Abstract
We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction \(\rho \) such that the micro phase separation results in an ensemble of small disks of one component. We consider the two dimensional case in this paper, whereas the three dimensional case was already considered in Niethammer and Oshita (Calc Var PDE 39:273–305, 2010). Starting from the free boundary problem restricted to disks we rigorously derive the heterogeneous mean-field equations on a time scale of the order of \({\mathcal {R}}^{3}\ln (1/\rho )\), where \({\mathcal {R}}\) is the mean radius of disks. On this time scale, the evolution is dominated by coarsening and stabilization of the radii of the disks, whereas migration of disks becomes only relevant on a larger time scale.
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1 Introduction
Diblock copolymer molecules consist of subchains of two different type of monomers, say A- and B-monomers. The different type of subchains tend to segregate, and hence the phase separation take place. However since the subchains are chemically bonded, the two subchains mix on a macroscopic scale, while on a molecular scale, A- and B-subchains still segregate and the micro-domains are formed. This is called micro phase separation. For more physical background on this phenomenon we refer to [2, 8].
In the strong segregation regime, energetically favorable configurations have been characterized in the Ohta–Kawasaki theory [18] by minimizers of an energy functional, which is in the two dimensional case of the form
Here \((0,L)^2 \subset {\mathbb {R}}^2\) is the domain covered by the copolymers, \(\Omega \subset [0,L)^2\) denotes the region covered by, say, A-monomers, \(\rho =\frac{|\Omega |}{L^2} \in (0,1)\) the average density, \(\sigma \in {\mathbb {R}}_+=(0,\infty )\) is a parameter related to the polymerization index, \(\chi \) is the characteristic function of \(\Omega \), and \(\mathcal{H}^1\) denotes one dimensional Hausdorff measure.
The first term in the energy prefers large blocks of monomers, the second favors a very fine mixture. Competition between these terms leads to minimizers of E which represent micro phase separation.
Starting with the pioneering work [15], where the Ohta–Kawasaki theory is formulated on a bounded domain as a singularly perturbed problem and the limiting sharp interface problem is identified, there has been a large body of analytical work. Minimizers of the energy functionals have been characterized in [1, 3, 4, 20], the existence/stability of stationary solutions has been investigated in [16, 17, 19, 21] and a time dependent model has been considered in [7, 9]. The mean field models in the three dimensional case have been derived in [6, 10, 12].
We consider the gradient flow of the energy, which is a standard way to set up a model for the evolution of the copolymer configuration that decreases energy and preserves the volume fraction. Then the evolution equation becomes the following extension of the Mullins–Sekerka evolution for phase separation in binary alloys [11]. The normal velocity v of the interface \(\partial \Omega =\partial \Omega (t)\) satisfies
where \([ \nabla w \cdot \vec n]\) denotes the jump of the normal component of the gradient of the potential across the interface. Here \({\vec {n}}\) denotes the outer normal to \(\Omega \) and
The potential w is for each time determined via
where \(\kappa \) is the curvature of \(\partial \Omega \). We are interested in the case that the volume of \(\Omega (t)\) is preserved in time and can thus impose Neumann or periodic boundary conditions for w on \(\partial (0,L)^2\). In what follows we will consider a periodic setting and hence always require that the potential w is \((0,L)^2\)-periodic. For local well-posedness of this evolution see [5].
The evolution defined by (2)–(4) has a formal interpretation as a gradient flow of the energy (1) on a Riemannian manifold. Indeed, consider the manifold of subsets of the 2-dimensional flat torus \({\mathbb {T}}\) of length L with fixed volume, that is \(\mathcal{M}:=\{ \Omega \subset {\mathbb {T}}\,;\, |\Omega |= L^2 \rho \}\), whose tangent space \(T_\Omega \mathcal{M}\) at an element \(\Omega \in \mathcal{M}\) is described by all kinematically admissible normal velocities of \(\partial \Omega \), that is,
The Riemannian structure is given by the following metric tensor on the tangent space:
where \(w^{\alpha } :{\mathbb {T}}\rightarrow {\mathbb {R}}\)\((\alpha =1,2)\) solves
for \(v^\alpha \in T_{\Omega }{\mathcal {M}}\)\((\alpha =1,2)\). The gradient flow of the energy (1) is now the dynamical system where at each time the velocity is the element of the tangent space in the direction of steepest descent of the energy. In other words, v is such that
for all \({\tilde{v}} \in T_{\Omega (t)}\mathcal{M}\). Choosing \({\tilde{v}} =v\) we immediately obtain the energy estimate associated with each gradient flow, which is
In what follows we consider the micro phase separation in the two dimensional case in the regime where the fraction of A-monomers is much smaller than the one of B-monomers. In this case A-phase consists of an ensemble of many small approximately circular particles. We reduce the evolution to the gradient flow on circular particles.
For that purpose we define the submanifold \(\mathcal{N}\subset \mathcal{M}\) of all sets \(\Omega \) which are the union of disjoint balls \(\Omega \;=\;\bigcup _i B_{R_i}(X_i)\), where the centers \(\{X_i\}_i\) and the radii \(\{R_i\}_i\) are variables. Hence \(\mathcal{N}\) can be identified with an open subspace of the hypersurface \( \{ {{\mathbf {Y}}}=\{R_i,X_i\}_i \,;\, (R_i,X_i) \in {\mathbb {R}}_+ \times {\mathbb {T}}, \pi \sum _i R_i^2=L^2 \rho \}\) in \({\mathbb {R}}^{3N}\), where N is the number and \(i=1,\ldots ,N\) an enumeration of the particles with centers in the torus \({\mathbb {T}}\). Since the normal velocity v satisfies \(v=\frac{d R_i}{d t}+\frac{d X_i}{d t}\cdot {\vec {n}}\) on \(\partial B_{R_i}(X_i)\), the tangent space can be identified with the hyperplane
such that \(V_i\) describes the rate of change of the radius of particle i and \(\xi _i\) the rate of change of its center. We use the abbreviation \({{\mathbf {Z}}}=\{V_i,\xi _i\}_i\) for \({{\mathbf {Z}}}=\sum _i (V_i \frac{\partial }{\partial R_i} + \xi _i \cdot \frac{\partial }{\partial X_i})\).
