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Analysis of a perturbed Cahn–Hilliard model for Langmuir–Blodgett films


An advective Cahn–Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn–Hilliard equation.

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The authors thank the hospitality of the departments of mathematics of the universities of Heidelberg and Vienna, of SISSA, and of the E. Schrödinger Institute in Vienna, where this research was developed. The authors are members of the GNAMPA group of INdAM. This work was partially supported by the 2015 GNAMPA project Fenomeni Critici nella Meccanica dei Materiali: un Approccio Variazionale. E.D. acknowledges the support of the Austrian Science Fund (FWF) project P 27052 and of the SFB project F65 Taming complexity in partial differential systems. The research of M.M. was partially supported by grant FCT-UTA_CMU/MAT/0005/2009 Thin Structures, Homogenization, and Multiphase Problems, by the ERC Advanced grant Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture (Grant agreement no.: 290888), by the ERC Starting grant High-Dimensional Sparse Optimal Control (Grant agreement no.: 306274), and by the DFG Project Identifikation von Energien durch Beobachtung der zeitlichen Entwicklung von Systemen (FO 767/7). Finally, the authors thank Stefano Melchionna for fruitful discussions.

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Appendix A

Appendix A

We collect here a few results of technical nature which are needed throughout the paper. The following lemma will be instrumental in the proof of Proposition 4.4.

Lemma A.1

Let \(q\in L^2(0,T;V)\) and let \(\tau :=\frac{T}{N}\) for \(N\in {\mathbb {N}}\). Then, as \(\tau \rightarrow 0\),

$$\begin{aligned} \frac{1}{\tau }\sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } q'(s)\,\mathrm ds \rightarrow q'(t) \qquad \text {weakly in } L^2(0,T;L^2(0,L)). \end{aligned}$$


We set

$$\begin{aligned} g_\tau (x,t):=\frac{1}{\tau }\sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } \bigl ( q'(x,t)-q'(x,s) \bigr )\,\mathrm ds. \end{aligned}$$

We can estimate

$$\begin{aligned} |g_\tau (x,t)|^2&= \frac{1}{\tau ^2} \sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \bigg | \int _{(k-1)\tau }^{k\tau } \bigl ( q'(x,t)-q'(x,s) \bigr )\,\mathrm ds \bigg |^2 \\&\leqslant \frac{1}{\tau } \sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } |q'(x,t)-q'(x,s)|^2 \,\mathrm ds \\&\leqslant 2|q'(x,t)|^2 + \frac{2}{\tau } \sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } |q'(x,s)|^2 \,\mathrm ds\,, \end{aligned}$$

so that by integrating on \((0,L)\times (0,T)\) we obtain

$$\begin{aligned} \int _0^T\int _0^L |g_\tau (x,t)|^2\,\mathrm dx\mathrm dt \leqslant 4\int _0^T\int _0^L |q'(x,t)|^2\,\mathrm dx \mathrm dt =:M <\infty . \end{aligned}$$

This implies that, up to subsequences, \(g_\tau \rightharpoonup g\) weakly in \(L^2((0,L)\times (0,T))\).

We shall now show that \(g=0\). The estimate

$$\begin{aligned} |g_\tau (x,t)| \leqslant \frac{1}{\tau } \int _{t-\tau }^{t+\tau } \big | q'(x,t)-q'(x,s) \big | \,\mathrm ds \end{aligned}$$

implies that \(\Vert g_\tau (t)\Vert _{L^2(0,L)}\rightarrow 0\) for almost every \(t\in (0,T)\) by [34, Theorem 8.19]; in turn, given \(\varepsilon >0\), by Egorov’s Theorem we can find a set \(A_\varepsilon \subset (0,T)\) with measure smaller than \(\varepsilon \) such that \(\Vert g_\tau (t)\Vert _{L^2(0,L)}\rightarrow 0\) uniformly on \((0,T)\setminus A_\varepsilon \). Therefore for any test function \(\varphi \in L^2((0,L)\times (0,T))\) we have

$$\begin{aligned} \limsup _{\tau \rightarrow 0}\bigg |\int _0^T\int _0^L g_\tau (x,t)\varphi (x,t)\,\mathrm dx\mathrm dt \bigg |&\leqslant \limsup _{\tau \rightarrow 0} \int _0^T \Vert g_\tau (t)\Vert _{L^2(0,L)} \Vert \varphi (t)\Vert _{L^2(0,L)} \,\mathrm dt\\&= \limsup _{\tau \rightarrow 0} \int _{A_\varepsilon } \Vert g_\tau (t)\Vert _{L^2(0,L)} \Vert \varphi (t)\Vert _{L^2(0,L)} \,\mathrm dt\\&\leqslant M^\frac{1}{2}\biggl ( \int _{A_\varepsilon } \Vert \varphi (t)\Vert _{L^2(0,L)}^2\,\mathrm dt \biggr )^\frac{1}{2}, \end{aligned}$$

and the right-hand side can be made arbitrarily small as \(\varepsilon \rightarrow 0\). This shows that \(g_\tau \rightharpoonup 0\) weakly in \(L^2((0,L)\times (0,T))\) and (A.1) holds. \(\square \)

We discuss in the following two lemmas the dependence on t of the function \(z_{\varphi (t)}\) introduced in (2.10), when \(\varphi \) depends on t.

