Asymptotic behavior of a model for order-disorder and phase separation

Abstract

Our aim in this article is to study the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system. In particular, we prove the existence of an exponential attractor and, as a consequence, the existence of the global attractor with finite fractal dimension.

Keywords

Allen–Cahn/Cahn–Hilliard system global attractor exponential attractor fractal dimension

1. Introduction

A. Novick-Cohen and J. Cahn introduced in [3] the following phase-field system: $\begin{array}{l} \frac{\partial u}{\partial t} = h^{2} Δ (f (u + v) + f (u - v) - h^{2} Δ u), \\ \frac{\partial v}{\partial t} = - f (u + v) + f (u - v) - α v + h^{2} Δ v, \end{array}$ to model simultaneous order-disorder and phase separation in binary alloys on a BCC lattice in the neighborhood of the triple point. These authors explored two phenomenological approaches leading to systems of coupled Allen–Cahn/Cahn–Hilliard (AC/CH) equations (see [3] for more details).

Another important application of coupled (AC/CH) equations is the following. Under appropriate compositional conditions, ordering can be induced in a previously homogeneous material. If the composition differs slightly from these conditions, the excess composition can emerge as droplets along the boundaries between the ordered regions. This phenomena can be modeled by a coupled (AC/CH) system with degenerate mobilities. In similar applications, surface diffusion coupled with motion by mean curvature appears quite naturally. There are additional effects which are often neglected and which arguably should be included. However, the coupled motion, by itself, is not overly well understood and it was thus reasonable to isolate it and study it, even given its limitations (see [5]).

Here, u denotes the concentration of one of the components and is a conserved quantity, while v is an order parameter. Furthermore, h is a (positive) parameter which represents the lattice spacing and the parameter α reflects the location of the system within the phase diagram and may be either positive or negative. We further note that the system is a gradient flow in ${(H^{1})}^{'} \times L^{2}$ for the free energy $\begin{matrix} J (u, v) = \int_{Ω} {G (u + v) + G (u - v) + \frac{α}{2} v^{2} + \frac{1}{2} h^{2} (| \nabla u |^{2} + | \nabla v |^{2})} d x, \end{matrix}$ where $f = G^{'}$ .

These equations, endowed with Neumann boundary conditions, have been studied in [2] by A. Novick-Cohen, D. Brochet, and D. Hilhorst who proved the well-posedness and the existence of maximal attractors and inertial sets (i.e., exponential attractors) for the usual cubic nonlinear term $f (s) = s^{3} - β s$ in three space dimensions.

Our main aim in this paper is to improve these results. In particular, taking initial conditions in $H^{2}$ allows us to prove the existence of exponential attractors (and, thus, of the finite-dimensional global attractor) for a large class of nonlinear terms containing polynomials of arbitrary odd degree with a strictly positive leading coefficient in three space dimensions.

A similar system, with a non-constant mobility, was treated by L. Giacomelli and A. Novick-Cohen in [4] who proved the existence of weak solutions for the Neumann problem for a degenerate parabolic system consisting of a fourth-order and a second-order equations with singular lower-order terms in one space dimension. In addition, asymptotics for a similar system with a non-constant mobility, proposed as a diffuse interface model for simultaneous order-disorder and phase separation, was studied in [10]. There, A. Novick-Cohen focused on motion in the plane. This framework yields both sharp interface and diffuse interface models of sintering of small grains and thermal grains boundary grooving in polycrystalline films. This work was extended in [11], where the authors studied the partial wetting case, and their analysis accounts for motion in three dimensions.

We also mention that numerical methods to solve coupled (AC/CH) systems were studied in, e.g., [3,8,12,14–16]. Furthermore, a NKS method for the implicit solution of a coupled (AC/CH) system was proposed in [17].

In this work, we take $h = 1$ and $α = 0$ and obtain the following system: $\begin{array}{l} (1) & \frac{\partial u}{\partial t} + Δ^{2} u - Δ (f (u + v) + f (u - v)) = 0, \\ (2) & \frac{\partial v}{\partial t} - Δ v + f (u + v) - f (u - v) = 0, \\ (3) & u = v = 0 on Γ, \\ (4) & u_{| t = 0} = u_{0}, v_{| t = 0} = v_{0}, \end{array}$ where Ω is a bounded domain of $R^{N}$ ( $N = 1, 2$ , or 3) with smooth boundary Γ.

As far as the nonlinear term is concerned, we make the following assumptions: $\begin{array}{l} (5) & f is of class C^{2}, f (0) = 0, \\ (6) & f^{'} (s) ⩾ - c_{0}, c_{0} > 0, s \in R, \\ (7) & f (s) s ⩾ c_{1} F (s) - c_{2}, F (s) ⩾ - c_{3}, c_{1} > 0, c_{2}, c_{3} ⩾ 0, s \in R, \end{array}$ where $F (s) = \int_{0}^{s} f (ξ) d ξ$ .

Throughout this work, the same letter c (and, sometimes, $c^{'}$ and $c^{″}$ ) denotes constants which may change from line to line, or even in a same line. Similarly, the same letter Q denotes monotone increasing (with respect to each argument) functions which may change from line to line, or even in a same line.

2. A priori estimates

In this section, we will establish a number of important inequalities that will be used later in the proof of the existence of the solution and the existence of finite-dimensional attractors.

In what follows, the Poincaré, Hölder and Young inequalities are extensively used, without further referring to them. We rewrite (1) in the equivalent form $\begin{matrix} (8) & {(- Δ)}^{- 1} \frac{\partial u}{\partial t} - Δ u + f (u + v) + f (u - v) = 0 . \end{matrix}$

We multiply (8) by u, integrate over Ω and have $\begin{matrix} (9) & \frac{1}{2} \frac{d}{d t} ‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + ((f (u + v) + f (u - v), u)) = 0, \end{matrix}$ where $‖ \cdot ‖_{- 1}^{2} = ‖ {(- Δ)}^{- \frac{1}{2}} \cdot ‖$ , $‖ \cdot ‖$ being the usual $L^{2}$ -norm, with associated scalar product $((\cdot, \cdot))$ .

We then multiply (2) by v and have $\begin{matrix} (10) & \frac{1}{2} \frac{d}{d t} ‖ v ‖^{2} + ‖ \nabla v ‖^{2} + ((f (u + v) - f (u - v), v)) = 0 . \end{matrix}$

The sum of (9) and (10) gives $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ u ‖_{- 1}^{2} + ‖ v ‖^{2}) + ‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2} \\ (11) & + ((f (u + v), u + v)) + ((f (u - v), u - v)) = 0, \end{array}$ which yields, owing to (7), $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ u ‖_{- 1}^{2} + ‖ v ‖^{2}) + ‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2} + c \int_{Ω} (F (u + v) + F (u - v)) d x ⩽ c^{'}, \\ (12) & c, c^{'} ⩾ 0 . \end{array}$

We multiply (8) by $\frac{\partial u}{\partial t}$ to obtain $\begin{matrix} (13) & {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + \frac{1}{2} \frac{d}{d t} ‖ \nabla u ‖^{2} + ((f (u + v) + f (u - v), \frac{\partial u}{\partial t})) = 0 . \end{matrix}$

We now multiply (2) by $\frac{\partial v}{\partial t}$ and find $\begin{matrix} (14) & {‖ \frac{\partial v}{\partial t} ‖}^{2} + \frac{1}{2} \frac{d}{d t} ‖ \nabla v ‖^{2} + ((f (u + v) - f (u - v), \frac{\partial v}{\partial t})) = 0 . \end{matrix}$

