On the Cahn–Hilliard equation with no-flux and strong anchoring conditions

Abstract

The Cahn–Hilliard equation is a common model to describe phase separation processes of a mixture of two components. We study the Cahn–Hilliard equation coupled with the homogeneous strong anchoring condition (i.e., homogeneous Dirichlet condition) on the relative concentration u of the two phases. Moreover, we adopt no-flux boundary condition to keep conservation of mass. With a specific quartic form of the double-well potential, we prove the existence and uniqueness of the weak solution to this model by interpreting the problem as a gradient flow of the Cahn–Hilliard free energy. Utilizing the minimizing movement scheme and time discretization method, we show that the approximation solutions converge to the weak solution of the Cahn–Hilliard equation. Finally, we prove that the weak solution satisfies an energy dissipation inequality.

1 Introduction

The Cahn–Hilliard equation

$$\begin{aligned} \partial _t u&=\Delta (-\kappa \Delta u + W'(u)), \quad x\in \Omega , t \in (0,\infty ), \end{aligned}$$

(1.1)

was proposed in 1958 to model phase separation that occurs in binary alloys [12, 13]. Here, $\Omega$ is a bounded domain in ${\mathbb {R}}^d, \; d=1,2,3,\dots$. u(x, t) is the relative concentration of the two phases, which evolves in time and space, W(u) is a double-well potential with two equal minima at $u^- < u^+$ corresponding to the two pure phases, and $\kappa > 0$ is a parameter whose square root $\sqrt{\kappa }$ is proportional to the thickness of the transition region between the two phases. The Cahn–Hilliard equation appears in modeling many other phenomena, including the dynamics of two populations [17], the biomathematical modeling of a bacterial film [40], phase separations in polymers [54], the growth of tumor tissues [60], and certain thin film problems [49, 57]. It has been found to help describe various phenomena ranging from nanoscale precipitation [54, 63] to the clumping of galaxies in the universe [52]. This equation also has applications in image processing, where it is used for image inpainting and segmentation[9, 61]. Furthermore, it plays an important role in phase field methods [48], where it is employed to model behavior of conserved order variables.

To model the two-phase system, different forms of double-well potential W(u) have been used. A common choice is the smooth double-well potential, for example,

$$\begin{aligned} W(u)=\gamma (u-u^+)^2(u-u^-)^2, \quad \gamma > 0, \end{aligned}$$

(1.2)

since it is convenient for theoretical analysis and numerical simulations. In this paper we will focus on double-well potentials of the form (1.2). Other choices for the double-well potential W(u) include double-barrier potentials and logarithmic potentials, see, e.g., [11, 20] for a comparison of these potentials.

The Cahn–Hilliard equation (1.1) is a fourth-order parabolic equation. To obtain a well-posed problem, we need to complement (1.1) with an initial condition

$$\begin{aligned} u(x,0) = \Phi (x) \quad \textrm{in}\; \Omega , \end{aligned}$$

(1.3)

and boundary conditions. A common choice for the boundary conditions is the homogeneous Neumann conditions (see, e.g., [4,5,6, 11, 28, 34, 42, 50, 58]),

$$\begin{aligned} \partial _{{\textbf{n}}} u&= 0, \end{aligned}$$

(1.4)

$$\begin{aligned} \partial _{{\textbf{n}}} (-\kappa \Delta u + W'(u))&= 0, \end{aligned}$$

(1.5)

where ${\textbf{n}}$ is the exterior unit vector normal to the boundary $\partial \Omega$. Since the flux is postulated to be [24]

$$\begin{aligned} {\textbf{J}} = -\nabla (-\kappa \Delta u + W'(u)), \end{aligned}$$

(1.6)

the condition (1.5) is also called the no-flux boundary condition, which guarantees the conservation of mass. The Neumann condition (1.4) for u is for mathematical convenience, but a consequence is that near the boundary $\partial \Omega$, the level surfaces of u are required to be perpendicular to $\partial \Omega$ [21]. Another common choice for the boundary conditions is the periodic boundary condition, which also conserves mass and has been widely used particularly in computational studies (cf. e.g., [14,15,16, 18, 20, 31, 37,38,39, 62]). Under either Neumann boundary condition (1.4)–(1.5) or the periodic boundary condition, it is easy to show that the Cahn–Hilliard energy functional

$$\begin{aligned} E (u) = \int _\Omega \frac{\kappa }{2}|\nabla u|^2 + W(u)\,dx. \end{aligned}$$

(1.7)

is decreasing in time.

In materials science, it is common that we control the assembly of complex structures through templated substrates or boundaries. The structure formed by self-assembly is strongly affected by the boundary surface pattern and interactions between monomers and the surface. Therefore, we can fine tune the self-assembled structures by altering either the surface pattern or the interactions (cf. [43, 59, 64] and references therein). Mathematically this means that we want u to match a prescribed function $\phi$ on $\partial \Omega$. There are different ways to measure how well u matches $\phi$ on $\partial \Omega$. The strongest match is a pointwise Dirichlet boundary condition

$$\begin{aligned} u=\phi \quad \text { on }\partial \Omega , \end{aligned}$$

(1.8)

and we call it the strong anchoring condition. We may also use weaker anchoring conditions. For instance, we may prescribe a tolerance for the $L^2(\partial \Omega )$ norm $\Vert u-\phi \Vert _{L^2(\partial \Omega )}$. Since for any given $t>0\ u(\cdot ,t)$ needs to be at least $H^1(\Omega )$ in space, the trace theorem requires that $u|_{\partial \Omega }(\cdot ,t)$ to be in $H^{1/2}(\partial \Omega )$. So for the strong anchoring condition to make sense, we need at least $\phi \in H^{1/2}(\partial \Omega )$. In contrast, the aforementioned $L^2$–weak anchoring condition is meaningful as long as $\phi \in L^2(\partial \Omega )$. The strong anchoring condition enjoys the advantage of being simple and relatively easy to enforce numerically, while the weak anchoring conditions have broader applications due to the less stringent restrictions on $\phi$.

There have been only a few studies that investigated Cahn–Hilliard type problems with Dirichlet boundary conditions for u. In some instances, the equation is complemented with a homogeneous Dirichlet boundary condition for $-\kappa \Delta u + W'(u)$, which can be used to model the propagation of a solidification front into an ambient medium which is at rest relative to the front [10, 27, 33]. Another work is [7], in which Bates and Han studied an integro-differential extension of the Cahn–Hilliard equation with Dirichlet boundary condition for u on a bounded domain. In addition, people have studied the Dirichlet boundary value problems for the biharmonic equation [26], the Schrödinger equation [25], and the regularized long wave equation [47].

We will explore the strong anchoring condition (1.8) coupled with the no-flux boundary condition (1.5). The latter is a natural requirement since in template modulated pattern formations, the system boundary is usually impermeable, i.e., no mass flows across the boundary. We want to emphasize that our setting is different from tumor growth problems (for example, [33]), in which there is no mass conservation since the tumor is growing (or shrinking). In the literature there are very few studies about problems in our setting. The only paper we found is [44], in which Li, Jeong, Shin, and Kim presented a conservative numerical method for the Cahn–Hilliard equation with Dirichlet boundary condition on u and Neumann boundary condition (1.5) in complex domains. The purpose of our work is to lay the theoretical foundation by establishing the wellposedness of the Cahn–Hilliard equation (1.1) with the no-flux condition (1.5) and the strong anchoring condition (1.8).

Due to nonlinearity, the solution u, if exists, depends not only on the boundary value $\phi$, but on a delicate relation between $\phi$ and the double well potential W. Indeed, in [21] we have proved that with a Dirichlet boundary condition $u=\phi$ on $\partial \Omega$ but without mass conservation, the properties of the minimizers for the Cahn–Hilliard energy functional (1.7) are determined by the symmetry of W. To be more precise, in the symmetric case when W is chosen as (1.2) and $\phi$ takes the well-mixed homogeneous value

$$\begin{aligned} \phi = \frac{u^+ + u^-}{2}, \end{aligned}$$

(1.9)

there is a bifurcation phenomena as the value of $\kappa$ varies. If $\phi$ is uniformly above or uniformly below the homogeneous value, the symmetry breaks and the bifurcation does not exist. This motivated us to start our study with the homogeneous boundary value (1.9), and leave the nonhomogeneous situation for future studies.

1.1 Main result

To simplify notations, we choose the smooth quartic double-well potential

$$\begin{aligned} W(u)=\frac{1}{4}(u^2-1)^2, \end{aligned}$$

(1.10)

which has two minima at $u^{\pm }=\pm 1$. In this setting, the homogeneous boundary value becomes $\phi =0$ and the corresponding strong anchoring condition for u becomes

$$\begin{aligned} u = 0 \text { on } \partial \Omega . \end{aligned}$$

(1.11)

We also set $\kappa =1$ since it does not play any role in the analysis in this paper. Therefore the system we study is

$$\begin{aligned} \partial _t u&=\Delta (-\Delta u + W'(u)) \quad (x,t) \in \Omega _T:= \Omega \times (0,T), \end{aligned}$$

(1.12)

$$\begin{aligned} u&= 0 \quad \textrm{on}\; \partial \Omega \times (0,T), \end{aligned}$$

(1.13)

$$\begin{aligned} \partial _{\textbf{n}} (-\Delta u + W'(u))&= 0 \quad \textrm{on}\; \partial \Omega \times (0,T), \end{aligned}$$

(1.14)

where $T \in (0,\infty )$, and $\Omega$ is a bounded domain in ${\mathbb {R}}^d$ with a $C^2$ boundary $\partial \Omega$. We concentrate on $d=2,3.$ Moreover, we assume the initial data to be

$$\begin{aligned} u(x,0) = \Phi (x) \quad \textrm{in}\; \Omega , \end{aligned}$$

(1.15)

where $\Phi \in H_0^1(\Omega )$ is a given function, with initial total mass

$$\begin{aligned} {\mathcal {M}}:=\int _\Omega \Phi (x) \,dx. \end{aligned}$$

(1.16)

Since we set $\kappa =1$, the Cahn–Hilliard energy functional is now defined by

$$\begin{aligned} E (u)&= \int _\Omega \frac{1}{2}|\nabla u|^2 + W(u)\,dx. \end{aligned}$$

(1.17)

Recall that $H^1(\Omega )/{\mathbb {R}}$ is the quotient space of equivalence classes of $H^1$ functions in which two functions are equivalent if they vary by only a spatial constant.

