nabla_{ε} w (T_{0}) ‖}_{L^{\infty}}^{2} = \frac{ν}{4} . \end{matrix}$ Since $\nabla_{ε} w (t) = \nabla_{ε} M w (t) + \nabla_{ε} (I - M) w (t)$ , we will estimate separately $‖ \nabla_{ε} M w (t) ‖_{L^{\infty}}$ and $‖ \nabla_{ε} (I - M) w (t) ‖_{L^{\infty}}$ .

First, by using continuous Sobolev embeddings and interpolation inequalities, we obtain for all $δ_{0} \in [0, 1]$ $\begin{array}{rcl} {‖ \nabla_{ε} M w (t) ‖}_{L^{\infty}} & ⩽ & c {‖ \nabla_{ε} M w (t) ‖}_{H^{1 + δ_{0}}} \\ ⩽ & c α^{- \frac{3 + δ_{0}}{4}} {‖ M w (t) ‖}_{V_{ε}^{1}}^{\frac{1 - δ_{0}}{2}} {‖ M w (t) ‖}_{V_{ε}^{3}}^{\frac{1 + δ_{0}}{2}} . \end{array}$ Since $w_{0} = (I - M) u_{0}$ , inequality (4.1) and Lemma 2.1, imply that $\begin{array}{rcl} {‖ w (t) ‖}_{V_{ε}^{1}}^{2} & ⩽ & c ε^{4} (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w_{0} ‖_{L^{2}}^{2} + {‖ \nabla_{ε} (I - M) f ‖}_{L^{\infty} (L^{2})}^{2}) e^{K_{3}} \\ ⩽ & c ε^{4} α^{- 1} ({‖ \nabla_{ε} (w_{0} - α Δ_{ε}) w_{0} ‖}_{L^{2}}^{2} + α {‖ \nabla_{ε} (I - M) f ‖}_{L^{\infty} (L^{2})}^{2}) e^{K_{3}} . \end{array}$ The above inequality and estimate (4.11) imply that $\begin{array}{rcl} α^{2} {‖ \nabla_{ε} M w (t) ‖}_{L^{\infty}}^{2} & ⩽ & c ε^{2 - 2 δ_{0}} ({‖ \nabla_{ε} (w_{0} - α Δ_{ε} w_{0}) ‖}_{L^{2}}^{2} + {(1 + α)}^{2} {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2} \\ + (1 + α^{3} + α^{5}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} . \end{array}$ We deduce from the Brezis-Gallouet inequality that $\begin{array}{rcl} {‖ \nabla_{ε} (I - M) w (t) ‖}_{L^{\infty}}^{2} & ⩽ & c {‖ Δ_{ε} (I - M) w (t) ‖}_{L^{2}}^{2} (1 + ln (1 + \frac{‖ Δ_{ε} \nabla_{ε} (I - M) w (t) ‖_{L^{2}}^{2}}{‖ Δ_{ε} (I - M) w (t) ‖_{L^{2}}^{2}})) . \end{array}$ Also, we have $\begin{matrix} {‖ Δ_{ε} (I - M) w (t) ‖}_{L^{2}}^{2} ⩽ c ε^{2} {‖ Δ_{ε} \nabla_{ε} (I - M) w (t) ‖}_{L^{2}}^{2} . \end{matrix}$ Arguing as in (4.9), we show that $\begin{array}{rcl} {‖ \nabla_{ε} (I - M) w (t) ‖}_{L^{\infty}}^{2} & ⩽ & C ε^{2} (1 + ln (1 + c ε^{- 2})) {‖ Δ_{ε} \nabla_{ε} (I - M) w (t) ‖}_{L^{2}}^{2} . \end{array}$ Estimate (4.11) implies that $\begin{array}{l} C α^{2} {‖ \nabla_{ε} (I - M) w (t) ‖}_{L^{\infty}}^{2} \\ ⩽ C ε^{2} (1 + ln (1 + c ε^{- 2})) ({‖ \nabla_{ε} (w_{0} - α Δ_{ε} w_{0}) ‖}_{L^{2}}^{2} + {(1 + α)}^{2} {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2} \\ + (1 + α^{3} + α^{5}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} . \end{array}$ Let $δ_{0} \in [0, \frac{1}{2}]$ . If we assume ε small enough such that $\begin{array}{rcl} (4.12) & (ε^{2 - 2 δ_{0}} + ε^{2} (1 + ln (1 + c ε^{- 2}))) R^{2} ⩽ \frac{ν}{4 C}, \end{array}$ then $\begin{matrix} C α {‖ \nabla_{ε} w (T_{0}) ‖}_{L^{\infty}} < \frac{ν}{4} . \end{matrix}$ This contradicts the fact that $C α ‖ \nabla_{ε} w (T_{0}) ‖_{L^{\infty}} = \frac{ν}{4}$ .