Then, we sum (13) and (14) and have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2} \\ (15) & + ((f (u + v), \frac{\partial}{\partial t} (u + v))) + ((f (u - v), \frac{\partial}{\partial t} (u - v))) = 0, \end{array}$ which implies $\begin{matrix} (16) & \frac{1}{2} \frac{d}{d t} (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2} + 2 \int_{Ω} F (u + v) d x + 2 \int_{Ω} F (u - v) d x) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2} = 0 . \end{matrix}$

Summing (12) and (16), we obtain $\begin{matrix} \begin{matrix} \frac{1}{2} \frac{d}{d t} (‖ u ‖_{- 1}^{2} + ‖ v ‖^{2} + ‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2} + \int_{Ω} (F (u + v) + F (u - v)) d x) + ‖ \nabla u ‖^{2} \\ + ‖ \nabla v ‖^{2} + \int_{Ω} (F (u + v) + F (u - v) d x) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2} ⩽ c^{'}, c^{'} ⩾ 0 . \end{matrix} \end{matrix}$ Let $\begin{matrix} E_{1} = ‖ u ‖_{- 1}^{2} + ‖ v ‖^{2} + ‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2} + \int_{Ω} (F (u + v) + F (u - v)) d x . \end{matrix}$ Then the previous inequality is equivalent to $\begin{matrix} (17) & \frac{1}{2} \frac{d}{d t} E_{1} + c (E_{1} + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) ⩽ c^{'}, \end{matrix}$ whence $‖ \frac{\partial u}{\partial t} ‖_{- 1}^{2} + ‖ \frac{\partial v}{\partial t} ‖^{2} \in L^{1} (0, T)$ .

In particular, we deduce from (17) the dissipative estimate $\begin{array}{rcl} E_{1} (t) & ⩽ & c e^{- c^{'} t} (‖ u_{0} ‖_{- 1}^{2} + ‖ v_{0} ‖^{2} + ‖ \nabla u_{0} ‖^{2} + ‖ \nabla v_{0} ‖^{2} \\ (18) & + \int_{Ω} (F (u_{0} + v_{0}) + F (u_{0} - v_{0})) d x) + c^{″}, c^{'} > 0, t ⩾ 0 . \end{array}$ Furthermore, for every $r > 0$ , $\begin{array}{l} \int_{t}^{t + r} ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) d τ \\ ⩽ c e^{- c^{'} t} (‖ u_{0} ‖_{- 1}^{2} + ‖ v_{0} ‖^{2} + ‖ \nabla u_{0} ‖^{2} + ‖ \nabla v_{0} ‖^{2} \\ (19) & + \int_{Ω} (F (u_{0} + v_{0}) + F (u_{0} - v_{0})) d x) + c^{″} (r), c^{'} > 0, t ⩾ 0 . \end{array}$

We multiply (1) by $\frac{\partial u}{\partial t}$ . It follows from the continuity of f and the continuous embedding $H^{2} (Ω) \subset C (\overline{Ω})$ that $\begin{matrix} (20) & \frac{d}{d t} ‖ Δ u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ Q (‖ u ‖_{H^{2} (Ω)} + ‖ v ‖_{H^{2} (Ω)}) . \end{matrix}$

We can obtain a similar inequality for v by multiplying (2) by $(- Δ) \frac{\partial v}{\partial t}$ , yielding $\begin{matrix} (21) & \frac{d}{d t} ‖ Δ v ‖^{2} + {‖ \frac{\partial \nabla v}{\partial t} ‖}^{2} ⩽ Q (‖ u ‖_{H^{2} (Ω)} + ‖ v ‖_{H^{2} (Ω)}) . \end{matrix}$

Summing then (20) and (21), we obtain $\begin{array}{l} \frac{d}{d t} (‖ Δ u ‖^{2} + ‖ Δ v ‖^{2}) + {‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial \nabla v}{\partial t} ‖}^{2} \\ (22) & ⩽ Q (‖ u ‖_{H^{2}} + ‖ v ‖_{H^{2}}) . \end{array}$

In particular, setting $\begin{matrix} y = ‖ Δ u ‖^{2} + ‖ Δ v ‖^{2}, \end{matrix}$ we deduce from (22) a differential inequality of the form $\begin{matrix} (23) & y^{'} ⩽ Q (y) . \end{matrix}$ Let z be the solution of the ordinary differential equation $\begin{matrix} (24) & z^{'} = Q (z), with z (0) = y (0) . \end{matrix}$ It follows from the comparison principle that there exists $T_{0} = T_{0} (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ v_{0} ‖_{H^{2} (Ω)})$ belonging to, say, $(0, \frac{1}{2})$ such that $\begin{matrix} (25) & y (t) ⩽ z (t), t \in [0, T_{0}], \end{matrix}$ whence $\begin{matrix} (26) & {‖ u (t) ‖}_{H^{2}}^{2} + {‖ v (t) ‖}_{H^{2}}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ v_{0} ‖_{H^{2}}), t ⩽ T_{0} . \end{matrix}$ Therefore, u and v a priori belong to $L^{\infty} (0, T_{0}; H^{2} (Ω))$ .

We now need to prove that u and v are in $L^{\infty} (T_{0}, T; H^{2} (Ω))$ , so that we can obtain the regularity $u, v \in L^{\infty} (0, T; H^{2} (Ω))$ .

To do so, we differentiate equations (2) and (8) with respect to time, $\begin{array}{l} (27) & {(- Δ)}^{- 1} \frac{\partial}{\partial t} \frac{\partial u}{\partial t} - Δ \frac{\partial u}{\partial t} + f^{'} (u + v) (\frac{\partial u}{\partial t} + \frac{\partial v}{\partial t}) + f^{'} (u - v) (\frac{\partial u}{\partial t} - \frac{\partial v}{\partial t}) = 0, \\ (28) & \frac{\partial}{\partial t} \frac{\partial v}{\partial t} - Δ \frac{\partial v}{\partial t} + f^{'} (u + v) (\frac{\partial u}{\partial t} + \frac{\partial v}{\partial t}) + f^{'} (u - v) (\frac{\partial u}{\partial t} - \frac{\partial v}{\partial t}) = 0, \\ (29) & \frac{\partial u}{\partial t} = \frac{\partial v}{\partial t} = 0 on Γ . \end{array}$

Summing (27) times $\frac{\partial u}{\partial t}$ and (28) times $\frac{\partial v}{\partial t}$ gives $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} \\ + ((f^{'} (u + v) (\frac{\partial u}{\partial t} + \frac{\partial v}{\partial t}), \frac{\partial u}{\partial t} + \frac{\partial v}{\partial t})) \\ (30) & + ((f^{'} (u - v) (\frac{\partial u}{\partial t} - \frac{\partial v}{\partial t}), \frac{\partial u}{\partial t} - \frac{\partial v}{\partial t})) = 0, \end{array}$ which yields, owing to (6), $\begin{matrix} (31) & \frac{1}{2} \frac{d}{d t} ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} ⩽ c ({‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) . \end{matrix}$ Employing the interpolation inequality $\begin{matrix} {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ c^{'} {‖ \frac{\partial u}{\partial t} ‖}_{- 1} ‖ \nabla \frac{\partial u}{\partial t} ‖, c^{'} > 0, \end{matrix}$ we deduce that $\begin{matrix} (32) & {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ c^{'} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + \frac{1}{2} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}, \end{matrix}$ whence $\begin{array}{l} \frac{d}{d t} ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + 2 {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} \\ (33) & ⩽ c ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) . \end{array}$ Applying Gronwall’s lemma to (33) we find that $‖ \frac{\partial u}{\partial t} ‖_{- 1}^{2} + ‖ \frac{\partial v}{\partial t} ‖^{2} \in L^{\infty} (0, T)$ .