Definition 1.1

A function $u \in L^2(0,T;H_0^1(\Omega ))$ is said to be a weak solution to the Cahn–Hilliard system (1.12) with strong anchoring condition (1.13), no-flux boundary condition (1.14), and initial value (1.15) with prescribed total mass (1.16), if there exists a corresponding chemical potential function $\mu \in L^2(0,T;H^1(\Omega )/{\mathbb {R}})$ uniquely determined by u such that the following conditions hold:

• (i) For any $\xi \in L^2(0,T;H^1(\Omega ))$ with $\partial _t\xi \in L^2(\Omega _T)$ and $\xi (T)=0$, the following integral equality holds:

  $$\begin{aligned} \int _0^T\int _\Omega (u - \Phi )\partial _t \xi \,dxdt =\int _0^T\int _\Omega \nabla \mu \cdot \nabla \xi \,dxdt. \end{aligned}$$

  (1.18)

• (ii) For any $\eta \in L^2(0,T;H_0^1(\Omega )) \cap L^\infty (\Omega _T)$ with $\int _\Omega \eta (x,t)\,dx = 0$ for all $t\in [0,T]$, the following integral equality holds:

  $$\begin{aligned} \int _0^T\int _\Omega \mu \eta \,dxdt = \int _0^T\int _\Omega \nabla u \cdot \nabla \eta + W'(u)\eta \,dxdt. \end{aligned}$$

  (1.19)

• (iii) $u(x,0) = \Phi (x)$ for a.e. $x \in \Omega$.

• (iv) $\int _\Omega u(x,t)\,dx = {\mathcal {M}}$ for a.e. $t\in [0,T]$.

Theorem 1.2

Let $T \in (0,\infty )$ and $\Phi \in H_0^1(\Omega )$ with $\int _\Omega \Phi \,dx = {\mathcal {M}}$ and $E (\Phi ) < \infty$. There exists a unique function $u \in L^\infty (0,T;H_0^1(\Omega ))\cap C^{0,\beta }([0,T];L^4(\Omega ))$, where $\beta = 1/8$ if $d=2$ and $\beta = 1/16$ if $d=3$, that is a weak solution to the Cahn–Hilliard equation (1.12)–(1.16) in the sense defined in Definition 1.1. In addition, u and its corresponding chemical potential function $\mu \in L^2(0,T;H^1(\Omega )/{\mathbb {R}})$ satisfy the following energy inequality:

$$\begin{aligned} E (u(\cdot ,t)) + \frac{1}{2}\int _0^t ||\nabla \mu (s)||_{L^2(\Omega )} ds \le E (\Phi ) \quad \text { for any } t\in [0,T]. \end{aligned}$$

(1.20)

For convenience we may pick a unique representative of $\mu$ by requiring $\int _\Omega \mu (x,t)\,dx = 0$ for all $t>0$.

Remark 1.3

The requirement for the test function $\eta$ in (1.19) to have zero average is due to the conservation of mass in the calculation of the variational derivative of the free energy E.

Remark 1.4

The solution u is global in time since $T>0$ is arbitrary.

Remark 1.5

We need the specific form (1.10) of W to prove the uniqueness of the weak solution. For the nonhomogeneous boundary condition $u=\phi$, we can make a shift $\tilde{u}:= u - \phi$ and write the following equations for $\tilde{u}$:

$$\begin{aligned} \tilde{u}_t&=\Delta (-\Delta \tilde{u} +\tilde{W}'(\tilde{u}) ), \\ \tilde{u}&= 0 \quad \text {on }\partial \Omega , \\ \partial _{{\textbf{n}}} (-\Delta \tilde{u} + \tilde{W}'(\tilde{u}))&= 0 \quad \text {on }\partial \Omega . \end{aligned}$$

Here $\tilde{W}(\tilde{u}):= W(\tilde{u}+\phi ) -\tilde{u}\Delta \phi .$ Following the same lines, we can still prove the existence of a weak solution $\tilde{u}$. However, since the new potential $\tilde{W}$ takes a different form, the uniqueness argument breaks down. We will explore the uniqueness of the nonhomogeneous problem in future studies.

The relation between u and $\mu$ is as follows. Suppose there are sufficiently regular functions u and $\mu$ that satisfy (1.19). Then

$$\begin{aligned} \int _\Omega \mu \eta \,dx = \int _\Omega (-\Delta u + W'(u))\eta \,dx \end{aligned}$$

(1.21)

for any $\eta \in H_0^1(\Omega ) \cap L^\infty (\Omega )$ with $\int _\Omega \eta \,dx = 0$. Since we require $\int _\Omega \eta \,dx = 0$, we can only conclude that

$$\begin{aligned} \mu =-\Delta u + W'(u) + g(t), \end{aligned}$$

(1.22)

where g(t) is an arbitrary spatial constant that may vary in time. That is, $\mu (\cdot , t)\in H^{-1}(\Omega )/{\mathbb {R}}$. The requirement $\int _\Omega \mu \,\,dx =0$ is for us to remove the uncertainty induced by the arbitrary special constant g(t). To satisfy $\int _\Omega \mu \,\,dx =0$, g(t) needs to be

$$\begin{aligned} g(t) = \frac{1}{|\Omega |}\int _\Omega (\Delta u - W'(u)) \,dx = \frac{1}{|\Omega |} \left( \int _{\partial \Omega } \frac{\partial u}{\partial {\textbf{n}}}\, dS -\int _\Omega W'(u)\,\,dx\right) \end{aligned}$$

(1.23)

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

The Cahn–Hilliard equation (1.12) is usually written in the form of a system of two equations:

$$\begin{aligned} \partial _t u&=\Delta \omega , \quad (x,t)\in \Omega _T, \end{aligned}$$

(1.24)

$$\begin{aligned} \omega&= -\Delta u + W'(u), \quad (x,t)\in \Omega _T. \end{aligned}$$

(1.25)

This formulation is fine if we choose the periodic boundary condition or the Neumann boundary condition (1.4) for u. Indeed, the precise form of the Cahn-Hilliard equation should be written as

$$\begin{aligned} u_t = \Delta \frac{\delta E}{\delta u}, \end{aligned}$$

(1.26)

where $\frac{\delta E}{\delta u}$ is the variational derivative of E. Under the periodic or Neumann condition (1.4) and without mass conservation, we do have

$$\begin{aligned} \frac{\delta E}{\delta u} = -\Delta u + W'(u) \quad \text { a.e. in } \Omega . \end{aligned}$$

However, with homogeneous Dirichlet condition $u=0$ on $\partial \Omega$ and mass conservation, all we can get is $\frac{\delta E}{\delta u} = -\Delta u + W'(u) + g(t)$, where g(t) is an arbitrary spatial constant, see Sect. 2.1 for details. In other words, $\frac{\delta E}{\delta u}$ is an equivalence class, which can be written in terms of elements in a quotient space,

$$\begin{aligned} \frac{\delta E}{\delta u} = -\Delta u + W'(u) \text { in the quotient space } H^{-1}(\Omega )/{\mathbb {R}}. \end{aligned}$$

Our $\mu$ defined by (1.22) with g(t) given by (1.23) is a particular representative of $\frac{\delta E}{\delta u}$ in the quotient space. Compared with $\omega$ defined by (1.25), we see that

$$\begin{aligned} \omega = \mu - g(t), \end{aligned}$$

(1.27)

where g is defined by (1.23). This spatial constant g does not play any role in the original Cahn–Hilliard equation (1.12) since $\nabla \omega = \nabla \mu$.

1.2 The minimizing movement scheme

Our proof of Theorem 1.2 utilizes the gradient flow structure and the minimizing movement scheme for the Cahn–Hilliard equation (1.12). We will give a brief introduction about the minimizing movement method.