The metric tensor is then given by
where the function \( w^{\alpha }:{\mathbb {T}}\rightarrow {\mathbb {R}}\) solves
for \({\mathbf {Z}}^\alpha =\{V_i^\alpha ,\xi _i^\alpha \}_i \in T_{{\mathbf {Y}}} {\mathcal {N}}\), \(\alpha =1,2\). For the following it will be convenient to split the metric tensor into the radial and shift part respectively. For any \(\{V_i,\xi _i\}_i\), we write
where u and \(\phi \) are harmonic in- and outside the particles and where
We consider the energy \(E({{\mathbf {Y}}})=E_{\text {surf}}({{\mathbf {Y}}})+\sigma E_{\text {nl}}({{\mathbf {Y}}})\), where
with \(\mu :{\mathbb {T}}\rightarrow {\mathbb {R}}\) solving \( - \Delta \mu = \chi _{\cup B_{R_i}} - \rho \). We obtain the differentials of the energies in the direction of a tangent vector \({\tilde{{{\mathbf {Z}}}}}=\{{\tilde{V_i}},{\tilde{\xi _i}}\}_i\) as
and
Here \({\tilde{w}}:{\mathbb {T}}\rightarrow {\mathbb {R}}\) is a function \(w^\alpha \) satisfying (9) for \({{\mathbf {Z}}}^\alpha ={\tilde{{{\mathbf {Z}}}}}\). The integration by parts yields
From now on we consider an arrangement of particles as described above which evolves according to the gradient flow equation. This means that for any \(t \ge 0\), it holds for \({{\mathbf {Z}}}(t)=\frac{d}{d t}{\mathbf {Y}}(t)\) that
that is,
for all \({\tilde{{{\mathbf {Z}}}}}\in T_{{{\mathbf {Y}}}}\mathcal{N}\). Since \({\tilde{{{\mathbf {Z}}}}}\) is an arbitrary element of the tangent space we conclude from (11) that w satisfies
and
for all i such that \(R_i>0\), with a Lagrange parameter \(\lambda (t)\) that ensures volume conservation. Equations (12) and (13) are the analogue of (4) in the restricted setting.
Our aim is to identify the evolution in the limit of vanishing volume fraction of particles. More precisely, we consider a sequence of systems characterized by the parameter
in the limit \(\varepsilon \rightarrow 0\). Here d is defined by
and \(\mathcal{R}\) by
Then \(\frac{1}{d^2}\) denotes the number density of particles, and \(\pi \,\mathcal{R}^2\) the average volume of particles. Here and throughout this paper we use the abbreviation \(\sum _i = \sum _{i:R_i>0}\).
Our main result informally says that when \(L \sim L_{sc}\), with
on the time scale of order \({\mathcal{R}}^3 \ln (1/\rho )\), the number density of particles with radius r and center x, denoted by \(\nu =\nu (t,r,x)\) (suitably normalized), satisfies
where \(\psi =\psi (t,x)\) satisfies for each t that
in the limit \(\varepsilon \rightarrow 0\). Here \(\sigma \) is also suitably normalized.
We remark that on the other hand, in the case that \(L \ll L_{sc}\), that is, in the very dilute case, one obtains a homogeneous version where \( \psi \) is constant in space, and is replaced by \(\lambda (t)\). More precisely that the number density of particles with radius r, denoted by \(\nu (t,r)\) (suitably normalized), satisfies
with
2 The result
In this section, we will introduce suitably rescaled variables, state the precise assumption on our initial particle arrangement, and present the statement of our main result.
We assume from now on that \(L=L_{sc}\) for the ease of presentation, and we will rescale the spatial variables by \(L_{sc}\) such that
Notice that \(\rho =\pi \alpha _{\varepsilon }^2\varepsilon ^{-2}\) and \(\ln (1/\rho ) \sim \varepsilon ^{-2}\). We introduce \({\hat{R_i}}\), \({\hat{t}}\), \({\hat{V_i}}\), \(\hat{\xi }\), \({\hat{w}}\), \({\hat{\sigma }}\) and \({\hat{\mu }}\) via
From now on we only deal with the rescaled quantities and drop the hats in the notation.
We denote the joint distribution of particle centers and radii at a given time t by \(\nu _t^\varepsilon \in (C^0_p)^*\), which is given by
where \(C^0_p\) stands for the space of continuous functions on \({\mathbb {R}}_+ \times {\mathbb {T}}\) with compact support contained in \({\mathbb {R}}_+ \times {\mathbb {T}}\). Here \({\mathbb {T}}\) denotes the unit flat torus, and \({\mathbb {R}}_+=(0,\infty )\). Note that since \(\zeta (r,x)=0\) for \(r=0\), particles which have vanished do not enter the distribution. The natural space for \(\nu _t^\varepsilon \) and its limit \(\nu _t\) is the space \((C^0_p)^*\) of Borel measures on \({\mathbb {R}}_+ \times {\mathbb {T}}\).
We are now going to make the assumptions on our initial particle arrangement precise. Notice first, that in view of (15) and (16) we have
It follows immediately, that
that is the surface energy of the initial particle arrangement is finite.
Furthermore it is natural to assume that initially the nonlocal energy is uniformly bounded in \(\varepsilon \), that is
where C is independent of \(\varepsilon \) and where \(\mu ^{\varepsilon }(0,x)\) satisfies \(-\Delta \mu ^{\varepsilon }(0,\cdot ) = \frac{\varepsilon ^2}{\alpha _{\varepsilon }^2} \chi _{\cup _i B_i(0)} - \pi \) and \(\int _{{\mathbb {T}}} \mu ^{\varepsilon }\,dx =0\).
