1. Introduction

In this paper, we consider a second-order time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity, and we prove that any sequence generated by the scheme converges to a steady state as time goes to infinity, provided that the time-step is chosen small enough. The Cahn-Hilliard equation ^[10] reads

$\begin{cases} u_t=\Delta w\\ w=-\gamma\Delta u+f(u) \end{cases} \mbox{in }\Omega\times(0, +\infty),$

(1.1)

where Ω is a bounded subset of ${\mathbb{R}^d}$ (1≤d≤3) with smooth boundary and γ > 0. A typical choice for the nonlinearity is

$f(s)=c(s^3-s)$

(1.2)

with c > 0. More general conditions on f are given in Section 2, see (2.3)-(2.5). Equation (1.1) is completed with Neumann boundary conditions and an initial datum.

The Cahn-Hilliard equation was analyzed by many authors and used in different contexts (see, e.g., ^[11,37] and references therein). In particular, it is a H^-1 gradient flow for the energy

$E(u)=\int_\Omega\left[\frac{\gamma}{2}|\nabla u|^2+F(u)\right] dx,$

where F is an antiderivative of f. Convergence of single trajectories to equilibrium for (1.1)-(1.2) has been proved in ^[42]. The proof uses the gradient flow structure of the equation and a Łojasiewicz-Simon inequality ^[44].

In one space dimension, the set of steady states corresponding to (1.1)-(1.2) is finite ^[24,32]. In this case, the use of a Łojasiewicz-Simon inequality can be avoided ^[51] but otherwise, the situation is highly complicated; if d=2 or 3, there may even be a continuum of stationary solutions (see, e.g., ^[47] and references therein). The Łojasiewicz-Simon inequality allows to prove convergence to an equilibrium without any knowledge on the set of steady states. This celebrated inequality is based on the analyticity of f (see ^[27] for a recent overview). In contrast, for the related semilinear parabolic equation, convergence to equilibrium may fail for a nonlinearity of class C^{∞ [39]}.

Using similar techniques, convergence to equilibrium for the non-autonomous Cahn-Hilliard equation was proved in ^[15], and the case of a logarithmic nonlinearity was considered in ^[1]. The Cahn-Hilliard equation endowed with dynamic or Wentzell boundary conditions was analyzed in ^{[14,40,48,49]}. Coupled systems were also considered (see, e.g., ^[18,30,41]).

Since many space and/or time discretizations of the Cahn-Hilliard equation are available in the literature (see, e.g., ^{[5,17,20,21,22,26,36,43,50]}), it is natural to ask whether convergence to equilibrium also holds for these discretizations, by using similar techniques.

If we consider only a space semi-discretization of (1.1), and if this discretization can be shown to preserve the gradient flow structure, then convergence to equilibrium is a consequence of Łojasiewicz’s classical convergence result ^[33] and its generalizations ^[8,27]. Thanks to the finite dimension, the Łojasiewicz-Simon inequality reduces to the standard Łojasiewicz inequality. The latter is a direct consequence of analyticity of the discrete energy functional.

Thus, the situation regarding the space discretization is well understood, and we believe that the focus should be put on the time discretization, in the specific case where the time scheme preserves the gradient flow structure. In this regard, convergence to equilibrium for a fully discrete version of (1.1)-(1.2) was first proved in ^[34]: the time scheme was the backward Euler scheme and the space discretization was a finite element method. Fully discretized versions of Cahn-Hilliard type equations were considered in ^[12,13,29], where this nice feature of the backward Euler scheme was again demonstrated ^[6,25]. In ^[4], convergence to equilibrium was proved for several fully discretized versions of the closely related Allen-Cahn equation; the time scheme was either first order or second order, conditionnally or unconditionnally stable, and the time-step could possibly be variable. In addition, general conditions ensuring convergence to equilibrium for a time discretization were given ^[7].

Therefore, the fully discrete case is now also well understood. The last stage is to study the time semi-discrete case. This is all the more interesting since this approach is independent of a choice of a specific space discretization. Convergence to equilibrium was proved for the backward Euler time semi-discretization of the Allen-Cahn equation in ^[34] (see also ^[9]). A related damped wave equation was considered in ^[38].

For schemes different from the backward Euler method, the situation is not so clear, and this is well illustrated by the second order case. Indeed, there exist several second-order time semidiscretizations of (1.1)-(1.2) which preserve the gradient flow structure (see, e.g., ^[43,50] and references therein). Most of these schemes are one-step methods, which can be seen as variants of the Crank-Nicolson scheme, such as the classical secant scheme ^[16,17] or the more recent scheme of Gomez and Hughes ^[21], which is a Crank-Nicolson scheme with stabilization.

However, we have not been able to prove convergence to equilibrium for any of these second-order one-step schemes. One difficulty is that the gradient of E (cf. (3.2)) is treated in an implicit/explicit way, and another difficulty is that the discrete dynamical system associated with the scheme is defined on a space of infinite dimension. The first difficulty can be circumvented in finite dimension, as recently shown in ^[23], where convergence to equilibrium was proved for a fully discrete approximation of the modified phase-field crystal equation using the second-order time discretization of Gomez and Hughes. A related difficulty has been pointed out in ^[46] where the stability of the Crank-Nicolson scheme for the Navier-Stokes equation was proved in a finite dimensional setting only.

In this paper, instead of a Crank-Nicolson type method, we use a standard two-step scheme with fixed time-step, namely the backward differentiation formula of order two. It is well-known ^[43,45] that this scheme enjoys a Lyapunov stability, namely, if the time-step is small enough, a so-called pseudoenergy (cf. (2.17)) is nonincreasing at every time iteration. Thanks to the implicit treatment of the gradient of E (cf. (2.13)), the proof of convergence is similar to the case of the backward Euler scheme in ^[34,38]. Using the Lyapunov stability, we first prove Lasalle’s invariance principle by a compactness argument (Proposition 3.1). Convergence to a steady state is then obtained as a consequence of an appropriate Łojasiewicz-Simon inequality (Lemma 3.2), which is the most technical point. In order to derive the convergence rate in H¹ norm, we also take advantage of the fact that the scheme is more dissipative than the original equation (see Remark 2.4).

