Layer dynamics for the one dimensional $\varvec{\varepsilon }$-dependent Cahn–Hilliard/Allen–Cahn equation

Abstract

We study the dynamics of the one-dimensional $\varepsilon$-dependent Cahn–Hilliard/Allen–Cahn equation within a neighborhood of an equilibrium of N transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a mass-conservation constraint in place of one of the zero-mass flux boundary conditions at $x=1$. Motivated by the study of Carr and Pego on the layered metastable patterns of Allen–Cahn in Carr and Pego in (Commun Pure Appl Math 42:523–576, 1989), and by this of Bates and Xun (J Differ Equ 111:421–457, 1994) for the Cahn–Hilliard equation, we implement an N-dimensional, and a mass-conservative $N-1$-dimensional manifold respectively; therein, a metastable state with N transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized Cahn–Hilliard/Allen–Cahn operator, and specify wide families of $\varepsilon$-dependent weights $\delta (\varepsilon )$, $\mu (\varepsilon )$, acting at each part of the operator, for which the dynamics are stable and rest exponentially small in $\varepsilon$. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to Allen–Cahn, or, when the mass is conserved, close to the Cahn–Hilliard solution.

Appendix

In Sects. 4.1–4.2 we prove various estimates for the non mass-conserving manifold approximation used throughout this paper, and collect together existing results thereof from the work of Carr and Pego [10]. Some of the estimates have been also proven in [4] and then used in their integrated version for the mass-conserving case. Then in Sect. 4.3 we derive certain a priori energy estimates for establishing the well-posedness of the mass-conserving problem considered in §3.

1.1 Estimates for the stationary Dirichlet problem (2.7)

As it is clear from the definition (2.10), many of our subsequent estimates involving $u^h,$ rest upon certain properties of the stationary states $\phi$ of (1.1), namely the solutions of the Dirichlet problem (2.7). In this section we record these properties and for their proof we refer to [10].

Since $\phi _{\scriptscriptstyle \varepsilon }(0, \ell , \pm 1)$ depends on $\varepsilon$ and $\ell$ only through the ratio $\texttt {r} = \varepsilon /\ell ,$ we may define

$$\begin{aligned} \alpha _{\pm }( \texttt {r}) := F\big (\phi _{\scriptscriptstyle \varepsilon }(0,\ell , \pm 1)\big ),\qquad \beta _{\pm }(\texttt {r}) := 1 {\mp } \phi _{\scriptscriptstyle \varepsilon }(0,\ell , \pm 1). \end{aligned}$$

(4.1)

In what follows, C will denote a positive constant not necessarily the same at each occurrence and we stress that C is independent of $\varepsilon , x, h_j$’s, j’s.

Proposition 4.1

([10, Proposition 3.4])There exists $r_0>0$ such that if $0< r < r_0,$ then

$$\begin{aligned} \alpha _{\pm }(r)= & {} \frac{1}{2} K_{\pm }^2 A_{\pm }^2 {\text {exp}}\Big (\frac{-A_{\pm }}{r}\Big ) \left[ 1+\mathcal {O}\left( r^{-1}{\text {exp}}\bigl (\frac{-A_{\pm }}{2r}\bigr )\right) \right] , \end{aligned}$$

(4.2)

$$\begin{aligned} \beta _{\pm }(r)= & {} K_{\pm } {\text {exp}}\Big (\frac{-A_{\pm }}{2r}\Big ) \left[ 1+\mathcal {O}\left( r^{-1}{\text {exp}}\bigl (\frac{-A_{\pm }}{2r}\bigr )\right) \right] , \end{aligned}$$

(4.3)

where

$$\begin{aligned} A_{\pm }:= & {} f'(\pm 1) > 0, \end{aligned}$$

(4.4)

$$\begin{aligned} K_{\pm }:= & {} 2 \, {\text {exp}} \left[ \int _0^1\Big (\frac{ A }{ \sqrt{2 F(\pm t)} } - \frac{1}{1-t}\Big ) \, dt \right] , \end{aligned}$$

(4.5)

with $A:=\min \{A_+, A_-\},$ and the asymptotic formulas (4.2), (4.3) also hold when they are differentiated a finite number of times, e.g.

$$\begin{aligned} \alpha _+'(r)= - A_+ \, r^{-2} \, \alpha _+(r) \, \left[ 1+\mathcal {O}\left( r^{-1}{\text {exp}}\bigl (\frac{-A_+}{2r}\bigr )\right) \right] . \end{aligned}$$

(4.6)

Proposition 4.2

([10, Lemma 7.4])Let $0< r < r_0.$ Then there exist constants $C_1 \in (0,1)$ and $C_2 > 0$ such that, for $|x\pm \frac{\ell }{2}|\le \varepsilon ,$

$$\begin{aligned} |\phi (x, \ell , \pm 1)|\le & {} C_1 , \end{aligned}$$

(4.7a)

$$\begin{aligned} F(\phi (x, \ell , \pm 1))\ge & {} C_2. \end{aligned}$$

(4.7b)

Proposition 4.3

([10, Lemma 7.5])For $|x|\le 2\varepsilon ,$ we have

$$\begin{aligned} |\phi (x, \ell , \pm 1) - (-1)^j|\le & {} C \beta (r) , \end{aligned}$$

(4.8a)

$$\begin{aligned} |\phi _x(x, \ell , \pm 1)|\le & {} C \varepsilon ^{-1} \beta (r). \end{aligned}$$

(4.8b)

Proposition 4.4

([10, Lemma 7.7])We have

$$\begin{aligned} \int _{-\ell /2}^{\ell /2} |\phi _x| \, dx \le 2, \qquad \int _{-\ell /2}^{\ell /2} |\phi _{xx}| \, dx \le C\varepsilon ^{-1}, \qquad \int _{-\ell /2}^{\ell /2} |\phi _{xx}|^2 \, dx \le C\varepsilon ^{-3}. \end{aligned}$$

(4.9)

Beside the above estimates for $\phi$ and its derivatives with respect to x,  we will also need estimates on the derivatives $\phi _\ell (x, \ell , \pm 1):=\tfrac{\partial }{\partial \ell }\phi (x, \ell , \pm 1).$