1.2.1 Introduction of the minimizing movement method

The concept of minimizing movement involves the recursive minimization

• (a) $\nu _\tau ^0:= \tilde{\nu } \in {\mathcal {S}}$ is given,

• (b) $\nu _\tau ^n$ is a minimizer for ${\mathcal {F}}(\cdot ,\nu _\tau ^{n-1},\tau )$ for any $n=1,2,\ldots$,

for a given functional ${\mathcal {F}}: {\mathcal {S}}\times {\mathcal {S}}\times (0,1) \rightarrow [-\infty ,\infty ]$ on a topological space $({\mathcal {S}},\sigma )$. The purpose of a minimizing movement scheme is to study the limit of $\{\nu _\tau \}_{0<\tau <1}$ as $\tau \searrow 0$. The parameter $\tau \in (0,1)$ plays the role of discrete time step size. If a sequence $\{\nu _\tau ^n\}_{n=1}^\infty$ satisfies the recursion (a) and (b), we call the corresponding piecewise constant interpolation

$$\begin{aligned} \nu _\tau (0)&:= \tilde{\nu }, \\ \nu _\tau (t)&:= \nu _\tau ^n \quad \textrm{for} \; \textrm{all} \; t\in ((n-1)\tau ,n\tau ], n=1,2,\ldots , \end{aligned}$$

a discrete solution. This method plays an important role in the theory of existence of solution of differential equations since it provides a time discrete approximation solution for the differential equation while not requiring the initial condition $\tilde{\nu }$ to be smooth. Moreover, the interpolation method guarantees the compactness of the family of discrete solutions [35, 36]. Therefore, the minimizing movement method has been used in many works to study the existence of solution for some classes of PDEs (cf. e.g., [1, 2, 32, 46, 51, 53, 56, 65]).

1.2.2 Comparison with the Galerkin approximation method

Another common way to handle the Cahn–Hilliard equation (1.24)–(1.25) with homogeneous Neumann boundary conditions $\partial _{{\textbf{n}}} u = \partial _{{\textbf{n}}} \omega = 0$ (or periodic boundary conditions) is to use the Galerkin approximation. The idea is to define

$$\begin{aligned} u^N(x,t)=\sum _{j=1}^{N}c^N_j(t)\phi _j(x), \quad \omega ^N(x,t) =\sum _{j=1}^{N}d^N_j(t)\phi _j(x), \quad N=1,2,3,\ldots , \end{aligned}$$

(1.28)

where $\phi _j (j=1,2,\ldots )$ are eigenfunctions of the eigenvalue problem $-\Delta u = \lambda u$ in $\Omega$ subject to homogeneous Neumann boundary condition (resp. periodic boundary condition), which form a complete orthonormal basis for $L^2(\Omega )$. We look for a pair of functions $(u^N,\omega ^N)$ that solves the following system of ODEs for $\{c^N_j\}_{j=1}^{N}$

$$\begin{aligned} \int _\Omega \partial _t u^N\phi _j \,dx&= -\int _\Omega \nabla \omega ^N \cdot \nabla \phi _j \,dx, \end{aligned}$$

(1.29)

$$\begin{aligned} \int _\Omega \omega ^N \phi _j \,dx&= \int _\Omega \left( \nabla u^N \cdot \nabla \phi _j + W'(u^N)\phi _j \right) \,dx, \end{aligned}$$

(1.30)

$$\begin{aligned} u^N(x,0)&= \sum _{j=1}^{N} \left( \int _\Omega \Phi \phi _j \,dx \right) \phi _j(x). \end{aligned}$$

(1.31)

The next step is to prove that the sequence $\{u^N\}_{N=1}^\infty$ converges to some function u (in some suitable sense, up to a subsequence) which is a weak solution to (1.24)–(1.25) in some suitable sense. This method is efficient and has been used in many studies (cf. e.g., [3, 8, 19, 22, 23, 29]).

However, the Galerkin approximation method has some limitations when it comes to mixed boundary conditions like (1.13)–(1.14). First, when defining $u^N$ and $\omega ^N$ in (1.28), we need to use different bases for $u^N$ and $\omega ^N$ corresponding to different boundary conditions for u and $\omega$. This would make the calculation more complicated. Furthermore, the Galerkin approximation does not capture the feature that allows an extra spatial constant in the chemical potential as in (1.22). In the equation (1.24)–(1.25) u and $\omega$ are always paired, and the Galerkin approxiation method only works with one specific choice $\omega$ of the chemical potential, not with the general setting where a spatial constant in the chemical potential, like (1.22), is allowed. The minimizing movement method, on the other hand, can handle this problem elegantly and in the most precise way using functional analysis knowledge. By utilizing the gradient flow structure and minimizing movement method, we obtain the framework that helps us deal with general cases where a constant in the chemical potential is allowed.

2 Preliminaries

In this section we introduce some preliminaries that will be used in the rest of this paper.

• (A1) For any real number ${\mathcal {M}}$, we define

  $$\begin{aligned} X&:=\{u\in H^1(\Omega ): \int _\Omega u \,dx = 0 \}, \\ X_0&:=\{u\in H^1_0(\Omega ): \int _\Omega u \,dx = 0 \},\\ X_{\mathcal {M}}&:=\{u\in H^1_0(\Omega ): \int _\Omega u \,dx = {\mathcal {M}}\}, \end{aligned}$$

  then X and $X_0$ are Hilbert spaces with respect to the inner product

  $$\begin{aligned} (u,v)_{H^1(\Omega )}:=\int _\Omega uv + \nabla u\cdot \nabla v \,dx, \quad u,v\in X. \end{aligned}$$

  (2.1)

  However, for any $u \in X$, by Poincaré’s inequality,

  $$\begin{aligned} ||\nabla u||_{L^2(\Omega )} \le ||u||_{H^1(\Omega )} \le C||\nabla u||_{L^2(\Omega )}. \end{aligned}$$

  (2.2)

  Hence, in X, the inner product (2.1) is equivalent to the following inner product

  $$\begin{aligned} (u,v)_X :=\int _\Omega \nabla u\cdot \nabla v \,dx, \quad u,v\in X, \end{aligned}$$

  (2.3)

  and X and $X_0$ are also Hilbert spaces with respect to the inner product (2.3).

• (A2) For any $f\in H^1(\Omega )'$ with $\int _\Omega f \,dx =0$, where $H^1(\Omega )'$ is the dual space of $H^1(\Omega )$, the Neumann problem

  $$\begin{aligned} -\Delta v&= f \quad \textrm{in}\; \Omega , \end{aligned}$$

  (2.4)
  $$\begin{aligned} \partial _{\textbf{n}} v&= 0 \quad \textrm{on}\; \partial \Omega , \end{aligned}$$

  (2.5)
  $$\begin{aligned} \int _\Omega v \,dx&= 0, \end{aligned}$$

  (2.6)

  has a unique weak solution $v_f\in H^1(\Omega )$. We denote $v_f$ as $(-\Delta )^{-1}f$. Moreover, if $f \in L^2(\Omega )$, then by regularity theory (see [30], §6.3), $v_f\in H^2(\Omega )$, that is,

  $$\begin{aligned} -\Delta ((-\Delta )^{-1}f) = -\Delta v_f = f \quad \mathrm{a.e}. \, \textrm{in}\; \Omega . \end{aligned}$$

  (2.7)

• (A3) Let $X_0'$ and $X'$ be the dual spaces of $X_0$ and X, respectively. By the Riesz Representation Theorem (see [30], §D.3), for any $f\in X_0'$, there exists a unique function $w_f\in H^1(\Omega )$ such that

  $$\begin{aligned} \langle f,\xi \rangle _{(X_0',X_0)}=\int _\Omega \nabla w_f \cdot \nabla \xi \,dx, \end{aligned}$$

  (2.8)

  for any $\xi \in X_0$. If $f\in H^1(\Omega )'$ with $\int _\Omega f \,dx = 0$, then the function $w_f$ is the same as $v_f$ defined in (A2). Thus, we also denote $w_f$ as $(-\Delta )^{-1}f$. Then we define an inner product in $X_0'$ by

  $$\begin{aligned} (f,g)_{X_0'}:=\int _\Omega \nabla (-\Delta )^{-1}f\cdot \nabla (-\Delta )^{-1}g \,dx, \quad f,g\in X_0'. \end{aligned}$$

  (2.9)

  We also define its induced norm $||u||_{X_0'}:=(u,u)_{X_0'}^{1/2}$. Since $X_0 \subset X \subset H^1(\Omega ) \subset X' \subset X_0'$, we can also use this inner product and its induced norm for functions in $X_0, X$ and $X'$.

2.1 Variational derivative of the Cahn–Hilliard energy functional in a quotient space

Since the total mass $\int _\Omega u\,dx$ is conserved, we need to consider the Cahn–Hilliard energy functional

$$\begin{aligned} E (u) = \int _\Omega \frac{1}{2}|\nabla u|^2 + W(u)\,dx \end{aligned}$$

(2.10)

in the admissible set $X_{\mathcal {M}}$. Provided that u is sufficiently regular, for any perturbation $\eta \in X_0\cap L^\infty (\Omega )$, the variational derivative $\delta E / \delta u$ is defined through

$$\begin{aligned} \left\langle \frac{\delta E }{\delta u},\eta \right\rangle _{(X_0',X_0)}&= \frac{d}{ds} \bigg |_{s=0} E (u + s\eta ) \nonumber \\&= \int _\Omega (-\Delta u + W'(u))\eta \,dx. \end{aligned}$$

(2.11)

Since $\int _\Omega \eta \,dx = 0$, $\delta E / \delta u$ is not necessarily equal to $-\Delta u + W'(u)$. Instead, we just need

$$\begin{aligned} \frac{\delta E }{\delta u} = -\Delta u + W'(u) + g, \end{aligned}$$

(2.12)

where g is a spatial constant that is independent on x. In other words, we can say that

$$\begin{aligned} \frac{\delta E }{\delta u} = -\Delta u + W'(u) \quad \textrm{in}\; \textrm{the}\; \textrm{quotient}\; \textrm{space}\; H^{-1}(\Omega )/{\mathbb {R}}. \end{aligned}$$