It would be interesting to extend our convergence result to first-order or second-order schemes where the nonlinearity is treated explicitly. In order for such schemes to preserve the gradient structure, the standard approach is to truncate the cubic nonlinearity f (cf. (1.2)) at±∞ so as to have a linear growth at most ^[43]. However, it is not known if the energy associated with such a nonlinearity satisfies a Łojasiewicz-Simon inequality, in contrast with the finite-dimensional case where it can be proved for certain space discretizations ^[4].

It could also be of interest to investigate whether a similar convergence result holds for the p-step backward differentiation formula (BDF), with p≥3. A favorable situation is the 3-step BDF method, which preserves the gradient flow structure, at least in finite dimension ^[45]. The paper is organized as follows. In Section 2, we introduce the scheme, we establish its wellposedness and we show that it is Lyapunov stable. In Section 3, we prove the convergence result.

---

2. The time semi-discrete scheme

---

2.1. Notation and assumptions

Let H=L²(Ω) be equipped with the L²(Ω) norm |·|₀ and the L²(Ω) scalar product (·, ·). We denote V=H¹(Ω) the standard Sobolev space based on the L²(Ω) space. We use the hilbertian semi-norm |·|₁=|▽·|₀ in V, and the norm in V is $\|v\|_1^2=|v|_0^2+|v|_1^2$. We denote $-\Delta : V\to V'$ the bounded operator associated with the inner product on V through

$\langle-\Delta u, v\rangle_{V', V}=(\nabla u, \nabla v), \quad\forall u, v\in V,$

where V′ is the topological dual of V. As usual, we will denote W^{k, p}(Ω) the Sobolev spaces based on the L^p(Ω) space ^[19].

For a function $u\in L^2(\Omega)$, we denote

$\langle u\rangle=\frac{1}{|\Omega|}\int_\Omega u\, dx\quad\mbox{ and }\quad \dot{u}=u-\langle u\rangle,$

where |Ω| is the Lebesgue measure of Ω. We also define

$\dot{H}=\{u\in L^2(\Omega), \ \langle u\rangle=0\}, \quad \dot{V}=V\cap \dot{H}.$

We will use the continuous and dense injections

$\dot{V}\subset\dot{H}=\dot{H}'\subset\dot{V}'.$

As a consequence of the Poincar′e-Wirtinger inequality, the norms ||v||₁ and

$v\mapsto (|v|_1^2+\langle v\rangle^2)^{1/2}$

(2.1)

are equivalent on V. The operator $-\dot{\Delta}:\dot{V}\to \dot{V}'$, that is the restriction of-Δ, is an isomorphism. The scalar product in $\dot{V}'$ is given by

$(\dot{u}, \dot{v})_{-1}=(\nabla (-\dot{\Delta})^{-1}\dot{u}, \nabla(-\dot{\Delta})^{-1}\dot{v})=\langle \dot{u}, (-\dot{\Delta})^{-1}\dot{v}\rangle_{\dot{V}', \dot{V}}$

and the norm is given by

$|\dot{u}|_{-1}^2=(\dot{u}, \dot{u})_{-1}=\langle\dot{u}, (-\dot{\Delta})^{-1}\dot{u}\rangle_{\dot{V}', \dot{V}}.$

We recall the interpolation inequality

$|\dot{u}|_0^2\le |\dot{u}|_{-1}|\dot{u}|_1, \quad\forall \dot{u}\in \dot{V}.$

(2.2)

We assume that the nonlinearity f: $\mathbb{R} \to \mathbb{R}$ is analytic and if d≥2, we assume in addition that there exist a constant C > 0 and a real number p≥0 such that

$|f'(s)|\le C(1+|s|^p), \quad\forall s\in\mathbb{R}, \label{hypf1}$

(2.3)

with p < 4 if d=3. No growth assumption is needed if d=1. We also assume that

$f'(s)\ge-c_f, \quad\forall s\in\mathbb{R},$

(2.4)

for some (optimal) nonnegative constant c_f, and that

$\liminf_{|s|\to+\infty} \frac{f(s)}{s}>0.$

(2.5)

We define the energy functional

$E(u)=\frac{\gamma}{2}|u|_1^2+(F(u), 1),$

(2.6)

where F(s) is a given antiderivative of f. The Sobolev injection $V\subset L^{p+2}(\Omega)$ and the growth assumption (2.3) ensure that $E(u) < +\infty$ and $f(u)\in V'$, for all $u \in V$. In fact, by [31, Corollaire 17.8], the functional E is of class C² on V. For any $u, v, w\in V$, we have

$\langle dE(u), v\rangle_{V', V}=\int_\Omega[\gamma\nabla u\cdot\nabla v +f(u)v]dx, \label{aux21}$

(2.7)

$\langle d^2E(u)v, w\rangle_{V', V}=\int_\Omega[\gamma\nabla v\cdot\nabla w+f'(u)vw]dx, \label{aux22}$

(2.8)

where $dE(u)\in V'$ is the first differential of E at u and $d^2E(u)\in\mathcal{L}(V, V')$ is the differential of order two of E at u.

If u is a regular solution of (1.1), on computing we see that

$\frac{d}{dt}E(u(t))=-|w|_1^2=-|u_t|_{-1}^2\quad t\ge 0,$

(2.9)

so that E is a Lyapunov functional associated with (1.1).