We thus have proved that there exists $ε_{0} > 0$ , such that for $0 < ε ⩽ ε_{0}$ , for any $u_{0}$ and f satisfying the assumption of Theorem 1.1, the solution w of equation (2.4) exists globally. Since $u = v + w$ and according to Proposition 3.1, Theorem 1.1 is proved. □

We now prove Theorem 1.2.

Proof Theorem 1.2: Case of small α .

We remark that when α is small, for example when $α λ_{1} ⩽ 1$ , estimate (4.11) can be improved.

Taking the inner product in $L^{2} {(Q)}^{3}$ of equation (2.4) with $Δ_{ε}^{2} w$ , we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ Δ_{ε} w ‖_{L^{2}}^{2}) + ν ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ = - \int_{Q} B_{ε} (w, w) Δ_{ε}^{2} w - \int_{Q} B_{ε} (v, w) Δ_{ε}^{2} w - \int_{Q} B_{ε} (w, v) Δ_{ε}^{2} w + \int_{Q} (I - M) f Δ_{ε}^{2} w \\ = A_{1} + A_{2} + A_{3} + A_{4} . \end{array}$ Performing several integrations by parts to the first term $A_{1}$ , using the fact that ${div}_{ε} w = 0$ , we obtain $\begin{array}{rcl} A_{1} & = & \sum_{i, j, m = 1}^{3} \int_{Q} (\partial_{m} w_{i} \partial_{i} w_{j} \partial_{m} Δ_{ε} w_{j} + \partial_{m} w_{i} \partial_{m} w_{j} \partial_{i} Δ_{ε} w_{j}) \\ - α \sum_{i, j, m = 1}^{3} [\int_{Q} (\partial_{m} w_{i} \partial_{i} Δ_{ε} w_{j} - \partial_{j} w_{i} \partial_{m} Δ_{ε} w_{i} - \partial_{m} w_{i} \partial_{j} Δ_{ε} w_{i}) \partial_{m} Δ_{ε} w_{j}] . \end{array}$ By the Hölder inequality, we get $\begin{array}{rcl} A_{1} & ⩽ & 2 ‖ \nabla_{ε} w ‖_{L^{4}}^{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} + 3 α ‖ \nabla_{ε} w ‖_{L^{\infty}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} . \end{array}$ Applying the Nirenberg inequality and interpolating $L^{4}$ between $L^{2}$ and $H^{1}$ , we obtain $\begin{array}{rcl} (4.13) & ‖ \nabla_{ε} w ‖_{L^{4}}^{2} ⩽ c ‖ w ‖_{L^{2}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} . \end{array}$ Therefore, using the above inequality and the Young inequality, we obtain $\begin{array}{rcl} A_{1} ⩽ (\frac{ν}{16} + C (‖ w ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{ε} w ‖_{L^{\infty}}^{2})) ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} . \end{array}$ Similarly, using several integrations by parts and the Hölder inequality, we get $\begin{array}{rcl} A_{2} & = & \sum_{i, j, m} \int \partial_{m} v_{i} \partial_{i} (w_{j} - α Δ_{ε} w_{j}) \partial_{m} Δ_{ε} w_{j} \\ + \sum_{i, j, m} \int (\partial_{m} (v_{i} - α Δ_{ε} v_{i}) \partial_{j} w_{i} - \partial_{j} (v_{i} - α Δ_{ε} v_{i}) \partial_{m} w_{i}) \partial_{m} Δ_{ε} w_{j} \\ ⩽ & 3 ‖ \nabla_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} + α ‖ \nabla_{ε} v ‖_{L^{\infty}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} . \end{array}$ By the Sobolev inequalities, the Nirenberg inequalities, inequality (4.13) and the Young inequality, we have $\begin{matrix} \begin{matrix} A_{2} ⩽ & (\frac{ν}{16} + C ‖ w ‖_{L^{2}}^{2}) ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} + C α^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ + C (‖ \nabla_{ε} v ‖_{L^{2}}^{2} ‖ Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2}) . \end{matrix} \end{matrix}$ Likewise, the estimate of the term $A_{3}$ also follows from integrations by parts and the Hölder inequalities $\begin{array}{rcl} A_{3} & = & \sum_{i, j, m = 1}^{3} \int_{Q} [\partial_{m} w_{i} \partial_{i} (v_{j} - α Δ_{ε} v_{j}) + w_{i} \partial_{m} \partial_{i} (v_{j} - α Δ_{ε} v_{j})] \partial_{m} Δ_{ε} w_{j} \\ + \sum_{i, j, m = 1}^{3} \int_{Q} [\partial_{m} (w_{i} - α Δ_{ε} w_{i}) \partial_{j} v_{i} - \partial_{j} (w_{i} - α Δ_{ε} w_{i}) \partial_{m} v_{i}] \partial_{m} Δ_{ε} w_{j} \\ ⩽ & (‖ \nabla_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} + α ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{4}} ‖ \nabla_{ε} w ‖_{L^{4}} + α ‖ \nabla_{ε} v ‖_{L^{\infty}} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} \\ + ‖ Δ_{ε} v ‖_{L^{4}} ‖ w ‖_{L^{4}} + α ‖ w ‖_{L^{\infty}} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}) ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}} . \end{array}$ By the Young inequality, we have $\begin{array}{rcl} A_{3} & ⩽ & (\frac{ν}{16} + C ‖ w ‖_{L^{2}}^{2}) ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} + C α^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ + C (‖ Δ_{ε} v ‖_{L^{2}}^{2} ‖ Δ_{ε} \nabla_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{4}) . \end{array}$ Finally, by the Hölder inequality and the Young inequality, we obtain $\begin{array}{rcl} A_{4} & ⩽ & \frac{ν}{16} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} + c {‖ \nabla_{ε} (I - M) f ‖}_{L^{2}}^{2} . \end{array}$ Collecting all these bounds, we obtain the following inequality $\begin{array}{l} \frac{d}{d t} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w ‖_{L^{2}}^{2}) + (\frac{3 ν}{2} - C (‖ w ‖_{L^{2}}^{2} + α^{2} ‖ \nabla_{ε} w ‖_{L^{\infty}}^{2})) ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ ⩽ c α^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ + C (‖ Δ_{ε} v ‖_{L^{2}}^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{4}) + c {‖ \nabla_{ε} (I - M) f ‖}_{L^{2}}^{2} . \end{array}$ Let R be a positive constant such that $\begin{array}{rcl} (4.14) & c (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + (1 + α) {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2} + (1 + α^{3}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} ⩽ R^{2} . \end{array}$ Therefore, we argue as in the case of arbitrary α. Since $w_{0} = (I - M) u_{0}$ and using inequality (4.1), we have for all $t ⩾ 0$ $\begin{array}{rcl} {‖ w (t) ‖}_{L^{2}}^{2} & ⩽ & c ε^{4} (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w_{0} ‖_{L^{2}}^{2} + (1 + α) {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2}) e^{K_{3}} ⩽ ε^{4} R^{2} . \end{array}$ On the other hand, we deduce from the Brezis-Gallouet inequality and inequality (4.7) that $\begin{array}{rcl} α^{2} ‖ \nabla_{ε} w_{0} ‖_{L^{\infty}}^{2} & ⩽ & c ε^{2} (1 + ln (1 + c ε^{- 2})) α^{2} ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} ⩽ α ε^{2} (1 + ln (1 + c ε^{- 2})) R^{2} . \end{array}$ We assume α and ε small enough such