We now rewrite equations (8) and (2) in elliptic form as follows: $\begin{array}{l} (34) & - Δ u = - {(- Δ)}^{- 1} \frac{\partial u}{\partial t} - f (u + v) - f (u - v), \\ (35) & - Δ v = - \frac{\partial v}{\partial t} - f (u + v) + f (u - v) . \end{array}$ We multiply (34) by $- Δ u$ and (35) by $- Δ v$ , sum the resulting equalities and obtain $\begin{matrix} (36) & ‖ Δ u ‖^{2} + ‖ Δ v ‖^{2} ⩽ {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2} + c (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}), c > 0 . \end{matrix}$ Therefore, u and v are now in $L^{\infty} (T_{0}, T; H^{2} (Ω))$ , and finally in $L^{\infty} (0, T; H^{2} (Ω))$ , $T > 0$ .

Our aim now is to find another inequality for u and v in $H^{2} (Ω)$ :

We start by multiplying (27) by $t \frac{\partial u}{\partial t}$ and (28) by $t \frac{\partial v}{\partial t}$ to obtain, summing the two resulting equalities, $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (t {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + t {‖ \frac{\partial v}{\partial t} ‖}^{2}) + c t {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + t {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} \\ (37) & ⩽ \frac{1}{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + \frac{1}{2} {‖ \frac{\partial v}{\partial t} ‖}^{2} + c^{'} t ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) . \end{array}$ We deduce from (19), (37), and Gronwall’s lemma that $\begin{matrix} (38) & {‖ \frac{\partial u}{\partial t} (T_{0}) ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} (T_{0}) ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ v_{0} ‖_{H^{2} (Ω)}) . \end{matrix}$

Then, we apply Gronwall’s lemma again to (37) and obtain $\begin{matrix} (39) & {‖ \frac{\partial u}{\partial t} (t) ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} (t) ‖}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ v_{0} ‖_{H^{2} (Ω)}), t ⩾ T_{0} . \end{matrix}$

We now rewrite (8) in the form $\begin{matrix} (40) & - Δ u + f (u + v) + f (u - v) = h_{u} (t), u = 0 on Γ, \end{matrix}$ for $t ⩾ T_{0}$ fixed, where $\begin{matrix} (41) & h_{u} (t) = - {(- Δ)}^{- 1} \frac{\partial u}{\partial t} \end{matrix}$ satisfies, owing to (39), $\begin{matrix} (42) & ‖ h_{u} (t) ‖ ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ v_{0} ‖_{H^{2} (Ω)}), t ⩾ T_{0} . \end{matrix}$

Furthermore, we rewrite (2) in the form $\begin{matrix} (43) & - Δ v + f (u + v) - f (u - v) = h_{v} (t), v = 0 on Γ, \end{matrix}$ for $t ⩾ T_{0}$ fixed, where $\begin{matrix} (44) & h_{v} (t) = - \frac{\partial v}{\partial t} \end{matrix}$ satisfies, owing to (39), $\begin{matrix} (45) & ‖ h_{v} (t) ‖ ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ v_{0} ‖_{H^{2} (Ω)}), t ⩾ T_{0} . \end{matrix}$

We multiply (40) by u and (43) by v and we sum the result. Then, noting that $f (s) s ⩾ - c, c ⩾ 0$ , we obtain $\begin{matrix} (46) & ‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2} ⩽ c ({‖ h_{u} (t) ‖}^{2} + {‖ h_{v} (t) ‖}^{2}) + c^{'} . \end{matrix}$

Multiplying now (40) by $- Δ u$ and (43) by $- Δ v$ , summing the two resulting equations, and since $f^{'} (s) ⩾ - c_{0}$ , we have $\begin{matrix} (47) & {‖ Δ u (t) ‖}^{2} + {‖ Δ v (t) ‖}^{2} ⩽ c ({‖ h_{u} (t) ‖}^{2} + {‖ h_{v} (t) ‖}^{2} + {‖ \nabla u (t) ‖}^{2} + {‖ \nabla v (t) ‖}^{2}) . \end{matrix}$ We thus deduce from (42) and (45)–(47) that $\begin{matrix} (48) & {‖ u (t) ‖}_{H^{2} (Ω)}^{2} + {‖ v (t) ‖}_{H^{2} (Ω)}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ v_{0} ‖_{H^{2} (Ω)}), t ⩾ 0 . \end{matrix}$

We now wish to find a dissipative estimate that is stronger than (18).

We multiply (8) by $- Δ u$ and (2) by $- Δ v$ and sum the two resulting equations to obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ u ‖^{2} + ‖ \nabla v ‖^{2}) + ‖ Δ u ‖^{2} + ‖ Δ v ‖^{2} \\ (49) & + ((f^{'} (u + v) \nabla (u + v), \nabla (u + v))) + ((f^{'} (u - v) \nabla (u - v), \nabla (u - v))) = 0, \end{array}$ which gives, owing to (6), $\begin{matrix} (50) & \frac{1}{2} \frac{d}{d t} (‖ u ‖^{2} + ‖ \nabla v ‖^{2}) + ‖ Δ u ‖^{2} + ‖ Δ v ‖^{2} ⩽ c (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}) . \end{matrix}$ Furthermore, $\begin{array}{rcl} ‖ u ‖_{H^{1} (Ω)} & ⩽ & c ‖ u ‖ ‖ u ‖_{H^{2} (Ω)}, c > 0, \\ ⩽ & \frac{1}{2} ‖ Δ u ‖^{2} + c ‖ u ‖^{2} . \end{array}$ Therefore, $\begin{matrix} (51) & \frac{d}{d t} (‖ u ‖^{2} + ‖ \nabla v ‖^{2}) + c (‖ Δ u ‖^{2} + ‖ Δ v ‖^{2}) ⩽ c^{'} (‖ u ‖^{2} + ‖ \nabla v ‖^{2}) . \end{matrix}$ Using now Gronwall’s lemma, we obtain $\begin{matrix} (52) & {‖ u (t) ‖}^{2} + {‖ \nabla v (t) ‖}^{2} ⩽ e^{c^{'} t} (‖ u_{0} ‖^{2} + ‖ \nabla v_{0} ‖^{2}) . \end{matrix}$ It follows from (51) that $\begin{matrix} (53) & \int_{0}^{1} (‖ u ‖_{H^{2} (Ω)}^{2} + ‖ v ‖_{H^{2} (Ω)}^{2}) d t ⩽ Q (‖ u_{0} ‖^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}) . \end{matrix}$ Hence, there exists $T \in (0, 1)$ such that $\begin{matrix} (54) & {‖ u (T) ‖}_{H^{2} (Ω)}^{2} + {‖ v (T) ‖}_{H^{2} (Ω)}^{2} ⩽ Q (‖ u_{0} ‖^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}) . \end{matrix}$

Repeating the estimates leading to (48), but starting from $t = T$ instead of $t = 0$ , we have the smoothing property $\begin{matrix} (55) & {‖ u (1) ‖}_{H^{2} (Ω)}^{2} + {‖ v (1) ‖}_{H^{2} (Ω)}^{2} ⩽ Q (‖ u_{0} ‖^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}) . \end{matrix}$