---

2.2. Existence, uniqueness and Lyapunov stability

Let $\tau>0$ denote the time-step. The second-order backward differentiation scheme for (1.1) reads ^[43,45]: let $(u_0, u_1)\in V\times V$ and for n=1, 2, ..., let $(u_{n+1}, w_{n+1})\in V\times V$ solve

$\left\{ \begin{gathered} \frac{1}{{2\tau }}(3{u_{n + 1}}-4{u_n} + {u_{n-1}}, \varphi) + (\nabla {w_{n + 1}}, \nabla \varphi)=0 \hfill \\ ({w_{n + 1}}, \psi)=\gamma (\nabla {u_{n + 1}}, \nabla \psi) + (f({u_{n + 1}}), \psi), \hfill \\ \end{gathered} \right.$

(2.10)

for all $(\varphi, \psi)\in V\times V$. For simplicity, we assume that

$\langle u_0\rangle=\langle u_1\rangle,$

(2.11)

so that, by induction, any sequence $(u_n)$ which complies with (2.10) satisfies $\langle u_n\rangle=\langle u_0\rangle$ for all $n$ (choose $\varphi=1/|\Omega|$ in (2.10)). We note that $w_0$ and $w_1$ need not be defined.

For later purpose, we note that if $\langle u_n\rangle=\langle u_{n-1}\rangle$, then (2.10) is equivalent to

$\begin{cases} \langle u_{n+1}\rangle=\langle u_n\rangle\\ (-\dot{\Delta})^{-1}\cfrac{(3u_{n+1}-4u_n+u_{n-1})}{2\tau}+\dot{w}_{n+1}=0\\ \dot{w}_{n+1}=-\gamma \Delta u_{n+1}+f(u_{n+1})-\langle f(u_{n+1})\rangle\\ \langle w_{n+1}\rangle=\langle f(u_{n+1})\rangle. \end{cases}$

(2.12)

Eliminating $w_{n+1}$ leads to

$(-\dot{\Delta})^{-1}\cfrac{(3u_{n+1}-4u_n+u_{n-1})}{2\tau}-\gamma \Delta u_{n+1}+f(u_{n+1})-\langle f(u_{n+1})\rangle=0.$

(2.13)

Proposition 2.1 (Existence for all $\tau$). For all $(u_0, u_1)\in V\times V$ such that $\langle u_0\rangle=\langle u_1\rangle$, there exists at least one sequence $(u_n, w_n)_{n}$ which complies with (2.10). Moreover, $\langle u_n\rangle=\langle u_0\rangle$ for all $n$.

Proof. Existence can be obtained by minimizing an appropriate functional. By induction, assume that for some $n\ge 1$, $(u_{n-1}, u_n)\in V\times V$ is defined, with $\langle u_n\rangle=\langle u_{n-1}\rangle=\langle u_0\rangle$. Then, by (2.13), $u_{n+1}$ can be obtained by solving

$\min\left\{\mathcal{G}_n(v)\ :\ v\in V, \ \langle v\rangle=\langle u_0\rangle\right\},$

(2.14)

where

$\mathcal{G}_n(v)=\frac{3}{4\tau}|\dot{v}|_{-1}^2+\frac{1}{2\tau}(-4\dot{u}_n+\dot{u}_{n-1}, \dot{v})_{-1}+E(v).$

By (2.5), there exist $\kappa_1>0$ and $\kappa_2\ge 0$ such that

$F(s)\ge \kappa_1s^2-\kappa_2, \quad\forall s\in\mathbb{R}.$

Thus, for all $v\in V$,

$(F(v), 1)\ge \kappa_1|v|_0^2-\kappa_2|\Omega|,$

and so

$E(v)\ge \kappa_3\|v\|_1^2-\kappa_2|\Omega|,$

(2.15)

with $\kappa_3=\min\{\gamma/2, \kappa_1\}>0$. Moreover, by the Cauchy-Schwarz inequality,

$|(-4\dot{u}_n+\dot{u}_{n-1}, \dot{v})_{-1}|\le |\dot{v}|_{-1}|-4\dot{u}_n+\dot{u}_{n-1}|_{-1}\le \frac{3}{2} |\dot{v}|_{-1}^2+C_n,$

for some constant $C_n$ which depends on $|\dot{u}_n|_{-1}$ and $|\dot{u}_{n-1}|_{-1}$. Summing up, we have proved that

$\mathcal{G}_n(v)\ge \kappa_3\|v\|_1^2-\kappa_2|\Omega|-\frac{C_n}{2\tau}.$

By considering a minimizing sequence $(v_k)$ for problem (2.14), we obtain a minimizer, i.e. $u_{n+1}$. Then $w_{n+1}$ can be recovered from $u_{n+1}$ by (2.12).

Proposition 2.2 (Uniqueness). If $1/\tau>c_f^2/(6\gamma)$, then for every $(u_0, u_1)\in V\times V$ such that $\langle u_0\rangle=\langle u_1\rangle$, there exists at most one sequence $(u_n, w_n)_{n}$ which complies with (2.10).

Proof. Assume that $(u_{n+1}, w_{n+1})$ and $(\tilde{u}_{n+1}, \tilde{w}_{n+1})$ are two solutions of (2.10), and denote $\delta u=u_{n+1}-\tilde{u}_{n+1}$, $\delta w=w_{n+1}-\tilde{w}_{n+1}$. On subtracting, we obtain

$\begin{gathered} 3(\delta u, \varphi)/(2\tau) + (\nabla \delta w, \nabla \varphi)=0, \hfill \\ (\delta w, \psi)=\gamma (\nabla \delta u, \nabla \psi) + (f({u_{n + 1}})-f({{\tilde u}_{n + 1}}), \psi), \hfill \\ \end{gathered}$

(2.16)

for all $(\varphi, \psi)\in V\times V$. Choosing $\varphi=\delta w$ and $\psi=\delta u$ yields

$-(2\tau/3)|\delta w|_1^2=\gamma|\delta u|_1^2+(f(u_{n+1})-f(\tilde{u}_{n+1}), \delta u).$

Using the mean value inequality and (2.4) yields

$(s-r)(f(s)-f(r))=f'(\xi)(s-r)^2\ge-c_f(s-r)^2,$

for all $r, s \in \mathbb{R}$, for some $\xi\in\mathbb{R}$ depending on $r, s$. Thus,

$c_f|\delta u|_0^2\ge \gamma |\delta u|_1^2+(2\tau/3)|\delta w|_1^2.$

Using now (2.16) with $\varphi=\delta u$, we obtain

$c_f|\delta u|_0^2=-(2\tau c_f/3)(\nabla \delta w, \nabla \delta u)\le \gamma |\nabla\delta u|_0^2+\frac{\tau^2c_f^2}{9\gamma}|\nabla \delta w|^2.$

If $\tau c_f^2 < 6\gamma$, then $\delta \dot{w}=0$, and by (2.16), $\delta u=0$ also. Uniqueness follows.

