\newsiamremark

remarkRemark \newsiamremarkhypothesisHypothesis \newsiamthmclaimClaim \headersWell-posedness of Cahn-Hilliard model for surface diffusionNiu, Xiang,Yan

Well-posedness of a modified degenerate Cahn-Hilliard model for surface diffusion

Xiaohua Niu School of Mathematics and Statistics, Xiamen University of Technology, Xiamen, China (). [email protected] Yang Xiang Department of Mathematics, The Hong Kong University of Science and Technollogy, Clear Water Bay, Kowloon, Hong Kong (). [email protected] Xiaodong Yan Department of Mathematics, The University of Connecticut, Storrs, CT, USA (). [email protected]

Abstract

We study the well-posedness of a modified degenerate Cahn-Hilliard type model for surface diffusion. With degenerate phase-dependent diffusion mobility and additional stabilizing function, this model is able to give the correct sharp interface limit. We introduce a notion of weak solutions for the nonlinear model. The existence result is obtained by approximations of the proposed model with nondegenerate mobilities. We also employ this method to prove existence of weak solutions to a related model where the chemical potential contains a nonlocal term originated from self-climb of dislocations in crystalline materials.

keywords:

Phase field model, degenerate Cahn-Hilliard equation, surface diffusion, well-posedness, weak solutions

{AMS}

35A01, 35G20, 35K25, 74N20, 82C26

1 Introduction

We consider the following modified degenerate Cahn-Hilliard type model

(1)		$\displaystyle g(u)\partial_{t}u$	$\displaystyle=$	$\displaystyle\nabla\cdot(M(u)\nabla\frac{\mu}{g(u)}),\ \ x\in\Omega\subset\mathbb{R}^{2},t\in[0,\infty)$
(2)		$\displaystyle\mu$	$\displaystyle=$	$\displaystyle-\Delta u+\frac{1}{\varepsilon^{2}}q^{\prime}(u).$

When $g\equiv 1$ , (1)-(2) becomes Cahn-Hilliard (CH) equation with degenerate mobility. The degenerate Cahn-Hilliard equation has been widely studied as a diffuse-interface model for phase separation in binary system [2, 3, 4, 5, 7, 10]. Over the years, the interface motion in the sharp limit has caught a lot of attention for various choice of mobility $M(u)$ and homogeneous free energy $q(u)$ . When $M(u)=1-u^{2}$ and $q$ being either the logarithmic free energy

q(u)=\frac{\theta}{2}\left[(1+u)\ln(1+u)+(1-u)\ln(1-u)\right]+\frac{1}{2}(1-u^{2})

with temperature $\theta=O(\varepsilon^{\alpha})$ or the double obstacle potential

q(u)=1-u^{2}\text{ for }|u|\leq 1,\text{ }q(u)=\infty\text{ otherwise},

Cahn, Elliott, and Novick-Cohen [19] showed via asymptotic expansions that the sharp-interface limit in the time scale $O(\varepsilon^{-2})$ is interface motion by surface diffusion. Sharp interface limits for different time scales were discussed in [5] for highly disparate diffusion mobility $M(u)=1+u$ and smooth double well $q(u)=\frac{1}{4}(1-u^{2})^{2}$ . In particular, the system evolves in $t=O(\varepsilon^{-2})$ time scale according to the combination of a one-sided modified Mullins–Sekerka problem in the phase with nonzero constant mobility and a nonlinear diffusion process that solves a quasi-stationary porous medium equation in the phase with small mobility. A later work by the same authors [6] derived sharp interface limit for $O(\varepsilon^{-2})$ time scale with diffusion mobility $M(u)=|1-u^{2}|$ and smooth double well potential $q(u)$ , noting the effect of the diffusion field on the interface motion as a jump of fluxes. The analysis was done on the (unphysical) solution branch with $|u|>1$ on some region. For $M(u)=1-u^{2}$ and $q=\frac{1}{4}(1-u^{2})^{2}$ , Lee, Münch and Süli [14] considered the physical branch of solution where $|u|<1$ everywhere and showed that there is an additional nonlinear bulk diffusion term appearing to leading order of the sharp interface limit. Further study in [15] indicates that the leading order sharp-interface motion depends sensitively on the choice of mobility.

The existence of weak solutions for degenerate Cahn-Hilliard equation was proved by Elliotte and Garcke [10] (see [24] for 1D case). Their results include the case $M(u)=1-u^{2}$ and $q$ being the logarithmic free energy. Dai and Du [7] introduced a different notion of weak solutions for degenerate Cahn-Hilliard equation with mobility $M(u)=|1-u^{2}|^{m}$ and smooth double well potentials; they showed that their model accommodates the Gibbs-Thomson effect, which was not by the method in [10].

There is a critical issue in modeling surface diffusion by the degenerate Cahn-Hilliard model [11, 21], due to the presence of incompatibility in the asymptotic matching between the outer and inner expansions. Rätz, Ribalta, and Voigt (RRV) [21] fixed this incompatibility by introducing a singular factor $1/g\left(u\right)$ in front of the chemical potential $\mu$ to force it to vanish in the far field. Their model essentially consists of equations (1)-(2) without the $g(u)$ term on the left side of (1), and other terms for modeling heteroepitaxial growth of thin films. The RRV model with the stabilizing function $g(u)$ has been validated by numerical simulations [21] and asymptotic analyses [11, 21]. It has been successfully generalized to many applications, e.g., growth of nanoscale membranes [1], dewetting of ultrathin films [17], and grain boundary formation in nanoporous metals [9]. Recently, a phase field model for dislocation self-climb by vacancy pipe diffusion was developed based on degenerate Cahn-Hilliard model with such stabilizing function [20]. However, to the best of our knowledge, well-posedness of these degenerate Cahn-Hilliard models with singular factor that give the correct sharp interface limit for surface diffusion has not been established in the literature.

In this paper, in order to prove the well-posedness of the RRV type Cahn-Hilliard model with correct sharp interface limit for surface diffusion, we propose a modified degenerated Cahn-Hilliard model as given in (1)-(2), and discuss its well-posedness and sharp interface limit. In particular, we have modified the original RRV model so that the equation can be written in the form of gradient flow of the total energy.

Our first result is a sharp interface limit equation for (1) and (2) via formal asymptotic analysis. We obtain the following sharp interface equation

(3)

v=\lambda\partial_{ss}(\alpha\kappa)

as $\varepsilon\rightarrow\infty$ . Here $\lambda<0$ , $\alpha<0$ are constants whose exact forms are derived in section 2. This validates this equation as a diffuse-interface model for surface diffusion.

Our main result concerns the well-posedness of the initial value problem of (1)-(2). For this purpose, we set $\Omega=[0,2\pi]^{n}$ and consider the following problem in a periodic setting when $n\leq 2$ .

(4)		$\displaystyle g(u)\partial_{t}u$	$\displaystyle=$	$\displaystyle\nabla\cdot(M(u)\nabla\frac{\mu}{g(u)}),\hskip 28.90755pt\text{ for }x\in\Omega,t\in[0,\infty)$
(5)		$\displaystyle\mu$	$\displaystyle=$	$\displaystyle-\Delta u+q^{\prime}(u).$

Here $g(u)=|1-u^{2}|^{m}$ for $2\leq m<\infty$ , $M(u)=M_{0}g(u)$ for some constant $M_{0}>0$ and $q(u)$ satisfies the following assumptions.

(i)

$q(u)\in C^{2}(\mathbb{R},\mathbb{R})$ and there exist constants $C_{i}>0$ , $i=1,\cdots,10$ such that for all $u\in\mathbb{R}$ and some $1\leq r<\infty$ , the following growth assumptions hold.

(6)	$\displaystyle C_{1}\|u\|^{r+1}-C_{2}\leq q(u)$	$\displaystyle\leq$	$\displaystyle C_{3}\|u\|^{r+1}+C_{4},$
(7)	$\displaystyle\|q^{\prime}(u)\|$	$\displaystyle\leq$	$\displaystyle C_{5}\|u\|^{r}+C_{6},$
(8)	$\displaystyle C_{t}\|u\|^{r-1}-C_{8}$	$\displaystyle\leq$	$\displaystyle q^{\prime\prime}(u)\leq C_{9}\|u\|^{r-1}+C_{10}.$

We see that the classical double well potential $q(u)=(1-u^{2})^{2}$ satisfies (6)-(8) with $r=3$ .

Our existence proof is obtained via approximations of the proposed model (4)-(5) with positive mobilities. Given any $\theta>0$ , we define

(9)

M_{\theta}(u):=M_{0}g_{\theta}(u)

with

(10)

g_{\theta}(u):=\left\{\begin{array}[]{cl}|1-u^{2}|^{m}&\text{ if }|1-u^{2}|>\theta,\\ \theta^{m}&\text{ if }|1-u^{2}|\leq\theta.\end{array}\right.

Our first step is to find a sufficiently regular solution for (4)-(5) with mobility $M_{\theta}(u)$ and stablizing function $g_{\theta}(u)$ together with a smooth potential $q(u)$ .

Theorem 1.1.

Let $M_{\theta},g_{\theta}$ be defined by (9) and (10), under the assumptions (6)-(8), for any $u_{0}\in H^{1}(\Omega)$ and any $T>0$ , there exists a function $u_{\theta}$ such that

a)

$u_{\theta}\in L^{\infty}(0,T;H^{1}(\Omega))\cap C([0,T];L^{p}(\Omega))\cap L^{2}(0,T;W^{3,s}(\Omega))$ , where $1\leq p<\infty$ , $1\leq s<2$ ,
b)

$\partial_{t}u_{\theta}\in L^{2}(0,T;(W^{1,q}(\Omega))^{\prime})$ for $q>2$ ,
c)

$u_{\theta}(x,0)=u_{0}(x)$ for all $x\in\Omega$ ,

which satisfies (4)-(5) in the following weak sense

	$\displaystyle\int_{0}^{T}<\partial_{t}u_{\theta},\phi>_{(W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega)}dt$
(11)		$\displaystyle=-\int_{0}^{T}\int_{\Omega}M_{\theta}(u_{\theta})\nabla\frac{-\Delta u_{\theta}+q^{\prime}(u_{\theta})}{g_{\theta}(u_{\theta})}\cdot\nabla\frac{\phi}{g_{\theta}(u_{\theta})}dxdt$

for all $\phi\in L^{2}(0,T;W^{1,q}(\Omega))$ with $q>2$ . In addition, the following energy inequality holds for all $t>0$ .