Since our equations are autonomous, we can make a translation in time and, repeating the estimates leading to (55), we have $\begin{matrix} (56) & {‖ u (t) ‖}_{H^{2} (Ω)}^{2} + {‖ v (t) ‖}_{H^{2} (Ω)}^{2} ⩽ Q ({‖ u (t - 1) ‖}^{2} + {‖ v (t - 1) ‖}_{H^{1} (Ω)}^{2}), t ⩾ 1, \end{matrix}$ which yields, owing to (18), $\begin{array}{rcl} {‖ u (t) ‖}_{H^{2} (Ω)}^{2} + {‖ v (t) ‖}_{H^{2} (Ω)}^{2} & ⩽ & e^{- c t} Q (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2} \\ (57) & + \int_{Ω} (F (u_{0} + v_{0}) + F (u_{0} - v_{0})) d x), t ⩾ 1 . \end{array}$

Combining the above estimate with (48) from 0 to 1, we obtain the dissipative estimate $\begin{matrix} (58) & {‖ u (t) ‖}_{H^{2} (Ω)}^{2} + {‖ v (t) ‖}_{H^{2} (Ω)}^{2} ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ v_{0} ‖_{H^{2} (Ω)}), t ⩾ 0 . \end{matrix}$

3. Existence and uniqueness of solutions

Theorem 3.1.
Let $T > 0$ be given and assume that $(u_{0}, v_{0}) \in H^{2} {(Ω)}^{2}$ . Then, ( 1 )–( 2 ) possesses a unique solution $(u, v)$ such that $(u, v) \in L^{\infty} (0, T; H^{2} {(Ω)}^{2})$ and $(\frac{\partial u}{\partial t}, \frac{\partial v}{\partial t}) \in L^{\infty} (0, T; H^{- 1} (Ω) \times L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} {(Ω)}^{2})$ .
Proof.
(a) Existence

The proof of existence (as well as the above (and the subsequent) a priori estimates) are based, e.g., on a classical Galerkin scheme.

Let A denote the minus Laplace operator associated with Dirichlet boundary conditions. This operator is a bounded, selfadjoint and strictly positive operator with compact inverse from $H_{0}^{1} (Ω)$ onto $H^{- 1} (Ω)$ . There is a set of eigenvectors ${φ_{i}, i ⩾ 1}$ for this operator that is regular as much as needed, associated with the eigenfunctions $0 < λ_{1} ⩽ λ_{2} ⩽ \dots$ such that it is orthonormal relative to the inner product in $L^{2} (Ω)$ and orthogonal relative to the one in $H_{0}^{1} (Ω)$ .

Setting $V_{m} = span {φ_{1}, \dots, φ_{m}}$ , we consider the following approximating problem, written in functional form: $\begin{array}{l} \frac{d u_{m}}{d t} + A^{2} u_{m} + A (f (u_{m} + v_{m}) + f (u_{m} - v_{m})) = 0, \\ \frac{d v_{m}}{d t} + A v_{m} + f (u_{m} + v_{m}) - f (u_{m} - v_{m}) = 0, \end{array}$ together with suitable initial conditions, namely, $\begin{matrix} u_{m} (0) = P_{m} u_{0}, v_{m} (0) = P_{m} v_{0}, \end{matrix}$ where $P_{m}$ is the orthogonal projector from $L^{2} (Ω)$ onto $V_{m}$ (for the $L^{2}$ -metric). This is equivalent to the following: $\begin{array}{l} \frac{d}{d t} ((u_{m}, ϕ)) - ((Δ u_{m}, Δ ϕ)) + ((\nabla f (u_{m} + v_{m}), \nabla ϕ)) - ((\nabla f (u_{m} - v_{m}), \nabla ϕ)) = 0, \\ \frac{d}{d t} ((v_{m}, ω)) + ((\nabla v_{m}, \nabla ω)) + ((f (u_{m} + v_{m}), ω)) - ((f (u_{m} - v_{m}), ω)) = 0, \end{array}$ $\forall ϕ, ω \in V_{m}$ , together with the above initial conditions. The proof of existence of a local (in time) solution to the approximating problem is standard (indeed, one has to solve a continuous system of ODEs).

Furthermore, we can write the equivalent of the previous estimates (with u and v replaced by $u_{m}$ and $v_{m}$ , respectively; this is now fully justified and no longer formal). Then (18) yields that this solution is actually global. Finally, the passage to the limit is based on classical (Aubin–Lions type) compactness results. Indeed, we have, in particular, $u_{m}$ bounded in $L^{\infty} (0, T; H^{2} (Ω))$ and $\frac{d u_{m}}{d t}$ bounded in $L^{\infty} (0, T; H^{- 1} (Ω))$ , independently of m, which yields that (at least for a subsequence which we do not relabel) $u_{m}$ converges strongly to, say, u in $C ([0, T]; H^{2 - δ} (Ω)), \forall δ > 0$ . In addition, $v_{m}$ is bounded in $L^{\infty} (0, T; H_{0}^{1} (Ω))$ and $\frac{d v_{m}}{d t}$ is bounded in $L^{\infty} (0, T; L^{2} (Ω))$ , independently of m, which also yields the strong convergence of $v_{m}$ to, say, v in $C ([0, T]; H^{1 - δ} (Ω)), \forall δ > 0$ .

We also note that it follows from (36) that $(u, v) \in L^{\infty} (0, T; H^{2} {(Ω)}^{2})$ , from (33) that $(\frac{\partial u}{\partial t}, \frac{\partial v}{\partial t}) \in L^{\infty} (0, T; H^{- 1} (Ω) \times L^{2} (Ω))$ and from (31) that $(\frac{\partial u}{\partial t}, \frac{\partial v}{\partial t}) \in L^{2} (0, T; H_{0}^{1} {(Ω)}^{2})$ , whence $(\frac{\partial u}{\partial t}, \frac{\partial v}{\partial t}) \in L^{\infty} (0, T; H^{- 1} (Ω) \times L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} {(Ω)}^{2})$ . Besides, our solution is weakly continuous with respect to time, i.e., it belongs to $C_{w} ([0, T]; H^{2} {(Ω)}^{2})$ .

(b) Uniqueness

Let $(u_{1}, v_{1})$ and $(u_{2}, v_{2})$ be two solutions with initial data $(u_{0, 1}, v_{0, 1})$ and $(u_{0, 2}, v_{0, 2})$ , respectively. We set $(u, v) = (u_{1} - u_{2}, v_{1} - v_{2})$ and $(u_{0}, v_{0}) = (u_{0, 1} - u_{0, 2}, v_{0, 1} - v_{0, 2})$ and have $\begin{array}{l} (59) & {(- Δ)}^{- 1} \frac{\partial u}{\partial t} - Δ u + f (u_{1} + v_{1}) - f (u_{2} + v_{2}) + f (u_{1} - v_{1}) - f (u_{2} - v_{2}) = 0, \\ (60) & \frac{\partial v}{\partial t} - Δ v + f (u_{1} + v_{1}) - f (u_{2} + v_{2}) - f (u_{1} - v_{1}) + f (u_{2} - v_{2}) = 0, \\ (61) & u = v = 0 on Γ, \\ (62) & u_{| t = 0} = u_{0}, v_{| t = 0} = v_{0} . \end{array}$