We define the following pseudo-energy

$\varepsilon (u, v)=E(u) + \frac{1}{{4\tau }}|\dot v|_{-1}^2, \quad \forall (u, v) \in V \times V'.$

(2.17)

For a sequence $(u_n)_n$, let also $\delta u_n=u_n-u_{n-1}$ denote the backard difference. The following relation will prove useful,

$3u_{n+1}-4u_n+u_{n-1}=2\delta u_{n+1}+(\delta u_{n+1}-\delta u_n).$

(2.18)

Proposition 2.3 (Lyapunov stability). Let $\varepsilon\in[0, 1)$. If $(u_n, w_n)_{n}$ is a sequence which complies with (2.10)-(2.11), then for all $n\ge 1$,

$\begin{gathered} \varepsilon ({u_{n + 1}}, \delta {u_{n + 1}}) + \frac{{\varepsilon \gamma }}{2}|{u_{n + 1}}-{u_n}|_1^2 + \left({\frac{1}{\tau }-\frac{{c_f^2}}{{8\gamma (1-\varepsilon)}}} \right)|{u_{n + 1}}-{u_n}|_{-1}^2 \hfill \\ + \frac{1}{{4\tau }}|\delta {u_{n + 1}}-\delta {u_n}|_{-1}^2 ≤ \varepsilon ({u_n}, \delta {u_n}). \hfill \\ \end{gathered}$

(2.19)

Proof. We take the $L^2$ scalar product of equation (2.13) by $\delta u_{n+1}$ and we use (2.18).

$\begin{gathered} \frac{1}{\tau }|\delta {u_{n + 1}}|_{-1}^2 + \frac{1}{{2\tau }}{(\delta {u_{n + 1}}-\delta {u_n}, \delta {u_{n + 1}})_{-1}} + \gamma (\nabla {u_{n + 1}}, \nabla ({u_{n + 1}}-{u_n})) \hfill \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad=(f({u_{n + 1}}), {u_n}-{u_{n + 1}}). \hfill \\ \end{gathered}$

By the Taylor-Lagrange formula, from (2.4), we deduce that

$F(r)-F(s)\ge f(s)(r-s)-\frac{c_f}{2}|r-s|^2, \quad\forall r, s\in\mathbb{R}.$

Thus,

$(f(u_{n+1}), u_n-u_{n+1})\le (F(u_n), 1)-(F(u_{n+1}), 1)+\frac{c_f}{2}|u_{n+1}-u_n|_0^2.$

Next, we use the well-known identity

$(a, a-b)_m=\frac{1}{2}|a|_m^2-\frac{1}{2}|b|_m^2+\frac{1}{2}|a-b|_m^2,$

for m=-1 and m=0. We find

$\begin{gathered} \hfill \frac{1}{\tau }|\delta {u_{n + 1}}|_{-1}^2 + \frac{1}{{4\tau }}\left({|\delta {u_{n + 1}}|_{-1}^2-|\delta {u_n}|_{-1}^2 + |\delta {u_{n + 1}}-\delta {u_n}|_{-1}^2} \right) \\ \hfill + \frac{\gamma }{2}\left({|{u_{n + 1}}|_1^2-|{u_n}|_1^2 + |{u_{n + 1}}-{u_n}|_1^2} \right) \\ \hfill ≤ (F({u_n}), 1)-(F({u_{n + 1}}), 1) + \frac{{{c_f}}}{2}|{u_{n + 1}}-{u_n}|_0^2. \\ \end{gathered}$

(2.20)

The interpolation inequality (2.2) and Young’s inequality yield

$\frac{c_f}{2}|u_{n+1}-u_n|_0^2\le \frac{\gamma(1-\varepsilon)}{2}|u_{n+1}-u_n|_1^2+\frac{c_f^2}{8\gamma(1-\varepsilon)}|u_{n+1}-u_n|_{-1}^2.$

Plugging this into (2.20) gives (2.19).

Remark 2.4. If $\tau$ is small enough, then by choosing $\varepsilon\in(0, 1)$, we see that the scheme (2.10) is more dissipative than the original equation (1.1), since the $H^1$ norm $|u_{n+1}-u_n|_1^2$ appears in (2.19); in contrast, only the $H^{-1}$ norm $|u_t|_{-1}^2$ appears in (2.9).

---

3. Convergence to equilibrium

For a sequence $(u_n)_n$ in $V$, we define its omega-limit set by

$\omega((u_n)_n):=\{u_\star\in V\ :\ \exists n_k\to \infty, \ u_{n_k}\to u_\star\mbox{ (strongly) in }V\}.$

Let $M\in\mathbb{R}$ be given and consider the following affine subspace of $V$,

$V_M=\{v\in V\ :\ \langle v\rangle=M\}=M+\dot{V}.$

(3.1)

The set of critical points of $E$ (see (2.6)) in $V_M$ is

$\mathcal{S}_M=\{u^\star\in V_M\ :-\gamma\Delta u^\star+f(u^\star)-\langle f(u^\star)\rangle=0\mbox{ in }\dot{V}'\}.$

Indeed, we already know that $E\in C^2(V_M; \mathbb{R})$. Observe that, for any $u\in V_M$, $\dot{v}\in\dot{V}$, we have (see (2.7))

$\begin{gathered} {\left\langle {dE\left(u \right), v} \right\rangle _{\dot V', \dot V}}=\int_\Omega {\left[{\gamma \nabla u \cdot \nabla v + f\left(u \right)v} \right]} dx \\=\int_\Omega {\left[{\gamma \nabla u \cdot \nabla v + \left({f\left(u \right)-\left\langle {f\left(u \right)} \right\rangle } \right)v} \right]} dx \\={\left\langle {-\gamma \Delta u + f\left(u \right)-\left\langle {f\left(u \right)} \right\rangle, v} \right\rangle _{\dot V', \dot V}}. \\ \end{gathered}$

(3.2)

By definition, $u^\star$ is a critical point of $E$ in $V_M$ if $dE(u^\star)=0$ in $\dot{V}'$. The definition of $\mathcal{S}_M$ follows.

