Approximation of the long-time dynamics of the dynamical system generated by the Ericksen

Abstract

In this article we study the long-time dynamics of the dynamical system generated by the Ericksen–Leslie model. More precisely, we discretize the Ericksen–Leslie equations in time using the implicit Euler scheme, and with the aid of the discrete Gronwall lemmas we prove that the scheme is uniformly bounded. Moreover, using the theory of the multi-valued attractors we prove in a particular case the convergence of the global attractors generated by the numerical scheme to the global attractor of the continuous system as the time-step approaches zero.

Keywords

Ericksen–Leslie equations implicit Euler scheme long-time stability attractors

1. .Introduction

Liquid crystals are considered either as an intermediate state between liquids and solids, or as the fourth state of matter (besides liquids, solids and gases). There are different types of liquid crystals phases, the nematic phase being the simplest one. In this phase, the molecules of the liquid crystal float around as in a liquid phase, but have the tendency of aligning along a preferred direction, in a crystal-like way.

The hydrodynamic theory of liquid crystals, now referred to as the Ericksen–Leslie (EL) dynamic theory, was developed in [10,11,25,26]. The EL model is now considered as one of the most successful theories used to model many dynamic phenomena in nematic liquid crystals (see, e.g., [17,49]).

The mathematical analysis of the EL models started with the work of [30–33], where the authors proved the existence of weak solutions to the Ginzburg–Landau approximation of the liquid crystal system. Since then, several studies have been devoted to the mathematical analysis of the EL models (see, e.g., [8,18,20,22,23,34,42,46]). In particular, it was shown that global weak solutions to liquid crystal system in 2D have at most finite many singular times, while the uniqueness of weak solutions to liquid crystal system in 2D was proved in [27,28,47,51]; global existence (but without uniqueness) of weak solutions to the liquid crystal system in 3D was recently established in [29], under the assumption that the initial director field takes value from the upper half unit sphere. If the initial data are suitably smooth, then the liquid crystal system has a unique local strong solution, see [21,23,46,49,50]. Moreover, if the initial data is suitably small, or the initial director field satisfies some geometrical condition in 2D, then the local strong solution to the liquid crystal system can be extended to be a global one, see [17,28]. It is worth mentioning that some mathematical analysis concerning the global existence of weak solutions and local or global well-posedness of strong solutions of the non-isothermal liquid crystal systems were addressed in [14,15].

Let us recall that some of the main difficulties in the mathematical analysis of the EL crystal flow is the presence of the term $| \nabla d |^{2} d$ in the dynamics of the director field d, as well as the constraint $| d | = 1$ that must be satisfied. To simplify the analysis, it is very common to approximate the EL crystal model by its Ginzburg–Landau approximation, in which the term $| \nabla d |^{2} d$ is replaced by a term of the form $- \frac{1}{α} ({| d |}^{2} - 1) d$ , where $α > 0$ is fixed a constant. In the model considered in the above references, the Ginzburg–Landau approximation is further simplified by replacing the term $- \frac{1}{α} ({| d |}^{2} - 1) d$ by a polynomial function of d and dropping the constraint $| d | = 1$ . The EL crystal model considered in this article is different from the model studied in [5,6,35] in the sense that our model includes both the term $| \nabla d |^{2} d$ and the constraint $| d | = 1$ . But using a transformation proposed in [17], this 2D EL crystal model takes a simpler form (with new unknown functions) in which the term $Δ d + | \nabla d |^{2} d$ simplifies to a linear term and the constraint $| d | = 1$ is automatically satisfied.

In this article, we discretize the Ericksen–Leslie equations in time using the implicit Euler scheme. Using the discrete Gronwall lemma and the discrete uniform Gronwall lemma we prove that the scheme is uniformly bounded. Moreover, using the theory of the multi-valued attractors we prove in a particular case the convergence of the global attractors generated by the numerical scheme to the global attractor of the continuous system as the timestep approaches zero. Our work has been inspired by previous results of the authors. In [14], for example, the authors considered the implicit Euler scheme for the 2D Navier–Stokes equations and proved that the numerical scheme was $H^{1}$ -uniformly stable in time. In [9] and [37] the authors applied the theory for multi-valued attractors to a homogeneous two-phase flow model and to the 2D Navier–Stokes equations, respectively, and proved the convergence of the discrete attractors to the global attractor of the continuous system as the time-step parameter approached zero.

The article is divided as follows. In the next section, we recall from [17] the Ericksen–Leslie model and its mathematical setting. In Section 3 we study the stability of a time discretization scheme for the model. More precisely, we prove that the scheme is uniformly bounded in the spaces $H$ and $U$ –defined in (2.12)–, provided that the time-step is small enough. In Section 4 we recall the theory of the so-called multi-valued attractors, and then we apply it to our model.

2. .The Ericksen–Leslie model and its mathematical setting

2.1. .Governing equations

We recall from [17] a reformulation of the simplified EL model. We assume that the domain $M$ of the fluid is the unit square $M = (0, 1)^{2}$ . The EL model is described by the following system

\begin{aligned} {\begin{cases} u_{t} - μ Δ u + (u \cdot \nabla) u + \nabla p = \nabla \cdot [λ \nabla d \otimes \nabla d + γ d \otimes (Δ d + | \nabla d |^{2} d)], \\ div u = 0, \\ d_{t} + u \cdot \nabla d - γ d \cdot \nabla u = α (Δ d + | \nabla d |^{2} d) - γ (d^{T} A d) d, \\ | d | = 1 \end{cases} \end{aligned}

(2.1)

M \times R_{+}

We complete these equations with the initial condition

\begin{aligned} (u, d) (0) = (u^{0}, d^{0}) in M, \end{aligned}

(2.2)

and with the boundary condition

\begin{aligned} u (x + e_{i}) = u (x), d (x + e_{i}) = d (x), \end{aligned}

(2.3)

where the unit vectors

e_{i}

(i = 1, 2)

are the canonical basis of

R^{2}

In the above system, the unknown functions are the velocity $u = (u_{1}, u_{2})^{T}$ of the fluid, the director field $d = (d_{1}, d_{2})^{T}$ – which stands for the averaged mocroscopic/continuum molecular orientation in $R^{2}$ –, and the pressure p of the fluid. The positive constants μ, λ, α are the viscosity of the fluid, the competition between the kinetic and the potential energy and the microscopic elastic relaxation time, respectively. The parameter γ is assumed to be non-negative, with the case $γ > 0$ denoting the presence of a stretching effect on the molecules of the crystal. The matrix $A = \frac{1}{2} (\nabla u + (\nabla u)^{T})$ denotes the rate of the strain tensor. The symbol ⊗ is the usual Kronecker product, e.g. $(a \otimes b)_{i j} = a^{i} b^{j}$ for $a, b \in R^{2}$ . The notation $\nabla d \otimes \nabla d$ denotes the $2 \times 2$ matrix, whose $(i, j)^{t h}$ entry is given by $\partial_{i} d \cdot \partial_{j} d$ .

Let us set $d = (\cos θ, \sin θ)^{T}$ . It follows that (see [13,17])

\begin{aligned} Δ d + | \nabla d |^{2} d & = Δ θ (\begin{matrix} - \sin θ \\ \cos θ \end{matrix}), \\ \nabla d \otimes \nabla d & = \nabla θ \otimes \nabla θ, \\ d \cdot \nabla u & = {(\cos θ \frac{\partial u_{1}}{\partial x_{1}} + \sin θ \frac{\partial u_{1}}{\partial x_{2}}, \cos θ \frac{\partial u_{2}}{\partial x_{1}} + \sin θ \frac{\partial u_{2}}{\partial x_{2}})}^{T}, \\ (d^{T} A d) d & = [\frac{\partial u_{1}}{\partial x_{1}} \cos^{2} θ + (\frac{\partial u_{2}}{\partial x_{1}} + \frac{\partial u_{1}}{\partial x_{2}}) \cos θ \sin θ + \frac{\partial u_{2}}{\partial x_{2}} \sin^{2} θ] (\cos θ, \sin θ)^{T} . \end{aligned}

(2.4)

Thus (2.1) becomes (see [17] for details)

\begin{aligned} {\begin{cases} u_{t} - μ Δ u + (u \cdot \nabla) u + \nabla p = λ \nabla \cdot [\nabla θ \otimes \nabla θ + γ M (θ) Δ θ], \\ div u = 0, \\ θ_{t} + u \cdot \nabla θ + γ F_{1} (u, θ) - α Δ θ = 0, \end{cases} \end{aligned}

(2.5)

where

M (θ)

and

F_{1} (u, θ)

are defined by

\begin{aligned} M (θ) & = (\begin{matrix} \cos θ \\ \sin θ \end{matrix}) \otimes (\begin{matrix} - \sin θ \\ \cos θ \end{matrix}) \\ F_{1} (u, θ) & = (\frac{\partial u_{1}}{\partial x_{1}} - \frac{\partial u_{2}}{\partial x_{2}}) \cos θ \sin θ - \frac{\partial u_{1}}{\partial x_{2}} \cos^{2} θ + \frac{\partial u_{2}}{\partial x_{1}} \sin^{2} θ . \end{aligned}

(2.6)

We endow system (2.6) with the initial condition

\begin{aligned} (u, θ) (0) = (u^{0}, θ^{0}) in M, \end{aligned}

(2.7)

and with the boundary condition

\begin{aligned} u (x + e_{i}) = u (x), θ (x + e_{i}) = θ (x) . \end{aligned}

(2.8)

We also impose

\begin{aligned} \int_{M} u d x = 0, \int_{M} θ d x = 0. \end{aligned}

(2.9)

2.2. .Mathematical setting

We recall from [43] the functional setting for periodic boundary value problems.

Let

\begin{aligned} V_{1} = {u : u = (u_{1}, u_{2}) is a trigonometric function defined on M, div u = 0, \int_{M} u d x = 0} . \end{aligned}

We define

H_{1}

and

V_{1}

as the closures of

V_{1}

(L^{2} (M))^{2}

and

(H^{1} (M))^{2}

, respectively. The inner product on

H_{1}

will be denoted by

(\cdot, \cdot)

, and the associated norm by

| \cdot |_{L^{2}}

. We equip

V_{1}

with the inner product

((u, v)) = (\nabla u, \nabla v)

, and the associated norm

‖ \cdot ‖

. We recall the Poincaré inequality

\begin{aligned} | u |_{L^{2}}^{2} ⩽ c_{M} ‖ u ‖^{2}, \forall u \in V_{1} . \end{aligned}

(2.10)

Let $P_{1} : {\dot{L}}^{2} (M) \to H_{1}$ be the Leray–Helmholtz projection. We define the Stokes operator

\begin{aligned} A_{0} = - P_{1} Δ u, \forall u \in D (A_{0}) = (H^{2} (M))^{2} \cap V_{1} . \end{aligned}

(2.11)

The Stokes operator

A_{0}

is a closed positive self-adjoint unbounded operator in

H_{1}

, with domain

D (A_{0})

. Its inverse

A_{0}^{- 1}

is compact in

H_{1}

and

| A_{0} \cdot |_{L^{2}}

is a norm on

D (A_{0})

, equivalent to the

H^{2}

-norm.

Let

\begin{aligned} V_{2} = {ϕ : ϕ is a trigonometric function defined on M, \int_{M} ϕ d x = 0} . \end{aligned}

We define

H_{2}

and

V_{2}

as the closures of

V_{2}

L^{2} (M)

and

H^{1} (M)

respectively. The inner product on

H_{2}

will be denoted by

(\cdot, \cdot)

, and the associated norm by

| \cdot |_{L^{2}}

. We equip

V_{2}

with the inner product

((ϕ, ψ)) = (\nabla ϕ, \nabla ψ)

, and the associated norm

‖ \cdot ‖

We denote by $A_{1}$ the periodic Laplacian operator with domain $D (A_{1}) = H^{2} (M) \cap V_{2}$ . Recall that

\begin{aligned} ⟨ A_{1} θ_{1}, θ_{2} ⟩ = \int_{M} \nabla θ_{1} \nabla θ_{2} d x, \forall θ_{1}, θ_{2} \in V_{2} . \end{aligned}

Now we define the Hilbert spaces $H$ and $U$ by

\begin{aligned} H = H_{1} \times H_{0}^{1} (M), U = V_{1} \times D (A_{1}), \end{aligned}

(2.12)

endowed with the scalar products whose associated norms are respectively

\begin{aligned} {| (u, θ) |}_{H}^{2} = | u |_{L^{2}}^{2} + | \nabla θ |_{L^{2}}^{2}, {| (u, θ) |}_{U}^{2} = | u |^{2} + | A_{1} θ |_{L^{2}}^{2} . \end{aligned}

(2.13)

Hereafter, we set

\begin{aligned} ⟨ (w_{1}, ψ_{1}), (w_{2}, ψ_{2}) ⟩_{H} = ⟨ w_{1}, w_{2} ⟩ + ⟨ ψ_{1}, A_{1} ψ_{2} ⟩, (w_{1}, ψ_{1}), (w_{2}, ψ_{2}) \in H . \end{aligned}

(2.14)

We introduce the bilinear operators $B_{0}$ and $B_{1}$ (and their associated trilinear forms $b_{0}$ , $b_{1}$ ) defined from $V_{1} \times V_{1}$ into $V_{1}^{*}$ (= the dual of $V_{1}$ ), and from $D (A_{0}) \times D (A_{1})$ into $L^{2} (M)$ , respectively, by:

\begin{aligned} ⟨ B_{0} (u, v), w ⟩ & = \int_{M} [(u \cdot \nabla) v] \cdot w d x = b_{0} (u, v, w), \forall u, v, w \in D (A_{0}), \\ ⟨ B_{1} (u, ϕ), ρ ⟩ & = \int_{M} [(u \cdot \nabla) ϕ] ρ d x = b_{1} (u, ϕ, ρ), \forall u \in D (A_{0}), ϕ, ρ \in D (A_{1}) . \end{aligned}