Lemma A.2

Let \(q\in L^2(0,T;V')\) and let \(z_{q(t)}\) be defined by (2.10). Then the map \(t\mapsto z_{q(t)}\) belongs to \(L^2(0,T;V)\).


For every \(\varphi \in V'\) the map

$$\begin{aligned} t \mapsto \left\langle \varphi ,z_{q(t)}\right\rangle _{V',V} {\mathop {=}\limits ^{(2.10)}} \int _0^L z_\varphi '(x)z_{q(t)}'(x)\,\mathrm dx = \left\langle q(t),z_\varphi \right\rangle _{V',V} \end{aligned}$$

is measurable thanks to the assumption \(q\in L^2(0,T;V')\). Hence the map \(t\mapsto z_{q(t)}\) is weakly measurable from (0, T) to V and in turn strongly measurable, by Pettis Theorem (see [34, Theorem 8.3]) and the separability of V. Moreover, by (2.10) we have

$$\begin{aligned} \int _0^T \left||z_{q(t)}\right||^2_V\,\mathrm dt = \int _0^T \left||q(t)\right||_{V'}^2 <\infty \end{aligned}$$

since \(q\in L^2(0,T;V')\), which shows that the map \(t\mapsto z_{q(t)}\) is Bochner integrable and belongs to \(L^2(0,T;V)\). \(\square \)

Notice that, if \(q\in L^2(0,T;L^2(0,L))\), then by (2.15) we have \(q\in L^2(0,T;V')\). Hence we can apply the previous lemma to deduce that also in this case \(z_{q(t)}\in L^2(0,T;V)\).

Lemma A.3

Let \(u\in H^1(0,T;V')\) and let \(z_{u(t)}\) be defined by (2.10). Then the map \(\psi (t) :=z_{u(t)}\) is in \(H^1(0,T;V)\), and

$$\begin{aligned} {\dot{\psi }}(t) = z_{{\dot{u}}(t)}. \end{aligned}$$


We first observe that \(\psi \in L^2(0,T;V)\), thanks to Lemma A.2. Similarly, the map \(t\mapsto z_{{\dot{u}}(t)}\) is in \(L^2(0,T;V)\), thanks to the same lemma applied to \(q={\dot{u}}\in L^2(0,T;V')\).

We now show equality (A.2), which will complete the proof of the lemma. By definition of weak derivative, (A.2) is equivalent to show

$$\begin{aligned} \int _0^T z_{{\dot{u}}(t)}\varphi (t)\,\mathrm dt = - \int _0^T z_{u(t)}{\dot{\varphi }}(t)\,\mathrm dt \end{aligned}$$

for every \(\varphi \in C^1_{\mathrm c}(0,T)\), where (A.3) is an equality between elements of V. For every \(\eta \in V\) we have

$$\begin{aligned} \begin{aligned} \Bigl (\int _0^T z_{u(t)}{\dot{\varphi }}(t)\,\mathrm dt \,,\, \eta \Bigr )_V&= \int _0^T \bigl (z_{u(t)}{\dot{\varphi }}(t), \eta \bigr )_V\,\mathrm dt = \int _0^T \int _0^L z'_{u(t)}(x){\dot{\varphi }}(t)\eta '(x) \,\mathrm dx \mathrm dt \\&{\mathop {=}\limits ^{(2.10)}} \int _0^T{\dot{\varphi }}(t) \left\langle u(t),\eta \right\rangle _{V',V}\,\mathrm dt = \left\langle \int _0^T u(t){\dot{\varphi }}(t)\,\mathrm dt,\eta \right\rangle _{V',V} \\&= - \left\langle \int _0^T {\dot{u}}(t)\varphi (t)\,\mathrm dt,\eta \right\rangle _{V',V} = -\int _0^T \varphi (t) \left\langle {\dot{u}}(t),\eta \right\rangle _{V',V}\,\mathrm dt \\&{\mathop {=}\limits ^{(2.10)}} -\int _0^T \int _{0}^L \varphi (t) z'_{{\dot{u}}(t)}(x)\eta '(x) \,\mathrm dx \mathrm dt = -\int _0^T \bigl ( \varphi (t) z_{{\dot{u}}(t)}, \eta \bigr )_V \mathrm dt \\&= - \Bigl (\int _0^T \varphi (t)z_{{\dot{u}}(t)}\,\mathrm dt \,,\, \eta \Bigr )_V \,. \end{aligned} \end{aligned}$$

In the previous chain of equalities we used several times the property that the Bochner integral commutes with linear operators, see for instance [34, Theorem 8.13]. This proves claim (A.3). \(\square \)

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Bonacini, M., Davoli, E. & Morandotti, M. Analysis of a perturbed Cahn–Hilliard model for Langmuir–Blodgett films. Nonlinear Differ. Equ. Appl. 26, 36 (2019).

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  • Evolution equations
  • Cahn–Hilliard equation
  • Langmuir–Blodgett transfer
  • Minimizing movements
  • Fixed point theorem
  • Thin films
  • Global attractor

Mathematics Subject Classification

  • 35K35
  • 49J40
  • 37L30
  • 74K35