We will see later [cf. (84)], that the nonlocal energy controls \( \sum _i \varepsilon ^2 R_i^{\,4}\). Hence, finiteness of the nonlocal energy initially also implies \(\sum _i \varepsilon ^2 R_i^{\, 4}(0) \le C\). For our analysis we need a little more than this. We need a certain tightness assumption which ensures, that not too much mass is contained in very large particles as \(\varepsilon \rightarrow 0\). More precisely, we assume that
Finally, we assume that initially particles are well separated in the sense that we assume that there is \(\gamma >0\), such that
In accordance with the notation in (22) we will use in the following the abbreviation \(\int \zeta \,d\nu _t:= \int _0^{\infty } \int _{{\mathbb {T}}} \zeta (r,x)\,d\nu _t(r,x)\) for \(\nu _t \in (C^0_p)^*\). Otherwise the domain of integration is specified.
The natural space for potentials of diffusion fields is \(H^{1}({\mathbb {T}})\). Furthermore we will denote by \(\mathring{H}^{1}({\mathbb {T}})\) the subspace of \(H^{1}({\mathbb {T}})\) of functions with mean value zero.
We can now state our main result which informally says that \(\nu _t^{\varepsilon }\) converges as \(\varepsilon \rightarrow 0\) to a weak solution of (18)–(19).
Theorem 2.1
Let \(T>0\) be given and assume that the assumptions in Sect. 2 are satisfied. Then there exists a subsequence, again denoted by \(\varepsilon \rightarrow 0\), and a weakly continuous map \([0,T]\ni t\mapsto \nu _t\in (C^0_p)^*\) with
\(\int r^2\,d\nu _t\;=\;1\) for all \(t\in [0,T]\). Furthermore, there exists a measurable map \((0,T)\ni t\mapsto \psi (t)\in H^{1}({\mathbb {T}})\) such that (18) and (19) hold in the following weak sense
distributionally on (0, T) for all \(\zeta \in C^0_p\) with \(\partial _r\zeta \in C^0_p\). Here
for all \(\zeta \in H^{1}({\mathbb {T}})\) and almost all \(t \in (0,T)\).
The proof of Theorem 2.1 goes similarly to the approach for the three dimensional case in [12]. However in contrast to the three dimensional case we need to estimate \(1/R_i\) term in the proof that the tightness property is preserved in time (see Lemmas 3.9 and 3.10) since the Lagrange multiplier diverges when particles disappear.
3 Proof of Theorem 2.1
We can deduce Theorem 2.1 by the homogenization of Rayleigh Principle (see Theorem 3.11). This will be obtained from the homogenization of metric tensor (Lemmas 3.5, 3.6) and the limit of the differential of the energy (Lemma 3.8). Also we need some a-priori estimates, which are given by a series of lemmas. The proof of Lemmas in this section will be given in Sect. 4. For readers convenience we will not abbreviate the arguments.
3.1 Gradient flow structure
In rescaled variables the submanifold \(\mathcal{N}\) turns into
and the tangent space
We use the abbreviation \({{\tilde{{{\mathbf {Z}}}}}}^{\varepsilon }=\{{\tilde{V_i}},{\tilde{\xi _i}}\}_i\) for \({{\tilde{{{\mathbf {Z}}}}}}^{\varepsilon }= \sum _i ({\tilde{V_i}}\frac{\partial }{\partial R_i}+\varepsilon \alpha _{\varepsilon }\ln (1/\rho ) {\tilde{\xi _i}}\cdot \frac{\partial }{\partial X_i})\), and regard \(\{{\tilde{V_i}},{\tilde{\xi _i}}\}_i\) as the component of a tangent vector \({{\tilde{{{\mathbf {Z}}}}}}^{\varepsilon }\) with respect to a basis
We will always denote by \({{\mathbf {Z}}}^{\varepsilon }=\{V_i,\xi _i\}_i\) the direction of steepest descent. Recall that \(V_i=\frac{d R_i}{d t }\), but \(\xi _i=(\varepsilon \alpha _{\varepsilon }\ln (1/\rho ))^{-1} \frac{d X_i}{d t}\). The notation \({{\tilde{{{\mathbf {Z}}}}}}^{\varepsilon }\) will be used for an arbitrary element of the tangent space. Furthermore we use the abbreviation \(B_i:=B_{\alpha _{\varepsilon }R_i}(X_i)\).
We define the energy in rescaled variables as
where \(\mu ^{\varepsilon }=\mu ^{\varepsilon }(t,x)\) solves \(-\Delta \mu ^{\varepsilon } = \frac{\varepsilon ^2}{\alpha _{\varepsilon }^2}\chi _{\cup B_i} - \pi \) and \(\int _{{\mathbb {T}}}\mu ^{\varepsilon }\,dx=0\), and the metric tensor for \({{\tilde{{{\mathbf {Z}}}}}}^{\varepsilon }\in T_{{{\mathbf {Y}}}^{\varepsilon }} \mathcal{N}^{\varepsilon }\) is computed via
where \({\tilde{w}}^\varepsilon ={\tilde{u}}^\varepsilon +{\tilde{\phi }}^\varepsilon \),
for all \(\zeta \in \mathring{H}^{1}({\mathbb {T}})\). Notice that the potentials are only determined up to additive constants. In what follows we fix this constant by requiring that \(\int _{{\mathbb {T}}} {\tilde{u}}^{\varepsilon }\,dx = \int _{{\mathbb {T}}} {\tilde{\phi }}^{\varepsilon }\,dx=0\). For the steepest descent directions \({\mathbf {Z}}^\varepsilon =\{V_i,\xi _i\}_i\), we define the potentials \(w^\varepsilon \), \(u^\varepsilon \), \(\phi ^\varepsilon \) analogously.
Equations (12) and (13) for the direction of steepest descent, turn into
for some \(\lambda ^\varepsilon (t) \in {\mathbb {R}}\) and
for all i such that \(R_i>0\). Here and in what follows we abbreviate, with some abuse of notations, for a disk \(B_R(X)\) the perimeter by \(|\partial B_R|\) and its area by \(|B_R|\). Now the energy estimate (8) reads
for all \(t_1 >0\). Finally, the Rayleigh principle says that \({\mathbf {Z}}^\varepsilon \) satisfies
for all \(\tilde{{\mathbf {Z}}}^\varepsilon \in T_{{\mathbf {Y}}} {\mathcal {N}}\) and nonnegative \(\beta \in C^\infty ([0,T])\).