(2.15)

Note that the trilinear form $b_{0}$ satisfies the following properties

\begin{aligned} | b_{0} (u, v, w) | ⩽ c_{b} | u |_{L^{2}}^{1 / 2} ‖ u ‖^{1 / 2} ‖ v ‖ | w |_{L^{2}}^{1 / 2} ‖ w ‖^{1 / 2}, \forall u, v, w \in V_{1}, \end{aligned}

(2.16)

\begin{aligned} | b_{0} (u, v, w) | ⩽ c_{b} | u |_{L^{2}}^{1 / 2} | A u |_{L^{2}}^{1 / 2} ‖ v ‖ | w |_{L^{2}}, \forall u \in D (A_{0}), v \in V_{1}, w \in H_{1}, \end{aligned}

(2.17)

\begin{aligned} | b_{0} (u, v, w) | ⩽ c_{b} | u |_{L^{2}}^{1 / 2} ‖ u ‖^{1 / 2} ‖ v ‖^{1 / 2} | A v |_{L^{2}}^{1 / 2} | w |_{L^{2}}, \forall u \in V_{1}, v \in D (A_{0}), w \in H_{1}, \end{aligned}

(2.18)

\begin{aligned} b_{0} (u, v, v) = 0, \forall u, v \in V_{1}, \end{aligned}

(2.19)

the last equation implying

\begin{aligned} b_{0} (u, v, w) = - b_{0} (u, w, v), \forall u, v, w \in V_{1} . \end{aligned}

(2.20)

Similar inequalities are valid for the trilinear form $b_{1}$ :

\begin{aligned} | b_{1} (u, ϕ, ψ) | ⩽ c_{b} | u |_{L^{2}}^{1 / 2} ‖ u ‖^{1 / 2} ‖ ϕ ‖ | ψ |_{L^{2}}^{1 / 2} ‖ ψ ‖^{1 / 2}, \forall u \in V_{1}, ϕ, ψ \in H^{1} (M), \end{aligned}

(2.21)

\begin{aligned} | b_{1} (u, ϕ, ψ) | ⩽ c_{b} | u |_{L^{2}}^{1 / 2} | A u |_{L^{2}}^{1 / 2} ‖ ϕ ‖ | ψ |_{L^{2}}, \forall u \in D (A_{0}), ϕ \in H^{1} (M), ψ \in L^{2} (M), \end{aligned}

(2.22)

\begin{aligned} | b_{1} (u, ϕ, ψ) | ⩽ c_{b} | u |_{L^{2}}^{1 / 2} ‖ u ‖^{1 / 2} ‖ ϕ ‖^{1 / 2} | A_{1} ϕ |_{L^{2}}^{1 / 2} | ψ |_{L^{2}}, \\ \forall u \in V_{1}, ϕ \in D (A_{1}), ψ \in L^{2} (M), \end{aligned}

(2.23)

\begin{aligned} b_{1} (u, ϕ, ϕ) = 0, \forall u \in V_{1}, ϕ \in H^{1} (M), \end{aligned}

(2.24)

\begin{aligned} b_{1} (u, ϕ, ψ) = - b_{1} (u, ψ, ϕ), \forall u \in V_{1}, ϕ, ψ \in H^{1} (M) . \end{aligned}

(2.25)

We also introduce the bilinear operator $R_{0}$ defined from $D (A_{1}) \times D (A_{1})$ into $V_{1}^{*}$ by:

\begin{aligned} ⟨ R_{0} (ϕ, ψ), u ⟩ = - \overset{2}{\sum_{i, j}} \int_{M} \frac{\partial ϕ}{\partial x_{i}} \frac{\partial ψ}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}} d x, \forall ϕ, ψ \in D (A_{1}), u \in V_{1} . \end{aligned}

(2.26)

Note that

\begin{aligned} | ⟨ R_{0} (ϕ, ψ), u ⟩ | ⩽ c ‖ ϕ ‖^{1 / 2} | A_{1} ϕ |_{L^{2}}^{1 / 2} ‖ ψ ‖^{1 / 2} | A_{1} ψ |_{L^{2}}^{1 / 2} ‖ u ‖, \forall ϕ, ψ \in D (A_{1}), u \in V_{1} . \end{aligned}

(2.27)

We recall from [4] that the operators $B_{1}$ and $R_{0}$ satisfy

\begin{aligned} ⟨ R_{0} (ϕ, ϕ), u ⟩ = - ⟨ B_{1} (u, ϕ), A_{1} ϕ ⟩, \forall (u, ϕ) \in U . \end{aligned}

(2.28)

We recall from [16] that

\begin{aligned} div (\nabla ϕ \otimes \nabla ϕ) = \nabla (\frac{1}{2} {| \nabla ϕ |}^{2}) - A_{1} ϕ (\nabla ϕ)^{T} . \end{aligned}

(2.29)

Since (see [4])

\begin{aligned} R_{0} (ϕ, ψ) = P_{1} [div (\nabla ϕ \otimes \nabla ψ)], \forall ϕ, ψ \in D (A_{1}), \end{aligned}

(2.30)

equations (2.29) and (2.30) then imply

\begin{aligned} R_{0} (ϕ, ϕ) = - P_{1} [A_{1} ϕ (\nabla ϕ)^{T}] . \end{aligned}

(2.31)

We also set

\begin{aligned} R_{1} (θ_{1}, θ_{2}) = - P_{1} \nabla \cdot [M (θ_{1}) Δ θ_{2}] . \end{aligned}

(2.32)

We can easily check that (see [17])

\begin{aligned} ⟨ R_{1} (θ, θ), v ⟩ = ⟨ F_{1} (v, θ), A_{1} θ ⟩, \end{aligned}

(2.33)

where

\begin{aligned} F_{1} : V_{1} \times H_{2} \to H_{2} \end{aligned}

(2.34)

has been defined in (2.6).

We note that

\begin{aligned} {| F_{1} (v, θ) |}_{L^{2}} ⩽ c ‖ v ‖, \forall v \in V_{1}, θ \in H_{1}, \\ {| R_{1} (θ_{1}, θ_{2}) |}_{V_{1}^{*}} ⩽ c | A_{1} θ_{2} |_{L^{2}}, \forall θ_{1} \in H_{1}, θ_{2} \in D (A_{1}) . \end{aligned}

(2.35)

Using the above notations, we rewrite the system (2.5) as

\begin{aligned} {\begin{cases} u_{t} + μ A_{0} u + B_{0} (u, u) + R_{0} (θ, θ) - γ R_{1} (θ, θ) = f, \\ θ_{t} + α A_{1} θ + B_{1} (u, θ) + γ F_{1} (u, θ) = g, \\ (u, θ) (0) = (u^{0}, θ^{0}) \in H, \end{cases} \end{aligned}

(2.36)

for some given functions

f \in L^{\infty} (R_{+}; H_{1})

and

g \in L^{\infty} (R_{+}; V_{2})

. Hereafter we let

‖ f ‖_{\infty} := ‖ f ‖_{L^{\infty} (R_{+}; H_{1})}

, and

‖ g ‖_{\infty} := ‖ g ‖_{L^{\infty} (R_{+}; V_{2})}

Note that the constraint $| d | = 1$ is automatically satisfied since $d = (\cos θ, \sin θ)^{T}$ , which is very important at least from the numerical point of view. The equivalence between the systems (2.1) and (2.36) is discussed in [13], where the authors studied the large-time behavior of (2.1) under some regularity conditions (referred to as a trigonometric condition) on the initial direction field. For additional details, the interested reader is also referred to [17], where a similar transformation is used to prove the existence of strong solutions to (2.1), and [24], where the 3D case is studied using a spherical coordinate system.

The solution $(u, θ)$ of (2.36) is uniformly bounded for all time (see [17]). More precisely,

\begin{aligned} {| (u, θ) |}_{H}^{2} & ⩽ M_{0} ({| (u^{0}, θ^{0}) |}_{H}, ‖ f ‖_{\infty}, ‖ g ‖_{\infty}), \end{aligned}

(2.37)

\begin{aligned} {| (u, θ) |}_{U}^{2} & ⩽ M_{1} ({| (u^{0}, θ^{0}) |}_{U}, ‖ f ‖_{\infty}, ‖ g ‖_{\infty}) . \end{aligned}

(2.38)

In this article we consider a time discretization of (2.36) using the fully implicit Euler scheme,

\begin{aligned} {\begin{cases} \frac{u^{n} - u^{n - 1}}{Δ t} + μ A_{0} u^{n} + B_{0} (u^{n}, u^{n}) + R_{0} (θ^{n}, θ^{n}) - γ R_{1} (θ^{n}, θ^{n}) = f^{n}, \\ \frac{θ^{n} - θ^{n - 1}}{Δ t} + α A_{1} θ^{n} + B_{1} (u^{n}, θ^{n}) + γ F_{1} (u^{n}, θ^{n}) = g^{n}, \\ u^{0} = u^{0}, θ^{0} = θ^{0}, \end{cases} \end{aligned}

(2.39)

and seek to obtain uniform bounds on

(u^{n}, θ^{n})

similar to (2.37) and (2.38). Once we have proved that the solution

(u^{n}, θ^{n})

of the above system is uniformly bounded in the

U

-norm, we show that in the case

γ = 0

the global attractors generated by the numerical scheme (2.39) converge to the global attractor of the continuous system (2.36) as the time-step approaches zero.

3. .V-uniform boundedness

We begin with the uniform boundedness of the approximate solution in $H$ .

Lemma 1.
For every $Δ t > 0$ , we have
$\begin{aligned} {| (u^{n}, θ^{n}) |}_{H}^{2} & ⩽ {(1 + \frac{μ + α}{c_{M}} Δ t)}^{- n} {| (u^{0}, θ^{0}) |}_{H}^{2} + \frac{c_{M}}{μ + α} (\frac{c_{M}}{μ} ‖ f ‖_{\infty}^{2} + \frac{1}{α} ‖ g ‖_{\infty}^{2}) \\ \cdot [1 - {(1 + \frac{μ + α}{c_{M}} Δ t)}^{- n}], \forall n ⩾ 0, \end{aligned}$
(3.1)
and there exists $K_{1} = K_{1} (| (u^{0}, θ^{0}) |_{H}, ‖ f ‖_{\infty}, ‖ g ‖_{\infty})$ such that
$\begin{aligned} {| (u^{n}, θ^{n}) |}_{H}^{2} ⩽ K_{1}, \forall n ⩾ 0, \end{aligned}$
(3.2)
and
$\begin{aligned} Δ t \overset{n}{\sum_{j = i}} (μ ‖ u^{n} ‖^{2} + α {| A_{1} θ^{n} |}_{L^{2}}^{2}) ⩽ {| (u^{i - 1}, θ^{i - 1}) |}_{H}^{2} + (\frac{c_{M}}{μ} ‖ f ‖_{\infty}^{2} + \frac{1}{α} ‖ g ‖_{\infty}^{2}) (n - i + 1) Δ t, \\ \forall i = 1, \dots, n . \end{aligned}$
(3.3)

Proof.
Taking the scalar product of the first equation of (2.39) with $2 Δ t u^{n}$ in $L^{2}$ and using the relation
$\begin{aligned} 2 (φ - ψ, φ)_{L^{2}} = | φ |_{L^{2}}^{2} - | ψ |_{L^{2}}^{2} + | φ - ψ |_{L^{2}}^{2}, \end{aligned}$
(3.4)
as well as the skew property (2.19), we obtain
$\begin{aligned} | u^{n} |_{L^{2}}^{2} - | u^{n - 1} |_{L^{2}}^{2} + | u^{n} - u^{n - 1} |_{L^{2}}^{2} + 2 μ Δ t ‖ u^{n} ‖^{2} + 2 Δ t ⟨ R_{0} (θ^{n}, θ^{n}), u^{n} ⟩ \\ - 2 γ Δ t ⟨ R_{1} (θ^{n}, θ^{n}), u^{n} ⟩ = 2 Δ t (f^{n}, u^{n}) . \end{aligned}$
(3.5)

Multiplying the second equation of (2.39) by $2 Δ t A_{1} θ^{n}$ , integrating, and recalling the skew property (2.24), we obtain
$\begin{aligned} ‖ θ^{n} ‖^{2} - ‖ θ^{n - 1} ‖^{2} + ‖ θ^{n} - θ^{n - 1} ‖^{2} + 2 α Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} + 2 Δ t b_{1} (u^{n}, θ^{n}, A_{1} θ^{n}) \\ + 2 γ Δ t ⟨ F_{1} (u^{n}, θ^{n}), A_{1} θ^{n} ⟩ = 2 Δ t (g^{n}, A_{1} θ^{n}) . \end{aligned}$
(3.6)
Adding (3.5) and (3.6), and recalling (2.33), we find
$\begin{aligned} | u^{n} |_{L^{2}}^{2} + ‖ θ^{n} ‖^{2} - (| u^{n - 1} |_{L^{2}}^{2} + ‖ θ^{n - 1} ‖^{2}) + | u^{n} - u^{n - 1} |_{L^{2}}^{2} + ‖ θ^{n} - θ^{n - 1} ‖^{2} \\ + 2 μ Δ t ‖ u^{n} ‖^{2} + 2 α Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} = 2 Δ t (f^{n}, u^{n}) + 2 Δ t (g^{n}, A_{1} θ^{n}) . \end{aligned}$
(3.7)
Using the Cauchy–Schwarz inequality and the Poincaré inequality (2.10), we majorize the right-hand side of (3.7) as follows
$\begin{aligned} 2 Δ t (f^{n}, u^{n}) & ⩽ 2 Δ t {| f^{n} |}_{L^{2}} > {| u^{n} |}_{L^{2}} ⩽ 2 Δ t \sqrt{c_{M}} Δ t {| f^{n} |}_{L^{2}} ‖ u^{n} ‖ \\ ⩽ μ Δ t {‖ u^{n} ‖}^{2} + \frac{c_{M}}{μ} Δ t {| f^{n} |}_{L^{2}}^{2}, \end{aligned}$
(3.8)

