Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics

TL;DR

This paper establishes existence, uniqueness, and regularity of solutions for a novel class of nonlocal Cahn-Hilliard equations with degenerate potentials, and demonstrates their convergence to local models as the nonlocal kernel approximates a delta function.

Contribution

It introduces a new existence theory for nonlocal Cahn-Hilliard equations with singular kernels not covered by previous frameworks, including convergence and regularity results.

Findings

Proved existence and uniqueness of solutions.

Demonstrated convergence to local Cahn-Hilliard equations.

Established higher regularity and stronger convergence topology.

Abstract

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.

Equations

$$E_{CH}(u)=\int_{\Omega}\Bigl{(}\frac{\tau^{2}}{2}|\nabla u(x)|^{2}+F(u(x))\Bigr{)}\,\textrm{d}x,$$

$$\begin{array}{rl}{\partial }_{t}u+\mathrm{div}{J}_{CH}& =0\text{,}\\ J_{CH}& =-m(u)\nabla {\mu }_{CH}\text{,}\\ {\mu }_{CH}& =\frac{\delta E_{CH}(u)}{\delta u}=-{\tau }^2\Delta u+F^{\prime }(u)\text{,}\end{array}$$

$$E_{NL}(u)$$

$$\begin{array}{rl}{\partial }_{t}u+\mathrm{div}{J}_{NL}& =0\text{,}\\ J_{NL}& =-m(u)\nabla {\mu }_{NL}\text{,}\\ {\mu }_{NL}& =\frac{\delta E_{NL}(u)}{\delta u}=(K\ast 1)u-K\ast u+F^{\prime }(u)\text{,}\end{array}$$

$$|K(x,y)|\le C|x-y{|}^{-\alpha }\text{with}0<\alpha <\frac{3}{2}$$

$$K_{\epsilon }(x,y)=\frac{{\rho }_{\epsilon }(|x-y|)}{|x-y{|}^2},$$

$$\frac{1}{4}{\int }_{\Omega }{\int }_{\Omega }{K}_{\epsilon }(x,y)(u(x)-u(y){)}^2\mathrm{d}x\text{d}y\to \frac{1}{2}{\int }_{\Omega }|\nabla u(x){|}^2\text{d}x.$$

$$F^{\prime \prime }\ge -B_1,$$

$$D(\mathbf{B}):=\{v\in L^2(\Omega ):(K\ast 1)v-(K\ast v)\in L^2(\Omega )\},$$

$$\mathbf{B}(v):=(K\ast 1)v-(K\ast v),\forall v\in D(\mathbf{B}),$$

$$F_{\log }(s)=\frac{\theta }{2}((1+s)\log (1+s)+(1-s)\log (1-s))+\frac{{\theta }_c}{2}-c{s}^2$$

$$F_{\mathrm{ob}}(s)=I_{[-1,1]}(s)+\frac{1}{2}(1-s^2),I_{[-1,1]}(s):=\{\begin{array}{ll}0& \text{if }s\in [-1,1]\\ +\infty & \text{otherwise}.\end{array}$$

$$\begin{array}{rl}{\partial }_{t}u+\mathrm{div}{J}_{NL}+\mathrm{div}(\beta u)& =0\text{,}\\ J_{NL}& =-\nabla {\mu }_{NL}\text{,}\\ {\mu }_{NL}& =\frac{\delta E_{NL}(u)}{\delta u}=(K\ast 1)u-K\ast u+F^{\prime }(u)\text{.}\end{array}$$

$$K_{\epsilon }(x,y):=\frac{1}{|x-y{|}^2}{\rho }_{\epsilon }(|x-y|),\text{for a.e. }(x,y)\in \Omega \times \Omega$$

$${\rho }_{\epsilon }(r)\ge 0$$

$$\mathrm{supp}({\rho }_{\epsilon })\subset [0,\mathrm{diam}(\Omega )]$$

$${\int }_0^{+\infty }{\rho }_{\epsilon }(r)r^{d-1}\text{d}r=\frac{1}{2M_d}$$

$$\underset{\epsilon \searrow 0}{\lim }{\int }_{\delta }^{+\infty }{\rho }_{\epsilon }(r)r^{d-1}\text{d}r=0$$

$${\partial }_t{u}_{\epsilon }-\Delta {\mu }_{\epsilon }+\mathrm{div}(\beta u_{\epsilon })=0$$

$${\mu }_{\epsilon }\in (K_{\epsilon }\ast 1)u_{\epsilon }-K_{\epsilon }\ast u_{\epsilon }+\gamma (u_{\epsilon })+\Pi (u_{\epsilon })$$

$$u_{\epsilon }(0)=u_{0,\epsilon }$$

$${\partial }_{t}u-\Delta \mu +\mathrm{div}(\beta u)=0$$

$$\mu \in -\Delta u+\gamma (u)+\Pi (u)$$

$$u(0)=u_0$$

$$(-\Delta {)}^{-1}:\{v\in (H^1(\Omega ){)}^{\ast }:(v{)}_{\Omega }=0\}\to \{w\in H^1(\Omega ):(w{)}_{\Omega }=0\}$$

$$(w{)}_{\Omega }=0,\text{and}{\int }_{\Omega }\nabla w(x)\cdot \nabla \phi (x)\text{d}x=⟨v,\phi {⟩}_{(H^1(\Omega ){)}^{\ast },H^1(\Omega )}\forall \phi \in H^1(\Omega ).$$

$$D({\mathbf{B}}_{\epsilon }):=\{v\in L^2(\Omega ):(K_{\epsilon }\ast 1)v-(K_{\epsilon }\ast v)\in L^2(\Omega )\},$$

$${\mathbf{B}}_{\epsilon }(v):=(K_{\epsilon }\ast 1)v-(K_{\epsilon }\ast v),\forall v\in D({\mathbf{B}}_{\epsilon }).$$

$${\mathbf{B}}_{\epsilon }(v)(x)={\int }_{\Omega }{\rho }_{\epsilon }(|x-y|)\frac{v(x)-v(y)}{|x-y{|}^2}\text{d}y\text{for a.e. }x\in \Omega .$$

$$\Vert {\mathbf{B}}_{\epsilon }(v){\Vert }_{L^2(\Omega )}\le C_{\epsilon ,\sigma }\Vert v{\Vert }_{C^{0,\sigma }(\overline{\Omega })}\forall v\in C^{0,\sigma }(\overline{\Omega }).$$

$$\Vert {\mathbf{B}}_{\epsilon }(v){\Vert }_{L^2(\Omega )}\le C_{\epsilon ,s}\Vert v{\Vert }_{H^s(\Omega )}\forall v\in H^s(\Omega ).$$

$$|{\mathbf{B}}_{\epsilon }(v)(x)|\le {\int }_{\Omega }{\rho }_{\epsilon }(|x-y|)\frac{|v(x)-v(y)|}{|x-y{|}^2}\text{d}y\le \Vert v{\Vert }_{C^{0,\sigma }(\overline{\Omega })}{\int }_{\Omega }\frac{{\rho }_{\epsilon }(|x-y|)}{|x-y{|}^{2-\sigma }}\text{d}y,$$

$$C_{\epsilon ,\sigma }:={\int }_{\Omega }\frac{{\rho }_{\epsilon }(|x-y|)}{|x-y{|}^{2-\sigma }}\text{d}y<+\infty$$

$$\Vert {\mathbf{B}}_{\epsilon }(v){\Vert }_{(H^1(\Omega ){)}^{\ast }}\le C\Vert \nabla v{\Vert }_{L^2(\Omega )}\forall v\in H^1(\Omega ).$$

$$\underset{n\to \infty }{\lim }⟨{\mathbf{B}}_{{\epsilon }_n}(v),\psi {⟩}_{(H^1(\Omega ){)}^{\ast },H^1(\Omega )}=⟨\mathbf{B}(v),\psi {⟩}_{(H^1(\Omega ){)}^{\ast },H^1(\Omega )}\forall v,\psi \in H^1(\Omega ).$$