3.2 A priori estimates and weak limits
It follows from definitions (15), (16) and the facts that volume of particles is conserved and the number can only decrease, that
On the other hand the uniform bound on the energy in (34) implies the following.
Lemma 3.1
Next to \(\nu _t^\varepsilon \), we introduce signed Borel measures \(\rho _t^\varepsilon \in (C^0_p)^*\) and \(\psi _t^{\varepsilon } \in ((C^0_p)^2)^*\) on \({\mathbb {R}}_+\times {\mathbb {T}}\) via
The measures satisfies \(\partial _t \nu ^{\varepsilon } + \partial _r \rho ^{\varepsilon } + \varepsilon \alpha _{\varepsilon }\ln (1/\rho ) \hbox {div}\, \psi ^{\varepsilon }=0\) in the sense of distributions, that is
for all \(\zeta \in C^0_p\cap C^\infty \) and \(\beta \in C_0^\infty ([0,T])\). As will be shown in Sect. 4, we have
which yields in particular
Bounds (39) and (40) yield a weak Hölder regularity in t of \(\{\nu _t^\varepsilon \}_t\):
Using Arzela–Ascoli’s Theorem, (36) and (42) imply that there exists a weakly continuous family \(\{\nu _t\}_t\) of nonnegative Borel measures on \({\mathbb {R}}_+ \times {\mathbb {T}}\) such that for a subsequence
for \(\zeta \) in a countable subset of \(C^0_p\cap C^\infty \). Again by (36), we see that we can extend the locally uniform convergence in (43) to all \(\zeta \in C^0_p\). Obviously, the bound (36) is conserved
and due to (37) and (38) we have
The uniform control of the signed Borel measures \(\{ r^2 \, d\rho _t^\varepsilon \, d t \}_\varepsilon \) and \(\{ r^2 \, d\psi _t^{\varepsilon }\, d t\}_\varepsilon \) on \({\mathbb {R}}_+ \times {\mathbb {T}}\times [0,T]\) implied by (40) ensures the weak convergence, where the limits can be regarded as bounded linear functionals on \(L^2(r^2\, d \nu _t \, d t)\) and \(L^2(r^2\, d \nu _t \, d t)^2\) respectively, and hence by Riesz Representation Theorem, there exist \(v\in L^2(r^2\,d\nu _t\,dt)\) and \(\xi \in L^2(r^2\,d\nu _t\,dt)^2\) with
such that for a subsequence
for all \(\beta \in C^0([0,T])\), \(\zeta \in C^0_p\) and \(\eta \in (C^0_p)^2\), and
for all nonnegative \(\beta \in C^0([0,T])\). Thus the limit of (39) is
for all \(\zeta \in C^0_p\cap C^\infty \) and \(\beta \in C_0^\infty ([0,T])\).
Definition 3.2
For \(\tilde{{\mathbf {Z}}}^\varepsilon \) satisfying \(\int _0^T g_{{\mathbf {Y}}^\varepsilon } (\tilde{{\mathbf {Z}}}^\varepsilon ,\tilde{{\mathbf {Z}}}^\varepsilon )\, d t \le C\), we can define \({\tilde{v}} \in L^2(r^3 \,d\nu _t\,d t)\) and \({\tilde{\xi }} \in L^2(r^3 \,d\nu _t\,d t)^2\) analogously. We say that \(\tilde{{\mathbf {Z}}}^\varepsilon \) converges weakly to \(({\tilde{v}},\tilde{\xi })\).
We are going to show below that for any \(T>0\), particles do not collide on a time interval [0, T] for sufficiently small \(\varepsilon \). More precisely
Lemma 3.3
For any \(T>0\) we can find \(\varepsilon _0>0\) such that
for all \(\varepsilon \in (0,\varepsilon _0]\).
Thus it follows that the marginal of \(\nu _t\) with respect to x has a bounded Lebesgue density. Hence it follows from (46) that the functional
is an element of \(H^{-1}({\mathbb {T}})=(H^{1}({\mathbb {T}}))^*\) for all \(t\in [0,T]\). Hence \(K(t,\cdot ) \in H^{1}({\mathbb {T}})\) is uniquely determined via (29) up to additive constants.
In order to prove Lemma 3.3 we show the following.
Lemma 3.4
(slow motion of the particle centers) As long as (51) is satisfied, we have
3.3 Homogenization of the metric tensor
We identify the \(\Gamma \)-limit for the metric tensor and provide the necessary results to pass to the limit in the metric tensor. The following is a lower semicontinuity result.
Lemma 3.5
(lower semicontinuity) For all nonnegative \(\beta =\beta (t)\in C^{\infty }([0,T])\) we have
where for almost all t the function \(u(t,\cdot )\in \mathring{H}^{1}({\mathbb {T}})\) is determined via
for all \(\zeta \in \mathring{H}^{1}({\mathbb {T}})\).
Furthermore we show that for any tangent vector of the limit manifold \(({\tilde{v}}, {\tilde{\xi }})\), there exists an approximating sequence along which the metric tensor is continuous.
Lemma 3.6
(construction) For any \({\tilde{v}}\in L^2(r^2 \,d\nu _t \, d t)\) with \( \int {\tilde{v}}\,r^2\,d\nu _t=0\) for almost all t and any \({\tilde{\xi }} \in L^2(r^2 \,d\nu _t \, d t)^3\) there exists a sequence \({{\tilde{{{\mathbf {Z}}}}}}^{\varepsilon }\) with \({\tilde{{{\mathbf {Z}}}}}^{\varepsilon } \in T_{{{\mathbf {Y}}}^{\varepsilon }}\mathcal{N}^{\varepsilon }\) such that \(\tilde{{\mathbf {Z}}}^\varepsilon \) weakly converges to \(({\tilde{v}},{\tilde{\xi }})\), and
for all nonnegative \(\beta =\beta (t) \in C^{\infty }([0,T])\) and with \({\tilde{u}}(t,\cdot ) \in \mathring{H}^{1}({\mathbb {T}})\) determined via
for all \(\zeta \in \mathring{H}^{1}({\mathbb {T}})\).
Note that the contribution from the drift term and the radial part do not interact in the limit \(\varepsilon \rightarrow 0\).