(12)		$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{\theta}(x,t)\|^{2}+q(u_{\theta}(x,t))\right)dx$
		$\displaystyle+\int_{0}^{t}\int_{\Omega}M_{\theta}(u_{\theta}(x,\tau)\left\|\nabla\frac{-\Delta u_{\theta}(x,\tau)+q^{\prime}(u_{\theta}(x,\tau)}{g_{\theta}(u_{\theta}(x,\tau))}\right\|^{2}dxd\tau$
	$\displaystyle\leq$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{0}(x)\|^{2}+q(u_{0}(x))\right)dx.$

Proof of theorem 1.1 is based on Galerkin approximations. Due to the presence of the stablizing function $g_{\theta}$ , it is not obvious how to pass to the limit in the nonlinear term of the Galerkin approximations. Our key observation in this step is strong convergence of $\nabla u^{N}$ (up to a subsequence) in $L^{2}(\Omega_{T})$ where $\Omega_{T}=\Omega\times[0,T])$ which allows us to pass to the limit in the nonlinear term.

To obtain the weak solution to (4), we consider the limit of $u_{\theta_{i}}$ for a sequence $\theta_{i}\downarrow 0$ . The key challenge is how to pass to the limit in both sides of (11). In the degenerate Cahn-Hilliard case, the estimates for the positive mobility approximations yield a uniform bound for $\partial_{t}u_{\theta_{i}}$ and it is straighforward to pass to the limit on the left hand side in the approximating equations. Moreover, the bound on $\partial_{t}u_{\theta_{i}}$ , together with bound on $u_{\theta_{i}}$ yields strong convergence of $\sqrt{M_{i}(u_{\theta_{i}})}$ in $C(0,T;L^{n}(\Omega))$ . By this and the weak convergence of $\sqrt{M_{i}(u_{\theta_{i}})}\nabla\mu_{\theta_{i}}$ in $L^{2}(\Omega_{T})$ , Dai and Du [7] showed (up to a subsequence) that $M_{\theta_{i}}(u_{\theta_{i}})\nabla\mu_{\theta_{i}}\rightharpoonup\sqrt{M(u)}\xi$ weakly in $L^{2}(0,T;L^{\frac{2n}{n+2}}(\Omega))$ where $\xi$ is the weak limit of $\sqrt{M_{i}(u_{\theta_{i}})}\nabla\mu_{\theta_{i}}$ . The main task left is to show $\sqrt{M(u)}\xi=M(u)(-\nabla\Delta u+q^{\prime\prime}(u)\nabla u)$ and the limit equation becomes a weak form Cahn-Hilliard equation. They [7] proved that this is almost true in the set where $u\neq\pm 1$ . Their main idea is the following. For small numbers $\delta_{j}$ monotonically decreasing to $0$ , they consider the limit in a subset $B_{j}$ of $\Omega_{T}$ where approximate solutions converges uniformly and $|\Omega_{T}\backslash B_{j}|<\delta_{j}$ . By decomposing $B_{j}=D_{j}\cup\tilde{D}_{j}$ where mobility is bounded from below uniformly in $D_{j}$ and controlled above in $\tilde{D}_{j}$ by suitable multiples of $\delta_{j}$ , they obtain the weak form equation for the limit function by passing to the limit of $u_{\theta_{i}}$ on $D_{j}$ then letting $j$ goes to $\infty$ . Under further regularity assumptions on $\nabla\Delta u$ , they obtained the explicit expression for $\xi$ in the weak form of the equation.

In this paper, we adapt their idea to our model. There are two main difficulties. The first obtacle is the bound estimate on $\partial_{t}u_{\theta_{i}}$ blows up when $\theta_{i}$ goes to zero and we can not pass to the limit on the left hand side of (11); secondly, due to the presence of the stablizing function $g$ on the right hand side, it is more complicated to derive an explicit expression of the weak limit of ${M_{i}(u_{\theta_{i}})}\nabla\frac{\mu_{\theta_{i}}}{g_{\theta_{i}}(u_{\theta_{i}})}$ in terms of $u$ on the right hand side of the limit equation. To overcome the first difficulty, we derive an alternative form of (11) by multiplying $g_{\theta}(u_{\theta})$ to both sides (valid due to regularity of $u_{\theta}$ , c.f. section 3.4 and equation (85)). From this, we obtain uniform estimates on $g_{\theta_{i}}(u_{\theta_{i}})\partial_{t}u_{\theta_{i}}$ which enables us to pass to the limit on the left hand side of the alternate equation (85). To find limit form on the right hand side of (85), we need convergence of $\sqrt{M_{\theta_{i}}(u_{\theta_{i}})}$ in $C(0,T;L^{p}(\Omega))$ . Due to the lack of control on $\partial_{t}u_{\theta_{i}}$ , such convergence can not be derived directly using Aubin-Lions Lemma [7]. Instead, we apply Aubin-Lions lemma to $G_{i}(u)=\int_{0}^{u}g_{\theta_{i}}(s)ds$ and derive convergence of $g_{\theta_{i}}(u_{\theta_{i}})$ (consequently on $M_{\theta_{i}}(u_{\theta_{i}})$ ) from convergence of $G_{i}$ through characterization of compact sets [22] in $L^{p}[0,T;B]$ . We then follow the idea in [7] to pass to the limit on the right hand side of (85). Finally, we identify an explicit expression of the weak limit of $\nabla\frac{\mu_{\theta_{i}}}{g_{\theta_{i}}(u_{\theta_{i}})}$ in terms of the weak limit $u$ under additional integrability assumptions on derivatives of $u$ .

Theorem 1.2.

For any $u_{0}\in H^{1}(\Omega)$ and $T>0$ , there exists a function $u:\Omega_{T}=\Omega\times[0,T]\rightarrow\mathbb{R}$ satisfying

i)

$u\in L^{\infty}(0,T;H^{1}(\Omega))\cap C([0,T];L^{s}(\Omega))$ , where $1\leq s<\infty$ ,
ii)

$g(u)\partial_{t}u\in L^{p}(0,T;(W^{1,q}(\Omega))^{\prime})$ for $1\leq p<2$ and $q>2$ .
iii)

$u(x,0)=u_{0}(x)$ for all $x\in\Omega$ ,

which solves (4)-(5) in the following weak sense

There exists a set $B\subset\Omega_{T}$ with $|\Omega_{T}\backslash B|=0$ and a function $\zeta:\Omega_{T}\rightarrow\mathbb{R}^{n}$ satisfying $\chi_{B\cap P}M(u)\zeta\in L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}}(\Omega,\mathbb{R}^{n}))$ such that

(13)

\int_{0}^{T}<g(u)\partial_{t}u,\phi>_{((W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega))}dt=-\int_{B\cap P}M(u)\zeta\cdot\nabla\phi dxdt

for all $\phi\in L^{p}(0,T;W^{1,q}(\Omega))$ with $p,q>2$ . Here $P:=\{(x,t)\in\Omega_{T}:|1-u^{2}|\neq 0\}$ is the set where $M(u),g(u)$ are nondegenerate and $\chi_{B\cap P}$ is the characteristic function of set $B\cap P$ .

Assume $u\in L^{2}(0,T;H^{2}(\Omega)).$ For any open set $U\in\Omega_{T}$ on which $g(u)>0$ and $\nabla\Delta u\in L^{p}(U)$ for some $p>1$ , we have

(14)

\zeta=\frac{-\nabla\Delta u+q^{\prime\prime}(u)\nabla u}{g(u)}-\frac{g^{\prime}(u)}{g^{2}(u)}\left(-\Delta u+q^{\prime}(u)\right)\nabla u\text{ a.e. in }U.

Moreover, the following energy inequality holds for all $t>0$

(15)				$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u(x,t)\|^{2}+q(u(x,t))\right)dz+\int_{\Omega_{r}\cap B\cap P}M(u(x,\tau))\|\zeta(x,\tau)\|^{2}dxd\tau$
(15)			$\displaystyle\leq$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{0}(x)\|^{2}+q(u_{0}(x))\right)dx.$

Adding an additional term $\frac{1}{\varepsilon}(-\Delta)^{\frac{1}{2}}u$ to the chemical potential in (2), we can apply the same method to derive existence of weak solutions of the modified model (see section 5 for further details). Such nonlocal model originates from the phase field model for self-climb of dislocation loops [20].

The paper is organized as follows. We shall derive sharp interface limit for (1) and (2) through formal asymptotic expansions in section 2. Section 3 is devoted to the proof of Theorem 1.1 and Theorem1.2 is proved in section 4. Similar existence theorems for the modified model with an additional nonlocal term added to the chemical potential are presented in section 5.

2 Sharp interface limit via asymptotic expansions

In this section, we perform a formal asymptotic analysis to obtain the sharp interface limit of the proposed phase field model (1)- (2)as $\varepsilon\rightarrow 0$ .

2.1 Outer expansions

We first perform expansion in the region far from the dislocations. Assume the expansion for $u$ is

(16)

u(x,y,t)=u^{(0)}(x,y,t)+u^{(1)}(x,y,t)\varepsilon+u^{(2)}(x,y,t)\varepsilon^{2}+\ldots

Correspondingly, we have

	$\displaystyle M(u)=M(u^{(0)})+M^{\prime}(u^{(0)})u^{(1)}\varepsilon+\left(M^{\prime}(u^{(0)})u^{(2)}+\frac{1}{2}M^{\prime\prime}\left(u^{(0)}\right)\left(u^{(1)}\right)^{2}\right)\varepsilon^{2}+\ldots,$
	$\displaystyle g(u)=g(u^{(0)})+g^{\prime}(u^{(0)})u^{(1)}\varepsilon+\left(g^{\prime}(u^{(0)})u^{(2)}+\frac{1}{2}g^{\prime\prime}(u^{(0)})\left(u^{(1)}\right)^{2}\right)\varepsilon^{2}+\ldots,$
	$\displaystyle q^{\prime}(u)=q^{\prime}(u^{(0)})+q^{\prime\prime}(u^{(0)})u^{(1)}\varepsilon+\left(q^{\prime\prime}(u^{(0)})u^{(2)}+\frac{1}{2}q^{(3)}(u^{(0)})\left(u^{(1)}\right)^{2}\right)\varepsilon^{2}+\ldots,$

We also expand chemical potential $\mu$ as

(17)

\mu=\frac{1}{\varepsilon^{2}}\left(\mu^{(0)}+\mu^{(1)}\varepsilon+\mu^{(2)}\varepsilon^{2}+\ldots\right).

and rewrite equation (1) as

(18)

g(u)\partial_{t}u=M_{0}\nabla\cdot(\nabla\mu-\mu\frac{g^{\prime}(u)}{g(u)}\nabla u).