$\begin{aligned} 2 Δ t (g^{n}, A_{1} θ^{n}) & ⩽ 2 Δ t {| g^{n} |}_{L^{2}} {| A_{1} θ^{n} |}_{L^{2}} \\ ⩽ α Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} + \frac{1}{α} Δ t {| g^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.9)

Relations (3.7)–(3.9) give
$\begin{aligned} {| u^{n} |}_{L^{2}}^{2} + ‖ θ^{n} ‖^{2} - ({| u^{n - 1} |}_{L^{2}}^{2} + ‖ θ^{n - 1} ‖^{2}) + {| u^{n} - u^{n - 1} |}_{L^{2}}^{2} + ‖ θ^{n} - θ^{n - 1} ‖^{2} \\ + μ Δ t ‖ u^{n} ‖^{2} + α Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} ⩽ \frac{c_{M}}{μ} Δ t {| f^{n} |}_{L^{2}}^{2} + \frac{1}{α} Δ t {| g^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.10)
Using again the Poincaré inequality (2.10) and neglecting some positive terms, we find from (3.10)
$\begin{aligned} {| (u^{n}, θ^{n}) |}_{H}^{2} ⩽ \frac{1}{β} {| (u^{n - 1}, θ^{n - 1}) |}_{H}^{2} + \frac{1}{β} (\frac{c_{M}}{μ} {| f^{n} |}_{L^{2}}^{2} + \frac{1}{α} {| g^{n} |}_{L^{2}}^{2}) Δ t, \end{aligned}$
(3.11)
where
$\begin{aligned} β = 1 + \frac{μ + α}{c_{M}} Δ t . \end{aligned}$
(3.12)
Using recursively (3.11), we find
$\begin{aligned} {| (u^{n}, θ^{n}) |}_{H}^{2} & ⩽ \frac{1}{β^{n}} {| (u^{0}, θ^{0}) |}_{H}^{2} + Δ t \overset{n}{\sum_{i = 1}} \frac{1}{β^{i}} (\frac{c_{M}}{μ} {| f^{n + 1 - i} |}_{L^{2}}^{2} + \frac{1}{α} {| g^{n + 1 - i} |}_{L^{2}}^{2}) \\ ⩽ {(1 + \frac{μ + α}{c_{M}} Δ t)}^{- n} {| (u^{0}, θ^{0}) |}_{H}^{2} + \frac{c_{M}}{μ + α} (\frac{c_{M}}{μ} ‖ f ‖_{\infty}^{2} + \frac{1}{α} ‖ g ‖_{\infty}^{2}) \\ \cdot [1 - {(1 + \frac{μ + α}{c_{M}} Δ t)}^{- n}], \end{aligned}$
(3.13)
which proves (3.1); (3.1) easily implies (3.2) with
$\begin{aligned} K_{1} ({| (u^{0}, θ^{0}) |}_{H}, ‖ f ‖_{\infty}, ‖ g ‖_{\infty}) := {| (u^{0}, θ^{0}) |}_{H}^{2} + \frac{c_{M}}{μ + α} (\frac{c_{M}}{μ} | ‖ f ‖_{\infty}^{2} + \frac{1}{α} ‖ g ‖_{\infty}^{2}) . \end{aligned}$
(3.14)

Adding inequalities (3.10) with n from i to m and dropping some positive terms, we obtain (3.3).

As a direct consequence of Lemma 1, we have
Corollary 3.1.
If
$\begin{aligned} 0 < Δ t ⩽ \frac{c_{M}}{μ + α} =: κ_{1}, \end{aligned}$
(3.15)
then
$\begin{aligned} {| u^{n} |}_{L^{2}}^{2} ⩽ 2 ρ_{0}^{2}, \forall n Δ t ⩾ T_{0} ({| u^{0} |}_{L^{2}}) := \frac{4 c_{M}}{μ + α} \ln (\frac{| (u^{0}, θ^{0}) |_{H}}{ρ_{0}}), \end{aligned}$
(3.16)
where $ρ_{0}^{2} := \frac{c_{M}}{μ + α} (\frac{c_{M}}{μ} ‖ f ‖_{\infty}^{2} + \frac{1}{α} ‖ g ‖_{\infty}^{2})$ .
Proof.
From the bound (3.1) on $| (u^{n}, θ^{n}) |_{H}^{2}$ , we infer that
$\begin{aligned} {| (u^{n}, θ^{n}) |}_{H}^{2} ⩽ {(1 + \frac{μ + α}{c_{M}} Δ t)}^{- n} {| (u^{0}, θ^{0}) |}_{H}^{2} + ρ_{0}^{2}, \end{aligned}$
and using assumption (3.15) on $Δ t$ and the fact that $1 + x ⩾ \exp (x / 2)$ if $x \in (0, 1)$ we obtain
$\begin{aligned} {| u^{n} |}_{L^{2}}^{2} ⩽ \exp (- \frac{μ + α}{2 c_{M}} n Δ t) {| (u^{0}, θ^{0}) |}_{H}^{2} + ρ_{0}^{2} . \end{aligned}$
For $n Δ t ⩾ T_{0}$ , the above inequality implies conclusion (3.16) of the corollary.

We now seek to obtain uniform bounds for the approximate solution $(u^{n}, θ^{n})$ in the space $U$ . In order to do this, we will first use the discrete Gronwall lemma to derive an upper bound on $‖ (u^{n}, θ^{n}) ‖_{U}$ , $n ⩽ N$ , for some $N > 0$ , and then we will use the discrete uniform Gronwall lemma to obtain an upper bound for $n ⩾ N$ .

We begin with some preliminary inequalities.
Lemma 2.
For every $Δ t > 0$ and for every $n ⩾ 1$ , we have
$\begin{aligned} {‖ (u^{n}, θ^{n}) ‖}_{U}^{2} - {‖ (u^{n - 1}, θ^{n - 1}) ‖}_{U}^{2} ⩽ c_{1} (1 + K_{1}) Δ t {‖ (u^{n}, θ^{n}) ‖}_{U}^{4} + \frac{5}{μ} Δ t ‖ f ‖_{\infty}^{2} + \frac{6}{α} Δ t ‖ g ‖_{\infty}^{2}, \end{aligned}$
(3.17)
where $c_{1}$ is given in (3.32) below. Moreover, for every $i = 1, \dots, n$ we have
$\begin{aligned} \overset{n}{\sum_{j = i}} (‖ u^{n} - u^{n - 1} ‖^{2} + {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2}) \\ ⩽ ‖ (u^{i - 1}, θ^{i - 1}) ‖_{U}^{2} + c_{1} (1 + K_{1}) Δ t \overset{n}{\sum_{j = i}} ‖ (u^{n}, θ^{n}) ‖_{U}^{4} \\ + (\frac{5}{μ} ‖ f ‖_{\infty}^{2} + \frac{6}{α} ‖ g ‖_{\infty}^{2}) (n - i + 1) Δ t . \end{aligned}$
(3.18)

Proof.
Multiplying the first equation of (2.39) by $2 Δ t A_{0} u^{n}$ , the second equation by $2 Δ t A_{1}^{2} θ^{n}$ , adding the resulting equations and integrating, we obtain (recalling (2.28) and (2.33)):
$\begin{aligned} ‖ u^{n} ‖^{2} + {| A_{1} θ^{n} |}_{L^{2}}^{2} - (‖ u^{n - 1} ‖^{2} + {| A_{1} θ^{n - 1} |}_{L^{2}}^{2}) + ‖ u^{n} - u^{n - 1} ‖^{2} + {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2} \\ + 2 μ Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + 2 α Δ t {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} + 2 Δ t b_{0} (u^{n}, u^{n}, A_{0} u^{n}) \\ - 2 Δ t b_{1} (A_{0} u^{n}, θ^{n}, A_{1} θ^{n}) - 2 γ Δ t ⟨ F_{1} (A_{0} u^{n}, θ^{n}), A_{1} θ^{n} ⟩ \\ + 2 Δ t b_{1} (u^{n}, θ^{n}, A_{1}^{2} θ^{n}) + 2 γ Δ t ⟨ F_{1} (u^{n}, θ^{n}), A_{1}^{2} θ^{n} ⟩ \\ = 2 Δ t (f^{n}, A_{0} u^{n}) + 2 Δ t (g^{n}, A_{1}^{2} θ^{n}) . \end{aligned}$
(3.19)
Using the Cauchy–Schwarz inequality, we estimate the right-hand side of the above equality as
$\begin{aligned} 2 Δ t (f^{n}, A_{0} u^{n}) & ⩽ 2 Δ t {| f^{n} |}_{L^{2}} {| A_{0} u^{n} |}_{L^{2}} ⩽ \frac{μ}{5} Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + \frac{5}{μ} Δ t {| f^{n} |}_{L^{2}}^{2}, \end{aligned}$
(3.20)

$\begin{aligned} 2 Δ t (g^{n}, A_{1}^{2} θ^{n}) & = 2 Δ t (A_{1}^{1 / 2} g^{n}, A_{1}^{3 / 2} θ^{n}) ⩽ 2 Δ t {| \nabla g^{n} |}_{L^{2}} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}} \\ ⩽ \frac{α}{6} Δ t {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} + \frac{6}{α} Δ t {| \nabla g^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.21)
We bound the nonlinear terms as follows:
$\begin{aligned} 2 Δ t | b_{0} (u^{n}, u^{n}, A_{0} u^{n}) | & ⩽ 2 c_{b} Δ t {| u^{n} |}_{L^{2}}^{1 / 2} ‖ u^{n} ‖ {| A_{0} u^{n} |}_{L^{2}}^{3 / 2} \\ ⩽ \frac{μ}{5} Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + \frac{c c_{b}^{4}}{μ^{3}} Δ t {| u^{n} |}_{L^{2}}^{2} ‖ u^{n} ‖^{4}, \end{aligned}$
(3.22)

$\begin{aligned} 2 Δ t | b_{1} (A_{0} u^{n}, θ^{n}, A_{1} θ^{n}) | & ⩽ 2 Δ t {| A_{0} u^{n} |}_{L^{2}} ‖ \nabla θ^{n} ‖_{L^{\infty}} {| A_{1} θ^{n} |}_{L^{2}} \\ ⩽ c Δ t {| A_{0} u^{n} |}_{L^{2}} ‖ θ^{n} ‖^{1 / 2} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{1 / 2} {| A_{1} θ^{n} |}_{L^{2}} \\ (by Agmo n^{'} s inequality) \\ ⩽ \frac{μ}{5} Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + \frac{α}{6} Δ t {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} + \frac{c}{μ^{2} α} Δ t ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{4}, \end{aligned}$
(3.23)