We multiply (59) by $\frac{\partial u}{\partial t}$ and (60) by $\frac{\partial v}{\partial t}$ and obtain, summing the two resulting equalities, $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2} \\ + ((f (u_{1} + v_{1}) - f (u_{2} + v_{2}), \frac{\partial}{\partial t} (u + v))) \\ (63) & + ((f (u_{1} - v_{1}) - f (u_{2} - v_{2}), \frac{\partial}{\partial t} (u - v))) = 0 . \end{array}$ Furthermore, $\begin{array}{l} ((f (u_{1} + v_{1}) - f (u_{2} + v_{2}), \frac{\partial}{\partial t} (u + v))) \\ = | (({(- Δ)}^{\frac{1}{2}} (f (u_{1} + v_{1}) - f (u_{2} + v_{2})), {(- Δ)}^{\frac{- 1}{2}} \frac{\partial}{\partial t} (u + v))) | \\ (64) & ⩽ c {‖ \frac{\partial}{\partial t} (u + v) ‖}_{- 1} ‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖, c > 0, \end{array}$ and similarly $\begin{array}{l} ((f (u_{1} - v_{1}) - f (u_{2} - v_{2}), \frac{\partial}{\partial t} (u - v))) \\ (65) & ⩽ c^{'} {‖ \frac{\partial}{\partial t} (u - v) ‖}_{- 1} ‖ \nabla (f (u_{1} - v_{1}) - f (u_{2} - v_{2})) ‖, c^{'} > 0 . \end{array}$ Therefore, $\begin{array}{l} \frac{d}{d t} (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}) + c {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + c {‖ \frac{\partial v}{\partial t} ‖}^{2} \\ (66) & ⩽ c {‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖}^{2} + c {‖ \nabla (f (u_{1} - v_{1}) - f (u_{2} - v_{2})) ‖}^{2} . \end{array}$ We can see that, owing to (58), $\begin{array}{l} ‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖ \\ = ‖ \nabla (\int_{0}^{1} f^{'} (u_{1} + v_{1} + s (u_{2} + v_{2} - u_{1} - v_{1})) d s (u - v)) ‖ \\ ⩽ ‖ \int_{0}^{1} f^{'} (u_{1} + v_{1} + s (u_{2} + v_{2} - u_{1} - v_{1})) d s \nabla (u - v) ‖ \\ + ‖ (u - v) \int_{0}^{1} f^{″} (u_{1} + v_{1} + s (u_{2} + v_{2} - u_{1} - v_{1})) (\nabla (u_{1} + v_{1}) \\ + s \nabla (u_{2} + v_{2} - u_{1} - v_{1})) d s ‖ \\ ⩽ Q (‖ u_{0, 1} ‖_{H^{2} (Ω)}, ‖ u_{0, 2} ‖_{H^{2} (Ω)}, ‖ v_{0, 1} ‖_{H^{2} (Ω)}, ‖ v_{0, 2} ‖_{H^{2} (Ω)}) (‖ \nabla (u - v) ‖ \\ + ‖ | u - v | | \nabla (u_{1} + v_{1}) | ‖ + ‖ | u - v | | \nabla (u_{2} + v_{2}) | ‖) \\ (67) & ⩽ Q (‖ u_{0, 1} ‖_{H^{2} (Ω)}, ‖ u_{0, 2} ‖_{H^{2} (Ω)}, ‖ v_{0, 1} ‖_{H^{2} (Ω)}, ‖ v_{0, 2} ‖_{H^{2} (Ω)}) (‖ \nabla u ‖ + ‖ \nabla v ‖) . \end{array}$ In the same way, $\begin{array}{l} ‖ \nabla (f (u_{1} - v_{1}) - f (u_{2} - v_{2})) ‖ \\ (68) & ⩽ Q (‖ u_{0, 1} ‖_{H^{2} (Ω)}, ‖ u_{0, 2} ‖_{H^{2} (Ω)}, ‖ v_{0, 1} ‖_{H^{2} (Ω)}, ‖ v_{0, 2} ‖_{H^{2} (Ω)}) (‖ \nabla u ‖ + ‖ \nabla v ‖) . \end{array}$ We deduce from (63)–(68) that $\begin{array}{l} \frac{d}{d t} (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}) + c {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + c {‖ \frac{\partial v}{\partial t} ‖}^{2} \\ ⩽ Q (‖ u_{0, 1} ‖_{H^{2} (Ω)}, ‖ u_{0, 2} ‖_{H^{2} (Ω)}, ‖ v_{0, 1} ‖_{H^{2} (Ω)}, ‖ v_{0, 2} ‖_{H^{2} (Ω)}) (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}) . \end{array}$ Now using Gronwall’s lemma, we obtain $\begin{matrix} (69) & {‖ \nabla u (t) ‖}^{2} + {‖ \nabla v (t) ‖}^{2} ⩽ e^{Q t} (‖ \nabla u_{0} ‖^{2} + ‖ \nabla v_{0} ‖^{2}), \end{matrix}$ whence the uniqueness (taking $(u_{0}, v_{0}) = (0, 0)$ ), as well as the continuous dependence with respect to the initial data. □
4. Existence of finite-dimensional attractors

Theorem 4.1.

The semigroup associated with ( 1 )–( 2 ) is dissipative in $H^{2} {(Ω)}^{2}$ , i.e., it possesses a bounded absorbing set $B_{0}$ in $H^{2} {(Ω)}^{2}$ .

Proof.

We can define the semigroup $\begin{array}{l} S (t) : H^{2} {(Ω)}^{2} ⟶ H^{2} {(Ω)}^{2} \\ (u_{0}, v_{0}) \mapsto (u (t), v (t)), \end{array}$ where $(u, v)$ is the unique solution to our system. The result then follows from (58). □

Remark 4.2.

It is easy to see that we can assume, without loss of generality, that $B_{0}$ is positively invariant by $S (t)$ , i.e., $S (t) B_{0} \subset B_{0}$ .

Theorem 4.3.

The semigroup S(t) possesses an exponential attractor $M \subset B_{0}$ , i.e.,

$M$ is compact in $H^{1} (Ω) \times H^{1} (Ω)$ ;

$M$ is positively invariant, $S (t) M \subset M$ , $\forall t ⩾ 0$ ;

$M$ has a finite fractal dimension in $H^{1} (Ω) \times H^{1} (Ω)$ ;

$M$ attracts exponentially fast the bounded subsets of $H_{0}^{1} {(Ω)}^{2}$ :

\forall B \subset H_{0}^{1} {(Ω)}^{2}

bounded,

{dist}_{H^{1} (Ω) \times H^{1} (Ω)} (S (t) B, M) ⩽ Q (‖ B ‖_{H_{0}^{1} {(Ω)}^{2}}) e^{- c t}, c > 0, t ⩾ 0

, where the constant c is independent of B and

{dist}_{H^{1} (Ω) \times H^{1} (Ω)}

denotes the Hausdroff semidistance between sets defined by

\begin{matrix} {dist}_{H^{1} (Ω) \times H^{1} (Ω)} (A, B) = sup_{a \in A} inf_{b \in B} ‖ a - b ‖_{H^{1} (Ω) \times H^{1} (Ω)} . \end{matrix}

Proof.