Set

w=-\mu\frac{g^{\prime}(u)}{g(u)}=\frac{1}{\varepsilon^{2}}\left(w^{(0)}+w^{(1)}\varepsilon+w^{(2)}\varepsilon^{2}+\ldots\right).

Plugging the expansions into (18) and (2) and matching the coefficients of $\varepsilon$ powers in both equations, the $O(\frac{1}{\varepsilon^{2}})$ of (18) and (2) yields

(19)		$\displaystyle 0=\nabla\cdot\left(\nabla\mu^{(0)}+w^{(0)}\nabla u^{(0)}\right)$
(20)		$\displaystyle\mu^{(0)}=q^{\prime}(u^{(0)}).$

Since

w^{(0)}=\mu^{(0)}\frac{g^{\prime}(u^{(0)})}{g(u^{(0)})},

then $u^{(0)}=1$ or $u^{(0)}=-1$ satisfies equations (19)-(20). In particular, such choice of $u^{(0)}$ implies $\mu^{(0)}=0$ .

The $O(\frac{1}{\varepsilon})$ equation of (18) and (2) reduces to

(21)		$\displaystyle 0=\nabla\cdot\left(\nabla\mu^{(1)}+w^{(0)}\nabla u^{(1)}+w^{(1)}\nabla u^{(0)}\right),$
(22)		$\displaystyle\mu^{(1)}=q^{\prime\prime}(u^{(0)})u^{(1)}.$

Since $u^{(0)}=1$ or $-1$ , $u^{(1)}=0$ satisfies (21)-(22). Moreover, such choice of $u^{(1)}$ guarantees $\mu^{(1)}=0$ .

The $O(1)$ equation of (18) and (2), taking into account of the fact $u^{(0)}=\pm 1$ , $\mu^{(0)}=\mu^{(1)}=0$ , reduces to

(23)		$\displaystyle 0=\nabla\cdot\left(\nabla\mu^{(2)}+w^{(0)}\nabla u^{(2)}\right),$
(24)		$\displaystyle\mu^{(2)}=q^{\prime\prime}(u^{(0)})u^{(2)}.$

Thus $u^{(2)}=0$ satisfies (23)-(24). Moreover, such choice of $u^{(1)}$ guarantees $\mu^{(2)}=0$ .

In general, if $u^{(0)}=\pm 1$ , $u^{(1)}=u^{(2)}=\ldots=u^{(k+1)}=0$ , the $O(\varepsilon^{k})$ for $k\geq 1$ equation of (18) and (2) yields

(25)		$\displaystyle 0=\nabla\cdot\left(\nabla\mu^{(k+2)}+w^{(0)}\nabla u^{(k+2)}\right)$
(26)		$\displaystyle\mu^{(k+2)}=q^{\prime\prime}(u^{(0)})u^{(k+2)}.$

Thus $u^{(k+2)}=0$ satisfies (25) and(26).

In summary, the $u=1$ or $u=-1$ in the outer region.

2.2 Inner expansions

For the small inner regions near the dislocations, we introduce local coordinates near the dislocations. Considering a dislocation $C$ parameterized by arc length parameter $s$ . We denote a point on the dislocation by ${\bf r}_{0}(s)$ with tangent unit vector $\mathbf{t}(s)$ and inward normal vector $\mathbf{n}(s)$ . A point near the dislocation $C$ is expressed as

{\bf r}(s,d)={\bf r}_{0}(s)+d\mathbf{n}(s),

where $d$ is the signed distance from point ${\bf r}$ to the dislocation. Since the gradients fields are of order $O(\frac{1}{\varepsilon})$ , we introduce $\rho=\frac{d}{\varepsilon}$ and use coordinates $(s,\rho)$ in the inner region. Under this setting, we write $u(x,y,t)=U(s,\rho,t)$ and equation (1) can be written as

(27)		$\displaystyle\hskip 36.135ptg(U)\left(\partial_{t}U+\frac{1}{\varepsilon}v_{n}\partial_{\rho}U\right)=\frac{M_{0}}{1-\varepsilon\rho\kappa}\partial_{s}\left(\frac{1}{1-\varepsilon\rho\kappa}\left(\partial_{s}\mu-\mu\frac{g^{\prime}(U)}{g(U)}\partial_{s}U\right)\right)$
	$\displaystyle\hskip 137.31255pt+\frac{1}{\varepsilon^{2}}\frac{M_{0}}{1-\varepsilon\rho\kappa}\partial_{\rho}\left((1-\varepsilon\rho\kappa)\left(\partial_{\rho}\mu-\mu\frac{g^{\prime}(U)}{g(U)}\partial_{\rho}U\right)\right),$
(28)		$\displaystyle\hskip 72.26999pt\mu=-\frac{1}{1-\varepsilon\rho\kappa}\partial_{s}\left(\frac{1}{1-\varepsilon\rho\kappa}\partial_{s}U\right)-\frac{1}{\varepsilon^{2}}\frac{1}{1-\varepsilon\rho\kappa}\partial_{\rho}\left((1-\varepsilon\rho\kappa)\partial_{\rho}U\right)+\frac{1}{\varepsilon^{2}}q^{\prime}(U).$

Assume $\mu$ takes the same form expansion as (17) and the following expansions hold for $U$ within dislocation core region:

(29)

U(s,\rho,t)=U^{(0)}(\rho)+\varepsilon U^{(1)}(s,\rho,t)+\varepsilon^{2}U^{(2)}+\ldots.

Here we assume the leading order solution $U^{(0)}$ , which describes the dislocation core profile, remains the same at all points on the dislocation at any time.

Set

W=\mu\frac{g^{\prime}(U)}{g(U)}=\frac{1}{\varepsilon^{2}}\left(W^{(0)}+W^{(1)}\varepsilon+W^{(2)}\varepsilon^{2}+\ldots\right),

the leading order for equation (27) and (28) is $O(\frac{1}{\varepsilon^{4}})$ , which yields

(30)		$\displaystyle 0=\partial_{\rho}\left(\partial_{\rho}\mu^{(0)}-W^{(0)}\partial_{\rho}U^{(0)}\right),$
(31)		$\displaystyle\mu^{(0)}=-\partial_{\rho\rho}U^{(0)}+q^{\prime}(U^{(0)}).$

Substituting $W^{(0)}=\mu^{(0)}\frac{g^{\prime}(U^{(0)})}{g(U^{(0)})}$ into (30), we can rewrite (30) as

0=\partial_{\rho}\left(\partial_{\rho}\mu^{(0)}-\mu^{(0)}\partial_{\rho}\ln g(U^{(0)})\right).

Integrating this equation, we have

(32)

\partial_{\rho}\mu^{(0)}-\mu^{(0)}\partial_{\rho}\ln g(U^{(0)})=C_{0}(s).

Since $\mu^{(0)}=0$ in the outer region, we must have $\mu^{(0)}\rightarrow 0$ and $\partial_{\rho}\mu^{(0)}\rightarrow 0$ as $\rho\rightarrow\pm\infty$ . Therefore $C_{0}(s)=0$ . Dividing (32) by $\mu^{(0)}$ and integrating, we have $\mu^{(0)}=\tilde{C}_{0}(s)g(U^{(0)})$ . Since $\mu^{(0)}$ is independent of $s$ and is $0$ in the outer region, we must have $\tilde{C}_{0}(s)=0$ . Thus

(33)

\mu^{(0)}=-\partial_{\rho\rho}U^{(0)}+q^{\prime}(U^{(0)})=0.

Solution $U^{(0)}$ to (33) subject to far field condition $U^{(0)}{(+\infty)}=-1$ and $U^{(0)}{(-\infty)}=1$ can be found numerically (see [20] for example). In particular, $\partial_{\rho}U^{(0)}<0$ for all $\rho$ .

Next, the $O(\frac{1}{\varepsilon^{3}})$ equation of (27) and (28) yields, using $\mu^{(0)}=0$ , that

(34)		$\displaystyle 0=\partial_{\rho}\left(\partial_{\rho}\mu^{(1)}-W^{(1)}\partial_{\rho}U^{(0)}\right),$
(35)		$\displaystyle\mu^{(1)}=-\partial_{\rho\rho}U^{(1)}+\kappa\partial_{\rho}U^{(0)}+q^{\prime\prime}(U^{(0)})U^{(1)}.$

When $\mu^{(0)}=0$ , we have $W^{(1)}=\mu^{(1)}\frac{g^{\prime}(U^{(0)})}{g(U^{(0)})}$ . Sustituting into (34) and integrating, we have

(36)

\partial_{\rho}\mu^{(1)}-\mu^{(1)}\partial_{\rho}\ln g(U^{(0)})=C_{1}(s).

Matching with the outer solutions ( $\partial_{\rho}\mu^{(1)},\mu^{(1)}\rightarrow 0$ as $\rho\rightarrow\pm\infty$ ), we conclude that $C_{1}(s)=0$ . Dividing (36) by $\mu^{(1)}$ and integrating, we have $\mu^{(1)}=\tilde{C}_{1}(s)g(U^{(0)})$ . Thus (35) can be written as

(37)

LU^{(1)}=-\kappa\partial_{\rho}U^{(0)}+\tilde{C}_{1}(s)g(U^{(0)})

where $L=-\partial_{\rho\rho}+q^{\prime\prime}(U^{(0)})$ is a linear operator whose kernal is span $\{\partial_{\rho}U^{(0)}\}$ . (37) is sovlable iff the right hand side is perpendicular to the kernal of $L$ , i.e.

\int_{-\infty}^{+\infty}\left(-\kappa\partial_{\rho}U^{(0)}+\tilde{C}_{1}(s)g(U^{(0)})\right)\partial_{\rho}U^{(0)}d\rho=0.

From this, we conclude

\tilde{C}_{1}(s)=\alpha\kappa,

where positive constants $\alpha$ and $\beta$ are given by

\alpha=\frac{\int_{-\infty}^{+\infty}\left(\partial_{\rho}U^{(0)}\right)^{2}d\rho}{\int_{-\infty}^{+\infty}g(U^{(0)})\partial_{\rho}U^{(0)}d\rho}<0.

Therefore

(38)

\mu^{(1)}=\alpha\kappa g(U^{(0)}).