Proposition 3.1. Assume that $1/\tau>c_f^2/(8\gamma)$ and let $(u_n, w_n)_n$ be a sequence which complies with (2.10)-(2.11). Then $\delta u_n\to 0$ in $V$ and $\omega((u_n)_n)$ is a nonempty compact and connected subset of $V$ which is included in $\mathcal{S}_M$ with $M=\langle u_0\rangle$. Moreover, $E$ is constant on $\omega((u_n)_n)$.

Proof. Using the assumption on $\tau$, we may choose $\varepsilon\in (0, 1)$ such that $1/\tau=c_f^2/(8\gamma(1-\varepsilon))$. Then (2.19) reads

$\varepsilon ({u_{n + 1}}, \delta {u_{n + 1}}) + \frac{{\varepsilon \gamma }}{2}|{u_{n + 1}}-{u_n}|_1^2 + \frac{1}{{4\tau }}|\delta {u_{n + 1}}-\delta {u_n}|_{-1}^2 ≤ \varepsilon ({u_n}, \delta {u_n}),$

(3.3)

for all $n\ge 1$. In particular, $(\varepsilon (u_n, \delta u_n))_n$ is non increasing. Moreover, by (2.15),

$\varepsilon (u, v) ≥ {\kappa _3}\left\| u \right\|_1^2 + \frac{1}{{4\tau }}|\dot v|_{-1}^2-{\kappa _2}|\Omega |, \quad \forall (u, v) \in V \times V'.$

(3.4)

Since $\varepsilon (u_1, \delta u_1) < +\infty$, we deduce from (3.4) that $(u_n, \delta u_n)$ is bounded in $V\times V'$ and that $\varepsilon (u_n, \delta u_n)$ is bounded from below. Thus, $\varepsilon (u_n, \delta u_n)$ converges to some $\varepsilon _\star$ in $\mathbb{R}$. By induction, from (3.3)-(3.4) we also deduce that

$\sum_{n=1}^\infty|u_{n+1}-u_n|_1^2\le \frac{2}{\varepsilon\gamma}(\varepsilon (u_1, \delta u_1)+\kappa_2|\Omega|) < +\infty.$

In particular, $\delta u_n\to 0$ in $V$. This implies that $E(u_n)\to \varepsilon _\star$, and so $E$ is equal to $\varepsilon _\star$ on $\omega((u_n)_n)$.

Next, we claim that the sequence $(u_n)$ is precompact in $V$. Let us first assume $d=3$. We deduce from the Sobolev imbedding ^[19] that $(u_n)$ is bounded in $L^6(\Omega)$. By the growth condition (2.3), there exists $2\ge q>6/5$ such that $\|f(u_{n+1})\|_{L^q(\Omega)}\le M_1$, where $M_1$ is independent of $n$. By elliptic regularity ^[3], we deduce from (2.13) that $(u_{n+1})_{n\ge 1}$ is bounded in $W^{2, q}(\Omega)$. Finally, from the Sobolev imbedding ^[19], $W^{2, q}(\Omega)$ is compactly imbedded in $V$, and the claim is proved.

In the case $d=1$ or $2$, we obtain directly from the Sobolev imbedding that $f(u_{n+1})$ is bounded in $L^q(\Omega)$, for any $q < +\infty$, and we conclude similarly.

As a consequence, $\omega((u_n)_n)$ is a nonempty compact subset of $V$. Since $|u_{n+1}-u_n|_1\to 0$, $\omega((u_n)_n)$ is also connected. Let finally $u_\star$ belong to $\omega((u_n)_n)$, with $n_k\to \infty$ such that $u_{n_k}\to u_\star$ in $V$. We let $n_k$ tend to $\infty$ in (2.11). Thanks to (2.13), the whole sequence $(u_n)$ belongs to $V_M$ and $u_\star$ as well, where $M=\langle u_0\rangle$. By (2.18), the term corresponding to the discrete time derivative tends to $0$ in $V$, and we obtain that $u_\star$ belongs to $\mathcal{S}_M$.

If the critical points of $E$ are isolated, i.e. $\mathcal{S}_M$ is discrete, then Proposition 3.1 ensures that the sequence $(u_n)_n$ converges in $V$. However, as pointed out in the introduction, the structure of $\mathcal{S}_M$ is generally not known, and there may even be a continuum of steady states. In such cases, the Łojasiewicz-Simon inequality which follows is needed to ensure convergence of the whole sequence $(u_n)$.

Lemma 3.2. Let $u^\star\in\mathcal{S}_M$. Then there exist constants $\theta\in(0, 1/2)$ and $\delta>0$ depending on $u^\star$ such that, for any $u\in V_M$ satisfying $|u-u^\star|_1 < \delta$, there holds

$|E(u)-E(u^\star)|^{1-\theta}\le |-\gamma \Delta u+f(u)-\langle f(u)\rangle|_{-1}.$

(3.5)

Proof. We will apply the abstract result of Theorem 11.2.7 in ^[27]. We introduce the auxiliary functional $E_M(v)=E(M+v)$ on $\dot{V}$. We will also use the auxiliary functions

$f_M(s)=f(M+s)\quad\mbox{ and }\quad F_M(s)=F(M+s).$

It is obvious that

$E_M(v)=\int_\Omega\left[\frac{\gamma}{2}|\nabla v|^2+F_M(v)\right] dx.$

The function $E_M$ is of class $C^2$ on $\dot{V}$ and by (3.2), for any $v\in \dot{V}$, we have