Full text

1. 1 Introduction

2. 2 Setting and main results

3. 2.1 Hypotheses

4. 2.2 Notation, preliminaries and comments

5. 2.3 Main results

6. 3 Proof of Theorem 2.1

7. 3.1 Approximation

8. 3.2 Uniform estimates

9. 3.3 Passage to the limit as $\lambda\searrow 0$

10. 3.4 Continuous dependence

11. 4 Proof of Theorem 2.2

12. 5 Proof of Theorems 2.3–2.4

1. Introduction

The Cahn-Hilliard equation was originally introduced in [13] in order to model the so-called “spinodal decomposition” phenomenon occurring during the phase separation processes in binary metallic alloys. Since then it has acquired fundamental importance in several diffuse-interface models in different fields, ranging from physics and engineering to biology.

This nonlinear parabolic PDE exhibits a gradient-flow structure (in the $H^{-1}$-metric) in terms of the free energy functional given by, cf. [13],

[TABLE]

where $\Omega$ is the $d$-dimensional flat torus, $F$ is a double-well potential, and $\tau$ is a small positive parameter related to the thickness of the transition region. The choice of the set $\Omega$ is classical in the literature, and corresponds to imposing periodic boundary conditions. The corresponding evolution problem reads as follows

[TABLE]

where $\mu_{CH}$ is the chemical potential associated to the energy $E_{CH}$, and the symbol ${\rm div}(\cdot)$ denotes the divergence operator. The function $m(\cdot)$ in (1.2) is known as mobility.

The mathematical literature on the classical Cahn-Hilliard equation has been widely developed in the last decades, in terms of well-posedness of the system with possibly degenerate potentials, viscosity terms and dynamic boundary conditions, but also in the direction of regularity, long-time behaviour of solutions, and optimal control problems. Among the extensive literature, we mention the works [14, 15, 16, 18, 19, 21, 37] dealing with existence-uniqueness of solutions, [20, 27, 38] for studies on the asymptotic behaviour of solutions, and [9, 46, 51] for analyses of the system incorporating possibly nonlinear viscosity terms. As far as optimal control problems are concerned, we point out the contributions [17, 22, 23, 28, 40].

In the early $90$’s in [36] G. Giacomin and J. Lebowitz considered the hydrodynamic limit of a microscopic model describing a $d$-dimensional lattice gas evolving via a Poisson nearest-neighbor process. In this seminal paper, the authors rigorously derived a nonlocal energy functional of the form

[TABLE]

where $K(x,y)$ is a positive and symmetric convolution kernel, and proposed the corresponding gradient flow as a model for binary alloys undergoing phase change.

The associated evolution problem, providing a nonlocal variant of the Cahn-Hilliard PDE, is given by the following system of equations:

[TABLE]

where $(K*1)(x):=\int_{\Omega}K(x,y)\textrm{d}y$ and $(K*u)(x):=\int_{\Omega}K(x,y)u(y)\,\textrm{d}y$, for $x\in\Omega$.

The study of such nonlocal Cahn-Hilliard equations has recently been the subject of an intense research activity (see, e.g. [1, 5, 33, 35, 39] and the references therein). All the available results in the literature dealing with nonlocal evolution of phase interfaces require the kernel $K$ to be symmetric and of class $W^{1,1}$. Such requirements are usually met by checking a condition in the following form

[TABLE]

(see [25, Remark 1]).

The interest in this nonlocal model is motivated by its atomistic justification and its generality. A further motivation for the study of models in the form (1.4) is the observation that, at least formally, when the interaction kernel $K$ is of the form $K(x,y)=K(|x-y|)$ and concentrates around the origin, then the behavior of the nonlocal interface evolution problems approaches that of the standard local Cahn-Hilliard equation.

This formal argument is enforced by the rigorous theory involving the variational convergence of nonlocal energies of the form (1.3) to local integral functionals as in (1.1). Building upon the seminal papers by J. Bourgain, H. Brezis, and P. Mironescu [10, 11], and of V. Mazy’a and T. Shaposhnikova [43, 44], a whole nonlocal-to-local framework has been developed for singular nonlocal kernels associated to fractional Sobolev spaces. This study has been complemented by the $\Gamma$-convergence analysis and Poincaré inequalities obtained by A. C. Ponce in [48, 49]. More specifically, considering the following family of convolution kernels, identified by a small positive parameter $\varepsilon$,

[TABLE]

where $(\rho_{\varepsilon})_{\varepsilon}$ is a suitable sequence of mollifiers, A. C. Ponce showed the variational convergence

[TABLE]

The first positive result towards rendering the formal nonlocal-to-local convergence of the Cahn-Hilliard models rigorously has been achieved in [45], where the authors have focused on convergence of weak solutions of the nonlocal Cahn-Hilliard equation (1.4) to weak solutions of its local counterpart (1.2), as the convolution kernel $K$ approximates a Dirac delta centered in the origin. In the aforementioned paper, the convergence is studied in the case of constant mobility, with a non-singular double-well potential satisfying a bounded-concavity assumption of the form

[TABLE]

for a positive constant $B_{1}$ small enough, (see [45, Assumption H3]).

Due to the above-mentioned variational convergence result, kernels in the form (1.6) are the most natural choice in the study of nonlocal phase transition problems. However, in general it is not true that these kernels enjoy a $W^{1,1}$ regularity, so that the available existence results in the literature do not apply. In addition, the usual condition (1.5) is not satisfied by $K_{\varepsilon}$ as in (1.6). This observation renders the analysis of this class of problems very delicate and several nontrivial difficulties arise. For example, the definition and regularity of the chemical potential $\mu_{NL}$ in (1.4) relies on the properties of the linear unbounded operator $({{\bf B}},D({\bf B}))$, defined as

[TABLE]

whose domain $D({\bf B})$ is, a priori, not explicitly characterizable and not even necessarily containing $H^{1}(\Omega)$ (see Subsection 2.2). Such endeavours are further enhanced when turning to the analysis of nonlocal diffusions driven by degenerate potentials.

The first contribution of this paper (see Theorem 2.1) is the development of a well-posedness theory for nonlocal Cahn-Hilliard equations having singular kernels $K_{\varepsilon}$ (for $\varepsilon>0$ being fixed) defined as in (1.6).