Letting $\overline{\mu}=\frac{\mu}{g(U)}$ , (27) can be written as

(39)				$\displaystyle g(U)\left(\partial_{t}U+\frac{1}{\varepsilon}v_{n}\partial_{\rho}U\right)$
(39)			$\displaystyle=$	$\displaystyle\frac{M_{0}}{1-\varepsilon\rho\kappa}\partial_{s}\left(\frac{g(U)}{1-\varepsilon\rho\kappa}\left(\partial_{s}\overline{\mu}\right)\right)+\frac{1}{\varepsilon^{2}}\frac{M_{0}}{1-\varepsilon\rho\kappa}\partial_{\rho}\left(\left(1-\varepsilon\rho\kappa\right)g(U)\partial_{\rho}\overline{\mu}\right)$

Using $\mu^{(0)}=0$ , $\partial_{\rho}\overline{\mu}^{(1)}=\partial_{\rho}\frac{\mu^{(1)}}{g(U^{(0)})}=0$ , the $O(\frac{1}{\varepsilon^{2}})$ order equation of (39) reduces to

\partial_{\rho}\left(g(U^{(0)})\partial_{\rho}U^{(2)}\right)=0.

Integrating with respect to $\rho$ , we have $g(U^{(0)})\partial_{\rho}U^{(2)}=C_{2}(s)$ . Matching with outer solutions, we must have $C_{2}(s)=0$ . Thus $\partial_{\rho}U^{(2)}=0$ which gives $U^{(2)}=\tilde{C}_{2}(s)$ .

Next we look at the $O(\frac{1}{\varepsilon})$ equation of (39). Using $\mu^{(0)}=0$ , $\partial_{\rho}\overline{\mu}^{(1)}=0$ and $\partial_{\rho}\overline{\mu}^{(2)}=0$ , we have

g(U^{(0)})v_{n}\partial_{\rho}U^{(0)}=M_{0}\partial_{s}\left(g(U^{(0)})\partial_{s}\overline{\mu}^{(1)}\right)+M_{0}\partial_{\rho}\left(g(U^{(0)})\partial_{\rho}\overline{\mu}^{(3)}\right).

Integrating this equation with respect to $\rho$ and matching with outer solutions yields

(40)

v_{n}=\lambda\partial_{ss}\overline{\mu}^{(1)}

where we used the fact that $g(U^{(0)})$ is independent of $s$ and

\lambda=\frac{M_{0}\int_{-\infty}^{+\infty}g(U^{(0)})d\rho}{\int_{-\infty}^{+\infty}g(U^{(0)})\partial_{\rho}U^{(0)}d\rho}<0.

By (38), we have $\overline{\mu}^{(1)}=\alpha\kappa$ , substitute this into (40), we obtain the sharp interface limit equation

(41)

v_{n}=\lambda\partial_{ss}\left(\alpha\kappa\right).

Remark 2.1.

Notice here the outer and inner expansions are similar to the expansions in [20]. We wrote out all details here for readers’ convenience.

3 Weak solution for phase field model with positive mobilities

In this section we prove existence of weak solutions for phase field model with positive mobilities. Let $\mathbb{Z}_{+}$ be the set of nonnegative integers and $\Omega=[0,2\pi]^{n}$ with $n\leq 2$ . We pick an orthonormal basis for $L^{2}(\Omega)$ as

\displaystyle\{\phi_{j}:j=1,2,\ldots\}=\left\{(2\pi)^{-{n/2}},\text{Re}\left(\pi^{-n/2}e^{i\xi\cdot x}\right),\text{Im}\left(\pi^{-n/2}e^{i\xi\cdot x}\right):\xi\in\mathbb{Z}^{n}_{+}\backslash\{0,\ldots,0\}\right\}.

Observe $\{\phi_{j}\}$ is also orthogonal in $H^{k}(\Omega)$ for any $k\geq 1$ . Here and throughout the paper, we denote $\Omega_{T}=(0,T)\times\Omega$ .

3.1 Galerkin approximations

Define

u^{N}(x,t)=\sum_{j=1}^{N}c_{j}^{N}(t)\phi_{j}(x),\hskip 36.135pt\mu^{N}(x,t)=\sum_{j=1}^{N}d^{N}_{j}(t)\phi_{j}(x),

where $\{c_{j}^{N},d^{N}_{j}\}$ satisfy

(42)	$\displaystyle\int_{\Omega}\partial_{t}u^{N}\phi_{j}dx$	$\displaystyle=$	$\displaystyle-\int_{\Omega}M_{\theta}(u^{N})\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\cdot\nabla\frac{\phi_{j}}{g_{\theta}(u^{N})}dx,$
(43)	$\displaystyle\int_{\Omega}\mu^{N}\phi_{j}dx$	$\displaystyle=$	$\displaystyle\int_{\Omega}\left(\nabla u^{N}\cdot\nabla\phi_{j}+q^{\prime}(u^{N})\phi_{j}\right)dx,$
(44)	$\displaystyle u^{N}(x,0)$	$\displaystyle=$	$\displaystyle\sum_{j=1}^{N}\left(\int_{\Omega}u_{0}\phi_{j}dx\right)\phi_{j}(x).$

(42)-(44) is an initial value problem for a system of ordinary equations for $\{c_{j}^{N}(t)\}$ . Since right hand side of (42) is continuous in $c_{j}^{N}$ , the system has a local solution.

Define energy functional

E(u)=\int_{\Omega}\left\{\frac{1}{2}|\nabla u|^{2}+q(u)\right\}dx.

Direct calculation yields

\frac{d}{dt}E(u^{N}(x,t))=-\int_{\Omega}M_{\theta}(u^{N})\left|\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right|^{2}dx,

integration over $t$ gives the following energy identity.

			$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u^{N}(x,t)\|^{2}+q(u^{N}(x,t))\right)dx$
			$\displaystyle+\int_{0}^{t}\int_{\Omega}M_{\theta}(u^{N}(x,\tau))\left\|\nabla\frac{\mu^{N}(x,\tau)}{g_{\theta}(u^{N}(x,\tau))}\right\|^{2}dxd\tau$
		$\displaystyle=$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u^{N}(x,0)\|^{2}+q(u^{N}(x,0))\right)dx$
		$\displaystyle\leq$	$\displaystyle\left\Arrowvert\nabla u_{0}\right\Arrowvert_{L^{2}(\Omega)}^{2}+C\left(\left\Arrowvert u_{0}\right\Arrowvert^{r+1}_{H^{1}{\Omega}}+\|\Omega\|\right)\leq C<\infty$

Here and throughout the paper, $C$ represents a generic constant possibly depending only on $T$ , $\Omega$ , $u_{0}$ but not on $\theta$ . Since $\Omega$ is bounded region, by growth assumption assumption (6) and Poincare’s inequality, the energy identity (3.1) implies $u^{N}\in L^{\infty}(0,T;H^{1}(\Omega))$ with

(46)

\left\Arrowvert u^{N}\right\Arrowvert_{L^{\infty}(0,T;H^{1}(\Omega))}\leq C\text{ for all }N,

and

(47)

\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega_{T})}\leq C\text{ for all }N.

By (46), the coefficients $\{c_{j}^{N}(t)\}$ are bounded in time, thus the system (42)-(44) has a global solution. In addition, by Sobolev embedding theorem and growth assumption (7) on $q^{\prime}(u)$ , we have

q^{\prime}(u^{N})\in L^{\infty}(0,T;L^{p}(\Omega)),\hskip 36.135ptM_{\theta}(u^{N})\in L^{\infty}(0,T;L^{p}(\Omega))

for any $1\leq p<\infty$ with

(48)		$\displaystyle\left\Arrowvert q^{\prime}(u^{N})\right\Arrowvert_{L^{\infty}(0,T;L^{p}(\Omega))}\leq C\text{ for all }N,$
(49)		$\displaystyle\left\Arrowvert M_{\theta}(u^{N})\right\Arrowvert_{L^{\infty}(0,T;L^{p}(\Omega))}\leq C\text{ for all }N.$

3.2 Convergence of $u^{N}$

Given $q>2$ and any $\phi\in L^{2}(0,T;W^{1,q}(\Omega))$ , let $\Pi_{N}\phi(x,t)=\sum_{j=1}^{N}\left(\int_{\Omega}\phi(x,t)\phi_{j}(x)dx\right)\phi_{j}(x)$ be the orthogonal projection of $\phi$ onto span $\{\phi_{j}\}_{j=1}^{N}$ . Then

			$\displaystyle\left\arrowvert\int_{\Omega}\partial_{t}u^{N}\phi dx\right\arrowvert=\left\arrowvert\int_{\Omega}\partial_{t}u^{N}\Pi_{N}\phi dx\right\arrowvert=\left\arrowvert\int_{\Omega}M_{\theta}(u^{N})\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\cdot\nabla\frac{\Pi_{N}\phi}{g_{\theta}(u^{N})}dx\right\arrowvert$
		$\displaystyle\leq$	$\displaystyle\left(\int_{\Omega}M_{\theta}(u^{N})\left\arrowvert\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\arrowvert^{2}dx\right)^{\frac{1}{2}}\left(\int_{\Omega}M_{\theta}(u^{N})\left\arrowvert\nabla\frac{\Pi_{N}\phi}{g_{\theta}(u^{N})}\right\arrowvert^{2}dx\right)^{\frac{1}{2}}.$

Since

\nabla\frac{\Pi_{N}\phi}{g_{\theta}(u^{N})}=\frac{1}{g_{\theta}(u^{N})}\nabla\Pi_{N}\phi-\Pi_{N}\phi\frac{g_{\theta}^{\prime}(u^{N})}{g^{2}_{\theta}(u^{N})}\nabla u^{N},

we have

			$\displaystyle\int_{\Omega}M_{\theta}(u^{N})\left\arrowvert\nabla\frac{\Pi_{N}\phi}{g_{\theta}(u^{N})}\right\arrowvert^{2}dx$
		$\displaystyle\leq$	$\displaystyle 2M_{0}\int_{\Omega}\left(\frac{1}{g_{\theta}(u^{N})}\left\arrowvert\nabla\Pi_{N}\phi\right\arrowvert^{2}+\frac{\|g^{\prime}_{\theta}(u^{N})\|^{2}}{g^{3}_{\theta}(u^{N})}\|\Pi_{N}\phi\|^{2}\|\nabla u^{N}\|^{2}\right)dx$
		$\displaystyle\leq$	$\displaystyle C(M_{0},\theta)\left(\left\Arrowvert\nabla\Pi_{N}\phi\right\Arrowvert^{2}_{L^{2}(\Omega)}+\left\Arrowvert\Pi_{N}\phi\right\Arrowvert^{2}_{L^{\infty}(\Omega)}\left\Arrowvert\nabla u^{N}\right\Arrowvert^{2}_{L^{2}(\Omega)}\right)\text{ here is where we need }q>2$
		$\displaystyle\leq$	$\displaystyle C(M_{0},\theta)\left(\left\Arrowvert\Pi_{N}\phi\right\Arrowvert^{2}_{W^{1,q}(\Omega)}\right)\leq C(M_{0},\theta)\left\Arrowvert\phi\right\Arrowvert^{2}_{W^{1,q}(\Omega)}.$

Therefore

(50)

\left\Arrowvert\partial_{t}u^{N}\right\Arrowvert_{L^{2}(0,T;(W^{1,q}(\Omega))^{\prime})}\leq C(M_{0},\theta)\text{ for all }N.