$dE_M(v)=-\gamma\Delta v+f_M(v)-\langle f_M(v)\rangle\mbox{ in }\dot{V}'.$

Similarly, by (2.8), for any $v, \varphi\in\dot{V}$, we have

$d^2E_M(v)\varphi=-\gamma\Delta\varphi+f_M'(v)\varphi-\langle f_M'(v)\varphi\rangle\mbox{ in }\dot{V}'.$

(3.6)

Let $v^\star\in \dot{V}$ be a critical point of $E_M$, i.e. a solution of $dE_M(v^\star)=0$ in $\dot{V}'$. Using (2.3) and elliptic regularity, we obtain that $v^\star\in C^0(\overline{\Omega})\subset L^\infty(\Omega)$. In particular, $f_M'(v)\in L^\infty(\Omega)$. The operator $A=d^2E_M(v^\star)\in\mathcal{L}(\dot{V}, \dot{V}')$ (cf.(3.6)) can be written

$A=-\gamma\Delta+P_0(f'_M(v^\star)Id),$

where $-\gamma\Delta:\dot{V}\to\dot{V}'$ is an isomorphism, $P_0:H\to\dot{H}$ is the $L^2$-projection operator, and $f_M'(v^\star)Id:\dot{V}\to H$ is a multiplication operator. Since $\dot{V}$ is compactly imbedded in $\dot{H}$ ^[19], $f_M'(v^\star)Id:\dot{V}\to H$ is compact, and $P_0(f_M'(v^\star)Id)$ as well. Using [27, Theorem 2.2.5], we obtain that $A$ is a semi-Fredholm operator.

Next, let $N(A)$ denote the kernel of $A$, and $\Pi:\dot{V}\to N(A)$ the $L^2$ projection. By [27, Corollary 2.2.6], $\mathcal{L}:=A+\Pi:\dot{V}\to\dot{V}'$ is an isomorphism. We choose $Z=\dot{H}$ and denote $W=\mathcal{L}^{-1}(Z)$; $W$ is a Banach space for the norm $\|w\|_W=|\mathcal{L}(w)|_0$. We claim that $W$ is continuously imbedded in $W^{2, 2}(\Omega)$.

Indeed, by definition, $w\in W$ if and only if $w\in\dot{V}$ and $\mathcal{L}(w)=g$ for some $g\in Z$, i.e.

$w\in\dot{V}\mbox{ and }-\gamma\Delta w+f_M'(v^\star)w-\langle f_M'(v^\star)w\rangle+\Pi w=g.$

Thus, $-\Delta w\in \dot{H}$. By elliptic regularity ^[3], $w\in W^{2, 2}(\Omega)$. Moreover, by the triangle inequality,

$\gamma |-\Delta w|_0\le C\|f_M'(v^\star)\|_{L^\infty}|w|_0+|\Pi w|_0+|\mathcal{L}(w)|_0\le C\|w\|_W,$

where $C$ is a constant independent of $w$. But, by elliptic regularity ^[3], we also know that $\|w\|_{W^{2, 2}}\le C|-\Delta w|_0$ for all $w\in\dot{V}$. This proves the claim.

The Nemytskii operator $f_M:v\mapsto f_M(v)$ is analytic from $L^\infty(\Omega)$ into $L^\infty(\Omega)$ (see [27, Example 2.3]). Using [27, Proposition 2.3.4], we find that the functional $v\mapsto \int_\Omega F_M(v)$ is real analytic from $L^\infty(\Omega)$ into $\mathbb{R}$. Thus, $E_M$, which is the sum of a continuous quadratic functional and of a functional which is real analytic on $W\subset W^{2, 2}(\Omega)\subset L^\infty(\Omega)$, is real analytic on $W$. We also obtain that $dE_M:W\to Z$ is real analytic.

We are therefore in position to apply the abstract Theorem 11.2.7 in ^[27], which shows that there exist $\theta\in (0, 1/2)$ and $\delta >0$ such that for all $v\in \dot{V}$,

$|v-v^\star|_1 < \delta\Rightarrow |E_M(v)-E_M(v^\star)|^{1-\theta}\le |dE_M(v)|_{-1}.$

(3.7)

Finally, we note that any $u^\star\in \mathcal{S}_M$ can be written $u^\star=M+v^\star$, where $v^\star$ is a critical point of $E_M$; by definition of $V_M$, any $u\in V_M$ can be written $u=M+v$ with $v\in \dot{V}$. The expected Łojasiewicz-Simon inequality (3.5) is exactly (3.7) written in terms of $u^\star$, $u$, $E$ and $f$.

Theorem 3.3. Assume that $1/\tau>c_f^2/(8\gamma)$ and let $(u_n, w_n)_n$ be a sequence which complies with (2.10)-(2.11). Then the whole sequence converges to $(u_\infty, w_\infty)$ in $V\times V$, with $u_\infty\in \mathcal{S}_M$, $M=\langle u_0\rangle$, and $w_\infty$ constant. Moreover, the following convergence rate holds

$\|u_n-u_\infty\|_1+\|w_n-w_\infty\|_1\le Cn^{-\frac{\theta}{1-\theta}},$

(3.8)

for all $n\ge 2$, where $C$ is a constant depending on $\|u_0\|_1$, $\|u_1\|_1$, $f$, $\gamma$, $\tau$, and $\theta$, while $\theta\in(0, 1/2)$ may depend on $u_\infty$.

Proof. Let $M=\langle u_0\rangle$. For every $u_\star\in \omega((u_n)_n)$, there exist $\theta\in(0, 1)$ and $\delta>0$ which may depend on $u_\star$ such that the inequality (3.5) holds for every $u\in B_\delta (u_\star)=\{u\in V_M\ :\ |u-u_\star| < \delta\}$. The union of balls $\{B_\delta(u_\star)\ :\ u_\star\in\omega((u_n)_n)\}$ forms an open covering of $\omega((u_n)_n)$ in $V_M$. Due to the compactness of $\omega((u_n)_n)$ in $V$, we can find a finite subcovering $\{B_{\delta^i}(u_\star^i)\}_{i=1}^m$ such that the constants $\delta^i$ and $\theta^i$ corresponding to $u_\star^i$ in Lemma 3.2 are indexed by $i$.