Proposition 4.5

([10, Lemma 7.8])For $x \in [-\ell , \ell ],$

$$\begin{aligned} \phi _\ell \big (x, \ell , \pm 1\big ) \,= \, - \frac{1}{2} {\text {sgn}}(x)\,\phi _x\big (x, \ell , \pm 1\big ) \,+\, \mathrm {w}\big (x, \ell , \pm 1\big ), \end{aligned}$$

(4.10)

where, for $x \ne 0,$

$$\begin{aligned} \mathrm {w}\big (x, \ell , \pm 1\big ) = \varepsilon ^{-1} \, \ell ^{-2} \, \alpha '_\pm (r) \,\phi _x(|x|, \ell , \pm 1)\, \int _{\ell /2}^{|x|} \phi _x\big (s, \ell , \pm 1\big )^{-2} \, ds, \end{aligned}$$

(4.11)

and

$$\begin{aligned} \mathrm {w}\big (0, \ell , \pm 1\big ) = \frac{-\varepsilon ^{-1} \, \ell ^{-2} \, \alpha '_\pm (r)}{\phi _{xx}(0, \ell , \pm 1)}. \end{aligned}$$

(4.12)

Proposition 4.6

([10, Lemma 7.9])Let $\mathrm {w}$ be defined in Proposition 4.5. There exists $r_0>0$ such that if $0< r < r_0,$ then

$$\begin{aligned} |\mathrm {w}\big (x, \ell , \pm 1\big )|\le & {} C \varepsilon ^{-1} \beta _\pm (r), \qquad \hbox {for}\quad x \in \big [-\frac{\ell }{2} - \varepsilon , \, \frac{\ell }{2} + \varepsilon \big ], \end{aligned}$$

(4.13)

$$\begin{aligned} \ |\mathrm {w}\big (x, \ell , \pm 1\big )|\le & {} C \varepsilon ^{-1} \alpha _\pm (r), \qquad \hbox {for} \quad \big |x \pm \frac{\ell }{2}\big | < \varepsilon . \end{aligned}$$

(4.14)

Lemma 4.7

([10, Lemma 7.10]) For $x \in [-\tfrac{\ell }{2} - \varepsilon , \, \tfrac{\ell }{2} + \varepsilon ], \, x\ne 0,$

$$\begin{aligned} \varvec{\big |} \mathrm {w}_x\big (x, \ell , \pm 1\big ) \varvec{\big |} = \left| \phi _{\ell x}\big (x, \ell , \pm 1\big ) + \frac{1}{2} {\text {sgn}}(x)\, \phi _{xx}\big (x, \ell , \pm 1\big ) \right| \le C \,\varepsilon ^{-2}\, r^{-1}\, \beta _{\pm }(r).\nonumber \\ \end{aligned}$$

(4.15)

One may show that $\mathrm {w}$ is $C^2$ on $[0, \ell ]$ and satisfies (see [10, (7.19)])

$$\begin{aligned} \varepsilon ^2 \mathrm {w}_{xx} = f'(\phi ) \mathrm {w}, \end{aligned}$$

(4.16)

which together with (4.13) yields

$$\begin{aligned} \varvec{\big |} \mathrm {w}_{xx}\big (x, \ell , \pm 1\big ) \varvec{\big |} \le C \,\varepsilon ^{-3}\, \beta _{\pm }(r). \end{aligned}$$

(4.17)

1.2 Estimates on the states $u^{\mathrm{h}}$

For $j =1 ,2, \ldots , N+1,$ and the $\ell _j$ that are given in (2.9), we set

$$\begin{aligned} r_j := \frac{\varepsilon }{\ell _j}, \end{aligned}$$

(4.18)

and

$$\begin{aligned} \alpha ^j := {\left\{ \begin{array}{ll} \alpha _+(r_j), &{} \hbox {for } j \hbox { even}, \\ \alpha _-(r_j), &{} \hbox {for } j \hbox { odd}, \end{array}\right. } \qquad \hbox {and}\qquad \beta ^j := {\left\{ \begin{array}{ll} \beta _+(r_j), &{} \hbox {for } j \hbox { even}, \\ \beta _-(r_j), &{} \hbox {for } j \hbox { odd}. \end{array}\right. } \end{aligned}$$

(4.19)

We also set

$$\begin{aligned} r := \max \limits _{\scriptscriptstyle 1\le j \le N+1} r_j = \frac{\varepsilon }{\min _j \ell _j}, \end{aligned}$$

(4.20)

and

$$\begin{aligned} \alpha (r) := \max \limits _{\scriptscriptstyle 1\le j \le N+1} \alpha ^j, \qquad \hbox {and}\qquad \beta (r) := \max \limits _{\scriptscriptstyle 1\le j \le N+1} \beta ^j. \end{aligned}$$

(4.21)

From the first estimate in (4.9) and the zero boundary values in (2.7) we deduce that $|\phi | \le 2$ on $[- \tfrac{\ell }{2}, \tfrac{\ell }{2}].$ Therefore, for each $j=1,\ldots ,N+1,$

$$\begin{aligned} |\phi ^j| \le 2 \qquad \hbox {on}\quad \bigl [m_j - \tfrac{\ell _j}{2}, \, m_j + \tfrac{\ell _j}{2}\bigr ], \end{aligned}$$

(4.22)

and as a consequence of the definition (2.10), $u^h$ is uniformly bounded on [0, 1],  thus $f(u^h)$ and $f'(u^h)$ are uniformly bounded too.