For $1\leq s<\infty$ , since $n\leq 2$ , by Sobolev embedding theorem and Aubin-Lions Lemma (see [22] and Remark 3.1) , the following embeddings are compact :

\left\{f\in L^{2}(0,T;H^{1}(\Omega)):\partial_{t}f\in L^{2}(0,T;(W^{1,q}(\Omega))^{\prime}\right\}\hookrightarrow L^{2}(0,T;L^{s}(\Omega)),

and

\left\{f\in L^{\infty}(0,T;H^{1}(\Omega)):\partial_{t}f\in L^{2}(0,T;(W^{1,q}(\Omega))^{\prime}\right\}\hookrightarrow C([0,T];L^{s}(\Omega)).

From this and the boundedness of $\{u^{N}\}$ and $\{\partial_{t}u^{N}\}$ , we can find a subsequence, and $u_{\theta}\in L^{\infty}(0,T;H^{1}(\Omega))$ such that as $N\rightarrow\infty$ ,

(51)	$\displaystyle u^{N}$	$\displaystyle\rightharpoonup$	$\displaystyle u_{\theta}\text{ weakly-* in }L^{\infty}(0,T;H^{1}(\Omega)),$
(52)	$\displaystyle u^{N}$	$\displaystyle\rightarrow$	$\displaystyle u_{\theta}\text{ strongly in }C([0,T];L^{s}(\Omega)),$
(53)	$\displaystyle u^{N}$	$\displaystyle\rightarrow$	$\displaystyle u_{\theta}\text{ strongly in }L^{2}(0,T;L^{s}(\Omega))\text{ and a.e. in }\Omega_{T},$
(54)	$\displaystyle\partial_{t}u^{N}$	$\displaystyle\rightharpoonup$	$\displaystyle\partial_{t}u_{\theta}\text{ weakly in }L^{2}(0,T;(W^{1,q}(\Omega))^{\prime})$

for $1\leq s<\infty$ . In addition

\left\Arrowvert u_{\theta}\right\Arrowvert_{L^{\infty}(0,T;H^{1}(\Omega))}\leq C,\hskip 36.135pt\left\Arrowvert\partial_{t}u_{\theta}\right\Arrowvert_{L^{2}(0,T;(W^{1,q}(\Omega))^{\prime})}\leq C(M_{0},\theta).

By (52), growth assumption (7) on $q^{\prime}(u^{N})$ , and general dominated convergence Theorem, we have for any $1\leq s<\infty$ ,

(55)		$\displaystyle M_{\theta}(u^{N})\rightarrow M_{\theta}(u_{\theta})\text{ strongly in }C([0,T];L^{s}(\Omega)),$
(56)		$\displaystyle\sqrt{M_{\theta}(u^{N})}\rightarrow\sqrt{M_{\theta}(u_{\theta})}\text{ strongly in }C([0,T];L^{s}(\Omega)),$
(57)		$\displaystyle q^{\prime}(u^{N})\rightarrow q^{\prime}(u_{\theta})\text{ strongly in }C([0,T];L^{s}(\Omega)).$

By (48) and (57), we have

(58)

q^{\prime}(u^{N})\rightharpoonup q^{\prime}(u_{\theta})\text{ weakly-* in }L^{\infty}([0,T];L^{s}(\Omega)).

Remark 3.1.

Let $X$ , $Y$ , $Z$ be Banach spaces with compact embedding $X\hookrightarrow Y$ and continuous embedding $Y\hookrightarrow Z$ . Then the embeddings

(59)

\{f\in L^{p}(0,T;X);\partial_{t}f\in L^{1}(0,T;Z)\}\hookrightarrow L^{p}(0,T;Y)

and

(60)

\{f\in L^{\infty}(0,T;X);\partial_{t}f\in L^{r}(0,T;Z)\}\hookrightarrow C([0,T];Y)

are compact for any $1\leq p<\infty$ and $r>1$ (Corollary 4, [22], see also [16]) . For convergence of $u^{N}$ , we apply this for $p=2=r$ with $X=H^{1}(\Omega)$ , $Y=L^{s}(\Omega)$ for $1\leq s<\infty$ and $Z=W^{1,q}(\Omega)^{\prime}$ .

3.3 Weak solution

By (47) and the lower bound on $M_{\theta}$ , we have

(61)

\left\Arrowvert\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega_{T})}\leq C\theta^{-\frac{m}{2}}.

By (43), (46) and (48), we have

			$\displaystyle\left\arrowvert\int_{\Omega}\frac{\mu^{N}\phi_{1}}{g_{\theta}(u_{N})}\right\arrowvert dx=\left\arrowvert\int_{\Omega}\mu^{N}\Pi_{N}\left(\frac{\phi_{1}}{g_{\theta}(u^{N})}\right)\right\arrowvert dx$
		$\displaystyle=$	$\displaystyle\left\arrowvert\int_{\Omega}\nabla u^{N}\cdot\nabla\Pi_{N}\left(\frac{\phi_{1}}{g_{\theta}(u^{N})}\right)dx+\int_{\Omega}q^{\prime}(u^{N})\Pi_{N}\left(\frac{\phi_{1}}{g_{\theta}(u^{N})}\right)dx\right\arrowvert$
		$\displaystyle=$	$\displaystyle\left\arrowvert\int_{\Omega}\nabla u^{N}\cdot\nabla\left(\frac{\phi_{1}}{g_{\theta}(u^{N})}\right)dx+\int_{\Omega}q^{\prime}(u^{N})\Pi_{N}\left(\frac{\phi_{1}}{g_{\theta}(u^{N})}\right)dx\right\arrowvert$
		$\displaystyle\leq$	$\displaystyle C\left(\theta^{-m-1}\left\Arrowvert\nabla u^{N}\right\Arrowvert_{L^{2}(\Omega)}^{2}+\theta^{-m}\left\Arrowvert q^{\prime}(u^{N})\right\Arrowvert_{L^{2}(\Omega)}\left\Arrowvert\phi_{1}\right\Arrowvert_{L^{2}(\Omega)}\right)$
		$\displaystyle\leq$	$\displaystyle C\theta^{-m-1}.$

(61),(3.3) and Poincare’s inequality yield

\left\Arrowvert\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(0,T;H^{1}(\Omega))}\leq C(\theta^{-m-1}+1).

Thus there exists a $w_{\theta}\in L^{2}(0,T;H^{1}(\Omega))$ and a subsequence of $\frac{\mu^{N}}{g_{\theta}(u^{N})}$ , not relabeled, such that

(63)

\frac{\mu^{N}}{g_{\theta}(u^{N})}\rightharpoonup w_{\theta}\text{ weakly in }L^{2}(0,T;H^{1}(\Omega)).

Therefore by (55), (63) and Sobolev embedding theorem, we have

(64)

\mu^{N}=g_{\theta}(u^{N})\cdot\frac{\mu^{N}}{g_{\theta}(u^{N})}\rightharpoonup\mu_{\theta}=g_{\theta}(u_{\theta})w_{\theta}\text{ weakly in }L^{2}(0,T;W^{1,s}(\Omega))

for any $1\leq s<2$ . Combining (56), (63)and (64), we have

(65)

\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\rightharpoonup\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\text{ weakly in }L^{2}(0,T;L^{q}(\Omega))

for any $1\leq q<2$ . By (47), we can improve this convergence to

(66)

\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\rightharpoonup\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\text{ weakly in }L^{2}(0,T;L^{2}(\Omega)).

By (43), we have

\int_{\Omega}\mu^{N}u^{N}dx=\int_{\Omega}\left(|\nabla u^{N}|^{2}dx+q^{\prime}(u^{N})u^{N}\right)dx.

Integrating with respect to $t$ from $0$ to $T$ , we have on $\Omega_{T}=\Omega\times[0,T]$ ,

\displaystyle\int_{\Omega_{T}}\mu^{N}(x,\tau)u^{N}(x,\tau)dxd\tau=\int_{\Omega_{T}}\left(\nabla u^{N}(x,\tau)|^{2}dx+q^{\prime}(u^{N}(x,\tau))u^{N}(x,\tau)\right)dxd\tau.

Passing to the limit in the equation above, by (53), (57) and (64), we have

(67)

\displaystyle\int_{\Omega_{T}}\mu_{\theta}u_{\theta}dxd\tau=\lim_{N\rightarrow\infty}\int_{\Omega_{T}}|\nabla u^{N}|^{2}dxd\tau+\int_{\Omega_{T}}q^{\prime}(u_{\theta})u_{\theta}dxd\tau

On the other hand,

			$\displaystyle\hskip 14.45377pt\int_{\Omega_{T}}\mu^{N}(x,\tau)u_{\theta}(x,\tau)dxd\tau=\int_{\Omega_{T}}\mu^{N}(x,\tau)\Pi_{N}u_{\theta}(x,\tau)dxd\tau$
		$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\left(\nabla u^{N}\cdot\nabla\Pi_{N}u_{\theta}(x,\tau)+q^{\prime}(u^{N})\Pi_{N}u_{\theta}(x,\tau)\right)dxd\tau$
		$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\left(\nabla u^{N}\cdot\nabla u_{\theta}(x,\tau)+q^{\prime}(u^{N})\Pi_{N}u_{\theta}(x,\tau)\right)dxd\tau.$

Since $\Pi_{N}u_{\theta}\rightarrow u_{\theta}$ strongly in $L^{2}(\Omega_{T})$ , by (51),(58) and (64), as $N\rightarrow\infty$ , (3.3) yields

(69)

\int_{\Omega_{T}}\mu_{\theta}u_{\theta}dxd\tau=\int_{\Omega_{T}}\left(|\nabla u_{\theta}|^{2}+q^{\prime}(u_{\theta}))u_{\theta}\right)dxd\tau.