$\begin{aligned} 2 Δ t | b_{1} (u^{n}, θ^{n}, A_{1}^{2} θ^{n}) | & = 2 Δ t | {(A_{1}^{1 / 2} B_{1} (u^{n}, θ^{n}), A_{1}^{3 / 2} θ^{n})}_{L^{2}} | \\ ⩽ 2 Δ t {| A_{1}^{1 / 2} B_{1} (u^{n}, ϕ^{n}) |}_{L^{2}} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}} \\ ⩽ c Δ t ‖ u^{n} ‖^{1 / 2} {| A_{0} u^{n} |}_{L^{2}}^{1 / 2} ‖ θ^{n} ‖^{1 / 2} {| A_{1} θ^{n} |}_{L^{2}}^{1 / 2} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}} \\ + c Δ t {| u^{n} |}_{L^{2}}^{1 / 2} ‖ u^{n} ‖^{1 / 2} {| A_{1} θ^{n} |}_{L^{2}}^{1 / 2} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{3 / 2} \\ (by Ladyzhenskaya's inequality) \\ ⩽ \frac{μ}{5} Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + \frac{α}{6} Δ t {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} + \frac{c}{μ α^{2}} Δ t ‖ u^{n} ‖^{2} ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} \\ + \frac{c}{α^{3}} Δ t {| u^{n} |}_{L^{2}}^{2} ‖ u^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.24)
Recalling (2.6), one can prove that
$\begin{aligned} ⟨ F_{1} (u, θ), A_{1} ψ ⟩ & = ⟨ F_{1} (A_{0} u, θ), ψ ⟩ + 2 \overset{2}{\sum_{i = 1}} \int_{M} (\frac{\partial^{2} u_{1}}{\partial x_{i} \partial x_{1}} - \frac{\partial^{2} u_{2}}{\partial x_{i} \partial x_{2}}) \cos 2 θ \frac{\partial θ}{\partial x_{i}} ψ d x \\ - 2 \overset{2}{\sum_{i = 1}} \int_{M} (\frac{\partial u_{1}}{\partial x_{1}} - \frac{\partial u_{2}}{\partial x_{2}}) \sin 2 θ {(\frac{\partial θ}{\partial x_{i}})}^{2} ψ d x \\ + \int_{M} (\frac{\partial u_{1}}{\partial x_{1}} - \frac{\partial u_{2}}{\partial x_{2}}) \cos 2 θ A_{1} θ ψ d x \\ + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial^{2} u_{1}}{\partial x_{i} \partial x_{2}} \sin 2 θ \frac{\partial θ}{\partial x_{i}} ψ d x + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial u_{1}}{\partial x_{2}} \cos 2 θ {(\frac{\partial θ}{\partial x_{i}})}^{2} ψ d x \\ + \int_{M} \frac{\partial u_{1}}{\partial x_{2}} \sin 2 θ A_{1} θ ψ d x + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial^{2} u_{2}}{\partial x_{i} \partial x_{2}} \sin 2 θ \frac{\partial θ}{\partial x_{i}} ψ d x \\ + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial u_{2}}{\partial x_{1}} \cos 2 θ {(\frac{\partial θ}{\partial x_{i}})}^{2} ψ d x + \int_{M} \frac{\partial u_{2}}{\partial x_{1}} \sin 2 θ A_{1} θ ψ d x, \end{aligned}$
(3.25)
and thus
$\begin{aligned} ⟨ F_{1} (u^{n}, θ^{n}), A_{1}^{2} θ^{n} ⟩ - ⟨ F_{1} (A_{0} u^{n}, θ^{n}), A_{1} θ^{n} ⟩ \\ = 2 \overset{2}{\sum_{i = 1}} \int_{M} (\frac{\partial^{2} u_{1}^{n}}{\partial x_{i} \partial x_{1}} - \frac{\partial^{2} u_{2}^{n}}{\partial x_{i} \partial x_{2}^{n}}) \cos 2 θ^{n} \frac{\partial θ^{n}}{\partial x_{i}} A_{1} θ^{n} d x \\ - 2 \overset{2}{\sum_{i = 1}} \int_{M} (\frac{\partial u_{1}^{n}}{\partial x_{1}} - \frac{\partial u_{2}^{n}}{\partial x_{2}}) \sin 2 θ^{n} {(\frac{\partial θ^{n}}{\partial x_{i}})}^{2} A_{1} θ^{n} d x \\ + \int_{M} (\frac{\partial u_{1}^{n}}{\partial x_{1}} - \frac{\partial u_{2}^{n}}{\partial x_{2}}) \cos 2 θ^{n} (A_{1} θ^{n})^{2} d x \\ + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial^{2} u_{1}^{n}}{\partial x_{i} \partial x_{2}} \sin 2 θ^{n} \frac{\partial θ^{n}}{\partial x_{i}} A_{1} θ^{n} d x + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial u_{1}^{n}}{\partial x_{2}} \cos 2 θ^{n} {(\frac{\partial θ^{n}}{\partial x_{i}})}^{2} A_{1} θ^{n} d x \\ + \int_{M} \frac{\partial u_{1}^{n}}{\partial x_{2}} \sin 2 θ^{n} (A_{1} θ^{n})^{2} d x + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial^{2} u_{2}^{n}}{\partial x_{i} \partial x_{2}} \sin 2 θ^{n} \frac{\partial θ^{n}}{\partial x_{i}} A_{1} θ^{n} d x \\ + 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial u_{2}^{n}}{\partial x_{1}} \cos 2 θ^{n} {(\frac{\partial θ^{n}}{\partial x_{i}})}^{2} A_{1} θ^{n} d x + \int_{M} \frac{\partial u_{2}^{n}}{\partial x_{1}} \sin 2 θ^{n} (A_{1} θ^{n})^{2} d x . \end{aligned}$
(3.26)
The first, fourth and seventh terms on the right-hand side are bounded as follows
$\begin{aligned} 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial^{2} u_{1}^{n}}{\partial x_{i} \partial x_{2}} \sin 2 θ^{n} \frac{\partial θ^{n}}{\partial x_{i}} A_{1} θ^{n} d x \\ ⩽ 2 {| A_{0} u^{n} |}_{L^{2}} ‖ \nabla θ^{n} ‖_{L^{4}} ‖ A_{1} θ^{n} ‖_{L^{4}} \\ ⩽ c {| A_{0} u^{n} |}_{L^{2}} {| \nabla θ^{n} |}_{L^{2}}^{1 / 2} ‖ \nabla θ^{n} ‖^{1 / 2} {| A_{1} θ^{n} |}_{L^{2}}^{1 / 2} ‖ A_{1} θ^{n} ‖^{1 / 2} \\ ⩽ c {| A_{0} u^{n} |}_{L^{2}} ‖ θ^{n} ‖^{1 / 2} {| A_{1} θ^{n} |}_{L^{2}} {| A_{1}^{3 / 2} θ^{n} |}^{1 / 2} \\ ⩽ \frac{μ}{5} {| A_{0} u^{n} |}_{L^{2}}^{2} + \frac{α}{6} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} + \frac{c}{μ^{2} α} ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{4} \end{aligned}$
(3.27)
The second, fifth and eighth terms on the right-hand side of (3.26) are bounded as follows
$\begin{aligned} 2 \overset{2}{\sum_{i = 1}} \int_{M} \frac{\partial u_{1}^{n}}{\partial x_{2}} \cos 2 θ^{n} {(\frac{\partial θ^{n}}{\partial x_{i}})}^{2} A_{1} θ^{n} d x & ⩽ 2 {| \nabla u^{n} |}_{L^{2}} {| \nabla θ^{n} |}_{L^{\infty}}^{2} {| A_{1} θ^{n} |}_{L^{2}} \\ ⩽ c ‖ u^{n} ‖ ‖ θ^{n} ‖ {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}} {| A_{1} θ^{n} |}_{L^{2}} \\ (by\;\; Agmon' s\;\; inequality) \\ ⩽ \frac{α}{6} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} + \frac{c}{α} ‖ u^{n} ‖^{2} ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.28)
The third, sixth and ninth terms on the right-hand side of (3.26) are bounded as follows
$\begin{aligned} \int_{M} \frac{\partial u_{1}^{n}}{\partial x_{2}} \sin 2 θ^{n} (A_{1} θ^{n})^{2} d x & ⩽ {| \nabla u^{n} |}_{L^{2}} ‖ A_{1} θ^{n} ‖_{L^{4}}^{2} ⩽ ‖ u^{n} ‖ {| A_{1} θ^{n} |}_{L^{2}} ‖ A_{1} θ^{n} ‖ \\ ⩽ c ‖ u^{n} ‖ {| A_{1} θ^{n} |}_{L^{2}} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}} \\ ⩽ \frac{α}{6} {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} + \frac{c}{α} ‖ u^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.29)
Combining (3.26)–(3.29) we obtain
$\begin{aligned} 2 γ Δ t | ⟨ F_{1} (u^{n}, θ^{n}), A_{1}^{2} θ^{n} ⟩ - ⟨ F_{1} (A_{0} u^{n}, θ^{n}), A_{1} θ^{n} ⟩ | \\ ⩽ \frac{μ}{5} Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + \frac{α}{2} Δ t {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} \\ + \frac{c}{μ^{2} α} Δ t ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{4} + \frac{c}{α} Δ t ‖ u^{n} ‖^{2} ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} + \frac{c}{α} Δ t ‖ u^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.30)
Relations (3.19)–(3.24) and (3.30) yield
$\begin{aligned} ‖ u^{n} ‖^{2} + {| A_{1} θ^{n} |}_{L^{2}}^{2} - (‖ u^{n - 1} ‖^{2} + {| A_{1} θ^{n - 1} |}_{L^{2}}^{2}) + ‖ u^{n} - u^{n - 1} ‖^{2} + {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2} \\ + μ Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + α Δ t {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} \\ ⩽ \frac{c c_{b}^{4}}{μ^{3}} Δ t {| u^{n} |}_{L^{2}}^{2} ‖ u^{n} ‖^{4} + \frac{c}{μ^{2} α} Δ t ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{4} \\ + \frac{c}{α} Δ t ‖ u^{n} ‖^{2} ‖ θ^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} + \frac{c}{α^{3}} Δ t {| u^{n} |}_{L^{2}}^{2} ‖ u^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} + + \frac{c}{α} Δ t ‖ u^{n} ‖^{2} {| A_{1} θ^{n} |}_{L^{2}}^{2} \\ + \frac{5}{μ} Δ t {| f^{n} |}_{L^{2}}^{2} + \frac{6}{α} Δ t {| \nabla g^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.31)
Taking
$\begin{aligned} c_{1} = c (\frac{c_{b}^{4}}{μ^{3}} + \frac{1}{μ^{2} α} + \frac{1}{α^{3}} + \frac{1}{α}), \end{aligned}$
(3.32)
and recalling (3.2), we obtain
$\begin{aligned} ‖ u^{n} ‖^{2} + {| A_{1} θ^{n} |}_{L^{2}}^{2} - (‖ u^{n - 1} ‖^{2} + {| A_{1} θ^{n - 1} |}_{L^{2}}^{2}) + ‖ u^{n} - u^{n - 1} ‖^{2} + {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2} \\ + μ Δ t {| A_{0} u^{n} |}_{L^{2}}^{2} + α Δ t {| A_{1}^{3 / 2} θ^{n} |}_{L^{2}}^{2} \\ ⩽ c_{1} (1 + K_{1}) Δ t ‖ (u^{n}, θ^{n}) ‖_{U}^{4} + \frac{5}{μ} Δ t ‖ f ‖_{\infty}^{2} + \frac{6}{α} Δ t ‖ g ‖_{\infty}^{2}, \end{aligned}$
(3.33)
from which (3.17) follows right away. Adding inequalities (3.33) with n from i to m and dropping some positive terms, we obtain (3.18).
Lemma 3.
For every $Δ t > 0$ and for every $n ⩾ 1$ , we have
$\begin{aligned} {‖ (u^{n}, θ^{n}) ‖}_{U}^{2} ⩽ K_{2} {‖ (u^{n - 1}, θ^{n - 1}) ‖}_{U}^{2} + c_{2} (‖ f ‖_{\infty}^{2} + ‖ g ‖_{\infty}^{2}), \end{aligned}$
(3.34)
where $K_{2} = K_{2} (| (u^{0}, θ^{0}) |_{H}, ‖ f ‖_{\infty}, ‖ g ‖_{\infty})$ and $c_{2}$ are given below in (3.43) and (3.44), respectively.
Proof.
Multiplying the first equation of (2.39) by $2 Δ t (u^{n} - u^{n - 1})$ , the second equation by $2 Δ t A_{1} (θ^{n} - θ^{n - 1})$ , adding the resulting equations and integrating, we obtain (recalling (2.28) and (2.33))
$\begin{aligned} 2 | u^{n} - u^{n - 1} |_{L^{2}}^{2} + 2 ‖ θ^{n} - θ^{n - 1} ‖^{2} + μ Δ t ‖ u^{n} ‖^{2} + α Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} \\ - μ Δ t ‖ u^{n - 1} ‖^{2} - α Δ t | A_{1} θ^{n - 1} |_{L^{2}}^{2} + μ Δ t ‖ u^{n} - u^{n - 1} ‖^{2} + α Δ t | A_{1} (θ^{n} - θ^{n - 1}) |_{L^{2}}^{2} \\ + 2 Δ t b_{0} (u^{n}, u^{n}, u^{n} - u^{n - 1}) - 2 Δ t b_{1} (u^{n} - u^{n - 1}, θ^{n}, A_{1} θ^{n}) \\ + 2 Δ t b_{1} (u^{n}, θ^{n}, A_{1} (θ^{n} - θ^{n - 1})) \\ - 2 γ Δ t ⟨ F_{1} (u^{n} - u^{n - 1}, θ^{n}), A_{1} θ^{n} ⟩ + 2 γ Δ t ⟨ F_{1} (u^{n}, θ^{n}), A_{1} (θ^{n} - θ^{n - 1}) ⟩ \\ = 2 Δ t (f^{n}, u^{n} - u^{n - 1}) + 2 Δ t (g^{n}, A_{1} (θ^{n} - θ^{n - 1})) . \end{aligned}$
(3.35)
Using the Cauchy–Schwarz inequality, we majorize the right-hand side of (3.35) by
$\begin{aligned} 2 Δ t (f^{n}, u^{n} - u^{n - 1}) & ⩽ 2 Δ t {| f^{n} |}_{L^{2}} {| u^{n} - u^{n - 1} |}_{L^{2}} \\ ⩽ 2 \sqrt{c_{M}} Δ t {| f^{n} |}_{L^{2}} ‖ u^{n} - u^{n - 1} ‖ (by (2 .10)) \\ ⩽ \frac{μ}{4} Δ t ‖ u^{n} - u^{n - 1} ‖^{2} + \frac{4 c_{M}}{μ} Δ t {| f^{n} |}_{L^{2}}^{2}, \end{aligned}$
(3.36)

$\begin{aligned} 2 Δ t {(g^{n}, A_{1} (θ^{n} - θ^{n - 1}))}_{L^{2}} | & ⩽ 2 Δ t {| g^{n} |}_{L^{2}} {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}} \\ ⩽ \frac{α}{4} Δ t {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2} + \frac{4}{α} Δ t {| g^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.37)
We bound the nonlinear terms as follows:
$\begin{aligned} 2 Δ t b_{0} (u^{n}, u^{n}, u^{n} - u^{n - 1}) & = - 2 Δ t b_{0} (u^{n}, u^{n}, u^{n - 1}) (by (2 .19)) \\ ⩽ 2 c_{b} Δ t {| u^{n} |}_{L^{2}} ‖ u^{n} ‖ ‖ u^{n - 1} ‖ (by (2 .16)) \\ ⩽ \frac{μ}{4} Δ t ‖ u^{n} ‖^{2} + \frac{4 c_{b}^{2}}{μ} K_{1} Δ t ‖ u^{n - 1} ‖^{2} (by (3 .2)), \end{aligned}$
(3.38)