Let $(u_{1}, v_{1})$ and $(u_{2}, v_{2})$ be two solutions to our system with initial data $(u_{0, 1}, v_{0, 1})$ and $(u_{0, 2}, v_{0, 2})$ , respectively. We note that it is sufficient here to take the initial data in the bounded absorbing set $B_{0}$ constructed in the previous section. We set $(u, v) = (u_{1}, v_{1}) - (u_{2}, v_{2})$ and $(u_{0}, v_{0}) = (u_{0, 1}, v_{0, 1}) - (u_{0, 2}, v_{0, 2})$ and have $\begin{array}{l} (70) & \frac{\partial u}{\partial t} + Δ^{2} u - Δ (f (u_{1} + v_{1}) - f (u_{2} + v_{2}) + f (u_{1} - v_{1}) - f (u_{2} - v_{2})) = 0, \\ (71) & \frac{\partial v}{\partial t} - Δ v + f (u_{1} + v_{1}) - f (u_{2} + v_{2}) - f (u_{1} - v_{1}) + f (u_{2} - v_{2}) = 0, \\ (72) & u = v = 0 on Γ, \\ (73) & u_{| t = 0} = u_{0}, v_{| t = 0} = v_{0} . \end{array}$

We recall inequality (69), $\begin{matrix} (74) & {‖ \nabla u (t) ‖}^{2} + {‖ \nabla v (t) ‖}^{2} ⩽ e^{Q t} (‖ \nabla u_{0} ‖^{2} + ‖ \nabla v_{0} ‖^{2}) . \end{matrix}$

We then multiply (70) by ${(- Δ)}^{- 1} \frac{\partial u}{\partial t}$ and (71) by v and have, summing the two resulting equations, $\begin{array}{rcl} \frac{d}{d t} (‖ \nabla u ‖^{2} + ‖ v ‖^{2}) + c {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + c ‖ \nabla v ‖^{2} & ⩽ & c {‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖}^{2} \\ (75) & + c {‖ \nabla (f (u_{1} - v_{1}) - f (u_{2} - v_{2})) ‖}^{2} . \end{array}$

We recall inequality (67), $\begin{array}{l} ‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖ \\ (76) & ⩽ Q (‖ u_{0, 1} ‖_{H^{2} (Ω)}, ‖ u_{0, 2} ‖_{H^{2} (Ω)}, ‖ v_{0, 1} ‖_{H^{2} (Ω)}, ‖ v_{0, 2} ‖_{H^{2} (Ω)}) (‖ \nabla u ‖ + ‖ \nabla v ‖) . \end{array}$ The same can be done to obtain $\begin{array}{l} ‖ \nabla (f (u_{1} - v_{1}) - f (u_{2} - v_{2})) ‖ \\ (77) & ⩽ Q (‖ u_{0, 1} ‖_{H^{2} (Ω)}, ‖ u_{0, 2} ‖_{H^{2} (Ω)}, ‖ v_{0, 1} ‖_{H^{2} (Ω)}, ‖ v_{0, 2} ‖_{H^{2} (Ω)}) (‖ \nabla u ‖ + ‖ \nabla v ‖) . \end{array}$

We deduce from (75)–(77) that $\begin{array}{l} \frac{d}{d t} (‖ \nabla u ‖^{2} + ‖ v ‖^{2}) + c {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + c ‖ \nabla v ‖^{2} \\ (78) & ⩽ Q (‖ u_{0, 1} ‖_{H^{2} (Ω)}, ‖ u_{0, 2} ‖_{H^{2} (Ω)}, ‖ v_{0, 1} ‖_{H^{2} (Ω)}, ‖ v_{0, 2} ‖_{H^{2} (Ω)}) (‖ \nabla u ‖^{2} + ‖ \nabla v ‖^{2}) . \end{array}$

It follows from (78) that $\begin{matrix} (79) & \int_{0}^{t} ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + ‖ \nabla v ‖^{2}) d x ⩽ c e^{c^{'} t} (‖ \nabla u_{0} ‖^{2} + ‖ \nabla v_{0} ‖^{2}), t ⩾ 0, \end{matrix}$ where the constants only depend on $B_{0}$ .

We then differentiate (70) with respect to time and have $\begin{array}{l} {(- Δ)}^{- 1} \frac{\partial}{\partial t} \frac{\partial u}{\partial t} - Δ \frac{\partial u}{\partial t} + f^{'} (u_{1} + v_{1}) \frac{\partial}{\partial t} (u + v) \\ + (f^{'} (u_{1} + v_{1}) - f^{'} (u_{2} + v_{2})) \frac{\partial}{\partial t} (u_{2} + v_{2}) + f^{'} (u_{1} - v_{1}) \frac{\partial}{\partial t} (u - v) \\ (80) & + (f^{'} (u_{1} - v_{1}) - f^{'} (u_{2} - v_{2})) \frac{\partial}{\partial t} (u_{2} - v_{2}) = 0 . \end{array}$

We multiply (80) by $(t - T_{0}) \frac{\partial u}{\partial t}$ , where $T_{0}$ is the same as in the previous section, and we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ((t - T_{0}) {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2}) + (t - T_{0}) {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} \\ ⩽ c (t - T_{0}) {‖ \frac{\partial u}{\partial t} ‖}^{2} + c (t - T_{0}) {‖ \frac{\partial v}{\partial t} ‖}^{2} \\ + (t - T_{0}) \int_{Ω} (| f^{'} (u_{1} + v_{1}) - f^{'} (u_{2} + v_{2}) | | \frac{\partial u}{\partial t} | | \frac{\partial}{\partial t} (u_{2} + v_{2}) |) d x \\ + (t - T_{0}) \int_{Ω} (| f^{'} (u_{1} - v_{1}) - f^{'} (u_{2} - v_{2}) | | \frac{\partial u}{\partial t} | | \frac{\partial}{\partial t} (u_{2} - v_{2}) |) d x + \frac{1}{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} . \end{array}$

Noting that $\begin{array}{l} \int_{Ω} | f^{'} (u_{1} + v_{1}) - f^{'} (u_{2} + v_{2}) | | \frac{\partial u}{\partial t} | | \frac{\partial}{\partial t} (u_{2} + v_{2}) | d x \\ ⩽ c \int_{Ω} | u + v | | \frac{\partial u}{\partial t} | | \frac{\partial}{\partial t} (u_{2} + v_{2}) | d x \\ ⩽ c ‖ \nabla (u + v) ‖ ‖ \nabla \frac{\partial u}{\partial t} ‖ ‖ \frac{\partial}{\partial t} (u_{2} + v_{2}) ‖, \end{array}$ and $\begin{array}{l} \int_{Ω} | f^{'} (u_{1} - v_{1}) - f^{'} (u_{2} - v_{2}) | | \frac{\partial u}{\partial t} | | \frac{\partial}{\partial t} (u_{2} - v_{2}) | d x \\ ⩽ c \int_{Ω} | u - v | | \frac{\partial u}{\partial t} | | \frac{\partial}{\partial t} (u_{2} - v_{2}) | d x \\ ⩽ c ‖ \nabla (u - v) ‖ ‖ \nabla \frac{\partial u}{\partial t} ‖ ‖ \frac{\partial}{\partial t} (u_{2} - v_{2}) ‖, \end{array}$ we obtain, owing to a proper interpolation inequality, $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ((t - T_{0}) {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2}) + (t - T_{0}) {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} \\ ⩽ c (t - T_{0}) ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2}) \\ + c^{'} (t - T_{0}) {‖ \frac{\partial}{\partial t} (u_{2} + v_{2}) ‖}^{2} {‖ \nabla (u + v) ‖}^{2} \\ (81) & + c^{'} (t - T_{0}) {‖ \frac{\partial}{\partial t} (u_{2} - v_{2}) ‖}^{2} {‖ \nabla (u - v) ‖}^{2} + \frac{1}{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} . \end{array}$

It follows from (19), (33), and (38) (for $(u, v) = (u_{1}, v_{1})$ and then $(u, v) = (u_{2}, v_{2})$ ) that $\begin{matrix} (82) & \int_{T_{0}}^{t} ({‖ \frac{\partial u_{1}}{\partial t} ‖}^{2} + {‖ \frac{\partial v_{1}}{\partial t} ‖}^{2}) d τ ⩽ c e^{c^{'} t}, t ⩾ T_{0}, \end{matrix}$ and $\begin{matrix} (83) & \int_{T_{0}}^{t} ({‖ \frac{\partial u_{2}}{\partial t} ‖}^{2} + {‖ \frac{\partial v_{2}}{\partial t} ‖}^{2}) d τ ⩽ c e^{c^{'} t}, t ⩾ T_{0}, \end{matrix}$ where the constants only depend on $B_{0}$ .