(67) and (69) gives

(70)

\lim_{N\rightarrow\infty}\int_{\Omega_{T}}|\nabla u^{N}|^{2}dxd\tau=\int_{\Omega_{T}}|\nabla u_{\theta}|^{2}dxd\tau.

By (46), $\nabla u^{N}\rightharpoonup\nabla u_{\theta}$ weakly in $L^{2}(\Omega_{T})$ , thus (70) implies

(71)

\nabla u^{N}\rightarrow\nabla u_{\theta}\text{ strongly in }L^{2}(\Omega_{T}).

Since $g_{\theta}\geq\theta^{m}$ , (53) implies

(72)

\frac{g^{\prime}(u^{N})}{g_{\theta}^{\frac{3}{2}}(u^{N})}\rightarrow\frac{g^{\prime}_{\theta}(u_{\theta})}{g_{\theta}^{\frac{3}{2}}(u_{\theta})}\text{ a.e in }\Omega_{T}.

In addition, $\frac{g^{\prime}(u^{N})}{g_{\theta}^{\frac{3}{2}}(u^{N})}$ is bounded by

(73)

\left\arrowvert\frac{g^{\prime}(u^{N})}{g_{\theta}^{\frac{3}{2}}(u^{N})}\right\arrowvert\leq C\theta^{-1-\frac{m}{2}}.

It follows from (71), (72), (73) and generalized dominated convergence theorem (see Remark 3.2) that

(74)

\frac{g^{\prime}(u^{N})}{g_{\theta}^{\frac{3}{2}}(u^{N})}\nabla u^{N}\rightarrow\frac{g^{\prime}_{\theta}(u_{\theta})}{g_{\theta}^{\frac{3}{2}}(u_{\theta})}\nabla u_{\theta}\text{ strongly in }L^{2}(\Omega_{T}).

Let

f^{N}(t)=\left\Arrowvert\frac{g^{\prime}(u^{N}(x,t))}{g_{\theta}^{\frac{3}{2}}(u^{N}(x,t))}\nabla u^{N}(x,t)-\frac{g^{\prime}_{\theta}(u_{\theta}(x,t))}{g_{\theta}^{\frac{3}{2}}(u_{\theta}(x,t))}\nabla u_{\theta}(x,t)\right\Arrowvert_{L^{2}(\Omega)},

by (74), we can extract a subsequence of $f^{N}$ , not relabeled, such that $f^{N}(t)\rightarrow 0$ a.e. in (0,T). By Egorov’s theorem, for any given $\delta>0$ , there exists $T_{\delta}\subset[0,T]$ with $|T_{\delta}|<\delta$ such that $f^{N}(t)$ converges to $0$ uniformly on $[0,T]\backslash T_{\delta}$ .

Given $\alpha(t)\in L^{2}(0,T)$ , for any $\varepsilon>0$ , there exists $T_{\delta}\subset[0,T]$ with $|T_{\delta}|<\delta$ such that

(75)

\int_{T_{\delta}}\alpha^{2}(t)dt<\varepsilon.

Multiplying (42) by $\alpha(t)$ and integrating in time yield

			$\displaystyle\int_{0}^{T}\alpha(t)\int_{\Omega}\partial_{t}u^{N}\phi_{j}dxdt=\int_{0}^{T}\alpha(t)\int_{\Omega}M_{\theta}(u^{N})\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\cdot\nabla\frac{\phi_{j}}{g_{\theta}(u^{N})}dxdt$
		$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}M_{0}\alpha(t)\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\cdot\nabla\phi_{j}dxdt$
			$\displaystyle-\int_{\Omega_{T}}\alpha(t)\sqrt{M_{0}}\phi_{j}\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}\cdot\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}dxdt$
			$\displaystyle=I-II.$

Since $\alpha(t)\nabla\phi_{j}\in L^{2}(0,T;L^{2}(\Omega))$ , by (63) and (64), we have

(77)

I=\int_{\Omega_{T}}M_{0}\alpha(t)\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\cdot\nabla\phi_{j}dxdt\rightarrow\int_{\Omega_{T}}M_{0}\alpha(t)\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla\phi_{j}dxdt.

To prove convergence on $II$ , observe

			$\displaystyle\int_{\Omega_{T}}\alpha(t)\phi_{j}\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}dxdt$
			$\displaystyle-\int_{\Omega_{T}}\alpha(t)\phi_{j}\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}dxdt$
		$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\alpha(t)\phi_{j}\left(\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right)\cdot\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}dxdt$
			$\displaystyle+\int_{\Omega_{T}}\alpha(t)\phi_{j}\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\cdot\left(\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}-\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\right)dxdt$
		$\displaystyle=$	$\displaystyle II_{1}+II_{2}$

From bound

			$\displaystyle\int_{\Omega_{T}}\left\arrowvert\alpha(t)\phi_{j}\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right\arrowvert^{2}dxdt$
		$\displaystyle\leq$	$\displaystyle C\theta^{-2-m}\left\Arrowvert\nabla u_{\theta}\right\Arrowvert_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}\int_{0}^{T}\alpha^{2}(t)^{2}dt,$

we conclude that $\alpha(t)\phi_{j}\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\in L^{2}(\Omega_{T})$ . By (66), we can pass to the limit in $II_{2}$ and conclude

II_{2}=\int_{\Omega_{T}}\alpha(t)\phi_{j}\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\cdot\left(\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}-\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\right)dxdt\rightarrow 0.

To pass to the limit in $II_{1}$ , we write

$\displaystyle II_{1}$	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}\alpha(t)\phi_{j}\left(\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right)\cdot\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}dxdt$
	$\displaystyle=$	$\displaystyle\int_{T_{\delta}}\int_{\Omega}\alpha(t)\phi_{j}\left(\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right)\cdot\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}dxdt$
		$\displaystyle+\int_{[0,T]\backslash T_{\delta}}\int_{\Omega}\alpha(t)\phi_{j}\left(\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right)\cdot\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}dxdt$
	$\displaystyle=$	$\displaystyle II_{11}+II_{12}.$

By (47), (51), (73) and (75), we can bound $II_{11}$ by

$\displaystyle\|II_{11}\|$	$\displaystyle\leq$	$\displaystyle\int_{T_{\delta}}\|\alpha(t)\|\left\Arrowvert\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right\Arrowvert_{L^{2}(\Omega)}\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega)}dt$
	$\displaystyle\leq$	$\displaystyle\left\Arrowvert\alpha(t)\right\Arrowvert_{L^{2}(T_{\delta})}\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega_{T})}\left\Arrowvert\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right\Arrowvert_{L^{\infty}(0,T;L^{2}(\Omega))}$
	$\displaystyle\leq$	$\displaystyle C(\theta)\varepsilon.$

For $II_{12}$ , we have

	$\displaystyle\|II_{12}\|$	$\displaystyle\leq$	$\displaystyle\int_{[0,T]\backslash T_{\delta}}\|\alpha(t)\|\left\Arrowvert\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right\Arrowvert_{L^{2}(\Omega)}\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega)}dt$
		$\displaystyle=$	$\displaystyle\int_{[0,T]\backslash T_{\delta}}\|\alpha(t)\|f^{N}(t)\|\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega)}dt.$

Since $f^{N}(t)$ converges to $0$ uniformly on $[0,T]\backslash T_{\delta}$ , $\alpha(t)\in L^{2}(0,T)$ and $\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega_{T})}\leq C$ , letting $N\rightarrow\infty$ in $II_{12}$ yields $II_{12}\rightarrow 0$ . Letting $\varepsilon\rightarrow 0$ , we conclude $II_{1}\rightarrow 0$ as $N\rightarrow\infty$ . Passing to the limit in (3.3), we have

(79)				$\displaystyle\int_{0}^{T}\alpha(t)\int_{\Omega}\left<\partial_{t}u_{\theta},\phi_{j}\right>_{(W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega))}dt$
(79)			$\displaystyle=$	$\displaystyle-\int_{\Omega_{T}}\alpha(t)M_{\theta}(u_{\theta})\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla\frac{\phi_{j}}{g_{\theta}(u_{\theta})}dxdt.$

Fix $q>2$ , given any $\phi\in L^{2}(0,T;W^{1,q}(\Omega))$ , its Fourier series $\sum_{j=1}^{\infty}a_{j}(t)\phi_{j}(x)$ converges strongly to $\phi$ in $L^{2}(0,T;W^{1,q}(\Omega))$ . Hence

			$\displaystyle\int_{\Omega_{T}}M_{\theta}(u_{\theta})\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla\frac{\phi-\Pi_{N}\phi}{g_{\theta}(u_{\theta})}dxdt$
		$\displaystyle=$	$\displaystyle\int_{\Omega_{T}}M_{0}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla(\phi-\Pi_{N}\phi)dxdt$
			$\displaystyle-\int_{\Omega_{T}}(\phi-\Pi_{N}\phi)\sqrt{M_{0}}\frac{g_{\theta}^{\prime}(u_{\theta})}{g_{\theta}^{\frac{3}{2}}(u_{\theta})}\nabla u_{\theta}\cdot\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}dxdt$
		$\displaystyle=$	$\displaystyle J_{1}-J_{2},$

where

J_{1}=\int_{\Omega_{T}}M_{0}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla(\phi-\Pi_{N}\phi)dxdt\rightarrow 0

by(63), (64) and strong convergence of $\Pi_{N}\phi$ to $\phi$ in $L^{2}(0,T;H^{1}(\Omega))$ . We can bound $J_{2}$ by

			$\displaystyle\|J_{2}\|=\left\arrowvert\int_{\Omega_{T}}(\phi-\Pi_{N}\phi)\sqrt{M_{0}}\frac{g_{\theta}^{\prime}(u_{\theta})}{g_{\theta}^{3/2}(u_{\theta})}\nabla u_{\theta}\cdot\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}dxdt\right\arrowvert$
		$\displaystyle\leq$	$\displaystyle\sqrt{M_{0}}\int_{0}^{T}\left\Arrowvert\phi-\Pi_{N}\phi\right\Arrowvert_{L^{\infty}(\Omega)}\left\Arrowvert\frac{g_{\theta}^{\prime}(u_{\theta})}{g_{\theta}^{3/2}(u_{\theta})}\nabla u_{\theta}\right\Arrowvert_{L^{2}(\Omega)}\left\Arrowvert\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\right\Arrowvert_{L^{2}(\Omega)}$
		$\displaystyle\leq$	$\displaystyle\sqrt{M_{0}}\left\Arrowvert\frac{g_{\theta}^{\prime}(u_{\theta})}{g_{\theta}^{3/2}(u_{\theta})}\nabla u_{\theta}\right\Arrowvert_{L^{\infty}(0,T;L^{2}(\Omega))}\left\Arrowvert\sqrt{M_{\theta}(u_{\theta})}\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\right\Arrowvert_{L^{2}(\Omega_{T})}\left\Arrowvert\phi-\Pi_{N}\phi\right\Arrowvert_{L^{2}(0,T;W^{1,q}(\Omega))}$
		$\displaystyle\rightarrow$	$\displaystyle 0\text{ as }N\rightarrow\infty.$

Consequently (79) and (3.3) imply

(81)

\int_{0}^{T}\left<\partial_{t}u_{\theta},\phi\right>_{(W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega))}dt=-\int_{\Omega_{T}}M_{\theta}(u_{\theta})\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla\frac{\phi}{g_{\theta}(u_{\theta})}dxdt

for all $\phi\in L^{2}(0,T;W^{1,q}(\Omega))$ with $q>2$ . Moreover, since $u^{N}(x,0)=\Pi_{N}u_{0}(x)\rightarrow u_{0}(x)$ in $H^{1}(\Omega)$ , we see that $u_{\theta}(x,0)=u_{0}(x)$ by (52).