$\begin{aligned} - 2 Δ t b_{1} (u^{n} - u^{n - 1}, θ^{n}, A_{1} θ^{n}) + 2 Δ t b_{1} (u^{n}, θ^{n}, A_{1} (θ^{n} - θ^{n - 1})) \\ = - 2 Δ t b_{1} (u^{n} - u^{n - 1}, θ^{n}, A_{1} θ^{n - 1}) + 2 Δ t b_{1} (u^{n - 1}, ϕ^{n}, A_{1} (θ^{n} - θ^{n - 1})) \\ ⩽ 2 c_{b} Δ t {| u^{n} - u^{n - 1} |}_{L^{2}}^{1 / 2} ‖ u^{n} - u^{n - 1} ‖^{1 / 2} ‖ θ^{n} ‖^{1 / 2} {| A_{1} θ^{n} |}_{L^{2}}^{1 / 2} {| A_{1} θ^{n - 1} |}_{L^{2}} \\ + 2 c_{b} Δ t {| u^{n - 1} |}_{L^{2}}^{1 / 2} ‖ u^{n - 1} ‖^{1 / 2} ‖ θ^{n} ‖^{1 / 2} {| A_{1} θ^{n} |}_{L^{2}}^{1 / 2} {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}} (by (2 .23)) \\ ⩽ \frac{μ}{4} Δ t ‖ u^{n} - u^{n - 1} ‖^{2} + \frac{α}{4} Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} + \frac{c}{\sqrt{μ α}} c_{b}^{2} K_{1} Δ t {| A_{1} θ^{n - 1} |}_{L^{2}}^{2} \\ + \frac{α}{4} Δ t {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2} + \frac{c}{α^{3}} c_{b}^{4} K_{1}^{2} Δ t ‖ u^{n - 1} ‖^{2} (by (3 .2)), \end{aligned}$
(3.39)

$\begin{aligned} - 2 γ Δ t ⟨ F_{1} (u^{n} - u^{n - 1}, θ^{n}), A_{1} θ^{n} ⟩ + 2 γ Δ t ⟨ F_{1} (u^{n}, θ^{n}), A_{1} (θ^{n} - θ^{n - 1}) ⟩ \\ = 2 γ Δ t ⟨ F_{1} (u^{n - 1}, θ^{n}), A_{1} θ^{n} ⟩ - 2 γ Δ t ⟨ F_{1} (u^{n}, θ^{n}), A_{1} θ^{n - 1} ⟩ \\ ⩽ 2 γ Δ t | F_{1} (u^{n - 1}, θ^{n}) | | A_{1} θ^{n} | + 2 γ Δ t | F_{1} (u^{n}, θ^{n}) | | A_{1} θ^{n - 1} | \\ ⩽ c γ Δ t ‖ u^{n - 1} ‖ | A_{1} θ^{n} | + c γ Δ t ‖ u^{n} ‖ | A_{1} θ^{n - 1} | (by (2 .35)) \\ ⩽ \frac{α}{4} Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} + \frac{c γ^{2}}{α} Δ t ‖ u^{n - 1} ‖^{2} + \frac{μ}{4} Δ t ‖ u^{n} ‖^{2} + \frac{c γ^{2}}{μ} Δ t {| A_{1} θ^{n - 1} |}^{2} . \end{aligned}$
(3.40)
Relations (3.35)–(3.40) yield
$\begin{aligned} 2 {| u^{n} - u^{n - 1} |}_{L^{2}}^{2} + 2 ‖ θ^{n} - θ^{n - 1} ‖^{2} + \frac{μ}{2} Δ t ‖ u^{n} ‖^{2} + \frac{α}{2} Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} \\ - (μ + \frac{4 c_{b}^{2}}{μ} K_{1} + \frac{c}{α^{3}} c_{b}^{4} K_{1}^{2} + \frac{c γ^{2}}{α}) Δ t ‖ u^{n - 1} ‖^{2} \\ - (α + \frac{c}{\sqrt{μ α}} c_{b}^{2} K_{1} + \frac{c γ^{2}}{μ}) Δ t {| A_{1} θ^{n - 1} |}_{L^{2}}^{2} \\ + \frac{μ}{2} Δ t ‖ u^{n} - u^{n - 1} ‖^{2} + \frac{α}{2} Δ t {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2} \\ ⩽ \frac{4 c_{M}}{μ} Δ t {| f^{n} |}_{L^{2}}^{2} + \frac{4}{α} Δ t {| g^{n} |}_{L^{2}}^{2}, \end{aligned}$
(3.41)
and neglecting some positive terms we obtain
$\begin{aligned} \frac{μ}{2} Δ t ‖ u^{n} ‖^{2} + \frac{α}{2} Δ t {| A_{1} θ^{n} |}_{L^{2}}^{2} & ⩽ (μ + \frac{4 c_{b}^{2}}{μ} K_{1} + \frac{c}{α^{3}} c_{b}^{4} K_{1}^{2} + \frac{c γ^{2}}{α}) Δ t ‖ u^{n - 1} ‖^{2} \\ + (α + \frac{c}{\sqrt{μ α}} c_{b}^{2} K_{1} + \frac{c γ^{2}}{μ}) Δ t {| A_{1} θ^{n - 1} |}_{L^{2}}^{2} \\ + \frac{4 c_{M}}{μ} Δ t {| f^{n} |}_{L^{2}}^{2} + \frac{4}{α} Δ t {| g^{n} |}_{L^{2}}^{2} . \end{aligned}$
(3.42)
Taking
$\begin{aligned} K_{2} & = K_{2} ({| (u^{0}, θ^{0}) |}_{H}, ‖ f ‖_{\infty}, ‖ g ‖_{\infty}) \\ = 2 (\frac{1}{μ} + \frac{1}{α}) (μ + \frac{4 c_{b}^{2}}{μ} K_{1} + \frac{c}{α^{3}} c_{b}^{4} K_{1}^{2} + \frac{c γ^{2}}{α} + α + \frac{c}{\sqrt{μ α}} c_{b}^{2} K_{1} + \frac{c γ^{2}}{μ}), \end{aligned}$
(3.43)

$\begin{aligned} c_{2} & = 8 (\frac{1}{μ} + \frac{1}{α}) (\frac{c_{M}}{μ} + \frac{1}{α}), \end{aligned}$
(3.44)
we obtain conclusion (3.34) of the lemma.

Combining the results of Lemma 2 and Lemma 3, we can prove the uniform boundedness of $‖ (u^{n}, ϕ^{n}) ‖_{U}$ . Before we do that, we recall the (discrete Gronwall) Lemma 4, as well as the (discrete uniform Gronwall) Lemma 4, whose proofs can be found in [41] or [45]:
Lemma 4.
Given $Δ t > 0$ and positive sequences $ξ_{n}$ , $η_{n}$ and $ζ_{n}$ such that
$\begin{aligned} ξ_{n} ⩽ ξ_{n - 1} (1 + Δ t η_{n - 1}) + Δ t ζ_{n}, f o r n ⩾ 1, \end{aligned}$
(3.45)
we have, for any $n ⩾ 2$ ,
$\begin{aligned} ξ_{n} ⩽ (ξ_{0} + Δ t \overset{n}{\sum_{i = 1}} ζ_{i}) \exp (Δ t \overset{n - 1}{\sum_{i = 0}} η_{i}) . \end{aligned}$
(3.46)

Lemma 5.
Given $Δ t > 0$ , positive integers $n_{0}$ , N, positive sequences $ξ_{n}$ , $η_{n}$ and $ζ_{n}$ such that
$\begin{aligned} ξ_{n} ⩽ ξ_{n - 1} (1 + Δ t η_{n - 1}) + Δ t ζ_{n}, for~ n ⩾ n_{0}, \end{aligned}$
(3.47)
and given the bounds
$\begin{aligned} Δ t \overset{N + k_{0}}{\sum_{n = k_{0}}} η_{n} ⩽ a_{1}, Δ t \overset{N + k_{0}}{\sum_{n = k_{0}}} ζ_{n} ⩽ a_{2}, Δ t \overset{N + k_{0}}{\sum_{n = k_{0}}} ξ_{n} ⩽ a_{3}, \end{aligned}$
(3.48)
for any $k_{0} ⩾ n_{0}$ , we have,
$\begin{aligned} ξ_{n} ⩽ (\frac{a_{3}}{N Δ t} + a_{2}) e^{a_{1}}, \forall n ⩾ N + n_{0} + 1. \end{aligned}$
(3.49)

We are now in a position to derive a uniform bound for $‖ (u^{n}, ϕ^{n}) ‖_{U}$ , for all $n ⩾ 1$ . Theorem 1.
Let $(u^{0}, ϕ^{0}) \in U$ and $(u^{n}, ϕ^{n})$ be a solution of the numerical scheme (2.39). Also, let $Δ t$ be such that $Δ t ⩽ κ_{1}$ , where $κ_{1}$ is given in (3.15). Then there exists $K_{5} (‖ (u^{0}, θ^{0}) ‖_{U})$ , such that
$\begin{aligned} ‖ (u^{n}, θ^{n}) ‖_{U} ⩽ K_{5} (‖ (u^{0}, θ^{0}) ‖_{U}), \forall n ⩾ 0, \end{aligned}$
(3.50)
and for all $i = 1, \dots, m$ , we have
$\begin{aligned} \overset{m}{\sum_{n = i}} (‖ u^{n} - u^{n - 1} ‖^{2} + {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2}) \\ ⩽ K_{5}^{2} + (c_{1} (1 + K_{1}) K_{5}^{4} + \frac{5}{μ} ‖ f ‖_{\infty}^{2} + \frac{6}{α} ‖ g ‖_{\infty}^{2}) (m - i + 1) Δ t . \end{aligned}$
(3.51)

Moreover, for any initial data $(u^{0}, θ^{0}) \in H$ , there exists $K_{4}$ , independent of the initial data, such that
$\begin{aligned} ‖ (u^{n}, θ^{n}) ‖_{U} ⩽ K_{4}, \forall n ⩾ N + N_{0} + 2, \end{aligned}$
(3.52)
where $N_{0} := ⌊ T_{0} / Δ t ⌋$ , with $T_{0}$ being given in (3.16).
Proof.
Let $T > 0$ be arbitrarily fixed and let $Δ t ⩽ κ_{1}$ . We set $N := ⌊ T / Δ t ⌋$ .

In order to derive a uniform bound for $‖ (u^{n}, θ^{n}) ‖_{U}$ for all $n ⩾ 1$ , we will apply (the discrete Gronwall) Lemma 4 to obtain a bound valid for $n = 1, \dots, N + N_{0} + 1$ , and then we will apply (the discrete uniform Gronwall) Lemma 5 to obtain a bound valid for $n ⩾ N + N_{0} + 2$ . In doing so, we first notice that using (3.34), (3.17) yields
$\begin{aligned} ‖ (u^{n}, θ^{n}) ‖_{U}^{2} ⩽ ‖ (u^{n - 1}, θ^{n - 1}) ‖_{U}^{2} + c_{1} (1 + K_{1}) Δ t ‖ (u^{n}, θ^{n}) ‖_{U}^{4} + \frac{5}{μ} Δ t ‖ f ‖_{\infty}^{2} + \frac{6}{α} Δ t ‖ g ‖_{\infty}^{2} \\ ⩽ ‖ (u^{n - 1}, θ^{n - 1}) ‖_{U}^{2} + c_{1} (1 + K_{1}) Δ t [K_{2} ‖ (u^{n - 1}, θ^{n - 1}) ‖_{U}^{2} + c_{2} (‖ f ‖_{\infty}^{2} + ‖ g ‖_{\infty}^{2})]^{2} \\ + \frac{5}{μ} Δ t ‖ f ‖_{\infty}^{2} + \frac{6}{α} Δ t ‖ g ‖_{\infty}^{2} \\ ⩽ ‖ (u^{n - 1}, θ^{n - 1}) ‖_{U}^{2} [1 + 2 c_{1} (1 + K_{1}) Δ t K_{2}^{2} ‖ (u^{n - 1}, θ^{n - 1}) ‖_{U}^{2}] \\ + 2 c_{1} c_{2}^{2} (1 + K_{1}) Δ t (‖ f ‖_{\infty}^{2} + ‖ g ‖_{\infty}^{2})^{2} + \frac{5}{μ} Δ t ‖ f ‖_{\infty}^{2} + \frac{6}{α} Δ t ‖ g ‖_{\infty}^{2}, \end{aligned}$
(3.53)
which we rewrite in the form
$\begin{aligned} ξ_{n} ⩽ ξ_{n - 1} (1 + Δ t η_{n - 1}) + Δ t ζ_{n}, \end{aligned}$
(3.54)
with
$\begin{aligned} ξ_{n} = ‖ (u^{n}, ϕ^{n}) ‖_{U}^{2}, η_{n} = 2 c_{1} (1 + K_{1}) K_{2}^{2} ‖ (u^{n}, θ^{n}) ‖_{U}^{2}, \\ ζ_{n} = 2 c_{1} c_{2}^{2} (1 + K_{1}) (‖ f ‖_{\infty}^{2} + ‖ g ‖_{\infty}^{2})^{2} + \frac{5}{μ} ‖ f ‖_{\infty}^{2} + \frac{6}{α} ‖ g ‖_{\infty}^{2} . \end{aligned}$
(3.55)
Applying Lemma 4 and recalling (3.3), we obtain
$\begin{aligned} ξ_{n} & = ‖ (u^{n}, ϕ^{n}) ‖_{U}^{2} ⩽ K_{3}^{2} (‖ (u^{0}, ϕ^{0}) ‖_{U}, ‖ f ‖_{\infty}, ‖ g ‖_{\infty}, (N + N_{0}) Δ t), \\ \forall n & = 1, \dots, N + N_{0} + 1, \end{aligned}$
(3.56)
for some continuous function $K_{3} (\cdot, \cdot, \cdot)$ , increasing in all its arguments.