In our analysis, we remove the small-concavity assumption on the potential that was required in [45], and include possibly degenerate double-well potentials $F$ defined on bounded domains. Indeed, while the classical choice for $F$ is the fourth-order polynomial $F_{\rm pol}(r):=\frac{1}{4}(r^{2}-1)^{2}$, $r\in\mathbb{R}$, with minima in $\pm 1$ (corresponding to the pure phases), it is well-known that, in view of the physical interpretation of the model, a more realistic description is given by the logarithmic double-well potential

[TABLE]

for $0<\theta<\theta_{c}$ and $c>0$, which by contrast is defined on the bounded domain $(-1,1)$ and possesses minima within the open interval $(-1,1)$. Another interesting example of $F$ which is included in our treatment is the so-called double-obstacle potential (see [7, 47]), having the form

[TABLE]

In this latter case, the derivative $F_{\rm ob}^{\prime}$ is not defined in the usual way, and has to be interpreted as the subdifferential $\partial F_{\rm ob}$ in the sense of convex analysis (see [4]). Analogously the equations defining the chemical potential must be read as a differential inclusion instead.

A further extension provided by our work is to consider a nonlocal Cahn-Hilliard equation augmented by a convection term in divergence form, i.e.

[TABLE]

Here, $\beta=\beta(t,x)$ denotes the velocity field, depending on time and space, which may be acting on the particular system in consideration. As a common choice in the literature, we considered constant mobility equal to one.

The interest in additional convective contributions is connected with applications in mixing and stirring of fluids, as well as in biological realizations of thin films via Langmuir-Blodgett transfer [6, 42]. We mention in this direction the contributions [8, 26, 31, 52] on the local Cahn-Hilliard with convection, [29, 30, 50] dealing with the nonlocal Cahn-Hilliard with local convection, and [32, 41] on the nonlocal case with nonlocal convection. A nonlocal convective Cahn-Hilliard type system modelling phase-separation has been analyzed in [24, 25]. Relevant studies in coupling the Cahn-Hilliard equation with a further equation for the velocity field have been the subject of [2, 3, 12, 34].

From a mathematical viewpoint, the presence of convection terms (i.e. when $\beta\not\equiv 0$) destroys the gradient-flow structure of the equation, causing the analysis to be even more delicate.

The proof strategy for Theorem 2.1 relies on three main ingredients: $(1)$ a suitable approximation of the nonlinearity and an existence analysis for the approximating equations based on a fixed point argument (see Subsection 3.1); $(2)$ the establishment of uniform estimates by ad-hoc multiplication of the equations with suitable test functions (see Subsection 3.2); $(3)$ a passage to the limit relying on nontrivial compactness and monotonicity arguments, falling outside the framework of classical Aubin-Lions embedding results (see Lemma 4 and Subsection 3.3). A delicate point is the proof of a uniform $H^{1}$-estimate, which strongly relies on the choice of periodic boundary conditions.

Our second contribution is established in Theorem 2.2, where we show convergence of solutions for the nonlocal convective Cahn-Hilliard equation with singular kernel to solutions of the associated local one. Our analysis extends the work in [45] to a wider class of double-well potentials, satisfying no bounded-concavity assumptions and being possibly degenerate. The nonlocal-to-local convergence in Theorem 2.2 relies in an essential way on the uniform a-priori estimates established in the proof of Theorem 2.1, and on showing the independence of the identified upper bounds from the non-locality parameter $\varepsilon$.

The third and fourth main results of the paper are a regularity analysis for solutions to (1.4). In particular, in Theorem 2.3 we show that, if the initial datum and the convection velocity satisfy additional integrability and differentiability assumptions, then solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity. In Theorem 2.4 we prove that they also converge to their local counterparts in stronger topologies. The regularity analysis in Theorems 2.3 and 2.4 is the byproduct of a time-differentiation of the nonlocal Cahn-Hilliard equation, and of the use of higher-order-in-time test functions.

The paper is organized as follows. Section 2 contains a description of the mathematical setting of the paper, the definition of weak solutions for the nonlocal and local convective Cahn-Hilliard equations, and the precise statements of the four main results. Sections 3 and 4 are devoted to the proof of Theorems 2.1 and 2.2, respectively. Eventually, in Section 5 we prove Theorems 2.3 and 2.4.

2. Setting and main results

2.1. Hypotheses

Throughout the paper we will assume the following:

**H1: **

$\Omega$ is the $d$-dimensional $(d=2,3)$ flat torus and $T>0$ is a fixed final time.

**H2: **

The kernel $K_{\varepsilon}:\Omega\times\Omega\to\mathbb{R}$ is defined as in (1.6):

[TABLE]

where $(\rho_{\varepsilon})_{\varepsilon>0}\subset L^{1}_{\rm{loc}}(0,+\infty)$ is a family of radial mollifiers on $\mathbb{R}$, satisfying

[TABLE]

with $M_{d}:=\int_{S^{d-1}}|e_{1}\cdot\sigma|^{2}\,d\mathcal{H}^{d-1}(\sigma)$.

**H3: **

$\gamma:\mathbb{R}\to 2^{\mathbb{R}}$ is a maximal monotone graph such that $0\in\gamma(0)$. This implies that $\gamma=\partial\hat{\gamma}$, where $\hat{\gamma}:\mathbb{R}\to[0,+\infty]$ is a proper, convex and lower semicontinuous function. The map $\Pi:\mathbb{R}\to\mathbb{R}$ is a Lipschitz-continuous function with Lipschitz constant $C_{\Pi}>0$. The double-well potential $F$ will be represented by $\hat{\gamma}+\hat{\Pi}$, where $\hat{\Pi}(t):=\int_{0}^{t}\Pi(r)\,\textrm{d}r\text{ for every }t\in\mathbb{R}$. Without restriction we will assume that $F$ is nonnegative.

**H4: **

The velocity $\beta$ depends on space and time, and satisfies $\beta\in L^{2}(0,T;L^{\infty}(\Omega;\mathbb{R}^{d}))$.

We point out that all assumptions collected in H2 correspond to the requirements in [49, 48].

For every $\varepsilon>0$, we consider the nonlocal Cahn-Hilliard equation with local convection

[TABLE]

and its local counterpart

[TABLE]

2.2. Notation, preliminaries and comments

In the sequel we will identify $L^{2}(\Omega)$ with its dual, so that $(H^{1}(\Omega),L^{2}(\Omega),(H^{1}(\Omega))^{*})$ will be a classical Hilbert triplet. We will use the symbol $(v)_{\Omega}$ for $\frac{1}{|\Omega|}\langle v,1\rangle_{(H^{1}(\Omega))^{*},H^{1}(\Omega)}$ for every $v\in(H^{1}(\Omega))^{*}$. Note that for $v\in L^{2}(\Omega)$, $(v)_{\Omega}$ coincides with the usual average. We recall that the operator

[TABLE]

is defined as the map assigning to every $v\in(H^{1}(\Omega))^{*}$ with null mean the unique element $w\in H^{1}(\Omega)$ such that

[TABLE]

It is well known that $(-\Delta)^{-1}$ is a linear isomorphism.

In this paper $C$ indicates a generic positive constant, possibly varying from line to line, depending only on the setting H1–H4. The dependence of constants on a specific parameter will be indicated explicitly through a subscript.