2.2 The gradient flow structure for the Cahn–Hilliard equation (1.12)–(1.14)

Provided that u is sufficiently regular, for any $\eta \in X_0\cap L^\infty (\Omega )$, using integration by parts, (1.12), (1.14) and (A2), we have

$$\begin{aligned} (\partial _t u,\eta )_{X_0'}&=\int _\Omega \nabla (-\Delta )^{-1} \partial _t u\cdot \nabla (-\Delta )^{-1}\eta \,dx \nonumber \\&=-\int _\Omega \nabla (-\Delta u + W'(u)) \cdot \nabla (-\Delta )^{-1}\eta \,dx \nonumber \\&=\int _\Omega (-\Delta u + W'(u)) \Delta ((-\Delta )^{-1}\eta ) \,dx \nonumber \\&=-\int _\Omega (-\Delta u + W'(u))\eta \,dx \nonumber \\&=-\left\langle \frac{\delta E }{\delta u},\eta \right\rangle _{(X_0',X_0)}. \end{aligned}$$

(2.13)

So the equation (1.12)–(1.14) is a gradient flow of the Cahn–Hilliard free energy E:

$$\begin{aligned} (\partial _t u,\eta )_{X_0'} = -\left\langle \frac{\delta E }{\delta u},\eta \right\rangle _{(X_0',X_0)} \quad \textrm{for}\; \textrm{all}\; \eta \in X_0\cap L^\infty (\Omega ). \end{aligned}$$

(2.14)

3 Implicit time discretization and preliminary estimates

In this section we introduce the implicit time discretization for the Cahn–Hilliard equation (1.12)–(1.16) and some estimates which are necessary to prove the main result in Theorem 1.2.

3.1 Implicit time discretization

To prepare for the proof of Theorem 1.2, we derive an implicit time discretization of the equation (1.12)–(1.16). Then we will prove that the corresponding time-discrete solution converges to a function u, which is a weak solution to the equation (1.12)–(1.16) in some suitable sense.

Let $N\in {\mathbb {N}}$ be arbitrary and let $\tau :=T/N$ denote the time step size. Without loss of generality, we assume that $\tau < 1$. We define $\phi ^n (n=0,1,\ldots ,N)$ recursively by the following construction:

• $\phi ^0:=\Phi$ (the initial data).

• If $\phi ^n$ is already constructed, we choose $\phi ^{n+1}$ to be a minimizer of the functional

  $$\begin{aligned} J_n(\zeta )=\frac{1}{2\tau }||\zeta -\phi ^n||_{X_0'}^2 + E (\zeta ) \end{aligned}$$

  (3.1)

  over the set $X_{\mathcal {M}}$.

The existence of such a minimizer is guaranteed by the following lemma.

Lemma 3.1

The functional $J_n$ has a global minimizer $\bar{\zeta }\in X_{\mathcal {M}}$, that is, for any $\zeta \in X_{\mathcal {M}}$,

$$\begin{aligned} J_n(\bar{\zeta }) \le J_n(\zeta ). \end{aligned}$$

Proof

Since $E \ge 0$, $J_n$ is bounded below by 0. Thus, $m:=\inf _{X_{\mathcal {M}}} J_n$ exists, and we can find a minimizing sequence $(\zeta _k)\subset X_{\mathcal {M}}\subset H_0^1(\Omega )$ such that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }J_n(\zeta _k) = m, \quad \textrm{and} \quad J_n(\zeta _k) \le m+1 \; \textrm{for} \; \textrm{all}\;k=1,2,3,\ldots \end{aligned}$$

(3.2)

Since $W(\zeta _k)\ge 0$, we have

$$\begin{aligned} m+1&\ge J_n(\zeta _k) = \frac{1}{2\tau }|| \zeta _k-\phi ^n||_{X_0'}^2 + \int _\Omega \frac{1}{2}|\nabla \zeta _k|^2 + W(\zeta _k)\,dx \nonumber \\&\ge \int _\Omega \frac{1}{2}|\nabla \zeta _k|^2\,dx, \end{aligned}$$

(3.3)

which implies that

$$\begin{aligned} ||\zeta _k||_{H_0^1(\Omega )} \le C. \end{aligned}$$

(3.4)

Hence $\{\zeta _k\}$ is bounded in $H_0^1(\Omega )\subset \subset L^2(\Omega )$. So there exists $\bar{\zeta } \in H_0^1(\Omega )$ and a subsequence of $\{\zeta _k\}$ (still denoted as $\{\zeta _k\}$) such that $\zeta _k \rightharpoonup \bar{\zeta }$ weakly in $H_0^1(\Omega )$, $\zeta _k \rightarrow \bar{\zeta }$ strongly in $L^2(\Omega )$, and $\zeta _k \rightarrow \bar{\zeta }$ a.e. in $\Omega$. Hence

$$\begin{aligned} \left| \int _\Omega \bar{\zeta }\,dx -{\mathcal {M}}\right| = \left| \int _\Omega (\bar{\zeta }-\zeta _k)\,dx \right| \le \int _\Omega |\bar{\zeta }-\zeta _k|\,dx \le C||\bar{\zeta }-\zeta _k||_{L^2(\Omega )} \rightarrow 0 \end{aligned}$$

as $k\rightarrow \infty$. This implies that $\int _\Omega \bar{\zeta }\,dx = {\mathcal {M}}$, hence $\bar{\zeta }\in X_{\mathcal {M}}$.

We rewrite the double well potential W as $W(\zeta ) = W_1(\zeta ) + W_2(\zeta )$, where $W_1(\zeta )=(\zeta ^4+1)/4$ convex and $W_2(\zeta )=-\zeta ^2/2$. We have

$$\begin{aligned} \int _\Omega W_2(\zeta _k) = -\frac{1}{2}\int _\Omega |\zeta _k|^2 \rightarrow -\frac{1}{2}\int _\Omega |\bar{\zeta }|^2 = \int _\Omega W_2(\bar{\zeta }) \end{aligned}$$

(3.5)

as $k \rightarrow \infty$. Since all other terms of $J_n$ can easily be handled by the weakly lower semicontinuity of convex functionals, we obtain that

$$\begin{aligned} J_n(\bar{\zeta }) \le \liminf \limits _{k\rightarrow \infty } J(\zeta _k) = m, \end{aligned}$$

(3.6)

which implies that $J_n(\bar{\zeta })=m$. $\square$

Since $\phi ^{n+1}$ is a minimizer of $J_n$, for any $\eta \in X_0\cap L^\infty (\Omega )$, we have

$$\begin{aligned} \frac{d}{ds} \bigg |_{s=0} J_n(\phi ^{n+1} + s\eta ) = 0, \end{aligned}$$

(3.7)

that is,

$$\begin{aligned} \left( \frac{\phi ^{n+1}-\phi ^n}{\tau }, \eta \right) _{X_0'} + \int _\Omega \nabla \phi ^{n+1} \cdot \nabla \eta + W'(\phi ^{n+1})\eta \,dx = 0 \end{aligned}$$

(3.8)

for any $\eta \in X_0\cap L^\infty (\Omega )$. The equation (3.8) can be interpreted as an implicit time discretization of the corresponding gradient flow equation (2.14).

Since $\phi ^n, \phi ^{n+1} \in X_{\mathcal {M}}\subset H^1(\Omega )'$, we have $\frac{\phi ^{n+1}-\phi ^n}{\tau } \in H^1(\Omega )'$ and $\int _\Omega \frac{\phi ^{n+1}-\phi ^n}{\tau } \,dx = 0$. Hence, by (A2), the equation

$$\begin{aligned} -\Delta \psi ^{n+1}&= -\frac{\phi ^{n+1}-\phi ^n}{\tau } \quad \textrm{in}\; \Omega , \end{aligned}$$

(3.9)

$$\begin{aligned} \partial _{\textbf{n}} \psi ^{n+1}&=0 \quad \textrm{on}\; \partial \Omega , \end{aligned}$$

(3.10)

$$\begin{aligned} \int _\Omega \psi ^{n+1} \,dx&= 0, \end{aligned}$$

(3.11)

has a unique solution $\psi ^{n+1} =(-\Delta )^{-1} \left( -\frac{\phi ^{n+1}-\phi ^n}{\tau }\right)$.

For any $\eta \in X_0\cap L^\infty (\Omega )$, using integration by parts, we have

$$\begin{aligned} \left( \frac{\phi ^{n+1}-\phi ^n}{\tau }, \eta \right) _{X_0'}&=-\int _\Omega \nabla (-\Delta )^{-1} \left( \frac{\phi ^{n+1}-\phi ^n}{\tau } \right) \cdot \nabla (-\Delta )^{-1}\eta \,dx \nonumber \\&=-\int _\Omega \nabla \psi ^{n+1} \cdot \nabla (-\Delta )^{-1}\eta \,dx \nonumber \\&= \int _\Omega \psi ^{n+1} \Delta ((-\Delta )^{-1}\eta )\,dx \nonumber \\&=-\int _\Omega \psi ^{n+1}\eta \,dx. \end{aligned}$$

(3.12)

Combining with (3.8) we get

$$\begin{aligned} \int _\Omega \psi ^{n+1}\eta \,dx = \int _\Omega \nabla \phi ^{n+1} \cdot \nabla \eta + W'(\phi ^{n+1})\eta \,dx \end{aligned}$$

(3.13)

for any $\eta \in X_0\cap L^\infty (\Omega )$. So $(\phi ^{n+1},\phi ^n, \psi ^{n+1})$ is a solution of

$$\begin{aligned} \int _\Omega \left( -\frac{\phi ^{n+1}-\phi ^n}{\tau } \right) \xi \,dx&= \int _\Omega \nabla \psi ^{n+1} \cdot \nabla \xi \,dx \quad \textrm{for}\; \textrm{all}\; \xi \in H^1(\Omega ), \end{aligned}$$

(3.14)

$$\begin{aligned} \int _\Omega \psi ^{n+1}\eta \,dx&= \int _\Omega \nabla \phi ^{n+1} \cdot \nabla \eta + W'(\phi ^{n+1})\eta \,dx \quad \textrm{for}\; \textrm{all} \; \eta \in X_0\cap L^\infty (\Omega ), \end{aligned}$$

(3.15)

$$\begin{aligned} \int _\Omega \psi ^{n+1} \,dx&= 0, \end{aligned}$$

(3.16)

which is an implicit time descretization of the equation (1.12)–(1.14).