Applying Gronwall’s lemma on (81) over $(T_{0}, t)$ (note that $T_{0} < 1$ ) and owing to (74), (79), (82), (83) we obtain $\begin{matrix} (84) & {‖ \frac{\partial u}{\partial t} (t) ‖}_{- 1}^{2} ⩽ c e^{c^{'} t} (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}), t ⩾ 1, \end{matrix}$ where the constants only depend on $B_{0}$ .

We now rewrite (70) in the form $\begin{matrix} (85) & - Δ u = {\tilde{h}}_{u} (t), u = 0 on Γ, \end{matrix}$ for $t ⩾ 1$ fixed, where $\begin{matrix} (86) & {\tilde{h}}_{u} (t) = - {(- Δ)}^{- 1} \frac{\partial u}{\partial t} - (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) - (f (u_{1} - v_{1}) - f (u_{2} - v_{2})), \end{matrix}$ satisfies, owing to (84), $\begin{matrix} (87) & {‖ {\tilde{h}}_{u} (t) ‖}^{2} ⩽ c e^{c^{'} t} (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}), t ⩾ 1, \end{matrix}$ where the constants only depend on $B_{0}$ .

Multiplying then (85) by $- Δ u$ , we have $\begin{matrix} ‖ Δ u (t) ‖ ⩽ ‖ {\tilde{h}}_{u} (t) ‖, t ⩾ 1, \end{matrix}$ whence $\begin{matrix} (88) & {‖ u (t) ‖}_{H^{2} (Ω)}^{2} ⩽ c e^{c^{'} t} (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}), t ⩾ 1, \end{matrix}$ where the constants only depend on $B_{0}$ .

Next, we differentiate (71) with respect to time and have $\begin{array}{l} \frac{\partial}{\partial t} \frac{\partial v}{\partial t} - Δ \frac{\partial v}{\partial t} + f^{'} (u_{1} + v_{1}) \frac{\partial}{\partial t} (u + v) + (f^{'} (u_{1} + v_{1}) - f^{'} (u_{2} + v_{2})) \frac{\partial}{\partial t} (u_{2} + v_{2}) \\ (89) & + f^{'} (u_{1} - v_{1}) \frac{\partial}{\partial t} (v - u) + (f^{'} (u_{2} - v_{2}) - f^{'} (u_{1} - v_{1})) \frac{\partial}{\partial t} (u_{2} - v_{2}) = 0 . \end{array}$

We multiply (89) by $(t - T_{0}) \frac{\partial v}{\partial t}$ , where $T_{0}$ is the same as above, and obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ((t - T_{0}) {‖ \frac{\partial v}{\partial t} ‖}^{2}) + (t - T_{0}) {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} \\ ⩽ (t - T_{0}) \int_{Ω} (| f^{'} (u_{1} + v_{1}) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u + v) |) d x \\ + (t - T_{0}) \int_{Ω} (| f^{'} (u_{1} + v_{1}) - f^{'} (u_{2} + v_{2}) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u_{2} + v_{2}) |) d x \\ + (t - T_{0}) \int_{Ω} (| f^{'} (u_{2} - v_{2}) - f^{'} (u_{1} + v_{1}) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u_{2} - v_{2}) |) d x \\ + (t - T_{0}) \int_{Ω} (| f^{'} (u_{1} - v_{1}) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u - v) |) d x + \frac{1}{2} {‖ \frac{\partial v}{\partial t} ‖}^{2} . \end{array}$

The terms $\begin{matrix} \int_{Ω} (| f^{'} (u_{1} + v_{1}) - f^{'} (u_{2} + v_{2}) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u_{2} + v_{2}) |) d x \end{matrix}$ and $\begin{matrix} \int_{Ω} (| f^{'} (u_{2} - v_{2}) - f^{'} (u_{1} + v_{1}) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u_{2} - v_{2}) |) d x \end{matrix}$ can be estimated as above. Furthermore, $\begin{array}{l} \int_{Ω} (| f^{'} (u_{1} + v_{1}) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u + v) |) d x \\ = \int_{Ω} (| f^{'} (u_{1} + v_{1}) - f^{'} (0) + f^{'} (0) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u + v) |) d x \\ ⩽ \int_{Ω} (| f^{'} (u_{1} + v_{1}) - f^{'} (0) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u + v) |) d x \\ + \int_{Ω} (| f^{'} (0) | | \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u + v) |) d x \\ ⩽ c (‖ u_{1} ‖_{L^{\infty} (Ω)} + ‖ v_{1} ‖_{L^{\infty} (Ω)}) \int_{Ω} (| \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u + v) |) d x \\ + c \int_{Ω} (| \frac{\partial v}{\partial t} | | \frac{\partial}{\partial t} (u + v) |) d x \\ ⩽ c (‖ u_{1} ‖_{H^{2} (Ω)} + ‖ v_{1} ‖_{H^{2} (Ω)}) (‖ \frac{\partial v}{\partial t} ‖ ‖ \frac{\partial}{\partial t} (u + v) ‖) + c (‖ \frac{\partial v}{\partial t} ‖ ‖ \frac{\partial}{\partial t} (u + v) ‖) \\ ⩽ c (‖ \frac{\partial v}{\partial t} ‖ ‖ \frac{\partial}{\partial t} (u + v) ‖) . \end{array}$

Therefore, we have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ((t - T_{0}) {‖ \frac{\partial v}{\partial t} ‖}^{2}) + (t - T_{0}) {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} \\ ⩽ c (t - T_{0}) {‖ \frac{\partial v}{\partial t} ‖}^{2} + c (t - T_{0}) {‖ \frac{\partial u}{\partial t} ‖}^{2} \\ + c (t - T_{0}) {‖ \frac{\partial}{\partial t} (u_{2} + v_{2}) ‖}^{2} {‖ \nabla (u + v) ‖}^{2} + \frac{1}{2} {‖ \frac{\partial v}{\partial t} ‖}^{2} \\ (90) & + c (t - T_{0}) {‖ \frac{\partial}{\partial t} (u_{2} - v_{2}) ‖}^{2} {‖ \nabla (u - v) ‖}^{2} . \end{array}$

We now apply Gronwall’s lemma to (90) and find, owing to (74), (82) and (83), $\begin{matrix} (91) & {‖ \frac{\partial v}{\partial t} (t) ‖}^{2} ⩽ c e^{c^{'} t} (‖ \nabla u_{0} ‖^{2} + ‖ \nabla v_{0} ‖^{2}), t ⩾ T_{0}, \end{matrix}$ where the constants only depend on $B_{0}$ .

In the same way, we rewrite (71) in the form $\begin{matrix} (92) & - Δ v = {\tilde{h}}_{v} (t), v = 0 on Γ, \end{matrix}$ for $t ⩾ 1$ fixed, where $\begin{matrix} (93) & {\tilde{h}}_{v} (t) = - \frac{\partial v}{\partial t} - (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) - (f (u_{2} - v_{2}) - f (u_{1} - v_{1})), \end{matrix}$ satisfies, owing to (74) and (91), $\begin{matrix} (94) & {‖ {\tilde{h}}_{v} (t) ‖}^{2} ⩽ c e^{c^{'} t} (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}), t ⩾ 1, \end{matrix}$ where the constants only depend on $B_{0}$ .