We collect here some useful properties of the nonlocal term. We define the operator $({{\bf B}_{\varepsilon}},D({\bf B}_{\varepsilon}))$ on $L^{2}(\Omega)$ in the following way:

[TABLE]

It is clear that ${\bf B}_{\varepsilon}$ is a linear unbounded operator in $L^{2}(\Omega)$, and that for every $v\in D({\bf B}_{\varepsilon})$ we have the representation

[TABLE]

We point out that the domain $D({\bf B}_{\varepsilon})$ is non-trivial. More specifically, we have the following result.

Lemma 1**.**

For every $\varepsilon,\sigma>0$, there holds $C^{0,\sigma}(\overline{\Omega})\subset D({\bf B}_{\varepsilon})$. Additionally, there exists a constant $C_{\varepsilon,\sigma}>0$ (only dependent on $\varepsilon$ and $\sigma$) such that

[TABLE]

In particular, for every $s>\frac{3}{2}$, $H^{s}(\Omega)\subset D({\bf B}_{\varepsilon})$ and there exists $C_{\varepsilon,s}>0$ such that

[TABLE]

Proof.

A direct computation shows that for every $v\in C^{0,\sigma}(\overline{\Omega})$ and for almost every $x\in\Omega$,

[TABLE]

where

[TABLE]

thanks to H2. The second part of the Lemma follows by the Sobolev embedding $H^{s}(\Omega)\hookrightarrow C^{0,\sigma}(\overline{\Omega})$ for every $s>\frac{3}{2}$ and $\sigma\in(0,s-\frac{3}{2})$. ∎

The operator $({\bf B}_{\varepsilon},D({\bf B_{\varepsilon}}))$ has been defined as a linear unbounded operator on $L^{2}(\Omega)$. Note that it is not necessarily true that $H^{1}(\Omega)\subset D({\bf B_{\varepsilon}})$. Nevertheless, we now show that actually $({\bf B}_{\varepsilon},D({\bf B_{\varepsilon}}))$ can be extended, uniformly in $\varepsilon$, to a linear bounded operator from $H^{1}(\Omega)$ to its dual.

Lemma 2**.**

For every $\varepsilon>0$ the operator $(D({\bf B_{\varepsilon}}),{\bf B}_{\varepsilon})$ can be uniquely extended to a linear continuous operator ${\bf B}_{\varepsilon}:H^{1}(\Omega)\to(H^{1}(\Omega))^{*}$. Additionally, there exists a positive constant $C$, independent of $\varepsilon$, such that

[TABLE]

In particular, the family $({\bf B}_{\varepsilon})_{\varepsilon}$ is uniformly bounded in $\mathscr{L}(H^{1}(\Omega),(H^{1}(\Omega))^{*})$ and there exists ${\bf B}\in\mathscr{L}(H^{1}(\Omega),(H^{1}(\Omega))^{*})$ and an infinitesimal sequence $(\varepsilon_{n})_{n}$ such that

[TABLE]

Proof.

By the Hölder inequality and [10, Theorem 1], we infer that

[TABLE]

for every $v,\psi\in H^{1}(\Omega)$. This implies that $(D({\bf B_{\varepsilon}}),{\bf B}_{\varepsilon})$ can be extended uniquely as required (the uniqueness follows Lemma 1, and from the density of $C^{0,\sigma}(\overline{\Omega})$ in $D({\bf B_{\varepsilon}})$). The second part of the lemma follows by observing that (2.9) implies the uniform boundedness of $({\bf B}_{\varepsilon})_{\varepsilon}$ in $\mathscr{L}(H^{1}(\Omega),(H^{1}(\Omega))^{*})$, and hence its precompactness in the weak operator topology of $\mathscr{L}(H^{1}(\Omega),(H^{1}(\Omega))^{*})$. ∎

In what follows, a crucial role is also played by the nonlocal energy contribution

[TABLE]

Owing to [10, Theorem 1], we have that $E_{\varepsilon}$ is well-defined, convex, and its differential $DE_{\varepsilon}:H^{1}(\Omega)\to(H^{1}(\Omega))^{*}$ is given by

[TABLE]

Moreover, by [10] the asymptotic behavior of $E_{\varepsilon}$ as $\varepsilon\to 0^{+}$ can be characterized as follows

[TABLE]

As a corollary, we deduce the following identification of the operator ${\bf B}$ in Lemma 3.

Lemma 3**.**

Let $(D(\bf B_{\varepsilon}),{\bf B}_{\varepsilon})_{\varepsilon}$ and ${\bf B}$ be as in Lemma 2. Then,

[TABLE]

where

[TABLE]

Proof.

By the characterization of the differential of $E_{\varepsilon}$, we have that

[TABLE]

for every $v_{1},v_{2}\in H^{1}(\Omega)$. Hence, for every subsequence $(\varepsilon_{n})_{n}$ as in Lemma 2, letting $n\to\infty$, by (2.10) we conclude that

[TABLE]

from which ${\bf{B}}=-\Delta$. In particular, this implies that the convergence holds along the entire sequence $\varepsilon$. ∎

We conclude this subsection with a lemma providing two fundamental compactness inequalities involving the family of operators $({\bf B}_{\varepsilon})_{\varepsilon}$. Such results are nontrivial, since they do not fall in the classical framework of the Aubin-Lions lemmas. The next lemma is a uniform counterpart to [45, Lemma 1].

Lemma 4**.**

For every $\delta>0$ there exist constants $C_{\delta}>0$ and $\varepsilon_{\delta}>0$ with the following properties:

1. (1)

For every sequence $(f_{\varepsilon})_{\varepsilon}\subset H^{1}(\Omega)$ there holds

[TABLE]

for every $0<\varepsilon_{1},\varepsilon_{2}<\varepsilon_{\delta}$. 2. (2)

For every sequence $(f_{\varepsilon})_{\varepsilon}\subset L^{2}(\Omega)$ there holds

[TABLE]

for every $0<\varepsilon_{1},\varepsilon_{2}<\varepsilon_{\delta}$.

Proof.

Assume by contradiction that (2.11) is false. Then, there exists $\bar{\delta}>0$ having the following property: for every $n\in\mathbb{N}$ we can find a sequence $(f_{\varepsilon}^{n})_{\varepsilon}\subset H^{1}(\Omega)$ and two parameters $\varepsilon_{n}^{1},\varepsilon_{n}^{2}<\frac{1}{n}$ such that

[TABLE]

Noting that $\|f_{\varepsilon_{n}^{1}}^{n}-f^{n}_{\varepsilon_{n}^{2}}\|_{H^{1}(\Omega)}>0$ for every $n$ and setting

[TABLE]

we have

[TABLE]

Hence, $g_{n}^{1}-g_{n}^{2}\to 0$ strongly in $L^{2}(\Omega)$ and the families $(\nabla g_{n}^{1})_{n}$ and $(\nabla g_{n}^{2})_{n}$ are relatively strongly compact in $L^{2}(\Omega;\mathbb{R}^{d})$ by [48, Theorem 1.2]. We deduce that $g_{n}^{1}-g_{n}^{2}\to 0$ strongly in $H^{1}(\Omega)$, but this is a contradiction since by definition we have $\|g_{n}^{1}-g_{n}^{2}\|_{H^{1}(\Omega)}=1$ for all $n$. The argument for (2.12) is entirely analogous. ∎

2.3. Main results

Before stating our main results, let us recall the notion of weak solutions to both the nonlocal as well as the local Cahn-Hilliard equation with local convection.