Similarly, from the second estimate in (4.9) we get that

$$\begin{aligned} |\phi ^j_x| \le C\varepsilon ^{-1} \qquad \hbox {on} \quad \bigl [m_j - \tfrac{\ell _j}{2},\, m_j + \tfrac{\ell _j}{2}\bigr ]. \end{aligned}$$

(4.23)

By (2.15) and (4.22) we get

$$\begin{aligned} |\phi ^j_{xx}| \le C\varepsilon ^{-2} \qquad \hbox {on}\quad \bigl [m_j - \tfrac{\ell _j}{2},\, m_j + \tfrac{\ell _j}{2}\bigr ], \end{aligned}$$

(4.24)

and a differentiation of (2.15) together with (4.22), (4.23) yields

$$\begin{aligned} |\phi ^j_{xxx}| \le C\varepsilon ^{-3} , \qquad \hbox {on} \quad \bigl [m_j - \tfrac{\ell _j}{2},\, m_j + \tfrac{\ell _j}{2}\bigr ], \end{aligned}$$

(4.25)

and in general, we may see that, see also in [4]

$$\begin{aligned} \Big | \partial _x^n \phi ^j \Big | \le C \varepsilon ^{-n}, \qquad \quad \hbox {on}\quad \bigl [m_j - \tfrac{\ell _j}{2}, m_j + \tfrac{\ell _j}{2}\bigr ]. \end{aligned}$$

(4.26)

By Proposition 4.3, we have

$$\begin{aligned} |\phi ^j(x) - (-1)^j|\le & {} C \beta (r) , \end{aligned}$$

(4.27a)

$$\begin{aligned} |\phi ^j_x(x)|\le & {} C \varepsilon ^{-1} \beta (r), \qquad \qquad \hbox {for}\quad |x-m_j| < 2 \varepsilon . \end{aligned}$$

(4.27b)

As a consequence of (4.27a), we have

$$\begin{aligned} |f(\phi ^j)| \,=\, |f(\phi ^j) - f((-1)^j)| \le C \beta (r), \qquad \qquad \hbox {for} \quad |x-m_j|<2\varepsilon , \end{aligned}$$

(4.28)

which on its turn together with (2.15) implies

$$\begin{aligned} | \phi _{xx}^j | \le C \varepsilon ^{-2} \beta (r), \qquad \qquad \hbox {for} \quad |x-m_j|<2\varepsilon . \end{aligned}$$

(4.29)

Proposition 4.8

([10, Lemma 8.2])Let $r_0 > 0$ be sufficiently small. There exist constants $C_1, C_2$ such that if we assume that $\varepsilon /\ell _j < r_0$ and $\varepsilon /\ell _{j+1} < r_0$ for $j \in \{1, 2, \ldots , N\},$ then

$$\begin{aligned} \varvec{\big |} \phi ^j(x) \, - \, \phi ^{j+1}(x) \varvec{\big |}\le & {} C_1\, \varvec{\big |} a^j \, - \, a^{j+1} \varvec{\big |}, \end{aligned}$$

(4.30a)

$$\begin{aligned} \varvec{\Big |} \phi ^j_x(x) \, - \, \phi ^{j+1}_x(x) \varvec{\Big |}\le & {} C_2 \,\varepsilon ^{-1}\, \varvec{\Big |} a^j \, - \, a^{j+1} \varvec{\Big |}, \qquad \qquad \hbox {for} \quad |x-h_j|<\varepsilon . \end{aligned}$$

(4.30b)

Moreover, by (2.15), mean value theorem and (4.30a) we have, with some $\theta _x$ between $\phi ^j(x)$ and $\phi ^{j+1}(x),$

$$\begin{aligned} \varvec{|} \phi ^j_{xx}(x) - \phi ^{j+1}_{xx}(x)\varvec{|}= & {} \varepsilon ^{-2} \varvec{|} f(\phi ^j(x)) -f(\phi ^{j+1}(x)) \varvec{|} \nonumber \\= & {} \varepsilon ^{-2} \varvec{|} f'(\theta _x) \varvec{|} \, \varvec{|}\phi ^{j+1} - \phi ^j\varvec{|} \nonumber \\\le & {} C \varepsilon ^{-2} \varvec{\big |} a^j \, - \, a^{j+1} \varvec{\big |}, \qquad \qquad \hbox {for} \quad |x-h_j|<\varepsilon . \end{aligned}$$

(4.31)

By differentiating (2.15), we may proceed recursively to get, for $n=1,2,3, \cdots ,$

$$\begin{aligned} \varvec{|} \partial ^n_x\phi ^j - \partial ^n_x\phi ^{j+1}\varvec{|}\le & {} C \varepsilon ^{-n} \varvec{\big |} a^j \, - \, a^{j+1} \varvec{\big |}, \qquad \qquad \hbox {for} \quad |x-h_j|<\varepsilon . \end{aligned}$$

(4.32)

Considering the smooth cut-off function $\chi ^j$ let us notice that, for $n=1,2,3,\cdots ,$

$$\begin{aligned} \Big | \frac{d^n }{dx^n} \chi ^j \Big | \le C \varepsilon ^{-n}. \end{aligned}$$

(4.33)

By (2.13), (4.23) and (4.33) we have

$$\begin{aligned} |u^h_{x}| \le C \varepsilon ^{-1}, \qquad \hbox {on} \quad [0, 1], \end{aligned}$$

(4.34)

and by (2.14), (4.24), (4.30), (4.33) we easily get

$$\begin{aligned} |u^h_{xx}| \le C \varepsilon ^{-2}, \qquad \hbox {on} \quad [0, 1]. \end{aligned}$$

(4.35)

Differentiating (2.14) we immediately get

$$\begin{aligned} u_{xxx}^h = {\left\{ \begin{array}{ll} \phi _{xxx}^j, m_j \le x \le h_j-\varepsilon , \\ \chi _{xxx}^j \, \big (\phi ^{j+1}-\phi ^j\big ) + 3 \chi _{xx}^j \, \big (\phi ^{j+1}_x-\phi ^j_x\big ) + 3 \chi _x^j \, \big (\phi _{xx}^{j+1}-\phi _{xx}^j\big ) \\ + (1-\chi ^j)\phi ^j_{xxx} + \chi ^j\phi ^{j+1}_{xxx}, \,|x-h_j| < \varepsilon , \\ \phi _{xxx}^{j+1}, h_j +\varepsilon \le x \le m_{j+1}. \end{array}\right. } \end{aligned}$$

(4.36)

By (4.25), (4.32), (4.33), (4.36), we easily obtain

$$\begin{aligned} |u^h_{xxx}| \le C \varepsilon ^{-3}, \qquad \hbox {on} \quad [0, 1]. \end{aligned}$$

(4.37)

Also for a smooth function ${\mathcal {F}} = {\mathcal {F}}(s), \, s \in [0,1],$ it is straightforward to show that the remainder $R(\chi )$ of the linear Lagrange interpolation of ${\mathcal {F}}$ at $s = 0$ and $s = 1,$

$$\begin{aligned} R(x) \,:=\, {\mathcal {F}}(x) \,-\, (1-x) {\mathcal {F}}(0) - x {\mathcal {F}}(1), \qquad x \in [0,1], \end{aligned}$$