Multiplying (92) by $- Δ v$ , we have $\begin{matrix} ‖ Δ v (t) ‖ ⩽ ‖ {\tilde{h}}_{v} (t) ‖, t ⩾ 1, \end{matrix}$ whence $\begin{matrix} (95) & {‖ v (t) ‖}_{H^{2} (Ω)}^{2} ⩽ c e^{c^{'} t} (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}), t ⩾ 1, \end{matrix}$ where the constants only depend on $B_{0}$ .

We deduce from (88) and (95) that $\begin{matrix} (96) & {‖ u (t) ‖}_{H^{2} (Ω)}^{2} + {‖ v (t) ‖}_{H^{2} (Ω)}^{2} ⩽ c e^{c^{'} t} (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{H^{1} (Ω)}^{2}), t ⩾ 1, \end{matrix}$ where the constants only depend on $B_{0}$ .

Finally, we derive a Hölder (both with respect to space and time) estimate. Actually, owing to (74), it suffices to prove the Hölder continuity with respect to time.

We have $\begin{array}{l} {‖ u (t_{1}) - u (t_{2}) ‖}_{H^{1} (Ω)} + {‖ v (t_{1}) - v (t_{2}) ‖}_{H^{1} (Ω)} \\ ⩽ c ‖ \nabla (u (t_{1}) - u (t_{2})) ‖ + c ‖ \nabla (v (t_{1}) - v (t_{2})) ‖ \\ ⩽ c (‖ \int_{t_{1}}^{t_{2}} \nabla \frac{\partial u}{\partial t} d τ ‖ + ‖ \int_{t_{1}}^{t_{2}} \nabla \frac{\partial v}{\partial t} d τ ‖) \\ (97) & ⩽ c | t_{1} - t_{2} |^{\frac{1}{2}} {| \int_{t_{1}}^{t_{2}} ({‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2}) d τ |}^{\frac{1}{2}} . \end{array}$ We note that it follows from (35), (39), and (19) that $\begin{matrix} (98) & | \int_{t_{1}}^{t_{2}} ({‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2}) d τ | ⩽ c, \end{matrix}$ where c only depends on $B_{0}$ and $T ⩾ T_{0}$ such that $t_{1}, t_{2} \in [T_{0}, T]$ . Actually, replacing $B_{0}$ by $S (1) B_{0}$ , we can assume that $T_{0} = 0$ .

Finally, we have $\begin{matrix} (99) & {‖ u (t_{1}) - u (t_{2}) ‖}_{H^{1} (Ω)} + {‖ v (t_{1}) - v (t_{2}) ‖}_{H^{1} (Ω)} ⩽ c | t_{1} - t_{2} |^{\frac{1}{2}}, \end{matrix}$ where c only depends on $B_{0}$ and T such that $t_{1}, t_{2} \in [0, T]$ and the result follows from (74), (96), and (99) (see [6,7]). □

Moreover, we can deduce from Theorem 4.3 and standard results the

Corollary 4.4.

The semigroup $S (t)$ possesses the finite-dimensional global attractor $A \subset B_{0}$ .

Remark 4.5.

We recall that the global attractor $A$ is the smallest (for the inclusion) compact set of the phase space which is invariant by the flow (i.e., $S (t) A = A$ , $\forall t ⩾ 0$ ) and attracts all bounded sets of initial data as time goes to infinity; it thus appears as a suitable object in view of the study of the asymptotic behavior of the system. Furthermore, the finite dimensionality means, roughly speaking, that, even though the initial phase space is infinite dimensional, the reduced dynamics is, in some proper sense, finite dimensional and can be described by a finite number of parameters. We refer the reader to [1,9,13] for more details and discussions on this.

Remark 4.6.

Compared to the global attractor, an exponential attractor is expected to be more robust under perturbations. Indeed, the rate of attraction of trajectories to the global attractor may be slow and it is very difficult, if not impossible, to estimate this rate of attraction with respect to the physical parameters of the problem in general. As a consequence, global attractors may change drastically under small perturbations. We refer the reader to [6,9] for discussions on this subject.

References

A.V.

Babin and

M.I.

Vishik , Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

Brochet ,

Hilhorst and

Novick-Cohen , Finite-dimensional exponential attractor for a model for order–disorder and phase separation, Appl. Math. Lett. 7 (1994), 83–87. doi:10.1016/0893-9659(94)90118-X.

J.W.

Cahn and

Novick-Cohen , Evolution equations for phase separation and ordering in binary alloys, Statistical Physics 76 (1994), 877–909. doi:10.1007/BF02188691.

Dal Passo ,

Giacomelli and

Novick-Cohen , Existence for an Allen–Cahn/Cahn–Hilliard system with degenerate mobility, Interfaces and Free Boundaries 1 (1999), 199–226.

Derkach ,

Novick-Cohen and

Vilenkin , Geometric interfacial motion: Coupling surface diffusion and mean curvature motion, submitted.

Eden ,

Foias ,

Nicolaenko and

Temam , Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Vol. 37, Wiley, New York, 1994.

Efendiev ,

Miranville and

Zelik , Exponential attractors for a nonlinear reaction-diffusion system in

R^{3}

, C. R. Acad. Sci., Paris Sér. I 330 (2000), 713–718. doi:10.1016/S0764-4442(00)00259-7.

P.C.

Millett ,

Rokkam ,

El-Azab ,

Tonks and

Wolf , Void nucleation and growth in irradiated polycrystalline metals: A phase-field model, Modelling Simul. Mater. Sci. Eng. 17 (2009), 0064003. doi:10.1088/0965-0393/17/6/064003.

Miranville and

Zelik , Attractors for dissipative partial differential equations in bounded and unbounded domains, in: Handbook of Differential Equations, Evolutionary Partial Differential Equations,

C.M.

Dafermos and

Pokorny , eds, Vol. 4, Elsevier, Amsterdam, 2008, pp. 103–200.

10.

Novick-Cohen , Triple-junction motion for an Allen–Cahn/Cahn–Hilliard system, Physica D 137 (2000), 1–24. doi:10.1016/S0167-2789(99)00162-1.

11.

Novick-Cohen and

Peres Hari , Geometric motion for a degenerate Allen–Cahn/Cahn–Hilliard system: The partial wetting case, Physica D 209 (2005), 205–235. doi:10.1016/j.physd.2005.06.028.

12.

Rokkam ,

El-Azab ,

Millett and

Wolf , Phase field modeling of void nucleation and growth in irradiated metals, Modelling Simul. Mater. Sci. Eng. 17 (2009), 0064002. doi:10.1088/0965-0393/17/6/064002.

13.

Temam : Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, 2nd edn, Springer, New York, 1997. doi:10.1007/978-1-4612-0645-3.

14.

M.R.

Tonks ,

Gaston ,

P.C.

Millett ,

Andrs and

Talbot , An object-oriented finite element framework for multiphysics phase field simulations, Comput. Mater. Sci. 51 (2012), 20–29. doi:10.1016/j.commatsci.2011.07.028.

15.

Wang ,

Lee ,

Anitescu ,

A.E.

Azab ,

L.C.

Mcinnes ,

Munson and

Smith , A differential variational inequality approach for the simulation of heterogeneous materials, in: Proc. SciDAC 2011, 2011.

16.

Xia ,

Xu and

C.W.

Shu , Application of the local discontinuous Galerkin method for the Allen–Cahn/Cahn–Hilliard system, Commun. Comput. Phys. 5 (2009), 821–835.

17.

Yang ,

X.C.

Cai ,

D.E.

Keyes and

Pernice , NKS method for the implicit solution of a coupled Allen–Cahn/Cahn–Hilliard system, in: Proceedings of the 21th International Conference on Domain Decomposition Methods, 2012.