Let $(\phi _N,\psi _N)$ denote the piecewise constant extension of the approximation solution $(\phi ^n,\psi ^n)_{n\in \{1,\ldots ,N\}}$ on the interval [0, T], that is, for any $n\in \{1,\ldots ,N\}, x\in \Omega$ and $t\in ((n-1)\tau ,n\tau ]$, we set

$$\begin{aligned} \phi _N(x,0)&:= \Phi (x), \end{aligned}$$

(3.17)

$$\begin{aligned} (\phi _N,\psi _N)(x,t)&:=(\phi ^n,\psi ^n)(x,n\tau ). \end{aligned}$$

(3.18)

By (3.16), $\int _\Omega \psi _N(x,t) \,dx = 0$ for all $t\in [0,T]$ and all $N=1,2,\ldots$ Similarly, we let $(\bar{\phi }_N,\bar{\psi }_N)$ denote the piecewise linear extension, that is, for any $n\in \{1,\ldots ,N\}, x\in \Omega$ and $t=(1-\alpha )(n-1)\tau +\alpha n\tau$ (where $0<\alpha \le 1$), we set

$$\begin{aligned} \bar{\phi }_N(x,0)&:= \Phi (x), \end{aligned}$$

(3.19)

$$\begin{aligned} (\bar{\phi }_N,\bar{\psi }_N)(x,t)&:=(1-\alpha )(\phi ^{n-1},\psi ^{n-1}) (x,(n-1)\tau )+\alpha (\phi ^n,\psi ^n)(x,n\tau ). \end{aligned}$$

(3.20)

3.2 Uniform bounds on the extensions

In this subsection we establish uniform bounds for the above extensions. Note that since X is a Hilbert space, $L^2(0,T;X)$ is also a Hilbert space [45].

Lemma 3.2

There exists a constant $C>0$ independent on $N,n,\tau$ such that

$$\begin{aligned} ||\phi _N||_{L^\infty (0,T;H_0^1(\Omega ))}&\le C, \end{aligned}$$

(3.21)

$$\begin{aligned} ||\bar{\phi }_N||_{C([0,T];H_0^1(\Omega ))}&\le C, \end{aligned}$$

(3.22)

$$\begin{aligned} ||\psi _N||_{L^2(0,T;X)}&\le C. \end{aligned}$$

(3.23)

Proof

For each $n\in \{0,\ldots ,N-1\}$, since $\phi ^{n+1}$ is a minimizer of $J_n$, then

$$\begin{aligned} E (\phi ^{n+1}) \le J_n(\phi ^{n+1}) \le J_n(\phi ^n) = E (\phi ^n). \end{aligned}$$

(3.24)

It follows inductively that

$$\begin{aligned} E (\phi ^{n+1}) \le E (\phi ^0) = E (\Phi ) \end{aligned}$$

(3.25)

for any $n\in \{0,\ldots ,N-1\}$. Thus

$$\begin{aligned} \int _\Omega \frac{1}{2} |\nabla \phi ^{n+1}|^2 \le E (\phi ^{n+1}) \le E (\Phi ). \end{aligned}$$

(3.26)

Since $\phi ^{n+1} \in X_{\mathcal {M}}\subset H_0^1(\Omega )$, (3.26) implies that

$$\begin{aligned} ||\phi ^{n+1}||_{H_0^1(\Omega )} \le C, \end{aligned}$$

(3.27)

where $C>0$ doesn’t depend on t and N. Recalling the definitions of $\phi _N$ and $\bar{\phi }_N$, we obtain (3.21) and (3.22).

Now pick an arbitrary number $n\in \{1,\ldots ,N\}$ and let $t:=n\tau$. For any $s\in (t-\tau ,\tau ]$, we have $\phi _N(s)=\phi _N(t)=\phi ^n(n\tau )$ and $\psi _N(s)=\psi _N(t)=\psi ^n(n\tau )$. Thus, by the definitions of $\psi _N$ and $||\cdot ||_{X_0'}$, we get

$$\begin{aligned}&E (\phi _N(t)) + \frac{1}{2}\int ^t_{t-\tau } ||\nabla \psi _N(s)||^2_{L^2(\Omega )} ds \nonumber \\&\quad = E (\phi _N(t)) + \frac{1}{2}\int ^t_{t-\tau } \frac{1}{\tau ^2} ||\phi _N(s)-\phi _N(s-\tau )||^2_{X'_0} ds \nonumber \\&\quad = E (\phi _N(t)) + \frac{1}{2}\int ^t_{t-\tau } \frac{1}{\tau ^2} ||\phi _N(t)-\phi _N(t-\tau )||^2_{X'_0} ds \nonumber \\&\quad = E (\phi _N(t)) + \frac{1}{2\tau } ||\phi _N(t)-\phi _N (t-\tau )||^2_{X'_0} \nonumber \\&\quad \le E (\phi _N(t-\tau )). \end{aligned}$$

(3.28)

It follows inductively that

$$\begin{aligned} E (\phi _N(t)) + \frac{1}{2}\int ^t_0 ||\nabla \psi _N(s)||^2_{L^2(\Omega )} ds \le E (\phi _1(t)) \le E (\phi ^0) = E (\Phi ). \end{aligned}$$

(3.29)

Since $E \ge 0$, we obtain that

$$\begin{aligned} \int ^t_0 ||\psi _N(s)||^2_X \,dx = \int ^t_0 ||\nabla \psi _N(s)||^2_{L^2(\Omega )} ds \le C. \end{aligned}$$

(3.30)

So (3.23) is established. $\square$

3.3 Hölder estimates for the piecewise linear extension

Now we show the Hölder continuity in time for the piecewise linear extension.

Lemma 3.3

There exists a constant $C>0$ independent on N such that for any $t_1, t_2 \in [0,T]$,

$$\begin{aligned} ||\bar{\phi }_N(t_1) - \bar{\phi }_N(t_2)||_{L^2(\Omega )} \le C |t_1 - t_2|^{1/4}. \end{aligned}$$

(3.31)

Proof

Let $t_1, t_2 \in [0,T]$ be arbitrary. Without loss of generality, we assume that $t_1 < t_2$. Since $\bar{\phi }_N$ is piecewise linear in t, it is weakly differentiable with respect to t. So we can rewrite (3.14) as

$$\begin{aligned} \int _\Omega \partial _t \bar{\phi }_N(t)\xi \,dx = -\int _\Omega \nabla \psi _N(t)\cdot \nabla \xi \,dx \end{aligned}$$

(3.32)

for all $\xi \in H^1(\Omega )$ and all $t\in [0,T]$. Choose $\xi = \bar{\phi }_N(t_2) - \bar{\phi }_N(t_1)$, integrating with respect to t from $t_1$ to $t_2$ and using the bounds in Lemma 3.2 we get

$$\begin{aligned} ||\bar{\phi }_N(t_2) - \bar{\phi }_N(t_1)||^2_{L^2(\Omega )}&= \left| \int _{t_1}^{t_2}\int _\Omega \nabla \psi _N(t) \cdot \nabla (\bar{\phi }_N(t_2) - \bar{\phi }_N(t_1))\,dxdt \right| \nonumber \\&\le C ||\bar{\phi }_N||_{L^\infty (0,T;H^1(\Omega ))} || \psi _N||_{L^2(0,T;X)} |t_1 - t_2|^{1/2} \nonumber \\&\le C |t_1 - t_2|^{1/2}. \end{aligned}$$

(3.33)

So Lemma 3.3 is established. $\square$

Lemma 3.4

There exists a constant $C>0$ independent on N such that for any $t_1, t_2 \in [0,T]$,

$$\begin{aligned} ||\bar{\phi }_N(t_1) - \bar{\phi }_N(t_2)||_{L^4(\Omega )} \le C |t_1 - t_2|^{\beta }, \end{aligned}$$

(3.34)

where $\beta = 1/8$ if $d=2$ and $\beta = 1/16$ if $d=3$.

Proof

Inequality (3.34) is slightly more complicated than the Sobolev embedding theorem. In fact we need to use an interpolation inequality such as Ladyzhenskaya’s inequality (see the Appendix): for any bounded Lipschitz domain $\Omega$ in ${\mathbb {R}}^d$ ($d=2$ or 3), there exists a constant $C > 0$ depending only on $\Omega$ such that if $f \in H_0^1(\Omega )$, then

$$\begin{aligned} ||f||_{L^4(\Omega )} \le C ||f||_{L^2(\Omega )}^{1/2} || \nabla f||_{L^2(\Omega )}^{1/2} \quad \textrm{if}\; d=2,\\ ||f||_{L^4(\Omega )} \le C ||f||_{L^2(\Omega )}^{1/4} || \nabla f||_{L^2(\Omega )}^{3/4} \quad \textrm{if}\; d=3. \end{aligned}$$

Then (3.34) is a consequence of these two inequalities and Lemmas 3.2 & 3.3.