We now apply Lemma 5, with $n_{0} = N_{0} + 1$ . In computing the sums $a_{1}$ , $a_{2}$ and $a_{3}$ that appear there, we note that since all those sums are taken for $n ⩾ N_{0} + 1$ and since, by hypothesis, $Δ t$ satisfies condition (3.15) of Corollary 3.1, we can replace $K_{1}$ , the bound on $| u^{n} |_{L^{2}}^{2}$ , by $2 ρ_{0}^{2}$ , whenever the former appears. We thus obtain
$\begin{aligned} ξ_{n} = {‖ (u^{n}, ϕ^{n}) ‖}_{U}^{2} ⩽ (\frac{a_{3}}{T} + a_{2}) e^{a_{1}} =: K_{4}^{2} (T), \forall n ⩾ N + N_{0} + 2, \end{aligned}$
(3.57)
which proves (3.52). Combining (3.57) and (3.56), we obtain (3.50).

Taking the sum of (3.31) with n from i to m and using (3.50) we obtain conclusion (3.51).

The theorem has been proved.

4. .Convergence of attractors

In this section we address the issue of the convergence of the attractors generated by the discrete system (2.39) to the attractor generated by the continuous system (2.36) when $γ = 0$ . Using Theorem 1, one can prove the uniqueness of the solution of (2.39) provided that $k ⩽ κ (‖ (u^{0}, θ_{0}) ‖_{U})$ , for some $κ (‖ (u^{0}, θ_{0}) ‖_{U}) > 0$ . This dependence of the time restriction on the initial data prevents us from defining a single-valued attractor in the classical sense. This is why we need to use the attractor theory for the so-called multi-valued mappings. Multi-valued dynamical systems have been investigated by many authors (see, e.g., [1,2,7,38–40]), but in this article we use the tools developed in [9] (see also, [12]) to study the convergence of the discrete (multi-valued) attractors to the continuous (single-valued) attractor.

The main tool in proving the convergence of the discrete attractors to the continuous attractor is the following theorem, whose proof can be found in [9] (see also [12,44,48]).

Theorem 2.
Let $S (t)$ be a closed multi-valued semigroup, possessing the global attractor $A$ , and for $κ_{0} > 0$ , let ${S_{k}, 0 < k ⩽ κ_{0}}_{n \in N}$ be a family of discrete closed multi-valued semigroups, with global attractor $A_{k}$ . Assume the following: (H1)
[Uniform boundedness]: there exists $κ_{1} \in (0, κ_{0}]$ such that the set
$\begin{aligned} K = ⋃_{k \in (0, κ_{1}]} A_{k} \end{aligned}$
is bounded in H;
(H2)
[Finite time uniform convergence]: there exists $t_{0} ⩾ 0$ such that for any $T^{⋆} > t_{0}$ ,
$\begin{aligned} lim_{k \to 0} sup_{x \in A_{k}, n k \in [t_{0}, T^{⋆}]} | S_{k}^{n} x - S (n k) x | = 0. \end{aligned}$

Then
$\begin{aligned} lim_{k \to 0} dist (A_{k}, A) = 0, \end{aligned}$
where dist denotes the Hausdorff semidistance defined as
$\begin{aligned} dist (B, C) = sup_{b \in B} inf_{c \in C} | b - c |, \forall B, C \subset H . \end{aligned}$
(4.1)

Hereafter we assume $γ = 0$ and we prove that there exists a time-step $κ_{1} > 0$ such that if $0 < k ⩽ κ_{1}$ , the system (2.39) generates a closed discrete m-semigroup ${S_{k}}_{k \in N}$ , with global attractors $A_{k}$ , that converges to $A$ in the sense of Theorem 2.

To that end, we define for $Δ t > 0$ the multi-valued map $S_{k} : 2^{H} \to 2^{H}$ as follows: for every $\tilde{v} = (\tilde{u}, \tilde{θ}) \in H$ ,
$\begin{aligned} S_{k} \tilde{v} = {v = (u, θ) \in U : v s o l v e s (4.3) b e l o w w i t h t i m e - s t e p k} : \end{aligned}$
(4.2)

$\begin{aligned} {\begin{cases} u + μ Δ t A_{0} u + Δ t B_{0} (u, u) + Δ t R_{0} (θ, θ) = \tilde{u} + Δ t f, \\ θ + α Δ t A_{1} θ + Δ t B_{1} (u, θ) = \tilde{θ} + Δ t g . \end{cases} \end{aligned}$
(4.3)

As in [9] (see also [12,36]), one can prove the following
Theorem 3.
For any $k > 0$ , the multivalued map $S_{k}$ associated to the implicit Euler scheme (2.39) generates a closed discrete multivalued semigroup ${S_{k}}_{n \in N}$ .

As a consequence of Theorem 1, we have
Theorem 4.
Let $k ⩽ κ_{1}$ , where $κ_{1}$ is given in Corollary 3.1. Then there exists a constant $R_{1} > 0$ such that for every $R ⩾ 0$ and $‖ (u^{0}, θ^{0}) ‖_{H} ⩽ R$ , there exists $N_{1} = N_{1} (R, Δ t) ⩾ 0$ such that
$\begin{aligned} ‖ S_{k}^{n} (u^{0}, θ^{0}) ‖_{U} ⩽ R_{1}, \forall n ⩾ N_{1} . \end{aligned}$
(4.4)
Thus, the set
$\begin{aligned} B_{1} = {(u, θ) \in U : ‖ (u, θ) ‖_{U} ⩽ R_{1}} \end{aligned}$
is a $U$ -bounded absorbing set for ${S_{k}}_{n \in N}$ , for $k \in (0, κ_{1}]$ .

Theorem 4 gives the existence of the discrete global attractors:
Proposition 1.
For every $k \in (0, κ_{1}]$ , there exists the global attractor $A_{k}$ of the m-semigroup ${S_{k}}_{n \in N}$ .

Since the global attractor $A_{k}$ is the smallest closed attracting set of H, Theorem 4 implies
$\begin{aligned} A_{k} \subset B_{1}, \forall k \in (0, κ_{1}], \end{aligned}$
(4.5)
and thus
$\begin{aligned} ⋃_{k \in (0, κ_{1}]} A_{k} \subset B_{1}, \end{aligned}$
(4.6)
and thus condition ( $H 1$ ) of Theorem 2 is satisfied. To prove that condition ( $H 2$ ) is satisfied, we define, for any $k = Δ t > 0$ , the following functions:
$\begin{aligned} u_{k} (t) & = u^{n}, t \in [(n - 1) k, n k), \end{aligned}$
(4.7)

$\begin{aligned} {\tilde{u}}_{k} (t) & = u^{n} + \frac{t - n k}{k} (u^{n} - u^{n - 1}), t \in [(n - 1) k, n k) . \end{aligned}$
(4.8)
Note that ${\tilde{u}}_{k} (n k) = u^{n}$ . Also, using the above notations, we can write the system (2.39) as follows, for $t \in [(n - 1) k, n k)$ :
$\begin{aligned} {\begin{cases} \frac{d {\tilde{u}}_{k} (t)}{d t} + μ A_{0} {\tilde{u}}_{k} (t) + B_{0} ({\tilde{u}}_{k} (t), {\tilde{u}}_{k} (t)) + R_{0} ({\tilde{θ}}_{k} (t), {\tilde{θ}}_{k} (t)) = f + f_{k} (t), \\ \frac{d {\tilde{θ}}_{k} (t)}{d t} + α A_{1} {\tilde{θ}}_{k} (t) + + B_{1} ({\tilde{u}}_{k} (t), {\tilde{θ}}_{k} (t)) = g + g_{k}, \end{cases} \end{aligned}$
(4.9)
where
$\begin{aligned} f_{k} (t) & = μ A ({\tilde{u}}_{k} (t) - u_{k} (t)) + B_{0} ({\tilde{u}}_{k} (t) - u_{k} (t), {\tilde{u}}_{k} (t)) + B_{0} (u_{k} (t), {\tilde{u}}_{k} (t) - u_{k} (t)) \\ + R_{0} ({\tilde{θ}}_{k} (t) - θ_{k} (t), {\tilde{θ}}_{k} (t)) + R_{0} (θ_{k} (t), {\tilde{θ}}_{k} (t) - θ_{k} (t)), \end{aligned}$
(4.10)

$\begin{aligned} g_{k} (t) & = α A_{1} ({\tilde{θ}}_{k} (t) - θ_{k} (t)) + B_{1} ({\tilde{u}}_{k} (t) - u_{k} (t), {\tilde{θ}}_{k} (t)) + B_{1} (u_{k} (t), {\tilde{θ}}_{k} (t) - θ_{k} (t)) . \end{aligned}$
(4.11)
Note that $f_{k}$ and $g_{k}$ satisfy the following
Lemma 6.
Let $T^{} > 0$ be arbitrarily fixed and let* $k < κ_{1}$ , where $κ_{1}$ is given in Corollary 3.1. Assume that $(u^{0}, θ^{0}) \in A_{k}$ and let $(u^{n}, θ^{n})$ be the solution of the numerical scheme (2.39). Then there exist $K_{6} (T^{})$ and* $K_{7} (T^{})$ such that*
$\begin{aligned} ‖ f_{k} ‖_{L^{2} (0, T^{}; {V_{1}}^{^{'}})}^{2} ⩽ Δ t K_{6} (T^{}), \end{aligned}$
(4.12)
and
$\begin{aligned} ‖ g_{k} ‖_{L^{2} (0, T^{}; D (A_{1})^{^{'}})}^{2} ⩽ Δ t K_{7} (T^{}) . \end{aligned}$
(4.13)

Proof.
Let $v \in V_{1}$ be such that $‖ v ‖ ⩽ 1$ , and let $t \in [(n - 1) k, n k)$ be fixed. Since $(u^{0}, θ^{0}) \in A_{k}$ , we have that $‖ (u^{0}, ϕ_{0}) ‖_{U} ⩽ R_{1}$ (by (4.6)) and then by Theorem 1 we obtain that
$\begin{aligned} ‖ (u^{n}, θ^{n}) ‖_{U} ⩽ K_{5} (‖ (u^{0}, θ^{0}) ‖_{U}) ⩽ K_{5} (R_{1}) =: C, \forall n ⩾ 0. \end{aligned}$
(4.14)
We thus have
$\begin{aligned} ‖ (u_{k} (t), θ_{k} (t)) ‖_{U} = ‖ (u^{n}, θ^{n}) ‖_{U} ⩽ C, \end{aligned}$
(4.15)
and (due to (4.8))
$\begin{aligned} ‖ {\tilde{u}}_{k} (t) ‖ ⩽ ‖ u^{n} ‖ + | \frac{t - n k}{k} | ‖ u^{n} - u^{n - 1} ‖ ⩽ C, \end{aligned}$
(4.16)

$\begin{aligned} {| A_{1} {\tilde{θ}}_{k} (t) |}_{L^{2}} ⩽ {| A_{1} θ^{n} |}_{L^{2}} + | \frac{t - n k}{k} | {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}} ⩽ C, \end{aligned}$
(4.17)
where the constant C in the last two lines above is different from C in Eq.’s (4.14) and (4.15).

As shown in [17], $S (t)$ satisfies the uniform energy estimate
$\begin{aligned} sup_{t ⩾ 0} sup_{(u^{0}, θ^{0}) \in B_{1}} ‖ S (t) (u^{0}, θ^{0}) ‖_{U} ⩽ C . \end{aligned}$
(4.18)

We thus have
$\begin{aligned} μ | (A ({\tilde{u}}_{k} (t) - u_{k} (t)), v) | ⩽ μ ‖ {\tilde{u}}_{k} (t) - u_{k} (t) ‖ ‖ v ‖ ⩽ μ ‖ u^{n} - u^{n - 1} ‖ . \end{aligned}$
(4.19)
Using (2.16) and (2.23), and recalling (3.50), we obtain
$\begin{aligned} | b_{0} ({\tilde{u}}_{k} (t) - u_{k} (t), {\tilde{u}}_{k} (t), v) + b_{0} (u_{k} (t), {\tilde{u}}_{k} (t) - u_{k} (t), v) | ⩽ C ‖ u^{n} - u^{n - 1} ‖ . \end{aligned}$
(4.20)
Recalling (2.27), we have
$\begin{aligned} | ⟨ R_{0} ({\tilde{θ}}_{k} (t) - θ_{k} (t), {\tilde{θ}}_{k} (t)) + R_{0} (θ_{k} (t), {\tilde{θ}}_{k} (t) - θ_{k} (t)), v ⟩ | \\ ⩽ c ({| A_{1} {\tilde{θ}}_{k} (t) |}_{L^{2}} + {| A_{1} θ_{k} (t) |}_{L^{2}}) {| A_{1} ({\tilde{θ}}_{k} (t) - θ_{k} (t)) |}_{L^{2}} \\ ⩽ C {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}} . \end{aligned}$
(4.21)
Relations (4.19)–(4.21) imply
$\begin{aligned} ‖ f_{k} (t) ‖_{{V_{1}}^{^{'}}} ⩽ C (‖ u^{n} - u^{n - 1} ‖ + {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}), \end{aligned}$
(4.22)
and thus, setting $N^{} = ⌊ T^{⋆} / k ⌋$ and recalling (4.14) and (3.18), we obtain
$\begin{aligned} ‖ f_{k} ‖_{L^{2} (0, T^{}; V_{1}^{'})}^{2} & = \int_{0}^{T^{⋆}} ‖ f_{k} (t) ‖_{V_{1}^{'}}^{2} d t ⩽ \overset{N^{} + 1}{\sum_{n = 1}} \int_{(n - 1) k}^{n k} ‖ f_{k} (t) ‖_{V_{1}^{'}}^{2} d t \\ ⩽ C k \overset{N^{⋆} + 1}{\sum_{n = 1}} (‖ u^{n} - u^{n - 1} ‖^{2} + {| A_{1} (θ^{n} - θ^{n - 1}) |}_{L^{2}}^{2}) ⩽ C k K_{6} (T^{}), \end{aligned}$
(4.23)
which proves (4.12).