3.4 The limit of the differential of the energy
We identify the limit of the differential of the energy. To that aim we identify the limit of the potentials \(\mu ^{\varepsilon }\) so as to prove the convergence for the nonlocal part of the energy.
Note that
Here, as also in the homogenization of the metric tensor, the key idea is that the potentials can be represented as a sum of monopoles, which represent the self-interaction of particles, plus a slowly varying field, which represents the interaction between different particles.
We set \(l= \gamma \varepsilon \), where \(\gamma >0\) is as in (51). We write \(\mu ^\varepsilon =\sum _i {\tilde{\mu }}_i + {\tilde{\mu }}^\varepsilon \) with
Here \({\tilde{\mu }}_i\) is chosen such that \(-\Delta {\tilde{\mu }}_i = \frac{\varepsilon ^2}{\alpha _{\varepsilon }^2}\) in \(B_i\) and \(\Delta {\tilde{\mu }}_i=0\) in \(B_l (X_i){\setminus } B_i\). Since
it holds that
The slowly varying field \({\tilde{\mu }}^\varepsilon \) converges strongly to K and this enables us to pass to the limit in the differential of the energy.
Lemma 3.7
as \(\varepsilon \rightarrow 0\). Here \(K(t,\cdot ) \in \mathring{H}^{1}({\mathbb {T}})\) satisfies for each t
for all \(\zeta \in H^{1}({\mathbb {T}})\).
Lemma 3.8
Assume that \({\tilde{{{\mathbf {Z}}}}}^\varepsilon \) satisfies
and converges weakly to \(({\tilde{v}},{\tilde{\xi }})\). Then for all \(\beta \in C^{\infty }([0,T])\),
where
3.4.1 Tightness
In order to prove Lemma 3.8 we first need to show that the tightness property (26) is preserved in time so that no mass is lost at infinity in the limit \(\varepsilon \rightarrow 0\).
Lemma 3.9
(tightness) For any \(t>0\) we have
This lemma is crucial to our proof. The proof is much more difficult than three dimensional case. In fact, the main idea of the proof is to show by asymptotics that, at least in some average sense, \(V_i\) satisfies approximately \(R_i^2 V_i \sim u R_i - 1 - \sigma R_i^3\) where u is the Lagrange multiplier that ensures the volume conservation. In contrast to the three dimensional case, the Lagrange multiplier may diverge. However we can still show the following a-priori estimate, and thus, at least on average, \(V_i \le 0\), if \(R_i\) is sufficiently large, and no mass can escape to infinity as \(\varepsilon \rightarrow 0\), from which one deduces Lemma 3.9.
Lemma 3.10
For any \(T>0\), there exist constants \(C_T>0\) and \(\varepsilon _1>0\) such that
for all \(\varepsilon \in (0,\varepsilon _1]\).
3.5 Homogenization of Rayleigh principle
The main task that remains to be done now is to determine the equation for the velocity function v. It will be characterized as the minimizer in the Rayleigh principle. Thus our task is to characterize the limits of \({\mathbf {Z}}^{\varepsilon }\) that satisfy (35).
Notice that we can use Lemma 3.8 with \({\mathbf {Z}}^{\varepsilon } - \tilde{{\mathbf {Z}}}^{\varepsilon }\) where \({\mathbf {Z}}^{\varepsilon }\) is the direction of steepest descent and \(\tilde{{\mathbf {Z}}}^{\varepsilon }\) is as in Lemma 3.6. The main result from which one deduces Theorem 2.1 is the following.
Theorem 3.11
For all nonnegative \(\beta \in C^{\infty }([0,T])\) we have
for all \({\tilde{v}}\in L^2(r^2 \, d\nu _t \, d t)\) and \({\tilde{\xi }} \in L^2(r^2 \, d\nu _t \, d t)^2\) such that \(\int r \, {\tilde{v}}\,d\nu _t=0\) for almost all t and such that \((v - {\tilde{v}})(t,x,\cdot ) =0\) in a neighborhood of \(r=0\). Here
for \({\tilde{v}}\) which vanish in a neighborhood of \(r=0\), and \(u(t,\cdot ) \), \({\tilde{u}}(t,\cdot )\), \(K(t,\cdot )\in \mathring{H}^{1}({\mathbb {T}})\) are determined for a.a. t via (53), (55), and (59) respectively.
The Euler–Lagrange equation for (62) becomes \(r^2 v= r(u-\sigma K-\lambda )-1-\sigma r^3\) with \(\lambda \) being a Lagrange multiplier that ensures the constraint \(\int r v \,d \nu _t=0\). Setting \(\psi (t,x)\equiv u-\sigma K -\lambda \), we can then derive (28)–(29). The proof is basically straightforward and goes similarly to the one in Chapter 6 of [13]. We omit the details here.
Notice that in the formulation (62) we need that \(v-{\tilde{v}}\) has compact support in the r-variable. This is due to the fact that we cannot guarantee that the term \(\int r v \,d\nu _t\) which appears in the differential of the surface energy is well-defined.