$\square$

4 The existence and uniqueness of the weak solution to the Cahn–Hilliard equation (1.12)–(1.16)

In this section we prove the existence and uniqueness of the weak solution to the equation (1.12)–(1.16) in the sense of Theorem 1.2.

4.1 Convergence of the approximation solutions

In this subsection we prove the convergence of the time-discrete solution.

Lemma 4.1

There exist functions

$$\begin{aligned} u&\in L^\infty (0,T;H_0^1(\Omega ))\cap C^{0,\beta }([0,T];L^4(\Omega )), \\ \mu&\in L^2(0,T;X), \end{aligned}$$

where $\beta = 1/8$ if $d=2$ and $\beta = 1/16$ if $d=3$, such that

$$\begin{aligned} \phi _N&\overset{*}{\rightharpoonup }\ u \quad \textrm{in} \; L^\infty (0,T;H_0^1(\Omega )), \end{aligned}$$

(4.1)

$$\begin{aligned} \bar{\phi }_N&\rightarrow u \quad \textrm{in}\; C^{0,\gamma }([0,T]; L^4(\Omega )) \quad \textrm{for} \; \textrm{any} \; \gamma \in (0,\beta ], \end{aligned}$$

(4.2)

$$\begin{aligned} \phi _N&\rightarrow u \quad \textrm{in}\; L^\infty (0,T;L^4(\Omega )), \end{aligned}$$

(4.3)

$$\begin{aligned} \phi _N&\rightarrow u \quad \mathrm{a.e}. \;\textrm{in}\; \Omega _T, \end{aligned}$$

(4.4)

$$\begin{aligned} \psi _N&\rightharpoonup \mu \quad \textrm{in}\; L^2(0,T;X), \end{aligned}$$

(4.5)

up to a subsequence as $N\rightarrow \infty$.

Proof

The bounds in Lemma 3.2 imply that there exist functions

$$\begin{aligned} u&\in L^\infty (0,T;H_0^1(\Omega )), \\ \mu&\in L^2(0,T;X), \end{aligned}$$

such that, after extraction of a subsequence,

$$\begin{aligned} \phi _N&\overset{*}{\rightharpoonup }\ u \quad \textrm{in}\; L^\infty (0,T;H_0^1(\Omega )), \\ \psi _N&\rightharpoonup \mu \quad \textrm{in}\; L^2(0,T;X), \end{aligned}$$

as $N\rightarrow \infty$.

Since $d=2$ or 3, by (3.22), $\{\bar{\phi }_N\}$ is bounded uniformly in $C([0,T];L^4(\Omega ))$. By Lemma 3.4, $\{\bar{\phi }_N\}$ is equicontinuous. So applying Arzelà–Ascoli theorem for Banach-valued functions (see Lemma 1 in [55]), we can extract of a subsequence of $\{\bar{\phi }_N\}$ (still denoted as $\{\bar{\phi }_N\}$) such that

$$\begin{aligned} \bar{\phi }_N \rightarrow u \quad \textrm{in}\; C([0,T];L^4(\Omega )) \quad \textrm{as}\; N\rightarrow \infty . \end{aligned}$$

(4.6)

Using Lemma 3.4, one can prove that $u\in C^{0,\beta }([0,T];L^4(\Omega ))$, for the value of $\beta$ indicated. For any $\gamma \in (0,\beta )$, we obtain by interpolation that

$$\begin{aligned} ||\cdot ||_{C^{0,\gamma }([0,T];L^4(\Omega ))} \le C || \cdot ||_{C^{0,\beta }([0,T];L^4(\Omega ))}^{\gamma /\beta } || \cdot ||_{C([0,T];L^4(\Omega ))}^{1-\gamma /\beta }. \end{aligned}$$

Hence it implies that

$$\begin{aligned} \bar{\phi }_N \rightarrow u \quad \textrm{in}\; C^{0,\gamma }([0,T];L^4(\Omega )) \quad \textrm{as}\; N\rightarrow \infty . \end{aligned}$$

This proves (4.2).

For any $t\in [0,T]$, we can find $n\in \{1,\ldots ,N\}$ and $\alpha \in (0,1]$ such that $t = (1-\alpha )(n-1)\tau + \alpha n\tau$. Then by the definitions of $\phi _N$ and $\bar{\phi }_N$, using Lemma 3.4 we have

$$\begin{aligned} ||\bar{\phi }_N(t) - \phi _N(t)||_{L^4(\Omega )}&= ||\alpha \phi ^n(n\tau ) + (1-\alpha )\phi ^{n-1}((n-1)\tau ) + \alpha \phi ^n(n\tau )||_{L^4(\Omega )} \nonumber \\&=(1-\alpha )||\phi ^n(n\tau ) - \phi ^{n-1} ((n-1)\tau )||_{L^4(\Omega )} \nonumber \\&\le C\tau ^{\beta }, \end{aligned}$$

(4.7)

for the value of $\beta$ indicated. Since $\tau = T/N$, taking limits as $N \rightarrow \infty$ in (4.7) we get

$$\begin{aligned} ||\bar{\phi }_N(t) - \phi _N(t)||_{L^4(\Omega )} \rightarrow 0 \quad \textrm{as}\; N \rightarrow \infty . \end{aligned}$$

(4.8)

Together with (4.2) we obtain (4.3). This also implies that $\phi _N \rightarrow u$ in $L^4(\Omega _T)$. Therefore we can extract a subsequence that converges almost everywhere in $\Omega _T$. This proves (4.4). $\square$

4.2 The existence of a weak solution

In this subsection we prove that the function u in Lemma 4.1 is a weak solution to the Eq. (1.12)–(1.16) in the sense of Theorem 1.2.

Pick any $\xi \in L^2(0,T;H^1(\Omega ))$ with $\partial _t\xi \in L^2(\Omega _T)$ and $\xi (T)=0$. Integrating both sides of the Eq. (3.32) with respect to t from 0 to T we get

$$\begin{aligned} \int _0^T\int _\Omega (\bar{\phi }_N - \Phi )\partial _t \xi \,dxdt = \int _0^T\int _\Omega \nabla \psi _N\cdot \nabla \xi \,dxdt. \end{aligned}$$

(4.9)

Using the convergence properties from Lemma 4.1 and taking limits as $N\rightarrow \infty$ in (4.9), we obtain (1.18).

By the definition of $\phi _N$, we have

$$\begin{aligned} \phi _N(x,0)&= \Phi (x) \quad \textrm{for}\; \textrm{all}\; x\in \Omega , \end{aligned}$$

(4.10)

$$\begin{aligned} \int _\Omega \phi _N(x,t) \,dx&= {\mathcal {M}}\quad \textrm{for}\; \textrm{all}\; t \in [0,T]. \end{aligned}$$

(4.11)

Since $\Omega$ is bounded, from (4.3) we get $\phi _N \rightarrow u$ in $L^\infty (0,T;L^1(\Omega ))$, which implies that

$$\begin{aligned} \int _\Omega u(x,t) \,dx = \lim \limits _{N \rightarrow \infty } \int _\Omega \phi _N(x,t) \,dx = {\mathcal {M}}\end{aligned}$$

(4.12)

for all $t \in [0,T]$. Moreover, we can extract a subsequence of $\{\phi _N(x,0)\}$ (still denoted as $\{\phi _N(x,0)\}$) such that $\phi _N (x,0) \rightarrow u(x,0)$ for a.e. $x \in \Omega$. Thus $u(x,0) =\Phi (x)$ for a.e. $x \in \Omega$.

Now we prove the relation (1.19) of u and $\mu$. Pick an arbitrary function $\eta \in L^2(0,T;X_0) \cap L^\infty (\Omega _T)$. By (3.15) and the definition of $\phi _N$, we have

$$\begin{aligned} \int _0^T\int _\Omega \psi _N\eta \,dxdt = \int _0^T \int _\Omega \nabla \phi _N \cdot \nabla \eta + W'(\phi _N)\eta \,dxdt. \end{aligned}$$

(4.13)

By (4.3) we have $\phi _N \rightarrow u$ in $L^\infty (0,T;L^3(\Omega ))$, which implies that $\phi _N \rightarrow u$ in $L^3(\Omega _T)$. Hence,

$$\begin{aligned} \int _0^T\int _\Omega \phi ^3_N \,dxdt \rightarrow \int _0^T\int _\Omega u^3 \,dxdt \quad \textrm{as}\; N\rightarrow \infty . \end{aligned}$$

(4.14)

Recalling that $W'(\phi _N) = \phi _N^3 - \phi _N$, using (4.14) and the convergence properties in Lemma 4.1, by taking limits as $N\rightarrow \infty$ in (4.13) we obtain (1.19).