From the definition of $\omega((u_n)_n)$, we know that there exists a sufficiently large $n_0$ such that $u_n\in \mathcal{U}=\cup_{i=1}^mB_{\delta^i}(u_\star^i)$ for all $n\ge n_0$. Taking $\theta=\min_{i=1}^m\{\theta^i\}$, we deduce from Lemma 3.2 and Proposition 3.1 that for all $n\ge n_0$,

$|E(u_n)-\varepsilon _\star|^{1-\theta}\le |-\gamma \Delta u_n+f(u_n)-\langle f(u_n)\rangle|_{-1},$

(3.9)

where $\varepsilon _\star$ is the value of $E$ on $\omega((u_n)_n)$. We may also assume (by taking a larger $n_0$ if necessary) that for all $n\ge n_0$, $|\delta u_n|_{-1}\le 1$.

We denote $\Phi_n=\varepsilon (u_n, \delta u_n)-\varepsilon _\star$, so that $\Phi_n\ge 0$ and $\Phi_n$ is nonincreasing. Let $n\ge n_0$. Using the inequality $(a+b)^{1-\theta}\le a^{1-\theta}+b^{1-\theta}$, valid for all $a, b\ge 0$, together with (3.9), we obtain

$\begin{gathered} \Phi _{n + 1}^{1-\theta } ≤ |E({u_{n + 1}})-{\varepsilon _ \star }{|^{1-\theta }} + {(4\tau)^{\theta-1}}|\delta {u_{n + 1}}|_{-1}^{2(1-\theta)} \\ ≤ |-\gamma \Delta {u_{n + 1}} + f({u_{n + 1}})-\langle f({u_{n + 1}})\rangle {|_{-1}} + {(4\tau)^{\theta-1}}|\delta {u_{n + 1}}{|_{-1}} \\ ≤ C\left({|{u_{n + 1}}-{u_n}{|_{-1}} + |\delta {u_{n + 1}}-\delta {u_n}{|_{-1}}} \right), \\ ≤ C{\left({|{u_{n + 1}}-{u_n}|_1^2 + |\delta {u_{n + 1}}-\delta {u_n}|_{-1}^2} \right)^{1/2}} \\ \end{gathered}$

(3.10)

where $C=C(\tau, \theta, \|(-\dot{\Delta})^{-1}\|_{\mathcal{L}(\dot{V}', \dot{V}')}, \|(-\dot{\Delta})^{-1}\|_{\mathcal{L}(\dot{V}, \dot{V})})$ (here and in the following, $C$ denotes a generic positive constant independent of $n$). For the third inequality, we have used (2.13) and (2.18). Next, we choose $\varepsilon\in (0, 1)$ such that $1/\tau=c_f^2/(8\gamma(1-\varepsilon))$. Then (3.3) holds, and it can be written

$\Phi_n-\Phi_{n+1}\ge C\left(|u_{n+1}-u_n|_1^2+|\delta u_{n+1}-\delta u_n|_{-1}^2\right),$

(3.11)

with $C=C(\tau, \gamma, \varepsilon)>0$.

Assume first that $\Phi_{n+1}>\Phi_n/2$. Then

$\Phi_n^{\theta}-\Phi_{n+1}^{\theta}=\theta\int_{\Phi_{n+1}}^{\Phi_n}x^{\theta-1}dx \ge \theta\frac{\Phi_n-\Phi_{n+1}}{\Phi_n} \ge 2^{\theta-1}\theta\frac{\Phi_n-\Phi_{n+1}}{\Phi_{n+1}^{1-\theta}}.$

Using (3.10) and (3.11), we find

$\Phi_n^\theta-\Phi_{n+1}^\theta\ge C\left(|u_{n+1}-u_n|_1^2+|\delta u_{n+1}-\delta u_n|_{-1}^2\right)^{1/2},$

where $C=C(\tau, \theta, \gamma, \varepsilon, \|(-\dot{\Delta})^{-1}\|_{\mathcal{L}(\dot{V}', \dot{V}')}, \|(-\dot{\Delta})^{-1}\|_{\mathcal{L}(\dot{V}, \dot{V})})$.

Now, if $\Phi_{n+1}\le \Phi_n/2$, then

$\Phi_n^{1/2}-\Phi_{n+1}^{1/2}\ge(1-1/\sqrt{2})\Phi_n^{1/2}\ge (1-1/\sqrt{2})(\Phi_n-\Phi_{n+1})^{1/2}$

and using (3.11) again, we find

$\Phi_n^{1/2}-\Phi_{n+1}^{1/2}\ge C\left(|u_{n+1}-u_n|_1^2+|\delta u_{n+1}-\delta u_n|_{-1}^2\right)^{1/2}.$

Thus, in both cases, we have

$|u_{n+1}-u_n|_1\le C(\Phi_n^\theta-\Phi_{n+1}^\theta)+C(\Phi_n^{1/2}-\Phi_{n+1}^{1/2}),$

(3.12)

for all $n\ge n_0$. Summing on $n\ge n_0$, we obtain

$\sum_{n=n_0}^{\infty}|u_{n+1}-u_n|_1\le C\Phi_{n_0}^\theta+C\Phi_{n_0}^{1/2} < +\infty.$

(3.13)

Using the Cauchy criterion, we find that the whole sequence $(u_n)$ converges to some $u_\infty$ in $V$. By Proposition 3.1, $u_\infty$ belongs to $\mathcal{S}_M$. Using the second equation in (2.12), we see that $\dot{w}_n\to 0$. For the term $\langle w_n\rangle$, we write

$\int_\Omega|f(u_n)-f(u_\infty)|dx=\int_\Omega\left|\int_0^1f'((1-s)u_n+su_\infty)(u_n-u_\infty)ds\right|dx.$

Using assumption (2.3), Hölder's inequality and Sobolev imbeddings, we find

$\int_\Omega|f(u_n)-f(u_\infty)|dx\le C(\|u_n\|_1, \|u_\infty\|_1)\|u_n-u_\infty\|_1.$

Since $(u_n)$ is bounded in $V$, this yields, for all $n\ge 2$,

$|\langle w_n\rangle-w_\infty|=|\langle f(u_n)\rangle-\langle f(u_\infty)|\rangle\le \langle |f(u_n)-f(u_\infty)|\rangle\le C\|u_n-u_\infty\|_1,$

(3.14)

where we have used the last equation in (2.12) and where $w_\infty=\langle f(u_\infty)\rangle$. This implies that $w_n\to w_\infty$ in $V$ (see (2.1)), and it concludes the proof of convergence.