4 Proof of Lemmas
Proof of Lemma 3.4
We set \(l=\gamma \varepsilon \), where \(\gamma \) is as in (51). Notice that due to (51) and \(\sum _i\varepsilon ^2 R_i^2=1\), the balls \(\{B_{l}(X_i)\}_i\) are disjoint. For given \(\{\xi _i\}_i\) define
For further use we collect some properties of \(\phi _i\). It is easily checked that \(\phi _i\) is continuous in \({\mathbb {T}}\), harmonic in \( B_{l}(X_i) {\setminus }\partial B_i \) and satisfies
Furthermore, \(\phi _i=0\) on \(\partial B_{l}(X_i)\) and
Then
Integration by parts yields
Together with (66), (67) and (33) we find
We estimate the last term on the right hand side of (68) via
We can write \(\mu ^\varepsilon = \sum _i \mu _i +{\bar{\mu }}^\varepsilon \) with
Notice that \(\mu _i\) is continuous in \({\mathbb {T}}\), and satisfies \(-\Delta \mu _i =\frac{\varepsilon ^2}{\alpha _{\varepsilon }^2}-\pi \) in \(B_i\), \(-\Delta \mu _i= -\pi \) in \(B_l (X_i) {\setminus } \overline{B_i}\). These imply that \({\bar{\mu }}^\varepsilon \), and hence \(\nabla {\bar{\mu }}^ \varepsilon \) also are harmonic in \( \cup _i B_l (X_i)\). Then, due to the properties of \(\mu _i\) and the mean value theorem, we have
But
so
Next we write \(u^\varepsilon =\sum _i u_i + {\bar{u}}^\varepsilon \) where
It is easily checked that \(u_i\) is continuous in \({\mathbb {T}}\), harmonic in \(B_l (X_i) {\setminus } \partial B_i \) and satisfies
Then from a similar argument as above, we see that
Thus, in summary we find
This completes the proof of Lemma. \(\square \)
Proof of Lemma 3.3
In Sect. 3.3 we will see that \(\int _0^{T} \int _{{\mathbb {T}}} | \nabla \phi ^{\varepsilon }|^2 + |\nabla u^{\varepsilon }|^2\,dx\,dt \le C\) (cf. (81)). Furthermore, due to (34), we have \(\int _{{\mathbb {T}}} |\nabla \mu ^{\varepsilon }|^2 \,dx \le C\). Then the statement follows from Lemma 3.4 and
\(\square \)
Proof of Lemma 3.5
We can prove
in the same way as in [13] since (51) is satisfied.
Hence, it remains to show that
and
Step 1: Monopoles. Our goal is to construct a good approximation of \(\phi ^{\varepsilon }\) which is based on cutting off the single monopole solutions. To that aim we define \(\phi _i\) as in (63) with \(\gamma \) is as in (51) and thus the balls \(\{B_l (X_i)\}_i\) are disjoint. Using (64) and (65), we find
Step 2: A lower bound. We will show that for any given \(\delta >0\) we have
if \(\varepsilon \) is sufficiently small. Indeed, due to the fact that \(\{B_l(X_i)\}_i\) are disjoint we have
On the other hand
where
for all \(\zeta \in C_0^{\infty }(B_l (X_i))\). From the corresponding Euler–Lagrange equation we see that the minimizer \({\hat{\psi }}\) is orthogonal to all divergence-free function and hence a gradient. We find that \({\hat{\psi }} = \nabla \phi _i\) and thus (77) follows from (76).
Step 3: Approximation of \(\phi ^{\varepsilon }\). With
we have
for any \(k>0\). In fact, for \(\bar{\phi }^{\varepsilon }:= \phi ^{\varepsilon }- \hat{\phi }^\varepsilon \) it follows from the definitions and (64) that
for all \(\zeta \in \mathring{H}^{1}({\mathbb {T}})\). We now define \(L_{\varepsilon } \in ({H^{1}({\mathbb {T}})})^*\) via
We observe that due to \( -\! \! \! \! \!\int _{\partial B_l(X_i)} \xi _i\cdot \vec n\,dS =0\) we can write
We can then estimate
where the last estimate follows from Trace Theorem and Poincaré’s inequality. This gives
Due to the simple estimate \(R_i \le C \varepsilon ^{-1/2}\) we find
which proves (79).
Step 4: Bounds on the individual terms.
We write
Now \(\nabla u^{\varepsilon }\) is orthogonal to \(\nabla {\hat{\phi }}^\varepsilon \) since
but \(\int _{\partial B_i} \phi _i\,dS=0\). Thus
Furthermore, for some small \(\delta >0\), using (77),
We choose \(\delta = \frac{1}{4}\) and then \(\varepsilon \) so small such that \(C_{\delta }\varepsilon ^4\le \frac{1}{4}\). Then (81) follows from (34) and (82).
Step 5: Lower semicontinuity. Since we have now established (81), we obtain the existence of the weak limit \(\xi \) as explained before in Sect. 3.2, and hence the assertion (74) follows from (49) and (77).
Step 6: The mixed term vanishes in the limit.
As an immediate consequence of (77), (79), (81) and (82) we find
and thus that (75) is valid. \(\square \)
Proof of Lemma 3.6
We need only to show that for given \({\tilde{\xi }}\) as above we can find \( {{\tilde{\phi }}}^{\varepsilon }\) such that
and
since the corresponding result for \({\tilde{u}}^\varepsilon \) can be proved in the same way as in [13]. The proof is in fact quite similar to the proof of Lemma 3.5, since there the minimization property of \(\{\xi _i\}_i\) is not used in the construction, which is henceforth quite general.
Step 1: Construction for smooth \({\tilde{\xi }}\). We first assume that \({\tilde{\xi }}\) is smooth. Then we define \({\tilde{\xi _i}}:= \tilde{\xi }(X_i)\) and construct \(\phi _i\) as in (63) for \(\xi _i={\tilde{\xi }}_i\), and a corresponding \({\hat{\phi }}^\varepsilon \) as in (78). The property (76) implies
and hence
as \(\varepsilon \rightarrow 0\). Here we used (43) and that \({\tilde{\xi }}\) has compact support.
Furthermore, we obtain also exactly as in the proof of Lemma 3.5 that \({\bar{\phi }}^\varepsilon := {\tilde{\phi }}^\varepsilon - \hat{\phi }^\varepsilon \) converges to zero strongly in \(L^2((0,T);H^{1}({\mathbb {T}}))\), and as a consequence the mixed term vanishes in the limit.
Step 2: Construction for general \({\tilde{\xi }}\).