(4.38)

is given by

$$\begin{aligned} R(x) = (1-x) \int _0^{\chi } s {\mathcal {F}}''(s)\, ds + x \int _{x}^1 (1-s) {\mathcal {F}}''(s)\, ds. \end{aligned}$$

(4.39)

We now use (2.13) and employ (4.38)–(4.39) for the function

$$\begin{aligned} {\mathcal {F}}(s) := f((1-s)\phi ^j + s \phi ^{j+1}), \end{aligned}$$

(4.40)

to get

$$\begin{aligned} {{L}}^b(u^h) = \varepsilon ^2 \chi _{xx}^j \, \big (\phi ^{j+1}-\phi ^j\big ) + 2 \varepsilon ^2 \chi _x^j \, \big (\phi _x^{j+1}-\phi _x^j\big ) + R^j, \qquad \hbox {for} \quad |x-h_j| < \varepsilon ,\nonumber \\ \end{aligned}$$

(4.41)

where

$$\begin{aligned} R^j = \big (\phi ^{j+1}-\phi ^j\big )^2 \left[ (1-\chi ^j) \int _0^{\chi ^j} s f''\big (\theta (s)\big ) \, ds + \chi ^j \int _{\chi ^j}^1 (1-s) f''\big (\theta (s)\big ) \, ds \right] ,\nonumber \\ \end{aligned}$$

(4.42)

and $\theta (s):=(1 - s) \phi ^j + s \phi ^{j+1}.$

We then combine (4.41)–(4.42) with (4.30), (4.33) to conclude that

$$\begin{aligned} |{{L}}^b(u^h)| \le C \alpha (r), \qquad \hbox {for} \quad |x-h_j| < \varepsilon ; \end{aligned}$$

(4.43)

cf. [10, Theorem 3.5].

At this point let us recall (2.16), i.e. ${L}^b(u^h)=0,$ for $|x-h_j| \ge \varepsilon ,$ which together with boundary values (2.19) and (4.43) show that $u^h$ “almost” satisfy the steady-state problem (2.2).

Remark 4.9

To show that

$$\begin{aligned} u_j^h \sim -u_x^h, \qquad \hbox {as} \quad r \rightarrow 0, \qquad \hbox {uniformly on} \quad I_j :=[m_j,\, m_{j+1}] \end{aligned}$$

(4.44)

first notice that only $\phi ^j$ and $\phi ^{j+1}$ depend on $h_j,$ so the support of $u_j^h$ is contained in $[h_{j-1}-\varepsilon , \, h_{j+1} + \varepsilon ].$

Applying (4.10) for the translate $\phi ^j$ of $\phi ,$

$$\begin{aligned} \phi ^j(x) := \phi \big (x - \tfrac{h_{j-1} + h_j}{2}, \, h_j - h_{j-1}, \, (-1)^j\big ) , \qquad x \in [h_{j-1} - \varepsilon , \, h_j+\varepsilon ], \end{aligned}$$

we have, for $x \in [h_{j-1} - \varepsilon , \, h_j+\varepsilon ],$

$$\begin{aligned} \frac{\partial }{\partial h_j} \phi ^j= & {} \phi ^j_x \, \frac{\partial }{\partial h_j}\left( x-\tfrac{h_{j-1} + h_j}{2}\right) \, + \, \phi ^j_\ell \,\frac{\partial }{\partial h_j} \big (h_j - h_{j-1}\big ) \nonumber \\= & {} - \, \frac{1}{2} \phi ^j_x \, - \, \frac{1}{2} {\text {sgn}}(x-m_j)\, \phi _x^j \,+\, \mathrm {w}^j , \end{aligned}$$

(4.45)

therefore

$$\begin{aligned} \frac{\partial }{\partial h_j} \phi ^j = - \phi ^j_x \,+\, \mathrm {w}^j , \qquad \hbox {in} \quad I_j:=[m_j,\, h_j+\varepsilon ] , \end{aligned}$$

(4.46)

since ${\text {sgn}}(x-m_j) > 0.$

Similarly, we obtain

$$\begin{aligned} \frac{\partial }{\partial h_j} \phi ^{j+1} \,=\, - \phi ^{j+1}_x \,-\, \mathrm {w}^{j+1} , \qquad \hbox {for} \quad x \in [h_j, \, h_{j+1}], \end{aligned}$$

(4.47)

and thus

$$\begin{aligned} \frac{\partial }{\partial h_j} \big (\phi ^{j+1} - \phi ^j\big ) \, = \, \phi ^j_x \, - \, \phi ^{j+1}_x \, - \, \mathrm {w}^j \,- \, \mathrm {w}^{j+1} , \qquad \hbox {in} \quad I_j. \end{aligned}$$

(4.48)

So recalling that $u^h = \big (1 - \chi ^j\big ) \, \phi ^j + \chi ^j \, \phi ^{j+1},$ and noticing that $\chi ^j_x = - \chi _{h_j}^j,$ it is straightforward to see that

(4.49)

For the translate $\phi ^{j+1}$ of $\phi ,$

$$\begin{aligned} \phi ^{j+1}(x) := \phi \big (x - \tfrac{h_j + h_{j+1}}{2}, \, h_{j+1} - h_j ; \, (-1)^{j+1}\big ) \qquad x \in [h_j - \varepsilon , \, h_{j+1} + \varepsilon ] \end{aligned}$$

we have, by (4.10),

$$\begin{aligned} \frac{\partial }{\partial h_j} \phi ^{j+1}= & {} \phi ^{j+1}_x \, \frac{\partial }{\partial h_j}\left( x-\tfrac{h_j + h_{j+1}}{2}\right) \, + \, \phi ^{j+1}_\ell \,\frac{\partial }{\partial h_j} \big (h_{j+1} - h_j\big ) \nonumber \\= & {} - \, \frac{1}{2} \phi ^{j+1}_x \, + \, \frac{1}{2} {\text {sgn}}(x-m_{j+1})\, \phi _x^{j+1} \, - \, \mathrm {w}^{j+1}\nonumber \\= & {} - \, \mathrm {w}^{j+1} , \qquad \hbox {in} \quad [m_{j+1} , \, h_{j+1} + \varepsilon ], \end{aligned}$$