Now let $ϕ \in D (A_{1})$ be such that $| A_{1} ϕ |_{L^{2}} ⩽ 1$ , and let $t \in [(n - 1) k, n k)$ be fixed. We have
$\begin{aligned} α | (A_{1} ({\tilde{θ}}_{k} (t) - θ_{k} (t)), ϕ) | ⩽ α {| {\tilde{θ}}_{k} (t) - θ_{k} (t) |}_{L^{2}} {| A_{1} ϕ |}_{L^{2}} ⩽ α {| θ^{n} - θ^{n - 1} |}_{L^{2}} . \end{aligned}$
(4.24)
Using (2.21) and recalling (4.15), we obtain
$\begin{aligned} | b_{1} ({\tilde{u}}_{k} (t) - u_{k} (t), {\tilde{θ}}_{k} (t), ϕ) | ⩽ C ‖ u^{n} - u^{n - 1} ‖, \end{aligned}$
(4.25)

$\begin{aligned} | b_{1} (u_{k} (t), {\tilde{θ}}_{k} (t) - θ_{k} (t), ϕ) | ⩽ C ‖ θ^{n} - θ^{n - 1} ‖ . \end{aligned}$
(4.26)
Relations (4.24)–(4.26) imply
$\begin{aligned} ‖ g_{k} (t) ‖_{D (A_{1})^{^{'}}} ⩽ C (‖ u^{n} - u^{n - 1} ‖ + ‖ θ^{n} - θ^{n - 1} ‖), \end{aligned}$
(4.27)
and recalling (4.14) and (3.18), we obtain (4.13). This completes the proof of the lemma.

We are now in a position to prove that condition (H2) of Theorem 2 is satisfied.
Proposition 2 Finite time uniform convergence

For any $T^{*} > 0$ we have

\begin{aligned} lim_{k \to 0} sup_{(u^{0}, ϕ^{0}) \in A_{k}, n k \in [0, T^{*}]} {‖ S_{k}^{n} (u^{0}, ϕ^{0}) - S (n k) (u^{0}, ϕ^{0}) ‖}_{H} = 0. \end{aligned}

(4.28)

Proof.

Subtracting (4.9) from (2.36) and setting

\begin{aligned} ξ_{k} (t) = u (t) - {\tilde{u}}_{k} (t), η_{k} (t) = θ (t) - {\tilde{θ}}_{k} (t), \end{aligned}

(4.29)

we obtain

\begin{aligned} {\begin{cases} \frac{d ξ_{k} (t)}{d t} + μ A_{0} ξ_{k} (t) + B_{0} (ξ_{k} (t), u (t)) + B_{0} ({\tilde{u}}_{k} (t), ξ_{k} (t)) \\ + R_{0} (η_{k} (t), θ (t)) + R_{0} ({\tilde{θ}}_{k} (t), η_{k} (t)) = - f_{k} (t), \\ \frac{d η_{k} (t)}{d t} + α A_{1} η_{k} (t) + B_{1} (ξ_{k} (t), θ (t)) + B_{1} ({\tilde{u}}_{k} (t), η_{k} (t)) = - g_{k} (t) . \end{cases} \end{aligned}

(4.30)

Multiplying the first equation of (4.30) by $ξ_{k} (t)$ and integrating we obtain

\begin{aligned} \frac{1}{2} \frac{d}{d t} {| ξ_{k} (t) |}_{L^{2}}^{2} + μ ‖ ξ_{k} (t) ‖^{2} + b_{0} (ξ_{k} (t), u (t), ξ_{k} (t)) \\ + ⟨ R_{0} (η_{k} (t), θ (t)) + R_{0} ({\tilde{θ}}_{k} (t), η_{k} (t)), ξ (t) ⟩ = - ⟨ f_{k} (t), ξ_{k} (t) ⟩ . \end{aligned}

(4.31)

Multiplying the second equation of (4.30) by $A_{1} η_{k} (t)$ and integrating we obtain

\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ η_{k} (t) ‖^{2} + α {| A_{1} η_{k} (t) |}_{L^{2}}^{2} + b_{1} (ξ_{k} (t), θ (t), A_{1} η_{k} (t)) + b_{1} ({\tilde{u}}_{k} (t), η_{k} (t), A_{1} η_{k} (t)) \\ = - ⟨ g_{k} (t), A_{1} η_{k} (t) ⟩ . \end{aligned}

(4.32)

Adding (4.31) and (4.32) we obtain

\begin{aligned} \frac{d}{d t} ‖ (ξ_{k} (t), η_{k} (t) ‖_{H}^{2} + 2 μ ‖ ξ_{k} (t) ‖^{2} + 2 α {| A_{1} η_{k} (t) |}_{L^{2}}^{2} + 2 b_{0} (ξ_{k} (t), u (t), ξ_{k} (t)) \\ + 2 b_{1} (ξ_{k} (t), θ (t), A_{1} η_{k} (t)) + 2 b_{1} ({\tilde{u}}_{k} (t), η_{k} (t), A_{1} η_{k} (t)) \\ + 2 ⟨ R_{0} (η_{k} (t), θ (t)) + R_{0} ({\tilde{θ}}_{k} (t), η_{k} (t)), ξ (t) ⟩ \\ = - 2 ⟨ f_{k} (t), ξ_{k} (t) ⟩ - 2 ⟨ g_{k} (t), A_{1} η_{k} (t) ⟩ . \end{aligned}

(4.33)

Using (2.16), (2.23) and (2.27) we bound the nonlinear terms as follows:

\begin{aligned} 2 | b_{0} (ξ_{k} (t), u (t), ξ_{k} (t)) | & ⩽ 2 c_{b} {| ξ_{k} (t) |}_{L^{2}} ‖ ξ_{k} (t) ‖ ‖ u (t) ‖ ⩽ \frac{μ}{5} ‖ ξ_{k} (t) ‖^{2} + c {| ξ_{k} (t) |}_{L^{2}}^{2} ‖ u (t) ‖^{2}, \end{aligned}

(4.34)

\begin{aligned} 2 | b_{1} (ξ_{k} (t), θ (t), A_{1} η_{k} (t)) | & ⩽ 2 c_{b} {| ξ_{k} (t) |}_{L^{2}}^{1 / 2} ‖ ξ_{k} (t) ‖^{1 / 2} ‖ θ (t) ‖^{1 / 2} {| A_{1} θ (t) |}_{L^{2}}^{1 / 2} {| A_{1} η_{k} (t) |}_{L^{2}} \\ ⩽ \frac{α}{5} {| A_{1} η_{k} (t) |}_{L^{2}}^{2} + \frac{μ}{5} ‖ ξ_{k} (t) ‖^{2} + c {| ξ_{k} (t) |}_{L^{2}}^{2} ‖ θ (t) ‖^{2} {| A_{1} θ (t) |}_{L^{2}}^{2}, \end{aligned}

(4.35)

\begin{aligned} 2 | b_{1} ({\tilde{u}}_{k} (t), η_{k} (t), A_{1} η_{k} (t)) | & ⩽ 2 c_{b} {| {\tilde{u}}_{k} (t) |}_{L^{2}}^{1 / 2} ‖ {\tilde{u}}_{k} (t) ‖^{1 / 2} ‖ η_{k} (t) ‖^{1 / 2} {| A_{1} η_{k} (t) |}_{L^{2}}^{3 / 2} \\ ⩽ \frac{α}{5} {| A_{1} η_{k} (t) |}_{L^{2}}^{2} + c {| {\tilde{u}}_{k} (t) |}_{L^{2}}^{2} ‖ {\tilde{u}}_{k} (t) ‖^{2} ‖ η_{k} (t) ‖^{2}, \end{aligned}

(4.36)

\begin{aligned} 2 ⟨ R_{0} (η_{k} (t), θ (t)), ξ (t) ⟩ & ⩽ c ‖ η_{k} (t) ‖^{1 / 2} {| A_{1} η_{k} (t) |}_{L^{2}}^{1 / 2} ‖ θ (t) ‖^{1 / 2} {| A_{1} θ (t) |}_{L^{2}}^{1 / 2} ‖ ξ (t) ‖ \\ ⩽ \frac{α}{5} {| A_{1} η_{k} (t) |}_{L^{2}}^{2} + \frac{μ}{5} ‖ ξ_{k} (t) ‖^{2} + c ‖ η_{k} (t) ‖^{2} ‖ θ (t) ‖^{2} {| A_{1} θ (t) |}_{L^{2}}^{2}, \end{aligned}

(4.37)

\begin{aligned} 2 ⟨ R_{0} ({\tilde{θ}}_{k} (t), η_{k} (t)), ξ (t) ⟩ & ⩽ c ‖ {\tilde{θ}}_{k} (t) ‖^{1 / 2} {| A_{1} {\tilde{θ}}_{k} (t) |}_{L^{2}}^{1 / 2} ‖ η_{k} (t) ‖^{1 / 2} {| A_{1} η_{k} (t) |}_{L^{2}}^{1 / 2} ‖ ξ (t) ‖ \\ ⩽ \frac{α}{5} {| A_{1} η_{k} (t) |}_{L^{2}}^{2} + \frac{μ}{5} ‖ ξ_{k} (t) ‖^{2} + c ‖ {\tilde{θ}}_{k} (t) ‖^{2} {| A_{1} {\tilde{θ}}_{k} (t) |}_{L^{2}}^{2} ‖ η_{k} (t) ‖^{2} . \end{aligned}

(4.38)

Using the Cauchy–Schwarz inequality, we bound the right-hand side of (4.33) as

\begin{aligned} | ⟨ f_{k} (t), ξ_{k} (t) ⟩ | ⩽ ‖ f_{k} (t) ‖_{V_{1}^{'}} ‖ ξ_{k} (t) ‖ ⩽ \frac{μ}{5} ‖ ξ_{k} (t) ‖^{2} + c ‖ f_{k} (t) ‖_{V_{1}^{'}}^{2}, \end{aligned}

(4.39)

\begin{aligned} | ⟨ g_{k} (t), A_{1} η_{k} (t) ⟩ | ⩽ ‖ g_{k} (t) ‖_{D {(A η)}^{'}} {| A_{1} η_{k} (t)) |}_{L^{2}} ⩽ \frac{α}{5} {| A_{1} η_{k} (t) |}_{L^{2}}^{2} + c ‖ g_{k} (t) ‖_{D {(A_{1})}^{'}}^{2} . \end{aligned}

(4.40)

Relations (4.33)–(4.40) give

\begin{aligned} \frac{d}{d t} ‖ (ξ_{k} (t), η_{k} (t) ‖_{H}^{2} + μ ‖ ξ_{k} (t) ‖^{2} + α {| A_{1} η_{k} (t) |}_{L^{2}}^{2} \\ ⩽ c {| ξ_{k} (t) |}_{L^{2}}^{2} ‖ u (t) ‖^{2} \\ + c {| ξ_{k} (t) |}_{L^{2}}^{2} ‖ θ (t) ‖^{2} {| A_{1} θ (t) |}_{L^{2}}^{2} + c {| {\tilde{u}}_{k} (t) |}_{L^{2}}^{2} ‖ {\tilde{u}}_{k} (t) ‖^{2} ‖ η_{k} (t) ‖^{2} \\ + c ‖ η_{k} (t) ‖^{2} ‖ θ (t) ‖^{2} {| A_{1} θ (t) |}_{L^{2}}^{2} + c ‖ {\tilde{θ}}_{k} (t) ‖^{2} {| A_{1} {\tilde{θ}}_{k} (t) |}_{L^{2}}^{2} ‖ η_{k} (t) ‖^{2} \\ + c ‖ f_{k} (t) ‖_{V_{1}^{'}}^{2} + c ‖ g_{k} (t) ‖_{D (A_{1})^{'}}^{2} . \end{aligned}

(4.41)

Neglecting some positive terms, the above relation implies

\begin{aligned} \frac{d}{d t} ‖ {(ξ_{k} (t), η_{k} (t) ‖}_{H}^{2} ⩽ G (t) ‖ {(ξ_{k} (t), η_{k} (t) ‖}_{H}^{2} + c {‖ f_{k} (t) ‖}_{{V_{1}}^{^{'}}}^{2} + c {‖ g_{k} (t) ‖}_{D (A_{1})^{^{'}}}^{2}, \end{aligned}