In order to finish the proof of the lemma we have to show that we can approximate \({\tilde{\xi }} \in L^2(r^2\,d\nu _t\,d t)^3\) by smooth functions \({\tilde{\xi }}_n\) with compact support such that
as \(n \rightarrow \infty \). But this follows from a density argument. \(\square \)
Proof of Lemma 3.1
We first prove (38). Set \(l=\gamma \varepsilon \), where \(\gamma \) is as in (51). Due to the fact that \(\{B_l (X_i)\}_i\) are disjoint we have
On the other hand, for each i,
where the infimum is taken over all \(\psi \in (L^2(B_l (X_i)))^2\) which satisfy
for all \(\zeta \in C_0^\infty (B_l (X_i))\). We see that the minimizer \({\hat{\psi }}\) is orthogonal to all divergence-free vector-valued functions. Hence \({\hat{\psi }}=\nabla \phi \), \(\phi =\text {const.}\) on \(\partial B_l (X_i)\), where \(-\Delta \phi =\tfrac{\varepsilon ^2}{\alpha _{\varepsilon }^2}\chi _{B_i}-\pi \). We find that \({\hat{\psi }}=\nabla \mu _i\), where \(\mu _i\) is defined as in (70), and we have
Then it follows from (34), \(\sum _i \varepsilon ^2 R_i^2 =1\) and (83) that
which proves (38). \(\square \)
Proof of Lemma 3.10
We can prove this in a similar way as in [14, section 3.4] by making use of (32) and Lemma A.1 although we are using Neumann boundary condition on \(\partial B_i\) instead of Dirichlet boundary condition.
Note \(0<\frac{1}{1-\varepsilon ^2 \ln R_i} \le \frac{1}{1+\varepsilon ^2 \ln \varepsilon } \le \frac{2e}{2e-1}\) by \(0<R_i \le 1/\varepsilon \) and \(|B_l(X_i)|/|A_i|\le 2\) for small \(\varepsilon \) with \(l=\gamma \varepsilon \) and
Setting \(v^\varepsilon =u^\varepsilon +\phi ^\varepsilon -2\sigma \mu ^\varepsilon -\lambda ^\varepsilon \), we have
since \(|B_l(X_i)|/|A_i|\le 2\) for small \(\varepsilon \). Using
we see that
Now we estimate the second term of the right hand side of (85). Note that \(v^\varepsilon =u^\varepsilon +\phi ^\varepsilon -2\sigma \mu ^\varepsilon -\lambda ^\varepsilon \) satisfies
for each i. Using Lemma A.1 for \(f=v^\varepsilon -\frac{1}{R_i}\), we have
and since \(|B_l(X_i)|/|A_i|\le 2\) for small \(\varepsilon \),
It follows from (85) and (86) that
We estimate the \(L^2\) norm of \(v^\varepsilon \).
Using
we see that
We estimate the second term of the right hand side of (90).
Note that due to
\(-\ln R_i \le \frac{1}{e\alpha R_i^\alpha }\), \(\sum _i \varepsilon ^4 R_i^2 \le C\) and \(\sum _i \varepsilon ^2 R_i^3 \le C\), there is some c, independent of \(\varepsilon \), such that
Then it follows from (90) and (91) that
Then by (88),
Thus we obtain from (87),
\(\square \)
Proof of Lemma 3.9
Step 1: An expression for \(V_i\). Similarly to \(\phi _i\) in the proof of Lemma 3.4 we introduce a suitable test function \(v_i\). Here it is the capacity potential of \(B_i\) with respect to \(B_l(X_i)\), with \(l = \gamma \varepsilon \). This gives that
with \(a^{\varepsilon } = (\ln \frac{l}{\alpha _{\varepsilon }R_i} )^{-1} \). With this definition we also have
Then
Due to (32) it follows that
Using that \(\mu ^\varepsilon = \sum _i \mu _i + {\bar{\mu }}^\varepsilon \) with \( \mu _i\) as in (70) we obtain
Similarly using that \(\phi ^\varepsilon =\sum _i \phi _i+{\bar{\phi }}^\varepsilon \) with \(\phi _i\) as in (63) we obtain
It follows that
Step 2: A bound on \(\int _{\partial B_l (X_i)} u^{\varepsilon } +\phi ^\varepsilon - 2 \sigma \mu ^\varepsilon \,d S\).
We define \(\psi (x)= \frac{1}{2} (l^2 - |x-X_i|^2)\) in \(B_l (X_i)\) such that
Thus, abbreviating \(v^{\varepsilon }= u^{\varepsilon } + \phi ^{\varepsilon } - 2 \sigma \mu ^{\varepsilon } \), we have
We multiply (97) by \(R_i^m a^\varepsilon \) with \(m\ge 0\), and sum over \(i \in I\) for some \(I \subset \{i:R_i>0\}\). Then due to \(a^{\varepsilon } \le 2 \varepsilon ^2\), we find, using \(l=\gamma \varepsilon \) and Hölder’s inequality, that
Due to Sobolev embedding and Poincare’s inequality (recall that \(v^{\varepsilon }\) has mean value zero) we finally find
Step 3: A bound for \(\lambda ^\varepsilon (t)\). We go back to (96) to estimate \(\lambda ^\varepsilon (t)\). For that purpose we multiply (96) by \(\varepsilon ^2 \) and sum over i. We find
First notice that due to
we have \(\left( \sum _i \frac{1}{ \ln \frac{l}{\alpha _{\varepsilon }R_i}} \right) ^{-1} \le \sum _i \varepsilon ^4 R_i^4 \ln \frac{l}{\alpha _{\varepsilon }R_i} \le C \) due to (84). As a consequence, we also have \(\frac{1}{\sum _i a^{\varepsilon }} \le C\).
Taking \(m=0\) in (98), we find
Using
and Lemma 3.10, it follows [cf. (81) in Sect. 3.3] that
Step 4: Completion of Proof. Now we go back to (96), multiply with \(\varepsilon ^2 R_i^2\) and sum, but only over i such that \(R_i \ge M\). We find
Here we used \(\frac{a^\varepsilon }{ \varepsilon ^{2}} =\frac{1}{1+ \varepsilon ^2 \ln \frac{\gamma }{ R_i}}=1+O(\varepsilon )\) as \(\varepsilon \rightarrow 0\) for i such that \(R_i\ge M\), \(\varepsilon R_i\le 1\) and \(M>\gamma \).