(4.50)

since ${\text {sgn}}(x-m_{j+1}) > 0.$

Recall that

$$\begin{aligned}u^h = \big (1 - \chi ^{j+1}\big ) \, \phi ^{j+1} + \chi ^{j+1} \, \phi ^{j+2}, \qquad x \in [m_{j+1}, m_{j+2}], \end{aligned}$$

so using (4.50) and noticing that $\chi _{h_j}^{j+1}=0=\phi _{h_j}^{j+2},$ it is straightforward to see that

(4.51)

Analogously, taking into account that $u^h = \big (1 - \chi ^{j-1}\big ) \, \phi ^{j-1} + \chi ^{j-1} \, \phi ^j \,$ for $x \in [m_{j-1}, m_j],$ using (4.45) and noticing that $\chi _{h_j}^{j-1}=0=\phi _{h_j}^{j-1},$ we obtain that

$$\begin{aligned} \frac{\partial }{\partial h_j} u^h = \chi ^{j-1} \frac{\partial }{\partial h_j} \phi ^j = \chi ^{j-1} \mathrm {w}^j, \qquad x \in [h_{j-1}-\varepsilon , \, m_j]. \end{aligned}$$

(4.52)

Gathering (4.49), (4.51), (4.52) we have that

$$\begin{aligned} \frac{\partial }{\partial h_j} u^h \, = \, {\left\{ \begin{array}{ll} \chi ^{j-1} \mathrm {w}^j , &{} \hbox {for} \quad h_{j-1}-\varepsilon \le x \le m_j, \\ - u^h_{x} \,+\, \mathrm {w}^j, &{} \hbox {for} \quad m_j \le x \le h_j-\varepsilon , \\ - u_x^h + \big (1-\chi ^j\big ) \mathrm {w}^j - \chi ^j \mathrm {w}^{j+1}, &{} \hbox {for} \quad |x-h_j| < \varepsilon , \\ - u^h_{x} \,-\, \mathrm {w}^{j+1} , &{} \hbox {for} \quad h_j +\varepsilon \le x \le m_{j+1}, \\ - \, (1 - \chi ^{j+1}) \mathrm {w}^{j+1} , &{} \hbox {for} \quad m_{j+1} \le x \le h_{j+1} +\varepsilon . \end{array}\right. } \end{aligned}$$

(4.53)

Then (4.44) follows from (4.53) combined with (4.13)–(4.14). See also in [4], for some analogous results.

1.3 A priori estimates for the problem (1.1)–(BC1)–(BC2)–(MC)

For the well-posedness of the initial and boundary value problem we may argue as in [14, §2]; next we derive the estimates needed in our case where we have replaced the b.c. at $x = 1$ with the mass conservation condition and added the Allen–Cahn lower order term in the pde.

Local in time existence may be proved by fixed-point theory, applying a Picard-type iteration scheme. In order to prove global existence, i.e. existence on [0, T] for any $T > 0,$ we need to derive certain a priori uniform estimates on u. To this aim, first notice that by (MC), (ACH) and (BC1)–(BC2) we have

$$\begin{aligned} 0= & {} \frac{d}{dt} \int _0^1 u(x, t) \, dx = \int _0^1 u_t(x, t) \, dx \\= & {} - \delta (\varepsilon )\int _0^1 \big (\varepsilon ^2 u_{xx} - W'(u)\big )_{xx} \, dx + \mu (\varepsilon )\int _0^1 \big (\varepsilon ^2 u_{xx} - W'(u)\big ) \, dx \\= & {} - \delta (\varepsilon )\varepsilon ^2 u_{xxx}(1,t) - \mu (\varepsilon )\int _0^1 W'(u) \, dx, \end{aligned}$$

so we have

$$\begin{aligned} \mu (\varepsilon )\int _0^1 W'(u) \, dx = - \delta (\varepsilon )\varepsilon ^2 u_{xxx}(1,t). \end{aligned}$$

(4.54)

Also, as in the proof of Lemma 2.7, we can see that for differentiable $\upsilon$ and any positive $\epsilon _1,$

$$\begin{aligned} \upsilon ^2(1,t) \le \Vert \upsilon \Vert _\infty ^2 \le 2\epsilon _1 \Vert \upsilon _x\Vert ^2 \, + \, \frac{2}{\epsilon _1} \, \Vert \upsilon \Vert ^2. \end{aligned}$$

(4.55)

For the special case of

$$\begin{aligned} W(u) = \frac{1}{4}(u^2-1)^2 , \qquad \hbox {thus}\qquad W'(u) = u^3 - u , \end{aligned}$$

(4.56)

we see that,

$$\begin{aligned} W'(u) \le c_1 W(u) + c_2 , \qquad \forall u \in {\mathbb {R}}, \end{aligned}$$

for some positive constants $c_1, c_2$ independent of u,  and so

$$\begin{aligned} \int _0^1 |W'(u)|\, dx \le c_1 \int _0^1 W(u) \, dx + c_2. \end{aligned}$$

(4.57)

Growth estimate for the energy: We set

$$\begin{aligned} E(t) := \int _0^1 \frac{\varepsilon ^2}{2} u_x^2 + W(u)\, dx, \end{aligned}$$

(4.58)

and we have

$$\begin{aligned} \frac{d}{dt} E(t)= & {} \int _0^1 \varepsilon ^2 u_x \, (u_t)_x \, + \, W'(u)\,u_t \, dx \\= & {} -\int _0^1 \big (\varepsilon ^2 u_{xx} \, - \, W'(u)\big ) \, u_t \, dx, \end{aligned}$$

where we integrated by parts the first term and applied (BC1). Then, by the pde (ACH) and integrations by parts combined with (BC1)–(BC2) we get