Definition 1** (Solution to the nonlocal Cahn-Hilliard equation).**

Let $\varepsilon>0$ and $T>0$ be fixed. A solution to the nonlocal Cahn-Hilliard equation (2.1)–(2.3) on $[0,T]$, and associated with the initial datum $u_{0,\varepsilon}\in L^{2}(\Omega)$, is a triplet $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ with the following properties

[TABLE]

satisfying $u_{\varepsilon}(0)=u_{0,\varepsilon}$, and such that

[TABLE]

for all $\varphi\in H^{1}(\Omega)$, and for almost every $t\in(0,T)$.

Definition 2** (Solution to the local Cahn-Hilliard equation).**

Let $T>0$ be fixed. A solution to the local Cahn-Hilliard equation (2.4)–(2.6) on $[0,T]$, and associated with the initial datum $u_{0}\in H^{1}(\Omega)$, is a triplet $(u,\mu,\xi)$ with the following properties

[TABLE]

satisfying $u(0)=u_{0}$, and such that

[TABLE]

for all $\varphi\in H^{1}(\Omega)$, for almost every $t\in(0,T)$.

Our first result is the well-posedness of solutions to the nonlocal Cahn-Hilliard equation.

Theorem 2.1**.**

Let assumptions H1–H4 be satisfied, and for every $\varepsilon>0$ let

[TABLE]

Then, there exists $\varepsilon_{0}>0$ having the following property: for every $\varepsilon<\varepsilon_{0}$ there exists a unique solution $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ to (2.1)–(2.3) associated with the initial datum $u_{0,\varepsilon}$, according to Definition 1. Furthermore, if $(\beta^{1},u_{0,\varepsilon}^{1})$ and $(\beta^{2},u_{0,\varepsilon}^{2})$ are two sets of data satisfying H4 and (2.15), with $(u_{0,\varepsilon}^{1})_{\Omega}=(u_{0,\varepsilon}^{2})_{\Omega}$, then there exists a positive constant $M_{\varepsilon}$, depending only on the setting H1–H3 and on the norms of the data $(\beta^{1},u_{0,\varepsilon}^{1})$ and $(\beta^{2},u_{0,\varepsilon}^{2})$ appearing in H4 and (2.15), such that, for any respective solution $(u_{\varepsilon}^{1},\mu_{\varepsilon}^{1},\xi_{\varepsilon}^{1})$ and $(u_{\varepsilon}^{2},\mu_{\varepsilon}^{2},\xi_{\varepsilon}^{2})$ to the nonlocal equation (2.1)–(2.3),

[TABLE]

The second result concerns nonlocal-to-local convergence.

Theorem 2.2**.**

Let assumptions H1–H4 be satisfied. Let $u_{0}\in H^{1}(\Omega)$, and for every $\varepsilon>0$ let $u_{0,\varepsilon}$ satisfy (2.15) and be such that

[TABLE]

Let $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ be the unique solution to (2.1)–(2.3) associated to $u_{0,\varepsilon}$ given by Theorem 2.1, and let $(u,\mu,\xi)$ be the unique solution to the local equation (2.4)–(2.6) associated to $u_{0}$, according to Definition 2.

Then, as $\varepsilon\searrow 0$,

[TABLE]

The last two results that we present deal with regularity of solutions to the nonlocal equation. In particular, we show that if the data are more regular, then the solution to the nonlocal equation inherits a further regularity, and the convergences to the local equation are obtained in stronger topologies.

Theorem 2.3**.**

Let assumptions H1–H4 be satisfied, and suppose also that

[TABLE]

For every $0<\varepsilon<\varepsilon_{0}$ let $u_{0,\varepsilon}$ satisfy (2.15) and

[TABLE]

Then the unique solution $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ to the nonlocal equation (2.1)–(2.3) with respect to the initial datum $u_{0,\varepsilon}$ also satisfies

[TABLE]

If also

[TABLE]

then in addition

[TABLE]

If also

[TABLE]

then in addition

[TABLE]

Theorem 2.4**.**

Let assumptions H1–H4 be satisfied. Let $u_{0}\in H^{1}(\Omega)$, and for every $\varepsilon>0$ let $u_{0,\varepsilon}$ satisfy (2.15), (2.16)–(2.18), (2.20) and

[TABLE]

Denoting by $(u,\mu,\xi)$ the unique solution to the local equation (2.4)–(2.6), if (2.19) holds then, in addition to the convergences in Theorem 2.2,

[TABLE]

If also (2.21) holds, then

[TABLE]

If also (2.22) holds, then

[TABLE]

3. Proof of Theorem 2.1

This section contains the proof of existence of a solution $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ to the nonlocal convective Cahn-Hilliard equation. We subdivide it in different steps. In this section, $\varepsilon>0$ is fixed.

3.1. Approximation

For every $\lambda>0$, let $\gamma_{\lambda}:\mathbb{R}\to\mathbb{R}$ be the Yosida approximation of $\gamma$, having Lipschitz constant $1/\lambda$, and set $\hat{\gamma}_{\lambda}(s):=\int_{0}^{s}\gamma_{\lambda}(r)\,\textrm{d}r$ for every $s\in\mathbb{R}$. We consider the approximated problem

[TABLE]

where $\beta_{\lambda}:=P_{\lambda}\beta$, $P_{\lambda}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is the projection on the closed ball of radius $\frac{1}{\lambda}$, and the initial datum $u_{0,\varepsilon}^{\lambda}$ satisfies

[TABLE]

for a certain $\lambda_{0}>0$ (possibly depending on $\varepsilon$).

Remark 3.1**.**

The existence of an approximating sequence $(u_{0,\varepsilon}^{\lambda})_{\lambda}$ satisfying (3.4)–(3.5) is guaranteed by (2.15). For example, let us consider the classical elliptic regularization given by the unique solution to the problem

[TABLE]

Note that we have not specified any boundary conditions for $u_{0,\varepsilon}^{\lambda}$ as we are working on the torus $\Omega$ (hence we have implicitly required periodic boundary conditions for $u_{0,\varepsilon}^{\lambda}$). Let us show that (3.4)–(3.5) are satisfied by this choice. Testing (3.6) by $u_{0,\varepsilon}^{\lambda}$ and using the Young inequality on the right-hand side we obtain

[TABLE]

This readily implies (3.4) and the first bound in (3.5). Moreover, testing (3.6) by $\gamma_{\lambda}(u_{0,\varepsilon}^{\lambda})$ we get

[TABLE]

Denoting by $\hat{\gamma}_{\lambda}^{*}$ the convex conjugate of $\hat{\gamma}_{\lambda}$, the first term on the left-hand side reads as

[TABLE]

the second term on the left-hand side is nonnegative by the monotonicity of $\gamma_{\lambda}$, while the right-hand side can be bounded through the Young inequality as

[TABLE]

Rearranging the terms we get $\|\hat{\gamma}_{\lambda}(u_{0,\varepsilon}^{\lambda})\|_{L^{1}(\Omega)}\leq\|\hat{\gamma}(u_{0,\varepsilon})\|_{L^{1}(\Omega)}$, from which the second bound in (3.5). Finally, testing (3.6) by ${\bf B}_{\varepsilon}(u_{0,\varepsilon}^{\lambda})$ we have

[TABLE]

where, thanks to the periodic boundary conditions, on the left-hand side we have

[TABLE]

and

[TABLE]

On the right-hand side, by the Hölder and Young inequalities, we have

[TABLE]

Rearranging the terms we get $E_{\varepsilon}(u_{0,\varepsilon}^{\lambda})\leq E_{\varepsilon}(u_{0,\varepsilon})$, from which the third bound in (3.5).