For the convergence rate, we will first show that

$0\le \Phi_n\le Cn^{-\frac{1}{1-2\theta}},$

(3.15)

for all $n\ge n_1$, for some $n_1> n_0$ large enough. The exponent $\theta$ is the same as above. If $\Phi_{n_1}=0$ for some $n_1\ge n_0$, then $\Phi_n=0$ for all $n\ge n_1$, and estimate 3.15 is obvious. So we may assume that $\Phi_n>0$ for all $n$. Let $n\ge n_0$ and denote $G(s)=\cfrac{1}{s^{1-2\theta}}$. The sequence $G(\Phi_n)$ is nondecreasing and tends to $+\infty$.

If $\Phi_{n+1}>\Phi_n/2$, then

$\begin{gathered} G({\Phi _{n + 1}})-G({\Phi _n})=\int_{{\Phi _{n + 1}}}^{{\Phi _n}} {\frac{{1-2\theta }}{{{s^{2-2\theta }}}}} ds \\ ≥ (1-2\theta){2^{2\theta-2}}\Phi _{n + 1}^{2\theta-2}[{\Phi _n}-{\Phi _{n + 1}}] \\ \mathop ≥ \limits^{\left({3.11} \right)} C\Phi _{n + 1}^{2\theta-2}\left({|{u_{n + 1}}-{u_n}|_1^2 + |\delta {u_{n + 1}}-\delta {u_n}|_{-1}^2} \right) \\ \mathop ≥ \limits^{\left({3.10} \right)} {C_1}, \\ \end{gathered}$

where $C_1$ is a positive constant independent of $n$.

If $\Phi_{n+1}\le \Phi_n/2$ and $\Phi_n\le 1$, then

$G(\Phi_{n+1})-G(\Phi_n)\ge \frac{2^{1-2\theta}-1}{\Phi_n^{1-2\theta}}\ge 2^{1-2\theta}-1.$

Let $n_0'\ge n_0$ be large enough so that $\Phi_{n_0'}\le 1$. Then, in both cases, for all $n\ge n_0'$, we have

$G(\Phi_{n+1})-G(\Phi_n)\ge C_2,$

where $C_2=\min\{C_1, 2^{1-2\theta}-1\}>0$. By summation on $n$, we obtain

$G(\Phi_n)-G(\Phi_{n_0'})\ge C_2(n-n_0'),$

for all $n\ge n_0'$. Thus, by choosing $n_1> n_0'$ large enough, we have

$G(\Phi_n)\ge \frac{C_2}{2}n,$

for all $n\ge n_1$, which yields (3.15).

Now, by summing estimate (3.12) on $n$, we find

$|u_n-u_\infty|_1\le \sum_{k=n}^\infty|u_{k+1}-u_k|_1\le C\Phi_n^\theta+C\Phi_n^{1/2}\le C\Phi_n^\theta,$

for all $n\ge n_1$. Using (3.15) yields

$\|u_n-u_\infty\|_1\le Cn^{-\frac{\theta}{1-2\theta}},$

(3.16)

for all $n\ge n_1$. We may change the constant $C$ in order for the estimate to hold for all $n\ge 2$. From (3.16) and the second equation in (3.12), we obtain the convergence rate for $(\dot{w}_n)$. The convergence rate for $\langle w_n\rangle$ is a consequence of (3.16) and (3.14). This concludes the proof.

Remark 3.4. It is possible to show that a local minimizer of $E$ in $V_M$ is stable uniformly with respect to $\tau$. More precisely, let $(u_n^\tau)_n$ denote a sequence which complies with (2.13) and corresponding to a time-step $\tau$. We assume $\tau\in(0, \tau^\star]$ where $\tau^\star>0$ is such that $1/\tau^\star>c_f^2/(8\gamma)$. If $u^\star\in V_M$ is a local minimizer of $E$ in $V_M$, and if $u_0^\tau=u_1^\tau$ is close enough to $u^\star$ in $V_M$, then the whole sequence $(u_n^\tau)_n$ remains close to $u^\star$, uniformly with respect to $\tau\in(0, \tau^\star]$. The proof of this stability result is based on the Łojasiewicz-Simon inequality (it may be false for a $C^\infty$ nonlinearity, see ^[2]). It is proved in ^[4] for several fully discrete approximations of the Allen-Cahn equation. The case of the semi-discrete scheme (2.13) is more involved. Indeed, dissipative estimates (uniform in $\tau$) are needed to obtain pre-compactness of the set $\{u_n^\tau\ :\ \tau\in(0, \tau^\star], \ n\in\mathbb{N}\}$ in $V_M$. Moreover, as $\tau\to 0^+$, the dissipation due to the scheme vanishes (cf. Remark 2.4). Thus, instead of the series $\sum_n|u_{n+1}^\tau-u_n^\tau|_1$ (cf. (3.13)), we have to deal with the series $\sum_n|u_{n+1}^\tau-u_n^\tau|_{-1}$. We refer the interested reader to ^[28,35] for the proof of stability of a local minimizer in an infinite dimensional setting (for continuous dynamical systems).

---

Acknowledgments

Paola F. Antonietti has been partially supported by SIR Project No. RBSI14VT0S“PolyPDEs: Non-conforming polyhedral finite element methods for the approximation of partial differential equations”funded by MIUR.

---