$$\begin{aligned} \frac{d}{dt} E(t) =&\int _0^1 \big (\varepsilon ^2 u_{xx} \, - \, W'(u)\big )\, \Big [\delta (\varepsilon )\, \big (\varepsilon ^2 u_{xx} - W'(u)\big )_{xx} - \mu (\varepsilon )\, \big (\varepsilon ^2 u_{xx} - W'(u)\big )\Big ] \, dx \nonumber \\ =&- \delta (\varepsilon )\int _0^1\left[ \big (\varepsilon ^2 u_{xx} - W'(u)\big )_x\right] ^2\, dx - \mu (\varepsilon )\int _0^1\left[ \varepsilon ^2 \, u_{xx} - W'(u)\right] ^2\, dx \nonumber \\&+ \delta (\varepsilon )\, \varepsilon ^2 \, u_{xxx}(1,t) \, \Big [ \varepsilon ^2 \, u_{xx}(1,t) - W'\big (u(1,t)\big )\Big ] \nonumber \\ {\mathop {=}\limits ^{\scriptscriptstyle (4.54)}}&- \delta (\varepsilon )\left\| \big (\varepsilon ^2 u_{xx} - W'(u)\big )_x \right\| ^2 - \mu (\varepsilon )\left\| \varepsilon ^2 \, u_{xx} - W'(u) \right\| ^2 \nonumber \\&- \mu (\varepsilon )\, \Big [\varepsilon ^2 \, u_{xx}(1,t) - W'\big (u(1,t)\big )\Big ] \, \int _0^1 W'(u) \, dx\nonumber \\ \le&- \delta (\varepsilon )\left\| \big (\varepsilon ^2 u_{xx} - W'(u)\big )_x\right\| ^2 - \mu (\varepsilon )\left\| \varepsilon ^2 \, u_{xx} - W'(u) \right\| ^2\nonumber \\&+\frac{\mu (\varepsilon )}{4 \epsilon } \Big [\varepsilon ^2 \, u_{xx}(1,t) - W'\big (u(1,t)\big )\Big ]^2 + \epsilon \, \mu (\varepsilon )\left( \int _0^1 W'(u)\,dx\right) ^2 . \end{aligned}$$

(4.59)

By (BC1) we have

$$\begin{aligned} \int _0^1 W'(u)\,dx = \int _0^1 \big (W'(u) - \varepsilon ^2 u_{xx}\big ) \,dx \le \left\| W'(u) - \varepsilon ^2 u_{xx}\right\| , \end{aligned}$$

(4.60)

and by (4.55) for $\upsilon =W'(u) - \varepsilon ^2 u_{xx}$ therein, we have

$$\begin{aligned} \frac{\mu (\varepsilon )}{4 \epsilon } \upsilon ^2(1,t) \le \frac{\mu (\varepsilon )\, \epsilon _1}{2 \epsilon } \Vert \upsilon _x\Vert ^2 \, + \, \frac{\mu (\varepsilon )}{2 \epsilon \, \epsilon _1} \, \Vert \upsilon \Vert ^2, \end{aligned}$$

(4.61)

so, choosing $\epsilon , \epsilon _1,$ so that

$$\begin{aligned} \frac{\mu (\varepsilon )\, \epsilon _1}{2 \epsilon } \le \delta (\varepsilon )\qquad \hbox {and}\qquad 0\le 1 -\epsilon - \frac{1}{2 \epsilon \, \epsilon _1}, \end{aligned}$$

(4.62)

we substitute (4.60), (4.61) into (4.59) to get that

$$\begin{aligned} \frac{d}{dt} E(t) \le 0. \end{aligned}$$

Therefore, $E(t) \le E(0)$ that is

$$\begin{aligned} \frac{\varepsilon ^2}{2} \Vert u_x\Vert ^2 \, + \, \int _0^1 W(u) \, dx \le E_0:= \int _0^1 \frac{\varepsilon ^2}{2} (u_0)_x^2 + W\big (u_0\big )\, dx, \end{aligned}$$

(4.63)

with $u_0(x):=u(x,0),$ and so

$$\begin{aligned} \frac{\varepsilon ^2}{2} \Vert u_x\Vert ^2 \le E_0, \end{aligned}$$

(4.64)

and

$$\begin{aligned} \int _0^1 W(u) \, dx \le E_0. \end{aligned}$$

(4.65)

Furthermore, integrating (4.59) we get

$$\begin{aligned} \int _0^t\left\| \varepsilon ^2 \, u_{xx} - W'(u) \right\| _{H^1}^2 \, d\tau \le C,\qquad 0\le t \le T, \end{aligned}$$

(4.66)

for some constant C depending on $u_0, T$ and $\delta (\varepsilon ), \mu (\varepsilon ), \varepsilon .$

Remark: A trivial calculus shows that the weakest condition (4.62) for $\delta (\varepsilon ), \mu (\varepsilon )$ is attained by choosing $\epsilon _1/\epsilon = 27/8$ for $\epsilon =2/3,$ and so $\tfrac{27}{16} \,\mu (\varepsilon )\,\le \,\delta (\varepsilon ).$ Let us also emphasize that the condition $c \, \mu (\varepsilon )\,\le \,\delta (\varepsilon )$ for some $c > 0,$ is weaker than the assumption (3.30) for establishing the slow evolution within the channel (3.52) (Theorem 3.2); e.g. take $\delta (\varepsilon ), \mu (\varepsilon )$ such that $\varepsilon ^{-3/2}\mu (\varepsilon )\,\ll \,\delta (\varepsilon )= {\mathcal {O}}(\varepsilon ^8).$

Growth estimate for $\Vert u\Vert ^2:$ Multiply (ACH) by u,  then integrate with respect to x and apply (BC1)–(BC2) to get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert u\Vert ^2&+ \delta (\varepsilon )\, \varepsilon ^2 \, \Vert u_{xx}\Vert ^2 + \mu (\varepsilon )\, \varepsilon ^2 \, \Vert u_{x}\Vert ^2 \nonumber \\ =&- \delta (\varepsilon )\, \varepsilon ^2 \, u_{xxx}(1,t) \, u(1,t) - \delta (\varepsilon )\int _0^1 \big (W'(u)\big )_x \, u_{x} \, dx - \mu (\varepsilon )\int _0^1 W'(u) \, u\, dx \nonumber \\ =&- \delta (\varepsilon )\, \varepsilon ^2 \, u_{xxx}(1,t) \, u(1,t) - \delta (\varepsilon )\int _0^1 W''(u) \, u^2_{x} \, dx - \mu (\varepsilon )\int _0^1 W'(u) \, u\, dx \nonumber \\ {\mathop {\le }\limits ^{\scriptscriptstyle W''\ge -1}}&- \delta (\varepsilon )\, \varepsilon ^2 \, u_{xxx}(1,t) \, u(1,t) + \delta (\varepsilon )\int _0^1 u^2_x \, dx - \mu (\varepsilon )\int _0^1 W'(u) \, u\, dx \nonumber \\ {\mathop {=}\limits ^{\scriptscriptstyle (4.54)}}&\delta (\varepsilon )\, \Vert u_x\Vert ^2 + \mu (\varepsilon )\left( u(1,t) \int _0^1 W'(u) \, dx - \int _0^1 W'(u) \, u\, dx \right) \nonumber \\ \le&\delta (\varepsilon )\, \Vert u_x\Vert ^2 + \mu (\varepsilon )\, \left( \Vert u\Vert _\infty ^2 + \left\| W'(u)\right\| ^2_1 \right) \nonumber \\ {\mathop {\le }\limits ^{\scriptscriptstyle (4.55)}}&\delta (\varepsilon )\, \Vert u_x\Vert ^2 + \mu (\varepsilon )\left( 2\, \Vert u_x\Vert ^2 \, + \, 2 \, \Vert u\Vert ^2 + \left\| W'(u)\right\| ^2_1 \right) . \end{aligned}$$