that $\begin{array}{rcl} (4.15) & (ε^{4} + α ε^{2} (1 + ln (1 + c ε^{- 2}))) R^{2} ⩽ \frac{ν}{4 C} . \end{array}$ Then $α^{2} ‖ \nabla_{ε} w_{0} ‖_{L^{\infty}}^{2} ⩽ \frac{ν}{4 C}$ and by the continuity of w there exists $T > 0$ such that for all $t \in [0, T]$ $\begin{array}{rcl} (4.16) & α^{2} {‖ \nabla_{ε} w (t) ‖}_{L^{\infty}}^{2} ⩽ \frac{ν}{4 C} . \end{array}$ We get that for all $t \in [0, T]$ $\begin{array}{rcl} C ({‖ w (t) ‖}_{L^{2}}^{2} + α^{2} {‖ \nabla_{ε} w (t) ‖}_{L^{\infty}}^{2}) ⩽ \frac{ν}{2} . \end{array}$ In particular, we obtain for $t \in [0, T]$ $\begin{array}{l} \frac{d}{d t} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w ‖_{L^{2}}^{2}) + ν ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ ⩽ c α^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ + C (‖ Δ_{ε} v ‖_{L^{2}}^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{4}) + C {‖ \nabla_{ε} (I - M) f ‖}_{L^{2}}^{2} . \end{array}$ When $α λ_{1} ⩽ 1$ , we have: $\begin{array}{l} \frac{d}{d t} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w ‖_{L^{2}}^{2}) + \frac{ν λ_{1}}{4} (‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w ‖_{L^{2}}^{2}) + \frac{ν}{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ ⩽ c α^{2} ‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} ‖ \nabla_{ε} Δ_{ε} w ‖_{L^{2}}^{2} \\ + C (‖ \nabla_{ε} Δ_{ε} v ‖_{L^{2}}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2}) ‖ Δ_{ε} v ‖_{L^{2}}^{2} + c {‖ \nabla_{ε} (I - M) f ‖}_{L^{2}}^{2} . \end{array}$ Let $β_{2} = min (\frac{ν λ_{1}}{4}, β_{1} = \frac{ν λ_{1}}{2 (1 + 2 λ_{1} α)})$ . Integrating the above inequality in time between 0 and t, we obtain $\begin{array}{l} ‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β_{2} (s - t)} {‖ \nabla_{ε} Δ_{ε} w (s) ‖}_{L^{2}}^{2} d s \\ ⩽ ‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + \frac{C}{β_{2}} {‖ (I - M) \nabla_{ε} f ‖}_{L_{t}^{\infty} (L^{2})}^{2} \\ + C (‖ \nabla_{ε} Δ_{ε} v ‖_{L_{t}^{\infty} (L^{2})}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L_{t}^{\infty} (L^{2})}^{2}) \int_{0}^{t} e^{β_{2} (s - t)} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} d s \\ + C \int_{0}^{t} e^{β_{2} (s - t)} α {‖ \nabla_{ε} Δ_{ε} v (s) ‖}_{L^{2}}^{2} ({‖ Δ_{ε} w (s) ‖}_{L^{2}}^{2} + α {‖ Δ_{ε} \nabla_{ε} w (s) ‖}_{L^{2}}^{2}) d s . \end{array}$ According to inequality (3.1) and (3.2), we have $\begin{matrix} C \int_{0}^{t} e^{β_{2} (s - t)} α {‖ \nabla_{ε} Δ_{ε} v (s) ‖}_{L^{2}}^{2} d s ⩽ K_{3}, \end{matrix}$ and $\begin{array}{rcl} (‖ \nabla_{ε} Δ_{ε} v ‖_{L_{t}^{\infty} (L^{2})}^{2} + α^{4} {‖ Δ_{ε}^{2} v ‖}_{L_{t}^{\infty} (L^{2})}^{2}) \int_{0}^{t} e^{β_{2} (s - t)} {‖ Δ_{ε}^{2} v ‖}_{L^{2}}^{2} d s ⩽ C (1 + α^{3}) K_{2}^{2} e^{2 K_{3}} . \end{array}$ By the Gronwall lemma, we obtain for all $t \in [0, T]$ $\begin{array}{l} ‖ Δ_{ε} w ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w ‖_{L^{2}}^{2} + \frac{ν}{2} \int_{0}^{t} e^{β_{2} (s - t)} {‖ \nabla_{ε} Δ_{ε} w (s) ‖}_{L^{2}}^{2} d s \\ (4.17) & ⩽ (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} + \frac{C}{β_{2}} {‖ (I - M) \nabla_{ε} f ‖}_{L_{t}^{\infty} (L^{2})}^{2} + C (1 + α^{3}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} . \end{array}$ Let us prove that inequality (4.16) is valid for all $t \in R^{+}$ . Let $[0, T_{0})$ denote the maximal time interval for which the above inequality is valid, then for $0 ⩽ t ⩽ T_{0}$ $\begin{matrix} C α {‖ \nabla_{ε} w (t) ‖}_{L^{\infty}} ⩽ \frac{ν}{4}, \end{matrix}$ and $\begin{matrix} C α {‖ \nabla_{ε} w (T_{0}) ‖}_{L^{\infty}} = \frac{ν}{4} . \end{matrix}$ As in the case of arbitrary α, we estimate separately $‖ \nabla_{ε} M w (t) ‖_{L^{\infty}}$ and $‖ \nabla_{ε} (I - M) w (t) ‖_{L^{\infty}}$ . By using continuous Sobolev embeddings and interpolation inequalities, we obtain $\begin{array}{rcl} {‖ \nabla_{ε} M w (t) ‖}_{L^{\infty}} & ⩽ & c {‖ \nabla_{ε} M w (t) ‖}_{H^{1 + δ_{0}}} \\ ⩽ & c α^{- \frac{1}{2}} {(α^{\frac{1}{2}} {‖ \nabla_{ε} M w (t) ‖}_{L^{2}})}^{\frac{1 - δ_{0}}{2}} {(α^{\frac{1}{2}} {‖ \nabla_{ε} Δ_{ε} M w (t) ‖}_{L^{2}})}^{\frac{1 + δ_{0}}{2}} . \end{array}$ Since $w_{0} = (I - M) u_{0}$ , the Sobolev inequalities and inequality (4.1) imply that $\begin{matrix} α {‖ \nabla_{ε} M w (t) ‖}_{L^{2}}^{2} ⩽ c ε^{4} (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w_{0} ‖_{L^{2}}^{2} + (1 + α) {‖ (I - M) f ‖}_{L^{2} (H^{1})}^{2}) e^{K_{3}} . \end{matrix}$ By inequality (4.17) and the above inequality, we have $\begin{array}{rcl} α^{2} {‖ \nabla_{ε} M w (t) ‖}_{L^{\infty}}^{2} & ⩽ & c α ε^{2 - 2 δ_{0}} (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ \nabla_{ε} Δ_{ε} w_{0} ‖_{L^{2}}^{2} + (1 + α) {‖ (I - M) f ‖}_{L_{t}^{\infty} (H^{1})}^{2} \\ + (1 + α^{3}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} . \end{array}$ Arguing as in (4.9), we show that $\begin{array}{rcl} {‖ \nabla_{ε} (I - M) w (t) ‖}_{L^{\infty}}^{2} & ⩽ & C ε^{2} (1 + ln (1 + c ε^{- 2})) {‖ Δ_{ε} \nabla_{ε} (I - M) w (t) ‖}_{L^{2}}^{2} . \end{array}$ Inequality (4.17) implies that $\begin{array}{rcl} α^{2} {‖ \nabla_{ε} (I - M) w (t) ‖}_{L^{\infty}}^{2} & ⩽ & C α ε^{2} (1 + ln (1 + c ε^{- 2})) (‖ Δ_{ε} w_{0} ‖_{L^{2}}^{2} + α ‖ Δ_{ε} \nabla_{ε} w_{0} ‖_{L^{2}}^{2} \\ + {‖ (I - M) \nabla_{ε} f ‖}_{L_{t}^{\infty} (L^{2})}^{2} + (1 + α^{3}) K_{2}^{2} e^{2 K_{3}}) e^{K_{3}} . \end{array}$ Therefore, by assumption (4.14), we obtain $\begin{array}{rcl} α^{2} {‖ \nabla_{ε} w (t) ‖}_{L^{\infty}}^{2} ⩽ α (ε^{2 - 2 δ_{0}} + ε^{2} (1 + ln (1 + c ε^{- 2}))) R^{2} . \end{array}$ If we assume α and ε small enough, then inequality (4.15) holds as well as the additional condition $\begin{array}{rcl} α (ε^{2 - 2 δ_{0}} + ε^{2} (1 + ln (1 + c ε^{- 2}))) R^{2} ⩽ \frac{ν}{4 C} . \end{array}$ Thus $\begin{matrix} C α {‖ \nabla_{ε} w (T_{0}) ‖}_{L^{\infty}} < \frac{ν}{4} . \end{matrix}$ Therefore the global existence of w solution of (2.4) is proved. Thus, Theorem 1.2 is a direct consequence of Proposition 3.2. □

Footnotes

Acknowledgement

We would like to thank Geneviève Raugel for helpful discussion, many suggestions and contributions.

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