The key feature of (102) is that we have a term of the form \(- \sum _{R_i \ge M} \varepsilon ^2 R_i^4\) on the right hand side and otherwise terms which converge to zero as \(M\rightarrow \infty \) uniformly in \(\varepsilon \). Now define
such that it follows from (102) together with (98) with \(m=2\) that
Now
To estimate the last term on the right-hand side of (103) we recall that we have the simple estimate \(R_i \le C\varepsilon ^{-1/2}\) such that
Thus, (103)–(107) imply that we have an estimate of the form
A simple comparison argument now implies that indeed \(y(t) \le C \big (y(0)+ M^{-\delta } \big )\) and thus the statement of the lemma follows with (26). \(\square \)
Proof of Lemma 3.7
We use an argument similar to Step 2 of the proof of [13, Lemma 5.4], and give only an outline of the proof. It follows from the definition of K (59) that
for all \(\zeta \in \mathring{H}^{1}({\mathbb {T}})\). Furthermore since \(\Delta {\tilde{\mu }}^\varepsilon = \pi \) in \({\mathbb {T}}{\setminus } \cup _i ( \partial B_l (X_i) \cup \partial B_i)\), \([\nabla {\tilde{\mu }}^\varepsilon \cdot {\vec {n}}]= 0\) on \(\partial B_i\), and \([\nabla {\tilde{\mu }}^\varepsilon \cdot {\vec {n}}]=- \frac{\varepsilon ^2 R_i^2}{2 l}\) on \(\partial B_l (X_i)\), we see that \({\tilde{\mu }}^\varepsilon \) solves
for all \(\zeta \in \mathring{H}^{1}({\mathbb {T}})\). We define
for \(\zeta \in \mathring{H}^{1}({\mathbb {T}})\). Then \(L_\varepsilon (t)\) and L(t) are elements of \(\mathring{H}^{-1}({\mathbb {T}})=(\mathring{H}^{1}({\mathbb {T}}))^*\) for all \(t \in [0,T]\). Since \(\langle L_\varepsilon (t),\zeta \rangle = 2 \int _{{\mathbb {T}}}\nabla {\tilde{\mu }}^\varepsilon \cdot \nabla \zeta \, d x\) and \( \langle L(t),\zeta \rangle =\int _{{\mathbb {T}}} \nabla K\cdot \nabla \zeta \, d x\), we only need to prove that \(L_\varepsilon \rightarrow L\) strongly in \(C^0([0,T];\mathring{H}^{-1}({\mathbb {T}}))\).
In order to prove this, define
Then we see that \({\tilde{L}}_\varepsilon (t)\in (L^{4}({\mathbb {T}}))^* \cong L^{4/3}({\mathbb {T}})\) and satisfies \(\Vert {\tilde{L}}_\varepsilon (t)\Vert _{L^{4/3}} \le C\) for all \(t \in [0,T]\) and small \(\varepsilon >0\). Here \(L^2\) estimate does not work. Furthermore for all \(t_1,t_2 \in [0,T]\) we have
These mean that \({\tilde{L}}_\varepsilon \in C^0([0,T];L^{4/3}({\mathbb {T}}))\) is uniformly bounded and equicontinuous. It follows from the compact embedding of \(L^{4/3}({\mathbb {T}})\) into \(\mathring{H}^{-1}({\mathbb {T}})\) and the generalized Arzela–Ascoli Theorem that \({\tilde{L}}_\varepsilon \) is relatively compact in \(C^0([0,T];\mathring{H}^{-1}({\mathbb {T}}))\).
Since we can deduce that \({\tilde{L}}_\varepsilon (t) \rightharpoonup L(t)\) weakly in \((C_p^0)^*\) for each \(t \in [0,T]\), and that \(L_\varepsilon -{\tilde{L}}_\varepsilon \rightarrow 0\) in \(C^0([0,T];\mathring{H}^{-1}({\mathbb {T}}))\), the proof of Lemma is completed. \(\square \)
Proof of Lemma 3.8
The differential of the nonlocal energy is given by
with \({\tilde{u}}^\varepsilon \) and \({\tilde{\phi }}^\varepsilon \) determined via (31).
Define \(u_i\) as in (72), however we take \(V_i={\tilde{V}}_i\), and set \({\tilde{u}}^\varepsilon =\sum _i u_i + {\bar{u}}^\varepsilon \). Note that
Also we use the same notations \({\tilde{\mu }}_i,{\tilde{\mu }}^\varepsilon \) as in Sect. 3.4.
We write
It follows from the condition (60) that
and that \({\bar{u}}^\varepsilon \) converges to \({\tilde{u}}\) strongly in \(L^2 ((0,T)\times {\mathbb {T}})\), where \({\tilde{u}}\) is determined via (53). (See the proof of [13, Lemma 5.4] for details.) Using
(113), Lemma 3.9, \(|R_i\ln 1/R_i|\le C \) for \(R_i\ll 1\) and \(\varepsilon R_i \le 1\), we see that
as \(\varepsilon \rightarrow 0\). In addition, we obtain
and it follows from Lemma 3.7 that
Finally from \(\sum _i |B_l (X_i)| \le C \gamma ^2 \sum _i \varepsilon ^2 \le C \gamma ^2\) and (111),
where
Since we can choose \(\gamma \) arbitrarily small, we obtain
from (112), (114), (115), (116) and (117).
Similarly we can prove that \(\int _0^T \beta \int _{{\mathbb {T}}} \nabla \tilde{\phi }^ \varepsilon \cdot \nabla \mu ^\varepsilon \,d x\,d t \rightarrow 0\) as \(\varepsilon \rightarrow 0\). This completes the proof. \(\square \)
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A Appendix
A Appendix
Lemma A.1
\(A=\{x\in {\mathbb {R}}^2\,;\, a<|x|<b\}\). There holds
for all \(f \in H^1(A)\) such that \(\int _{\partial B_a} f\, d S=0\).
Proof
Using the polar coordinate \((r,\theta )\), we write \(f=f(r,\theta )\). Let
for \(r \in [a,b]\). Then by
and
for \(s \in [a,b]\), we get
Note that \(g(r)=\int _a^r g'(s)\, d s\) by \(g(a)=0\). Hence
We find
\(\square \)
Remark: Similarly we can also show that
where
This statement is of interest in itself, but since we do not need it here we omit the proof for the sake of brevity.
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Niethammer, B., Oshita, Y. A rigorous derivation of mean-field models describing 2D micro phase separation. Calc. Var. 59, 54 (2020). https://doi.org/10.1007/s00526-020-1706-x
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DOI: https://doi.org/10.1007/s00526-020-1706-x