(4.67)

Regarding the term $\left\| W'(u)\right\| _1$ in (4.67), we combine (4.57) and (4.65) to see that

$$\begin{aligned} \left\| W'(u)\right\| _1 \le c_1 \left\| W(u)\right\| _1 + c_2 \le C. \end{aligned}$$

(4.68)

In view of (4.64) and (4.68), the estimate (4.67) yields

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert u\Vert ^2 + \delta (\varepsilon )\, \varepsilon ^2 \, \Vert u_{xx}\Vert ^2 + \mu (\varepsilon )\, \varepsilon ^2 \, \Vert u_{x}\Vert ^2 \le C_1 \Vert u\Vert ^2 \,+\, C_2, \end{aligned}$$

(4.69)

the constants $C_1, C_2$ depending only on $u_0$ and $\delta (\varepsilon ), \mu (\varepsilon ).$

In particular, (4.69) implies

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert u\Vert ^2 \le C_1 \Vert u\Vert ^2 \,+\, C_2, \end{aligned}$$

and integrating this inequality we get

$$\begin{aligned} \Vert u(\cdot ,t)\Vert ^2 \le e^{2 C_1 t} \Vert u_0\Vert ^2 + C_2 (e^{2 C_1 t} - 1) / C_1, \end{aligned}$$

(4.70)

and so

$$\begin{aligned} \Vert u(\cdot ,t)\Vert ^2\le c_1 \Vert u_0\Vert ^2 + c_2, \qquad 0\le t\le T, \end{aligned}$$

with $c_1= e^{2C_1 T},\, c_2=C_2 (e^{2C_1 T} - 1)/C_1, \,$ that is

$$\begin{aligned} \Vert u\Vert \le C, \qquad 0\le t \le T, \end{aligned}$$

(4.71)

with a constant C depending only on $u_0, T$ and $\delta (\varepsilon ), \mu (\varepsilon ).$

By (4.64) and (4.71) we get that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _\infty \le C, \qquad 0\le t \le T. \end{aligned}$$

(4.72)

Now we return to (4.69), ignore the positive term $\Vert u_{xx}\Vert ,$ then integrate and employ (4.70) to get

$$\begin{aligned}\left\| u(\cdot ,T)\right\| ^2 + 2 \, \mu (\varepsilon )\, \varepsilon ^2 \, \int _0^t\Vert u_x\Vert ^2 \, d\tau \le \Vert u_0\Vert ^2 \, e^{2 C_1 t} \, + \, C_3 \big (e^{2 C_1 t}-1\big ),\end{aligned}$$

therefore

$$\begin{aligned} \int _0^t\Vert u_{x}\Vert ^2 \, d\tau \le C,\qquad 0\le t \le T, \end{aligned}$$

(4.73)

for some positive constant C depending only on $u_0, T$ and $\delta (\varepsilon ), \mu (\varepsilon ), \varepsilon .$

Returning once more to (4.69), we get as above that

$$\begin{aligned} \int _0^t\Vert u_{xx}\Vert ^2 \, d\tau \le C,\qquad 0\le t \le T, \end{aligned}$$

(4.74)

as well.

For improving the regularity of the weak solution to be a classical one we may use a bootstrap argument; see e.g. (2-20)–(2-25) of [14]. Let us also remark that in view of (3.6), the $H^k$-regularity of u implies the $H^{k+1}$-regularity for the solution ${\tilde{u}}$ of the integrated problem.

Uniqueness: Let u, v be solutions of the problem (ACH)–(BC1)–(BC2)–(MC) and consider the difference $\mathrm {v} = u - v.$ In view of (ACH), we have

$$\begin{aligned} \mathrm {v}_t= & {} - \delta (\varepsilon )\, \varepsilon ^2 \mathrm {v}_{xxxx} \, + \, \delta (\varepsilon )\, \big ( W'(u) \,-\, W'(v) \big )_{xx} \nonumber \\&+ \mu (\varepsilon )\, \varepsilon ^2 \mathrm {v}_{xx} \, - \, \mu (\varepsilon )\, \big ( W'(u) \, - \, W'(v) \big ),\nonumber \\ \end{aligned}$$

(4.75)

the (BC1)–(BC2) yield the boundary conditions

$$\begin{aligned}&\mathrm {v}_{x}(0, \, t) = \mathrm {v}_{x}(1 , \, t) = 0 , \end{aligned}$$

(4.76)

$$\begin{aligned}&\mathrm {v}_{xxx}(0, \, t) = 0, \end{aligned}$$

(4.77)

and (MC) implies

$$\begin{aligned} \int _0^1 \mathrm {v}(x, t) \, dx = 0, \qquad t>0. \end{aligned}$$

(4.78)

Multiply the pde (4.75) by $\mathrm {v},$ then integrate with respect to x and apply (4.76)–(4.77) to get