Remark 3.2.

(Generalized dominated convergence theorem) Assume $E\subset\mathbb{R}^{n}$ is measurable. $g_{n}\rightarrow g$ strongly in $L^{q}(E)$ for $1\leq q<\infty$ and $f_{n}$ , $f$ : $E\rightarrow\mathbb{R}^{n}$ are measurable functions satisfying

f_{n}\rightarrow f\text{ a.e. in }E;\hskip 7.22743pt|f_{n}|^{p}\leq|g_{n}|^{q}\text{ a.e. in }E

with $1\leq p<\infty$ , then $f_{n}\rightarrow f$ in $L^{p}(E)$ .

3.4 Regularity of $u_{\theta}$

We now consider the regularity of $u_{\theta}$ . Given any $a_{j}(t)\in L^{2}(0,T)$ , $a_{j}(t)\phi_{j}\in L^{2}(0,T;C(\overline{\Omega})$ ). Integrating (43) from $0$ to $T$ , by (58),(64) and (71), we have

\displaystyle\int_{\Omega_{T}}\mu_{\theta}(x,t)a_{j}(t)\phi_{j}(x)dxdt=\int_{\Omega_{T}}\left(\nabla u_{\theta}\cdot a_{j}(t)\nabla\phi_{j}+q^{\prime}(u_{\theta})a_{j}(t)\phi_{j}\right)dxdt

for all $j\in{\bf N}$ . Given any $\phi\in L^{2}(0,T;H^{1}(\Omega))$ , its Fouirier series strongly converges to $\phi$ in $L^{2}(0,T;H^{1}(\Omega))$ , therefore

(82)

\displaystyle\int_{\Omega_{T}}\mu_{\theta}(x,t)\phi(x)dxdt=\int_{\Omega_{T}}\left(\nabla u_{\theta}\cdot\nabla\phi+q^{\prime}(u_{\theta})\phi\right)dxdt.

Recall $\mu_{\theta}\in L^{2}(0,T;L^{p}(\Omega))$ and $q^{\prime}(u_{\theta})\in L^{\infty}(0,T;L^{p}(\Omega))$ for any $1\leq p<\infty$ , regularity theory implies $u_{\theta}\in L^{2}(0,T;W^{2,p}(\Omega))$ . Hence

(83)

\mu_{\theta}=-\Delta u_{\theta}+q^{\prime}(u_{\theta})\text{ a.e. in }\Omega_{T}.

Since growth assumption on $q$ implies $|q^{\prime\prime}(u)|\leq C(1+|u|^{r-1})$ , pick $p>2$ , we have

			$\displaystyle\int_{\Omega}\|\nabla q^{\prime}(u_{\theta})\|^{2}dx=\int_{\Omega}\|q^{\prime\prime}(u_{\theta})\|^{2}\|\nabla u_{\theta}\|^{2}dx$
		$\displaystyle\leq$	$\displaystyle\left\Arrowvert q^{\prime\prime}(u_{\theta})\right\Arrowvert_{L^{\frac{2p}{p-2}}(\Omega)}^{2}\left\Arrowvert\nabla u_{\theta}\right\Arrowvert_{L^{p}(\Omega)}^{2}$
		$\displaystyle\leq$	$\displaystyle C\left(1+\left\Arrowvert u_{\theta}\right\Arrowvert_{L^{\frac{2p}{p-2}(r-1)}(\Omega)}^{2(r-1)}\right)\left\Arrowvert\nabla u_{\theta}\right\Arrowvert_{L^{p}(\Omega)}^{2}$
		$\displaystyle\leq$	$\displaystyle C\left(1+\left\Arrowvert u_{\theta}\right\Arrowvert^{2(r-1)}_{L^{\infty}(0,T;H^{1}(\Omega))}\right)\left\Arrowvert\nabla u_{\theta}\right\Arrowvert_{L^{p}(\Omega)}^{2}$

Therefore $\nabla q^{\prime}(u_{\theta})=q^{\prime\prime}(u_{\theta})\nabla u_{\theta}\in L^{2}(\Omega_{T})$ with

\displaystyle\int_{\Omega_{T}}|\nabla q^{\prime}(u_{\theta})|^{2}dxdt\leq\left(1+\left\Arrowvert u_{\theta}\right\Arrowvert^{2(r-1)}_{L^{\infty}(0,T;H^{1}(\Omega))}\right)\left\Arrowvert\nabla u_{\theta}\right\Arrowvert_{L^{2}(0,T;L^{p}(\Omega))}^{2}.

Hence $q^{\prime}(u_{\theta})\in L^{2}(0,T;H^{1}(\Omega))$ , combined with $\mu_{\theta}\in L^{2}(0,T;W^{1,s}(\Omega))$ for any $1\leq s<2$ , we have $u_{\theta}\in L^{2}(0,T;W^{3,s}(\Omega))$ and

(84)

\nabla\mu_{\theta}=-\nabla\Delta u_{\theta}+q^{\prime\prime}(u_{\theta})\nabla u_{\theta}\text{ a.e. in }\Omega_{T}.

Regularity of $u_{\theta}$ implies $\nabla u_{\theta}\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;L^{\infty}(\Omega))$ . A simple interpolation shows $\nabla u_{\theta}\in L^{\frac{2\mu}{\mu-2}}(0,T;L^{\mu}(\Omega))$ for any $\mu>2$ . Given any $\phi\in L^{p}(0,T;W^{1,q}(\Omega))$ with $p>2$ and $q>2$ , we have $g_{\theta}(u_{\theta})\phi\in L^{2}(0,T;W^{1.r}(\Omega))$ for any $r<\min(p,q)$ . From this, we can pick $g_{\theta}(u_{\theta})\phi$ as a test function in (81), we have

(85)

\int_{\Omega_{T}}\partial_{t}u_{\theta}g_{\theta}(u_{\theta})\phi dxdt=-\int_{\Omega_{T}}M_{\theta}(u_{\theta})\nabla\frac{\mu_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla\phi dxdt

for any $\phi\in L^{p}(0,T;W^{1,q}(\Omega))$ with $p,q>2$ .

Remark 3.3.

In fact, since $M_{\theta}(u_{\theta})\in L^{\infty}(0,T;L^{p}(\Omega))$ for $1\leq p<\infty$ , the right hand side of (85) is well defined for any $\phi\in L^{2}(0,T,W^{1,q}(\Omega))$ for $q>2$ and we can extend (85) to hold for all $\phi\in L^{2}(0,T,W^{1,q}(\Omega))$ .

3.5 Energy Inequality

Since $u^{N}$ and $\mu^{N}$ satisfies energy identity (3.1), passing to the limit as $N\rightarrow\infty$ and using strong convergence of $u^{N}(x,0)$ to $u_{0}$ in $H^{1}(\Omega)$ , together with the weak convergence of $u^{N}$ , $q^{\prime}(u^{N})$ and $\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}$ , the energy inequality (12) follows.

4 Phase field model with degenerate mobility

In this section, we prove theorem 1.2. Fix initial data $u_{0}\in H^{1}(\Omega)$ . We pick a montone decreasing positive sequence $\theta_{i}$ with $\lim_{i\rightarrow\infty}\theta_{i}=0$ . By theorem 1.1 and (85), for each $\theta_{i}$ , there exists

u_{i}\in L^{\infty}(0,T;H^{1}(\Omega))\cap L^{2}(0,T;W^{3,s}(\Omega))\cap C([0,T];L^{p}(\Omega))

with weak derivative

\partial_{t}u_{i}\in L^{2}(0,T;(W^{1,q}(\Omega))^{\prime}),

where $1\leq p<\infty$ , $1\leq s<2$ , $q>2$ such that $u_{\theta_{i}}(x,0)=u_{0}(x)$ and for all $\phi\in L^{2}(0,T;W^{1,q}(\Omega))$ ,

(86)		$\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}u_{i}\phi dxdt$	$\displaystyle=$	$\displaystyle-\int_{0}^{T}\int_{\Omega}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\nabla\frac{\phi}{g_{i}(u_{i})}dxdt,$
(87)		$\displaystyle\mu_{i}$	$\displaystyle=$	$\displaystyle-\Delta u_{i}+q^{\prime}(u_{i}).$

Moreover, for all $\psi\in L^{p}(0,T;W^{1,q}(\Omega))$ with $p,q>2$ , the following holds:

(88)

\int_{0}^{T}\int_{\Omega}g_{i}(u_{i})\partial_{t}u_{i}\psi dxdt=-\int_{0}^{T}\int_{\Omega}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\nabla\psi dxdt

Here we write $u_{i}=u_{\theta_{i}}$ , $M_{i}(u_{i})=M_{\theta_{i}}(u_{\theta_{i}})$ , $g_{i}(u_{i})=g_{\theta_{i}}(u_{\theta_{i}})$ for simplicity of notations.