4.3 The uniqueness of the weak solution

Before we prove that the weak solution is unique, we prove the following lemma:

Lemma 4.2

Let v be a function defined on $\Omega$ with $\int _\Omega v \,dx = 0$. For any two sequences $\{a_n\}_{n=1}^\infty , \{b_n\}_{n=1}^\infty$ with $a_n, b_n > 0$ for all $n=1,2,\ldots$, we define the cutoff function ${\mathcal {P}}_n$ to be

$$\begin{aligned} {\mathcal {P}}_n(s):= \left\{ \begin{array}{l} s, \; -a_n \le s\le b_n, \\ -a_n, \; s<-a_n, \\ b_n, \; s>b_n. \end{array} \right. \end{aligned}$$

(4.15)

Then there exist two sequences $\{a_n\}_{n=1}^\infty , \{b_n\}_{n=1}^\infty$ such that $\int _\Omega {\mathcal {P}}_n(v) \,dx = 0$ for all $n=1,2,\ldots$, and $\lim _{n \rightarrow \infty } {\mathcal {P}}_n(v(x)) = v(x)$ for a.e. $x \in \Omega$.

Proof

The case where v is bounded a.e. in $\Omega$ is trivial. We will prove Lemma 4.2 for the case ${{\,\mathrm{ess\,sup}\,}}_\Omega v =+\infty$. The case ${{\,\mathrm{ess\,inf}\,}}_\Omega v = -\infty$ is similar.

Assume that ${{\,\mathrm{ess\,sup}\,}}_\Omega v = +\infty$. For each $x \in \Omega$, we define

$$\begin{aligned} v_+(x) = \max \{v(x), 0\} \quad \textrm{and} \quad v_-(x) = \max \{-v(x), 0\}. \end{aligned}$$

(4.16)

Then $v = v_+ - v_-$, and since $\int _\Omega v \,dx = 0$, we have

$$\begin{aligned} \int _\Omega v_+ \,dx = \int _\Omega v_- \,dx. \end{aligned}$$

(4.17)

For each $\lambda > 0$, we define

$$\begin{aligned} {\mathcal {S}}_\lambda (s):= \left\{ \begin{array}{l} s, \; s\le \lambda , \\ \lambda , \; s>\lambda . \end{array} \right. \end{aligned}$$

(4.18)

Since ${{\,\mathrm{ess\,sup}\,}}_\Omega v = +\infty$, we have

$$\begin{aligned} \int _\Omega {\mathcal {S}}_n(v_+) \,dx < \int _\Omega v_+ \,dx \end{aligned}$$

(4.19)

for any $n=1,2,\ldots$, and we can choose $n_0$ large enough so that $\int _\Omega {\mathcal {S}}_{n_0}(v_+) \,dx > 0$. Since $\int _\Omega {\mathcal {S}}_\lambda (v_+) \,dx$ is increasing with respect to $\lambda$, then $\int _\Omega {\mathcal {S}}_n(v_+) \,dx > 0$ for all $n \ge n_0$. Moreover,

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _\Omega {\mathcal {S}}_n(v_+) \,dx = \int _\Omega v_+ \,dx. \end{aligned}$$

(4.20)

On the other hand,

$$\begin{aligned} \lim _{\lambda \rightarrow 0^+} \int _\Omega {\mathcal {S}}_\lambda (v_-) \,dx&= 0, \end{aligned}$$

(4.21)

$$\begin{aligned} \lim _{\lambda \rightarrow \infty } \int _\Omega {\mathcal {S}}_\lambda (v_-) \,dx&= \int _\Omega v_- \,dx = \int _\Omega v_+ \,dx. \end{aligned}$$

(4.22)

Thus, for each $n \ge n_0$, we have

$$\begin{aligned} \lim _{\lambda \rightarrow 0^+} \int _\Omega {\mathcal {S}}_\lambda (v_-) \,dx< \int _\Omega {\mathcal {S}}_n(v_+) \,dx < \lim _{\lambda \rightarrow \infty } \int _\Omega {\mathcal {S}}_\lambda (v_-) \,dx. \end{aligned}$$

(4.23)

Since $\int _\Omega {\mathcal {S}}_\lambda (v_-) \,dx$ is continuous with respect to $\lambda$, there exists $0< \lambda _n < \infty$ with

$$\begin{aligned} \int _\Omega {\mathcal {S}}_{\lambda _n}(v_-) \,dx = \int _\Omega {\mathcal {S}}_n(v_+) \,dx. \end{aligned}$$

(4.24)

Since $v_+$ and $v_-$ are nonnegative functions, we see that $\{\lambda _n\}_{n=n_0}^\infty$ is an increasing sequence. Then there are two cases:

Case 1: $\lim _{n \rightarrow \infty } \lambda _n = +\infty$. In this case, it is obvious that $-\lim _{n \rightarrow \infty } \lambda _n \le v(x) \le \lim _{n \rightarrow \infty } n$ for all $x \in \Omega$, and so the conclusion follows.

Case 2: $\lim _{n \rightarrow \infty } \lambda _n = \Lambda < +\infty$. We will prove that ${{\,\mathrm{ess\,sup}\,}}_\Omega v_- = \Lambda$.

If ${{\,\mathrm{ess\,sup}\,}}_\Omega v_- < \Lambda$, then there exists $n_1 > n_0$ such that ${{\,\mathrm{ess\,sup}\,}}_\Omega v_- < \lambda _{n_1}$. Thus,

$$\begin{aligned} \int _\Omega {\mathcal {S}}_{\lambda _{n_1}}(v_-) \,dx = \int _\Omega v_- \,dx, \end{aligned}$$

(4.25)

which is a contradiction since

$$\begin{aligned} \int _\Omega {\mathcal {S}}_{\lambda _n}(v_-) \,dx = \int _\Omega {\mathcal {S}}_n(v_+) \,dx < \int _\Omega v_+ \,dx = \int _\Omega v_- \,dx \end{aligned}$$

(4.26)

for all $n \ge n_0$.

If $\Lambda < {{\,\mathrm{ess\,sup}\,}}_\Omega v_-$, then $|\{x \in \Omega : \Lambda < v_- \le {{\,\mathrm{ess\,sup}\,}}_\Omega v_- \}| > 0$. Hence,

$$\begin{aligned} \int _\Omega {\mathcal {S}}_\Lambda (v_-) \,dx&= \int _{\{v_- \le \Lambda \}} v_- \,dx + \int _{\{v_-> \Lambda \}} \Lambda \,dx \nonumber \\&< \int _{\{v_- \le \Lambda \}} v_- \,dx + \int _{\{v_- > \Lambda \}} v_- \,dx = \int _\Omega v_- \,dx. \end{aligned}$$

(4.27)

Note that $\Lambda = \sup \{\lambda _n: n \ge n_0\}$, and since $\int _\Omega {\mathcal {S}}_\lambda (v_-) \,dx$ is increasing with respect to $\lambda$, we have $\int _\Omega {\mathcal {S}}_{\lambda _n} (v_-) \,dx \le \int _\Omega {\mathcal {S}}_\Lambda (v_-) \,dx$ for all $n \ge n_0$. Thus,

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _\Omega {\mathcal {S}}_{\lambda _n} (v_-) \,dx \le \int _\Omega {\mathcal {S}}_\Lambda (v_-) \,dx < \int _\Omega v_- \,dx, \end{aligned}$$

(4.28)

which is a contradiction since

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _\Omega {\mathcal {S}}_{\lambda _n} (v_-) \,dx = \lim _{n \rightarrow \infty } \int _\Omega {\mathcal {S}}_n(v_+) \,dx = \int _\Omega v_+ \,dx = \int _\Omega v_- \,dx. \end{aligned}$$

(4.29)

So we have ${{\,\mathrm{ess\,sup}\,}}_\Omega v_- = \Lambda$, that is, ${{\,\mathrm{ess\,inf}\,}}_\Omega v = -\Lambda$.

We relabel $\{\lambda _n\}_{n=n_0}^\infty$ and $\{n\}_{n=n_0}^\infty$ as $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$, respectively. By the above argument, we have $-\lim _{n \rightarrow \infty } a_n \le v(x) \le \lim _{n \rightarrow \infty } b_n$ for a.e. $x \in \Omega$, which implies that $\lim _{n \rightarrow \infty } {\mathcal {P}}_n(v(x)) = v(x)$ for a.e. $x \in \Omega$, where ${\mathcal {P}}_n$ is defined by (4.15). Moreover, for each $n=1,2,\ldots$, since ${\mathcal {P}}_n (v) = {\mathcal {S}}_{b_n} (v_+) - {\mathcal {S}}_{a_n} (v_-)$, by (4.24), we have $\int _\Omega {\mathcal {P}}_n(v) \,dx = 0$. This completes the proof of Lemma 4.2. $\square$

Now we prove the uniqueness of the weak solution of the Eq. (1.12)–(1.16). Assume that there are weak solutions $u_1, u_2 \in L^\infty (0,T;H_0^1(\Omega ))\cap C^{0,\beta }([0,T];L^4(\Omega ))$, where $\beta = 1/8$ if $d=2$ and $\beta = 1/16$ if $d=3$, in the sense of Theorem 1.2 with corresponding functions $\mu _1, \mu _2 \in L^2(0,T;H^1(\Omega ))$. We define

$$\begin{aligned} \bar{u}:= u_1 - u_2 \quad \textrm{and} \quad \bar{\mu }:= \mu _1 - \mu _2. \end{aligned}$$

(4.30)

By (1.18), we have

$$\begin{aligned} \int _0^T\int _\Omega \bar{u}\partial _t \xi \,dxdt = \int _0^T\int _\Omega \nabla \bar{\mu }\cdot \nabla \xi \,dxdt \end{aligned}$$