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \Vert \mathrm {v}\Vert ^2 + \delta (\varepsilon )\, \varepsilon ^2 \, \Vert \mathrm {v}_{xx}\Vert ^2 + \mu (\varepsilon )\, \varepsilon ^2 \, \Vert \mathrm {v}_x\Vert ^2 = - \delta (\varepsilon )\, \varepsilon ^2 \, \mathrm {v}_{xxx}(1,t) \, \mathrm {v}(1,t) \nonumber \\&\quad + \delta (\varepsilon )\int _0^1 \big ( W'(u) \, - \, W'(v) \big ) \, \mathrm {v}_{xx} \, dx - \mu (\varepsilon )\int _0^1 \big ( W'(u) \, - \, W'(v) \big ) \, \mathrm {v}\, dx. \end{aligned}$$

(4.79)

Let us next estimate the terms in the RHS of (4.79). To this aim, we set $K_{\scriptscriptstyle T} := \sup \big \{\Vert u(\cdot ,t)\Vert _\infty , \Vert \upsilon (\cdot ,t)\Vert _\infty :\, 0\le t \le T\big \}$ and $L = \max \big \{|W''(w)|: \, |w|\le K_{\scriptscriptstyle T}\big \}.$ In view of (4.72) we have $K_{\scriptscriptstyle T} < \infty ,$ and L depends on $u, \upsilon , W, T,$ but it is independent of t.

We then have that

$$\begin{aligned} \left| W'\big (\upsilon (x,t)\big ) \,-\, W'\big (u(x,t)\big ) \right| \le L \, |\upsilon (x,t) - u(x,t)| , \qquad 0 \le t \le T. \end{aligned}$$

(4.80)

Regarding the first term in the RHS of (4.79), we clearly have

$$\begin{aligned} \mathrm {v}(1,t) = \int _y^1 \mathrm {v}_x \,dx \,+\, \mathrm {v}(y,t) \le \int _0^1 |\mathrm {v}_x|\,dx \,+\, \mathrm {v}(y,t), \end{aligned}$$

and integrate this inequality with respect to y,  to get, by virtue of (4.78),

$$\begin{aligned} \mathrm {v}(1,t) \le \int _0^1 |\mathrm {v}_x|\,dx \,+\, \int _0^1 \mathrm {v}(x,t) \, dx = \left\| \mathrm {v}_x(\cdot ,t)\right\| _1 . \end{aligned}$$

(4.81)

Moreover, by (4.54) and (4.80) we get

$$\begin{aligned} \delta (\varepsilon )\, \varepsilon ^2 \, \mathrm {v}_{xxx}(1,t)= & {} \mu (\varepsilon )\int _0^1 \big ( W'(\upsilon ) \,-\, W'(u) \big ) \, dx \le L \, \mu (\varepsilon )\int _0^1 |\upsilon - u| \, dx\nonumber \\= & {} L \, \mu (\varepsilon )\, \left\| \mathrm {v}\right\| _1 . \end{aligned}$$

(4.82)

Consequently, by (4.81)–(4.82) we obtain that

$$\begin{aligned} \delta (\varepsilon )\, \varepsilon ^2 \, \mathrm {v}_{xxx}(1,t) \, \mathrm {v}(1,t)\le & {} \frac{L^2 \, \mu (\varepsilon )}{4 \epsilon } \, \left\| \mathrm {v}\right\| _1^2 \, + \, \epsilon \, \mu (\varepsilon )\, \left\| \mathrm {v}_x\right\| _1^2 \nonumber \\\le & {} \frac{L^2 \, \mu (\varepsilon )}{4 \epsilon } \, \left\| \mathrm {v}\right\| ^2 \, + \, \epsilon \, \mu (\varepsilon )\, \left\| \mathrm {v}_x\right\| ^2, \end{aligned}$$

(4.83)

for an arbitrary positive $\epsilon < 1.$

As for the second term in the RHS of (4.79), again we use (4.80) to see that

$$\begin{aligned} \delta (\varepsilon )\int _0^1 \big ( W'(u) \, - \, W'(v) \big ) \, \mathrm {v}_{xx} \, dx\le & {} \delta (\varepsilon )\, L \, \int _0^1 \big |u-v\big | \, |\mathrm {v}_{xx}| \, dx \nonumber \\\le & {} \frac{L^2 \, \delta (\varepsilon )}{4 \epsilon } \, \left\| \mathrm {v}\right\| ^2 \, + \, \epsilon \, \delta (\varepsilon )\, \left\| \mathrm {v}_{xx}\right\| ^2 , \end{aligned}$$

(4.84)

and for the last term in (4.79), estimate (4.80) yields the bound

$$\begin{aligned} \mu (\varepsilon )\int _0^1 \big ( W'(u) \, - \, W'(v) \big ) \, \mathrm {v}\, dx \le \mu (\varepsilon )\, L \, \int _0^1 \big |u-v\big | \, |\mathrm {v}| \, dx = \mu (\varepsilon )\, L \,\left\| \mathrm {v}\right\| ^2. \end{aligned}$$

(4.85)

We apply (4.83), (4.84), (4.85) into (4.79) to obtain

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \Vert \mathrm {v}\Vert ^2 + \delta (\varepsilon )\, (\varepsilon ^2 - \epsilon ) \, \Vert \mathrm {v}_{xx}\Vert ^2 + \mu (\varepsilon )\, (\varepsilon ^2 - \epsilon ) \, \Vert \mathrm {v}_x\Vert ^2\nonumber \\&\quad \le \left( \frac{L^2 \, \mu (\varepsilon )}{4 \epsilon } + \frac{L^2 \, \delta (\varepsilon )}{4 \epsilon } + \mu (\varepsilon )\, L \right) \, \left\| \mathrm {v}\right\| ^2. \end{aligned}$$

(4.86)

Therefore

$$\begin{aligned} \frac{d}{dt} \Vert \mathrm {v}\Vert ^2 \le c \, \left\| \mathrm {v}\right\| ^2 , \qquad 0 \le t \le T, \end{aligned}$$

(4.87)

for some constant c depending on $\delta (\varepsilon ), \mu (\varepsilon ), T,$ but independent of t and integrating with respect to t,  we obtain

$$\begin{aligned} \left\| \mathrm {v}(\cdot ,t)\right\| ^2 \le e^{ct} \, \left\| \mathrm {v}(\cdot ,0)\right\| ^2 = 0, \qquad 0 \le t \le T, \end{aligned}$$

that is $u \equiv \upsilon ,$ so the solution of (ACH)–(BC1)–(BC2)–(MC) is unique.