In this subsection, we show existence of an approximated solution $(u_{\varepsilon}^{\lambda},\mu_{\varepsilon}^{\lambda})$ for every $\lambda>0$ fixed. The proof strategy relies on the use of a fixed-point argument.

For every $w\in L^{2}(0,T;H^{s}(\Omega))$ with $s\in\left(\frac{3}{2},2\right)$, Lemma 1 ensures that

[TABLE]

so that we can study the auxiliary problem

[TABLE]

which can be seen as a local convective viscous Cahn-Hilliard equation with an additional source term in the definition of the chemical potential. It is well-known (see [26] for example) that such problem admits a unique weak solution $(v,\mu_{v})$ with

[TABLE]

satisfying (3.7)–(3.9) for example in the sense of distributions. Hence, the map

[TABLE]

associating to every $w\in L^{2}(0,T;H^{s}(\Omega))$ the solution $v$ to (3.7)–(3.9) is well-defined. We proceed by showing that $\Gamma_{\varepsilon}^{\lambda}$ has also some continuity properties. For $i=1,2$ let $w_{i}\in L^{2}(0,T;H^{s}(\Omega))$, and set $v_{i}:=\Gamma_{\varepsilon}^{\lambda}(w_{i})$. Then taking the difference of the corresponding equations (3.7) and (3.8) for $i=1,2$, we obtain

[TABLE]

Noting that $(v_{1}-v_{2})_{\Omega}=0$ by integrating (3.10), testing (3.10) by $(-\Delta)^{-1}(v_{1}-v_{2})$, equation (3.11) by $v_{1}-v_{2}$, and taking the difference, estimate (2.9) and assumption H4 yield

[TABLE]

for every $t\in[0,T]$.

Testing (3.10) by $v_{1}-v_{2}$, equation (3.11) by $-\Delta(v_{1}-v_{2})$, taking the difference, and using Lemma 1, a similar argument yields

[TABLE]

To handle the last term on the right-hand side we use the following compactness result: since $H^{2}(\Omega)\hookrightarrow H^{1}(\Omega)$ compactly, for every $\eta>0$ there is $C_{\eta}>0$ such that

[TABLE]

Hence, summing the two inequalities, using the Lipschitz-continuity of $\gamma_{\lambda}$ and $\Pi$, choosing $\eta>0$ sufficiently small, and applying the Gronwall’s Lemma, we deduce that there exists a positive constant $C_{\varepsilon,\lambda}$ such that

[TABLE]

In particular, $\Gamma_{\varepsilon}^{\lambda}$ is continuous from $L^{2}(0,T;H^{s}(\Omega))$ to $L^{2}(0,T;H^{s}(\Omega))$.

Fix $T_{0}>0$. By repeating the argument leading to (3.13) we deduce the estimate

[TABLE]

for every $w\in L^{2}(0,T_{0};H^{s}(\Omega))$, and $v=\Gamma_{\varepsilon}^{\lambda}(w)$. Now, since $s\in(\frac{3}{2},2)$, if $\vartheta\in(0,1)$ is such that

[TABLE]

by interpolation we get that

[TABLE]

which in turn yields that

[TABLE]

Consequently, we have that

[TABLE]

where $\frac{4}{s}>2$ since $s<2$. Hence, we infer that

[TABLE]

and we can choose $T_{0}$ sufficiently small such that $T_{0}^{\frac{1}{2}-\frac{s}{4}}C_{\varepsilon,\lambda,s}<1$. Thus,

[TABLE]

Banach fixed point theorem ensures the existence of a unique weak solution $(u_{\varepsilon}^{\lambda},\mu_{\varepsilon}^{\lambda})$ to the approximated problem (3.1)-(3.3) in $(0,T_{0})\times\Omega$, with

[TABLE]

Note that the choice of $T_{0}$ is independent of the initial time. Moreover, since $u_{\varepsilon}^{\lambda}\in H^{1}(0,T_{0};L^{2}(\Omega))\cap L^{\infty}(0,T_{0};H^{1}(\Omega))$, then $u_{\varepsilon}^{\lambda}$ is weakly continuous with values in $T_{0}$: this allows us to obtain the pointwise regularity $u_{\varepsilon}^{\lambda}(T_{0})\in H^{1}(\Omega)$. Such regularity is then enough to extend the solution to the next subinterval $[T_{0},2T_{0}]$ (see [26]): using a standard patching argument in time allows to extend the solution to the whole interval $[0,T]$.

3.2. Uniform estimates

In this subsection we show that there exists $\varepsilon_{0}>0$ independent of $\lambda$, and such that for $\varepsilon<\varepsilon_{0}$ the approximated solutions fulfill some uniform estimates independently of $\lambda$ and $\varepsilon$. In what follows we will always assume that $\lambda\in[0,1]$.

Step 1. We start by fixing $t\in[0,T]$, testing (3.1) with $\mu_{\varepsilon}^{\lambda}$, (3.2) with $\partial_{t}u_{\varepsilon}^{\lambda}$, taking the difference, and integrating the resulting equation on $(0,t)$. We obtain

[TABLE]

Using assumption H3, the uniform bound (3.5) and as well as Young’s inequality, we get

[TABLE]

for every $t\in[0,T]$.

We point out that, due to the periodic boundary conditions, and the fact that $\Omega$ is the $d$-dimensional torus, we formally have

[TABLE]

for almost every $s\in[0,T]$. Testing (3.1) with $u_{\varepsilon}^{\lambda}$ and (3.2) with $-\Delta u_{\varepsilon}^{\lambda}$, by considering the difference between the two resulting equation and by integrating in the time interval $(0,t)$, from H3 we deduce the estimate

[TABLE]

where the latter inequality holds for $\varepsilon$ smaller than a suitable constant $\varepsilon_{0}$ in view of Lemma 4. Noticing that the third term in the left-hand side of the above estimate is positive owing to the monotonicity of $\gamma_{\lambda}$, by [48, Theorem 1.1] we infer the bound

[TABLE]

By the Hölder inequality we deduce the estimate

[TABLE]

Thus, summing (3.14), (3.15), and (3.16), recalling H4 we obtain

[TABLE]

Recalling assumption H4 and applying Gronwall’s lemma, from the arbitrariness of $t\in[0,T]$ we deduce that there exists a constant $C_{\varepsilon}$ such that

[TABLE]

Testing equation (3.1) with a function $\varphi\in L^{2}(0,T;H^{1}(\Omega))$, integrating in time, and using (3.18)–(3.20) gives

[TABLE]

Step 2. In order to obtain an $L^{2}(0,T;H^{1}(\Omega))$-estimate on the chemical potential $\mu_{\varepsilon}^{\lambda}$, we need a bound on the $L^{2}(0,T)$-norm of the spatial mean of $\mu_{\varepsilon}^{\lambda}$. Thanks to the symmetry of the kernel $K$, the mean of the convolution terms vanishes, i.e.