(4.31)

for any $\xi \in L^2(0,T;H^1(\Omega ))$ with $\partial _t\xi \in L^2(\Omega _T)$ and $\xi (T)=0$. For any $t_0 \in [0,T]$ and any $\zeta \in L^2(0,T;H^1(\Omega ))$, we define

$$\begin{aligned} \xi (x,t):=\left\{ \begin{array}{l} \int _t^{t_0} \zeta (x,s)ds, \; t\le t_0, \\ 0, \; t > t_0, \end{array}\right. \end{aligned}$$

(4.32)

for all $x \in \Omega$. Since $\xi$ is an admissible test function for (4.31), we have

$$\begin{aligned} -\int _0^{t_0}\int _\Omega \bar{u}\zeta \,dxdt&= \int _0^{t_0} \int _\Omega \nabla \bar{\mu }\cdot \nabla \left( \int _t^{t_0} \zeta ds \right) \,dxdt \nonumber \\&= \int _0^{t_0}\int _\Omega \nabla \left( \int _0^t \bar{\mu } ds \right) \cdot \nabla \zeta \,dxdt \end{aligned}$$

(4.33)

Since $t_0 \in [0,T]$ is arbitrary, this implies

$$\begin{aligned} -\int _\Omega \bar{u}\zeta \,dx = \int _\Omega \nabla \left( \int _0^t \bar{\mu } ds \right) \cdot \nabla \zeta \,dx \end{aligned}$$

(4.34)

for a.e. $t \in [0,T]$, which implies

$$\begin{aligned} (-\Delta )^{-1}\bar{u} = -\int _0^t \bar{\mu }ds + c \quad \textrm{and} \quad \partial _t(-\Delta )^{-1}\bar{u} = -\bar{\mu } \end{aligned}$$

(4.35)

for any $t \in [0,T]$, with some constant $c \in {\mathbb {R}}$. Choosing $\zeta = \bar{\mu }$ yields

$$\begin{aligned} -\int _0^{t_0}\int _\Omega \bar{u} \bar{\mu } \,dxdt&= \int _0^{t_0}\int _\Omega \nabla (-\Delta )^{-1} \bar{u} \cdot \nabla \partial _t(-\Delta )^{-1}\bar{u} \,dxdt \nonumber \\&= \frac{1}{2} \int _0^{t_0}\int _\Omega \frac{d}{dt} \left( \nabla (-\Delta )^{-1}\bar{u} \cdot \nabla (-\Delta )^{-1}\bar{u} \right) \,dxdt \nonumber \\&= \frac{1}{2} \int _\Omega \nabla (-\Delta )^{-1}\bar{u}(t_0) \cdot \nabla (-\Delta )^{-1}\bar{u}(t_0) \,dx \nonumber \\&= \frac{1}{2} ||\bar{u}(t_0)||^2_{X_0'}. \end{aligned}$$

(4.36)

From the weak formulation (1.19) we have

$$\begin{aligned} \int _0^T\int _\Omega \bar{\mu }\eta \,dxdt = \int _0^T \int _\Omega \nabla \bar{u} \cdot \nabla \eta + (W'(u_1) - W'(u_2))\eta \,dxdt \end{aligned}$$

(4.37)

for any $\eta \in L^2(0,T;X) \cap L^\infty (\Omega _T)$.

Now we consider the cutoff function ${\mathcal {P}}_n$ defined by (4.15). For each $t \in [0,t_0]$, since $\int _\Omega \bar{u}(x,t) \,dx = 0$, by Lemma 4.2, there exist two sequences $\{a_n(t)\}_{n=1}^\infty , \{b_n(t)\}_{n=1}^\infty$ with $\int _\Omega {\mathcal {P}}_n(\bar{u}(x,t)) dx = 0$ for all $n=1,2,\ldots$, and $\lim _{n \rightarrow \infty } {\mathcal {P}}_n(\bar{u}(x,t)) = \bar{u}(x,t)$ for a.e. $x \in \Omega$. Hence $\eta = \chi _{[0,t_0]}{\mathcal {P}}_n(\bar{u})$ is an admissible test function for (4.37). Recall that we write $W(u) = W_1(u) + W_2(u)$, where $W_1(u)=(u^4+1)/4$ and $W_2(u)=-u^2/2$. Since $W'_1(u) = u^3$ is an increasing function, then $(W'_1(u_1) - W'_1(u_2)){\mathcal {P}}_n(\bar{u}) \ge 0$ a.e. in $\Omega _T$. Plugging this estimate into (4.37) with $\eta = \chi _{[0,t_0]}{\mathcal {P}}_n(\bar{u})$, we have

$$\begin{aligned} \int _0^{t_0}\int _\Omega \bar{\mu }{\mathcal {P}}_n(\bar{u}) \,dxdt \ge \int _0^{t_0}\int _\Omega \nabla \bar{u} \cdot \nabla {\mathcal {P}}_n(\bar{u}) + (W'_2(u_1) - W'_2(u_2)){\mathcal {P}}_n(\bar{u}) \,dxdt. \end{aligned}$$

(4.38)

Taking the limits as $n \rightarrow \infty$ we obtain

$$\begin{aligned} \int _0^{t_0}\int _\Omega \bar{\mu }\bar{u} \,dxdt&\ge ||\nabla \bar{u}||^2_{L^2(\Omega _{t_0})} + \int _0^{t_0}\int _\Omega (W'_2(u_1) - W'_2(u_2))\bar{u} \,dxdt\nonumber \\&\ge ||\nabla \bar{u}||^2_{L^2(\Omega _{t_0})} - || \bar{u}||^2_{L^2(\Omega _{t_0})}. \end{aligned}$$

(4.39)

Combining with (4.36) we get

$$\begin{aligned} \frac{1}{2} ||\bar{u}(t_0)||^2_{X_0'} + ||\nabla \bar{u}||^2_{L^2(\Omega _{t_0})} \le ||\bar{u}||^2_{L^2(\Omega _{t_0})}. \end{aligned}$$

(4.40)

Using integration by parts, Hölder’s inequality and Young’s inequality we get

$$\begin{aligned} ||\bar{u}||^2_{L^2(\Omega )}&= \int _\Omega \nabla (-\Delta )^{-1} \bar{u} \cdot \nabla \bar{u} \,dx \nonumber \\&\le ||\bar{u}||_{X'_0} ||\nabla \bar{u}||_{L^2(\Omega )} \nonumber \\&\le \frac{1}{4}||\bar{u}||^2_{X'_0} + ||\nabla \bar{u}||^2_{L^2(\Omega )}. \end{aligned}$$

(4.41)

Combining (4.40) and (4.41) we obtain

$$\begin{aligned} ||\bar{u}(t_0)||^2_{X_0'} \le \frac{1}{2} \int _0^{t_0} ||\bar{u}(t)||^2_{X_0'}dt. \end{aligned}$$

(4.42)

Then Gronwall’s inequality implies that $||\bar{u}(t_0)||^2_{X_0'} = 0$. Since $t_0 \in [0,T]$ is arbitrary, we have

$$\begin{aligned} ||\bar{u}(t)||^2_{X_0'} = 0 \quad \textrm{for}\; \textrm{all}\; t \in [0,T], \end{aligned}$$

which implies that $\bar{u} = 0$ for all $t \in [0,T]$ and a.e. $x \in \Omega$. Thus $u_1 = u_2$ for all $t \in [0,T]$ and a.e. $x \in \Omega$.

4.4 The energy inequality

The last part of the proof is to prove the energy inequality (1.20).

Let $t\in [0,T]$ be arbitrary. From (4.3) we get $\phi _N \rightarrow u$ in $L^\infty (0,T;L^2(\Omega ))$, which implies that $\phi _N(t) \rightarrow u(t)$ in $L^2(\Omega )$ and a.e. in $\Omega$. Recall that $W(\psi _N) = W_1(\psi _N) + W_2(\psi _N)$, where $W_1(\psi _N)=(\psi _N^4+1)/4$ convex and $W_2(\psi _N)=-\psi _N^2/2$. Since the remaining term of E is convex and $\phi _N$, $\psi _N$ are piecewise constant in t, we obtain from (3.29) that

$$\begin{aligned}&E (u(t)) + \frac{1}{2}\int _0^t ||\nabla \mu (s)||_{L^2(\Omega )} ds \nonumber \\&\quad \le \liminf \limits _{N\rightarrow \infty } \left( E (\phi _N(t)) + \frac{1}{2}\int _0^t ||\nabla \psi _N(s)||_{L^2(\Omega )} ds \right) \le E (\Phi ). \end{aligned}$$

(4.43)

So (1.20) is established. This completes the proof of Theorem 1.2.

Appendix: Ladyzhenskaya’s inequality

The Ladyzhenskaya’s inequality was introduced by O. A. Ladyzhenskaya [41] to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in ${\mathbb {R}}^2$ when the initial data is sufficiently smooth. This inequality is a member of a class of inqualities known as interpolation inequalities.

Lemma 4.3

(Ladyzhenskaya’s inequality) Let $\Omega$ be a Lipschitz domain in ${\mathbb {R}}^d$ ($d=2$ or 3) and $f \in H_0^1(\Omega )$. Then there exists a constant $C > 0$ depending only on $\Omega$ such that

$$\begin{aligned} ||f||_{L^4(\Omega )} \le C ||f||_{L^2(\Omega )}^{1/2} ||\nabla f||_{L^2(\Omega )}^{1/2} \quad \textrm{if}\; d=2, \\ ||f||_{L^4(\Omega )} \le C ||f||_{L^2(\Omega )}^{1/4} ||\nabla f||_{L^2(\Omega )}^{3/4} \quad \textrm{if}\; d=3. \end{aligned}$$