(4.42)

where

\begin{aligned} G (t) = c (‖ u (t) ‖^{2} + ‖ θ (t) ‖^{2} {| A_{1} θ (t) |}_{L^{2}}^{2} + {| {\tilde{u}}_{k} (t) |}_{L^{2}}^{2} ‖ {\tilde{u}}_{k} (t) ‖^{2} \\ + ‖ θ (t) ‖^{2} {| A_{1} θ (t) |}_{L^{2}}^{2} + ‖ {\tilde{θ}}_{k} (t) ‖^{2} {| A_{1} {\tilde{θ}}_{k} (t) |}_{L^{2}}^{2}) . \end{aligned}

(4.43)

By the Gronwall Lemma and using the fact that

ξ_{k} (0) = η_{k} (0) = 0

, we obtain

\begin{aligned} ‖ {(ξ_{k} (t), η_{k} (t) ‖}_{H}^{2} ⩽ c \int_{0}^{t} \exp (\int_{τ}^{t} G (s) d s) ({‖ f_{k} (τ) ‖}_{{V_{1}}^{^{'}}}^{2} + {‖ g_{k} (τ) ‖}_{D (_{1})^{^{'}}}^{2}) d τ . \end{aligned}

(4.44)

Recalling (4.18), (4.8) and (3.50), we obtain

\begin{aligned} \int_{τ}^{t} G (s) d s ⩽ c, \end{aligned}

(4.45)

for some constant

c = c (T^{*}) > 0

Relations (4.44), (4.45), (4.12) and (4.13) give

\begin{aligned} ‖ {(ξ_{k} (t), η_{k} (t) ‖}_{H}^{2} ⩽ c Δ t, \end{aligned}

(4.46)

and thus

\begin{aligned} lim_{k \to 0} sup_{(u^{0}, ϕ^{0}) \in A_{k}, n k \in [0, T^{*}]} ‖ S_{k}^{n} (u^{0}, ϕ^{0}) - S (n k) (u^{0}, ϕ^{0}) ‖_{H} \\ = lim_{k \to 0} sup_{(u^{0}, ϕ^{0}) \in A_{k}, n k \in [0, T^{*}]} sup_{(u^{n}, ϕ^{n}) \in S_{k}^{n} (u^{0}, ϕ^{0})} ‖ (u^{n}, ϕ^{n}) - (u (n k), ϕ (n k)) ‖_{H} \\ = lim_{k \to 0} sup_{(u^{0}, ϕ^{0}) \in A_{k}, n k \in [0, T^{*}]} sup_{(u^{n}, ϕ^{n}) \in S_{k}^{n} (u^{0}, ϕ^{0})} ‖ ({\tilde{u}}_{k} (n k), {\tilde{ϕ}}_{k} (n k)) - (u (n k), ϕ (n k)) ‖_{H} \\ = lim_{k \to 0} sup_{(u^{0}, ϕ^{0}) \in A_{k}, n k \in [0, T^{*}]} sup_{(u^{n}, ϕ^{n}) \in S_{k}^{n} (u^{0}, ϕ^{0})} ‖ (ξ_{k} (n k), η_{k} (n k)) ‖_{H} = 0, \end{aligned}

(4.47)

which concludes the proof of the lemma.

Having proved that conditions (H1) and (H2) of Theorem 2 are satisfied we also obtain that the discrete attractors converge to the continuous attractor as the time-step approaches zero. More precisely, we have the following:

Theorem 5.

The family of attractors ${A_{k}}_{k \in (0, κ_{1}]}$ converges, as $k \to 0$ , to $A$ , in the following sense:

\begin{aligned} lim_{k \to 0} dist (A_{k}, A) = 0, \end{aligned}

where dist denotes the Hausdorff semidistance in H, namely

\begin{aligned} dist (A_{k}, A) = sup_{x_{k} \in A_{k}} inf_{x \in A} ‖ x_{k} - x ‖_{L^{2}} . \end{aligned}

Proof.

The proof follows from Theorem 2 and Proposition 2.

Footnotes

Acknowledgements

The authors would like to thank AIM for the invitation to attend one of its workshops, which was the starting point for this project; they also thank the referees for helpful comments.

In this section, we recall some results on the existence of the global attractor $A$ . Some details can be found for instance in [3,19]. More precisely, the existence of $A$ follows from the propositions below and standard arguments (see for instance [43]). For $γ = 0$ , the system (2.36) reduces to (A.1)

\begin{aligned} {\begin{cases} u_{t} + μ A_{0} u + B_{0} (u, u) + R_{0} (θ, θ) = f, \\ θ_{t} + α A_{1} θ + B_{1} (u, θ) = g, \\ (u, θ) (0) = (u^{0}, θ^{0}) \in H . \end{cases} \end{aligned}

References

Ball

, Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations, J. Nonlinear Sci. 7 (1997), 475–502. doi:10.1007/s003329900037.

Barbu

, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leiden, 1976.

Bosia

, Well-posedness and long term behavior of a simplified Ericksen–Leslie nonautonomous system for nematic liquid crystal flow, Comm. Pure Appl. Anal. 21 (2012), 407–441. doi:10.3934/cpaa.2012.11.407.

Brzeźniak

Hausenblas

Razafimandimby

P.A.

, Some results on the penalised nematic liquid crystals driven by multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput. 7 (2019), 417–475.

Brzeźniak

Manna

Panda

A.A.

, Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form, J. Differential Equations 266 (2019), 6204–6283. doi:10.1016/j.jde.2018.11.001.

Brzeźniak

Manna

Panda

A.A.

, Large deviations for stochastic nematic liquid crystals driven by multiplicative Gaussian noise, Potential Anal. 53 (2020), 799–838. doi:10.1007/s11118-019-09788-6.

Caraballo

Langa

Melnik

Valero

, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal. 2 (2003), 153–201. doi:10.1023/A:1022902802385.

Cavaterra

Rocca

, Global weak solution and blow-up criterion of the general Ericksen–Leslie system for nematic liquid crystal flows, J. Differential Equations 255 (2013), 24–57. doi:10.1016/j.jde.2013.03.009.

Coti-Zelati

Tone

, Multivalued attractors and their approximation: Applications to the Navier–Stokes equations, Numerische Mathematik 122 (2012), 421–441. doi:10.1007/s00211-012-0463-y.

10.

Ericksen

J.L.

, Conservation laws for liquid crystals, Trans. Soc. Rheology 5 (1961), 23–34. doi:10.1122/1.548883.

11.

Ericksen

J.L.

, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal. 9 (1962), 371–378. doi:10.1007/BF00253358.

12.

Ewald

Tone

, Approximation of the long-term dynamics of the dynamical system generated by the two-dimensional thermohydraulics equations, International Journal of Numerical Analysis and Modeling 10 (2013), 509–535.

13.

Fan

Jiang

, Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions, Commun. Pure Appl. Anal. 15 (2016), 73–90. doi:10.3934/cpaa.2016.15.73.

14.

Feireisl

Frémond

Rocca

, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal. 205 (2012), 651–672. doi:10.1007/s00205-012-0517-4.

15.

Feireisl

Rocca

Schimperna

, On a non-isothermal model for nematic liquid crystals, Nonlinearity 24 (2011), 243–257. doi:10.1088/0951-7715/24/1/012.

16.

Gal

C.G.

Grasselli

, Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 401–436. doi:10.1016/j.anihpc.2009.11.013.

17.

Gong

Huang

Liu

, Global strong solutions of the 2D simplified Ericksen–Leslie system, Nonlinearity 28 (2015), 3677–3694. doi:10.1088/0951-7715/28/10/3677.

18.

Gong

, Local well-posedness of strong solutions to density-dependent liquid crystal system, Nonlinear Anal. 147 (2016), 26–44. doi:10.1016/j.na.2016.08.014.

19.

Grasselli

, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow, Z. Angew. Math. Phys. 62 (2011), 979–992. doi:10.1007/s00033-011-0157-9.

20.

Hong

M.C.

, Global existence of solutions of the simplified Ericksen–Leslie system in dimension two, Calc. Var. Partial Differential Equations 40 (2011), 15–36. doi:10.1007/s00526-010-0331-5.

21.

Hong

M.C.

Xin

, Blow-up criteria of strong solutions to the Ericksen–Leslie system in

R^{3}

, Comm. Partial Differential Equations 39 (2014), 1284–1328. doi:10.1080/03605302.2013.871026.

22.

Hong

M.C.

Xin

Z.P.

, Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in

R^{2}

, Adv. Math. 231 (2012), 1364–1400. doi:10.1016/j.aim.2012.06.009.

23.

Huang

Lin

Wang

, Regularity and existence of global solutions to the Ericksen–Leslie system in

R^{2}

, Comm. Math. Phys. 331 (2014), 805–850. doi:10.1007/s00220-014-2079-9.

24.

Huang

, A representation formula of incompressible liquid crystal flow and its applications, Nonlinearity 30 (2017), 1911–1919. doi:10.1088/1361-6544/aa63a5.

25.

Leslie

F.M.

, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math. 19 (1966), 357–370. doi:10.1093/qjmam/19.3.357.

26.

Leslie

F.M.

, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28 (1968), 265–283. doi:10.1007/BF00251810.

27.

Titi

E.S.

Xin

, On the uniqueness of weak solutions to the Ericksen–Leslie liquid crystal model in

R^{2}

, Math. Models Methods Appl. Sci. 26 (2016), 803–822. doi:10.1142/S0218202516500184.

28.

Lin

Wang

, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B 31 (2010), 921–938. doi:10.1007/s11401-010-0612-5.

29.

Lin

Wang

, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math. 69 (2016), 1532–1571. doi:10.1002/cpa.21583.

30.

Lin

F.H.

, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math. 42 (1989), 789–814. doi:10.1002/cpa.3160420605.

31.

Lin

F.H.

Liu

, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math. 48 (1995), 501–537. doi:10.1002/cpa.3160480503.

32.

Lin

F.H.

Liu

, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems 2 (1996), 1–22. doi:10.3934/dcds.1996.2.1.

33.

Lin

F.H.

Liu

, Existence of solutions for the Ericksen–Leslie system, Arch. Ration. Mech. Anal. 154 (2000), 135–156. doi:10.1007/s002050000102.

34.

Lin

F.H.

Liu

J.Y.

Wang

C.Y.

, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal. 197 (2010), 297–336. doi:10.1007/s00205-009-0278-x.

35.

Manna

Panda

A.A.

, Well-posedness and large deviations for 2D stochastic constrained Navier–Stokes equations driven by Lévy noise in the Marcus canonical form, J. Differential Equations 302 (2021), 64–138. doi:10.1016/j.jde.2021.08.035.

36.

Medjo

T.T.

Tone

, Long-time stability of the implicit Euler scheme for a family of regularized models for incompressible two-phase flows, Asymptot. Anal. 94 (2015), 125–160.

37.

Medjo

T.T.

Tone

, Long-times tability of a classical efficient scheme for an incompressible two-phase flow model, Asymptot. Anal. 95 (2015), 101–127.

38.

Melnik

Valero

, On attractors of multivalued semi-flows and differential inclusion, Set-Valued Anal. 6 (1998), 83–111. doi:10.1023/A:1008608431399.

39.

Rossi

Segatti

Stefanelli

, Attractors for gradient flows of nonconvex functionals and applications, Arch. Ration. Mech. Anal. 187 (2008), 91–135. doi:10.1007/s00205-007-0078-0.

40.

Schimperna

Segatti

Stefanelli

, Well-posedness and long-time behavior for a class of doubly nonlinear equations, Discrete Contin. Dyn. Syst. 18 (2007), 15–38. doi:10.3934/dcds.2007.18.15.

41.

Shen

, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal. 38 (1990), 201–229. doi:10.1080/00036819008839963.

42.

Sun

Liu

, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst. 23 (2009), 455–475. doi:10.3934/dcds.2009.23.455.

43.

Temam

, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New-York, 1988.

44.

Tone

Wang

, Approximation of the stationary statistical properties of the dynamical system generated by the two-dimensional Rayleigh–Benard convection problem, Analysis and Applications 09 (2011), 421–446. doi:10.1142/S0219530511001935.

45.

Tone

Wirosoetisno

, On the long-time stability of the implicit Euler scheme for the 2D Navier–Stokes equations, SIAM Journal on Numerical Analysis 44 (2006), 29–40. doi:10.1137/040618527.

46.

Wang

, Global existence of weak solution for the 2-D Ericksen–Leslie system, Calc. Var. Partial Differential Equations 51 (2014), 915–962. doi:10.1007/s00526-013-0700-y.

47.

Wang

Zhang

, On the uniqueness of weak solution for the 2-D Ericksen–Leslie system, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), 919–941. doi:10.3934/dcdsb.2016.21.919.

48.

Wang

, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp. 79 (2010), 259–280. doi:10.1090/S0025-5718-09-02256-X.

49.

Liu

, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations 45 (2012), 319–345. doi:10.1007/s00526-011-0460-5.

50.

Liu

, On the general Ericksen–Leslie system: Parodi’s relation, well-posedness and stability, Arch. Ration. Mech. Anal. 208 (2013), 59–107. doi:10.1007/s00205-012-0588-2.

51.

Zhang

, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differential Equations 252 (2012), 1169–1181. doi:10.1016/j.jde.2011.08.028.

Approximation of the long-time dynamics of the dynamical system generated by the Ericksen–Leslie equations

Abstract

Keywords

1. .Introduction

2. .The Ericksen–Leslie model and its mathematical setting

2.1. .Governing equations

Footnotes

Acknowledgements

References