[TABLE]

Since also $(\Delta u_{\varepsilon}^{\lambda})_{\Omega}=0$, owing to (3.19) and the Lipschitz continuity of $\Pi$, we get

[TABLE]

Hence $\{(\mu_{\varepsilon}^{\lambda})_{\Omega}\}_{\varepsilon}$ is uniformly bounded in $L^{2}(0,T)$ if $\{\gamma_{\lambda}(u_{\varepsilon}^{\lambda})\}_{\varepsilon}$ is uniformly bounded in $L^{2}(0,T;L^{1}(\Omega))$. We test (3.1) by $(-\Delta)^{-1}(u_{\varepsilon}-(u_{0,\varepsilon}^{\lambda})_{\Omega})$ and (3.2) by $u_{\varepsilon}-(u_{0,\varepsilon}^{\lambda})_{\Omega}$, obtaining

[TABLE]

We proceed by estimating each integral in the left-hand side of the above equation separately.

It is readily seen that $I_{1}+I_{2}$ is uniformly bounded in $L^{2}(0,T)$ due to (3.19), (3.21) and (3.5).

Regarding $I_{3}$, since $({\bf B}_{\varepsilon}(u_{\varepsilon}^{\lambda}))_{\Omega}=0$ we have that

[TABLE]

which is clearly bounded in $L^{2}(0,T)$ by (3.20).

To estimate $I_{4}$ we observe that in view of (2.15) and (3.4) there exist constants $M_{1},M_{2}>0$ depending only on the position of $(u_{0,\varepsilon})_{\Omega}$ in $\operatorname{Int}D(\gamma)$, such that

[TABLE]

cf. for example [21, p. 984] and the references within, while

[TABLE]

is bounded in $L^{\infty}(0,T)$ thanks to (3.19).

Eventually, $I_{5}$ can be estimated as follows

[TABLE]

where the right-hand side is bounded due to H4 and (3.19).

Combining this information, we conclude by difference that $\{\gamma_{\lambda}(u_{\varepsilon}^{\lambda})\}$ is uniformly bounded in $L^{2}(0,T;L^{1}(\Omega))$. Thus, from (3.18) and (3.22) we infer that

[TABLE]

Step 3. We proceed by proving that $\{\gamma_{\lambda}(u_{\varepsilon}^{\lambda})\}$ is uniformly bounded in $L^{2}(0,T;L^{2}(\Omega))$.

We test (3.2) with $\gamma_{\lambda}(u_{\varepsilon}^{\lambda})$. This gives

[TABLE]

We observe that the second term on the left-hand side is nonnegative owing to the monotonicity of $\gamma_{\lambda}$. Analogously, the third term on the left-hand side can be rewritten as

[TABLE]

which is also nonnegative due to the monotonicity of $\gamma_{\lambda}$. Applying Young’s inequality we deduce the bound

[TABLE]

which, together with H3, (3.19) and (3.23), implies the following estimate

[TABLE]

3.3. Passage to the limit as $\lambda\searrow 0$

We perform here the passage to the limit as $\lambda\searrow 0$, with $0<\varepsilon<\varepsilon_{0}$ still fixed. In view of the uniform bounds identified in Section 3.2 and the Aubin-Lions lemma, up to the extraction of (not relabeled) subsequences we have the following convergences:

[TABLE]

for some

[TABLE]

The strong convergence (3.25), the weak convergence (3.30) and the strong-weak closure of the maximal monotone graph $\gamma$ readily implies that $\xi_{\varepsilon}\in\gamma(u_{\varepsilon})$ almost everywhere in $(0,T)\times\Omega$. The Lipschitz continuity of $\Pi$ yields also

[TABLE]

Furthermore, for every $\varphi\in L^{2}(0,T;H^{2}(\Omega))$ by the triangle inequality we have that

[TABLE]

By the Hölder inequality, the fact that $\beta_{\lambda}\to\beta$ strongly in $L^{2}(0,T;L^{3}(\Omega))$ and the embedding $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$, for the first term on the right-hand side we have

[TABLE]

For the second term on the right-hand side note that $\beta\cdot\nabla\varphi\in L^{1}(0,T;L^{2}(\Omega))$ thanks to assumption H4, the fact that $\varphi\in L^{2}(0,T;H^{2}(\Omega))$ and the inclusion $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$, so that from (3.26)

[TABLE]

Hence, we conclude that

[TABLE]

From (3.26) and the fact that ${\bf B}_{\varepsilon}\in\mathscr{L}(H^{1}(\Omega),(H^{1}(\Omega))^{*})$, it is readily seen that

[TABLE]

By (3.29)–(3.30) and (3.31), by comparison it follows that the sequence $({\bf B}_{\varepsilon}(u_{\varepsilon}^{\lambda}))_{\lambda}$ is bounded in $L^{2}(0,T;L^{2}(\Omega))$, hence we also conclude that ${\bf B}_{\varepsilon}(u_{\varepsilon})\in L^{2}(0,T;L^{2}(\Omega))$

[TABLE]

Now, passing to the limit in (3.1)–(3.2) as $\lambda\searrow 0$, we obtain, in the sense of distributions,

[TABLE]

and

[TABLE]

Finally, the strong convergence (3.25) implies also that $u_{\varepsilon}(0)=u_{0,\varepsilon}$, so that $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ is a solution to the nonlocal Cahn-Hilliard equation (1.7) according to Definition 1. This completes the proof of the first assertion of Theorem 2.1.

3.4. Continuous dependence

Let $(\beta^{1},u_{0,\varepsilon}^{1})$ and $(\beta^{2},u_{0,\varepsilon}^{2})$ satisfy H4 and (2.15), with $(u_{0,\varepsilon}^{1})_{\Omega}=(u_{0,\varepsilon}^{2})_{\Omega}$, and let $(u_{\varepsilon}^{1},\mu_{\varepsilon}^{1},\xi_{\varepsilon}^{1})$ and $(u_{\varepsilon}^{2},\mu_{\varepsilon}^{2},\xi_{\varepsilon}^{2})$ be any corresponding solutions to the nonlocal equation (2.1)–(2.3). Then we have

[TABLE]

Noting that $(u_{\varepsilon}^{1}-u_{\varepsilon}^{2})_{\Omega}=0$ by the assumption on the initial data, we test the first equation by $(-\Delta)^{-1}(u_{\varepsilon}^{1}-u_{\varepsilon}^{2})$, the second by $u_{\varepsilon}^{1}-u_{\varepsilon}^{2}$, and take the difference: by performing classical computations we get

[TABLE]

By the Lipschitz-continuity of $\Pi$ we have

[TABLE]

while the Hölder and Young inequalities yield

[TABLE]

and

[TABLE]

The continuous-dependence property stated in Theorem 2.1 follows from Lemma 4 and the Gronwall lemma.

4. Proof of Theorem 2.2

In this section we perform the limit as $\varepsilon\searrow 0$.

First of all, going back to the arguments performed in the previous section to obtain estimates (3.18)–(3.24), we observe that assumptions (2.16)–(2.17) guarantee that the sequence of constants $(C_{\varepsilon})_{\varepsilon}$ is uniformly bounded for every $\varepsilon\in(0,\varepsilon_{0})$. Consequently, we deduce that there exists $C>0$ such that

[TABLE]

Hence, by comparison

[TABLE]

By Aubin-Lions compactness results we infer that, up to the extraction of (not relabeled) subsequences,

[TABLE]

for some

[TABLE]

We proceed by showing in addition that

[TABLE]

Indeed, Lemma 4 implies that for every $\delta>0$, there exist $C_{\delta}>0$ and $\varepsilon_{\delta}>0$ such that

[TABLE]

for every $0<\varepsilon<\varepsilon_{\delta}$. Thanks to (4.1), we infer that

[TABLE]

for a constant $C>0$. Similarly, using the second inequality in Lemma 4 and (4.1), the same argument ensures also that

[TABLE]

The strong convergence (4.8) follows then from the arbitrariness of $\delta$, and from (4.2).

From the strong convergence of $(u_{\varepsilon})_{\varepsilon}$ and the strong-weak closure of maximal monotone graphs it is readily seen that $\xi\in\gamma(u)$ and that

[TABLE]

Let us now identify the limit $\eta$ as $-\Delta u$. As $DE_{\varepsilon}={\bf B}_{\varepsilon}$, we have that

[TABLE]

for all $z_{1},z_{2}\in H^{1}(\Omega)$. Hence, for all $z\in L^{2}(0,T;H^{1}(\Omega))$ we deduce that

[TABLE]

The results in [10] and the dominated convergence theorem yield

[TABLE]

Owing to the convergences (4.8) and (4.5), we have that

[TABLE]

Finally, following the exact same steps as in [45], there holds

[TABLE]

Hence, letting $\varepsilon\to 0$ in (4.10), we obtain the inequality

[TABLE]

for every $z\in L^{2}(0,T;H^{1}(\Omega))$, so that $\eta=-\Delta u\in L^{2}(0,T;L^{2}(\Omega))$. By elliptic regularity we infer that $u\in L^{2}(0,T;H^{2}(\Omega))$.

Finally, Hölder’s inequality, the Sobolev embedding $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$, and the strong convergence (4.8) yield

[TABLE]

Thus, letting $\varepsilon\searrow 0$ in Definition 1 (of solution for the nonlocal Cahn-Hilliard) we obtain

[TABLE]

in the sense of distributions, as well as

[TABLE]

This implies that $u$ is a solution to the local Cahn-Hilliard equation (2.4)–(2.6), and concludes the proof of Theorem 2.2.

5. Proof of Theorems 2.3–2.4

We show first that under the additional assumption (2.20), the solution $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ to the nonlocal equation is more regular. Note that here $\varepsilon\in(0,\varepsilon_{0})$ is fixed.

The idea is to argue in a classical way, performing some additional estimates on the approximate solutions $(u_{\varepsilon}^{\lambda},\mu_{\varepsilon}^{\lambda})$ constructed in Section 3. To this end, note that by (2.20), the approximating sequence $(u_{0,\varepsilon}^{\lambda})_{\lambda}$ of initial data satisfying (3.4)–(3.5) can be chosen with the additional property

[TABLE]

First of all we need some preparatory work. Note that the elliptic problem corresponding to (3.1)–(3.3) at time [math], i.e.

[TABLE]

admits a unique solution $(u_{0,\varepsilon,\lambda}^{\prime},\mu_{0,\varepsilon,\lambda})\in(H^{1}(\Omega))^{*}\times H^{1}(\Omega)$. Testing the first equation by $\mu_{0,\varepsilon,\lambda}$, the second by $u_{0,\varepsilon,\lambda}^{\prime}$ and taking the difference yields

[TABLE]

From the first equation it is readily seen that

[TABLE]

with

[TABLE]

Hence the Young inequality, (2.19), (3.4)–(3.5), and (5.1) imply that

[TABLE]

We are now ready to perform the additional estimate on the approximated solutions. Again, we proceed formally in order to avoid heavy notations and since everything can be proved rigorously through a further regularization on the problem. The idea is to (formally) test the time derivative of (3.1) by $(-\Delta)^{-1}(\partial_{t}u_{\varepsilon}^{\lambda})$, the time derivative of (3.2) by $\partial_{t}u_{\varepsilon}^{\lambda}$ and then to take the difference: the resulting inequality is

[TABLE]

Now, note that by Hölder’s inequality and (2.19) we have

[TABLE]

and

[TABLE]

Thanks to Lemma 4 there holds

[TABLE]

for $\delta$ sufficiently small. Hence, putting this information together, using the Lipschitz-continuity of $\Pi$, the monotonicity of $\gamma_{\lambda}$, condition (5.2) and the already proved estimates (3.19) and (3.21), we are left with

[TABLE]

Since $s\mapsto\|\beta(s,\cdot)\|_{L^{\infty}(\Omega)}^{2}$ and $s\mapsto\|\partial_{t}\beta(s,\cdot)\|_{L^{3}(\Omega)}^{2}$ belong to $L^{1}(0,T)$ due to (2.19) and H4, using the Gronwall lemma and recalling [49, Theorem 1.1] we infer that

[TABLE]

Now, if (2.21) holds, we also have

[TABLE]

yielding by (3.19) and by comparison in (3.1),

[TABLE]

At this point, going back to the proof of Theorem 2.1, we repeat exactly the same arguments of Step 2 and Step 3: using the additional estimates (5.3)–(5.4), we deduce

[TABLE]

Furthermore, if also (2.22) holds we have

[TABLE]

so that from (3.21) and by comparison in (3.1) we infer that

[TABLE]

Hence, (5.3)–(5.6) ensure that the limit solution $(u_{\varepsilon},\mu_{\varepsilon},\xi_{\varepsilon})$ inherits the additional regularity stated in Theorem 2.3.

The proof of Theorem 2.4 follows now as in Section 4, noting that the assumption (2.23) implies that the family $(C_{\varepsilon})_{\varepsilon}$ appearing in (5.3)–(5.6) is uniformly bounded in $\varepsilon$.

Acknowledgements

The authors are very grateful to the anonymous referee for the constructive suggestions and remarks. E.D, H.R., and L.T. have been funded by the Austrian Science Fund (FWF) project F 65. The work of E.D. has been supported by the Austrian Science Fund (FWF) through projects I 4052-N32, and V 662-N32, as well as from BMBWF through the OeAD-WTZ project CZ04/2019. L.T. acknowledges partial support from the Austrian Science Fund (FWF) project P27052. L.S. has been funded by Vienna Science and Technology Fund (WWTF) through Project MA14-009.