4.1 Convergence of $u_{i}$ and equation for the limit function

Noticing the bound in (46) and (47) only depends on $u_{0}$ , we can find a constant $C$ , independent of $\theta_{i}$ such that

(89)		$\displaystyle\left\Arrowvert u_{i}\right\Arrowvert_{L^{\infty}(0,T;H^{1}(\Omega))}\leq C,$
(90)		$\displaystyle\left\Arrowvert\sqrt{M_{i}(u_{i})}\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\right\Arrowvert_{L^{2}(\Omega_{T})}\leq C.$

Growth condition on $q^{\prime}$ , and Sobolev embedding theorem give

	$\displaystyle\left\Arrowvert q^{\prime}(u_{i})\right\Arrowvert_{L^{\infty}(0,T;L^{p}(\Omega))}\leq C,$
	$\displaystyle\left\Arrowvert M_{i}(u_{i})\right\Arrowvert_{L^{\infty}(0,T;L^{p}(\Omega))}\leq C$

for any $1\leq p<\infty$ . By (88), for any $\phi\in L^{p}(0,T;W^{1,q}(\Omega))$ with $p,q>2$ ,

			$\displaystyle\left\arrowvert\int_{0}^{T}\int_{\Omega}g_{i}(u_{i})\partial_{t}u_{i}\phi dxdt\right\arrowvert=\left\arrowvert\int_{0}^{T}\int_{\Omega}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\nabla\phi dxdt\right\arrowvert$
		$\displaystyle\leq$	$\displaystyle\int_{0}^{T}\left(\left\Arrowvert\sqrt{M_{i}(u_{i})}\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\right\Arrowvert_{L^{2}(\Omega)}\left\Arrowvert\sqrt{M_{i}(u_{i})}\right\Arrowvert_{L^{\frac{2q}{q-2}}(\Omega)}\left\Arrowvert\nabla\phi\right\Arrowvert_{L^{q}(\Omega)}\right)dt$
		$\displaystyle\leq$	$\displaystyle\left\Arrowvert M_{i}(u_{i})\right\Arrowvert^{\frac{1}{2}}_{L^{\frac{p}{p-2}}(0,T;L^{\frac{q}{q-2}}(\Omega))}\left\Arrowvert\sqrt{M_{i}(u_{i})}\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\right\Arrowvert_{L^{2}(\Omega_{T})}\left\Arrowvert\nabla\phi\right\Arrowvert_{L^{p}(0,T;L^{q}(\Omega))}$
		$\displaystyle\leq$	$\displaystyle C\left\Arrowvert\phi\right\Arrowvert_{L^{p}(0,T;W^{1,q}(\Omega))}.$

Let

(92)

G_{i}(u_{i})=\int_{0}^{u_{i}}g_{i}(a)da.

Thus (4.1) yields $\partial_{t}G_{i}(u_{i})=g_{i}(u_{i})\partial_{t}u_{i}\in L^{p^{\prime}}(0,T;(W^{1,q}(\Omega))^{\prime})$ with $p^{\prime}=\frac{p}{p-1}$ and

(93)

\left\Arrowvert\partial_{t}G_{i}(u_{i})\right\Arrowvert_{L^{p^{\prime}}(0,T;(W^{1,q}(\Omega))^{\prime})}\leq C\text{ for all }i.

Moreover, by growth assumption on $g$ and estimates on $u_{i}$ , we have

(94)

\left\Arrowvert G_{i}(u_{i})\right\Arrowvert_{L^{\infty}(0,T;W^{1,s}(\Omega))}\leq C.

for $1\leq s<2$ . By (89), (90), (93)-(94) and Remark 3.1 we can find a subsequence, not relabeled, a function $u\in L^{\infty}(0,T;H^{1}(\Omega))$ , a function $\xi\in L^{2}(\Omega_{T})$ and a function $\eta\in L^{\infty}(0,T;W^{1,s}(\Omega))$ such that as $i\rightarrow\infty$ ,

(95)		$\displaystyle u_{i}\rightharpoonup u\text{ weakly-* in }L^{\infty}(0,T;H^{1}(\Omega)),$
(96)		$\displaystyle\sqrt{M_{i}(u_{i})}\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\rightharpoonup\xi\text{ weakly in }L^{2}(\Omega_{T}),$
(97)		$\displaystyle G_{i}(u_{i})\rightharpoonup\eta\text{ weakly-* in }L^{\infty}(0,T;W^{1,s}(\Omega))$
(98)		$\displaystyle G_{i}(u_{i})\rightarrow\eta\text{ strongly in }L^{\alpha}(0,T;L^{\beta}(\Omega))\text{ and a.e. in }\Omega_{T},$
(99)		$\displaystyle G_{i}(u_{i})\rightarrow\eta\text{ strongly in }C(0,T;L^{\beta}(\Omega)),$
(100)		$\displaystyle\partial_{t}G_{i}(u_{i})\rightharpoonup\partial_{t}\eta\text{ weakly in }L^{p^{\prime}}(0,T;(W^{1,q}\Omega))^{\prime}).$

where $1\leq\alpha,\beta<\infty$ . By (99) and (105) from Remark 4.1, we have

\left\Arrowvert G_{i}(u_{i}(x,t+h))-G_{i}(u_{i}(x,t))\right\Arrowvert_{C([0,T];L^{\beta}(\Omega))}\rightarrow 0\text{ uniformly in }i\text{ as }h\rightarrow 0.

Thus given any $\varepsilon>0$ , there exists $h_{\varepsilon}>0$ such that for all $0<h<h_{\varepsilon}$ and all $i$ ,

\left\Arrowvert G_{i}(u_{i}(x,t+h))-G_{i}(u_{i}(x,t))\right\Arrowvert_{C([0,T];L^{\beta}(\Omega))}^{\beta}<\varepsilon.

Given any $\delta>0$ , let $I_{\delta}=(1-\delta,1+\delta)\cup(-1-\delta,-1+\delta)$ . Consider the interval having $u_{i}(x,t)$ and $u_{i}(x,t+h)$ as end points. Denote this interval by $J_{i}(x,t;h)$ . We consider three cases.

Case I: $J_{i}(x,t;h)\cap I_{\delta}=\varnothing$ .

In this case, $g_{i}(s)\geq\max\{\theta_{i}^{m},\delta^{2m}\}$ for any $s\in J_{i}(x,t;h)$ and by (92)

\left|G_{i}(u_{i}(x,t+h))-G_{i}(u_{i}(x,t))\right|=\left|\int_{u_{i}(x,t)}^{u_{i}(x,t+h)}g_{i}(s)ds\right|\geq\delta^{2m}|u_{i}(x,t+h)-u_{i}(x,t)|.

Case II: $J_{i}(x,t;h)\cap I_{\delta}\neq\varnothing$ and $|u_{i}(x,t+h)-u_{i}(x,t)|\geq 3\delta$ .

In this case, we have

|J_{i}(x,t;h)\cap I_{\delta}^{c}|\geq\frac{1}{3}|J_{i}(x,t;h)|

and

	$\displaystyle\left\|G_{i}(u_{i}(x,t+h))-G_{i}(u_{i}(x,t))\right\|$	$\displaystyle\geq$	$\displaystyle\left\|\int_{J_{i}(x,t;h)\cap I_{\delta}^{c}}g_{i}(s)ds\right\|$
		$\displaystyle\geq$	$\displaystyle\frac{\delta^{2m}}{3}\|u_{i}(x,t+h)-u_{i}(x,t)\|.$

Case III: $J_{i}(x,t;h)\cap I_{\delta}\neq\varnothing$ and $|u_{i}(x,t+h)-u_{i}(x,t)|<3\delta$

In this case, we have

g_{i}(s)\leq\max\{(8\delta+16\delta^{2})^{m},\theta_{i}^{m}\}\text{ for any }s\in J_{i}(x,t;h).

Thus

\displaystyle\left|G_{i}(u_{i}(x,t+h))-G_{i}(u_{i}(x,t))\right|\leq 3\delta\max\{(8\delta+16\delta^{2})^{m},\theta_{i}^{m}\}.

Pick $\delta=\varepsilon^{\frac{1}{4m\beta}}$ and fix $t$ . Let

\Omega_{i}^{t}=\{x\in\Omega:J_{i}(x,t:h)\text{ satisfies case I or II}\}.

Then

			$\displaystyle\int_{\Omega}\left\arrowvert u_{i}(x,t+h)-u_{i}(x,t)\right\arrowvert^{\beta}dx$
		$\displaystyle=$	$\displaystyle\int_{\Omega_{i}^{t}}\left\arrowvert u_{i}(x,t+h)-u_{i}(x,t)\right\arrowvert^{\beta}dx+\int_{\Omega\backslash\Omega_{i}^{t}}\left\arrowvert u_{i}(x,t+h)-u_{i}(x,t)\right\arrowvert^{\beta}dx$
		$\displaystyle\leq$	$\displaystyle 3^{\beta}\varepsilon^{-\frac{1}{2}}\int_{\Omega_{i}^{t}}\left\arrowvert G_{i}(u_{i}(x,t+h))-G_{i}(u_{i}(x,t))\right\arrowvert^{\beta}dx+\int_{\Omega\backslash\Omega_{i}^{t}}\left\arrowvert u_{i}(x,t+h)-u_{i}(x,t)\right\arrowvert^{\beta}dx$
		$\displaystyle\leq$	$\displaystyle 3^{\beta}\varepsilon^{\frac{1}{2}}+C\varepsilon^{\frac{1}{4m}}$

Taking maximum on the left side, we have for all $i$ , any $h<h_{\varepsilon}$ ,

\left\Arrowvert(u_{i}(x,t+h)-u_{i}(x,t)\right\Arrowvert_{C([0,T];L^{\beta}(\Omega))}^{\beta}\leq 3^{\beta}\varepsilon^{\frac{1}{2}}+C\varepsilon^{\frac{1}{4m}}.

Thus

\left\Arrowvert u_{i}(x,t+h)-u_{i}(x,t)\right\Arrowvert_{C([0,T];L^{\beta}(\Omega))}^{\beta}\rightarrow 0\text{ uniformly as }h\rightarrow 0.

In addition, for any $0<t_{1}<t_{2}<T$ , (89) implies that for $1\leq\beta<\infty$ , we have

\int_{t_{1}}^{t_{2}}u_{i}(x,t)dt\text{ is relatively compact in }L^{\beta}(\Omega).

Therefore we conclude from Remark 4.1 that

(101)

u_{i}\rightarrow u(x,t)\text{ strongly in }C([0,T];L^{\beta}(\Omega))\text{ for }1\leq\beta<\infty.

Similarly. we can prove

(102)

u_{i}\rightarrow u(x,t)\text{ strongly in }L^{\alpha}(0,T;L^{\beta}(\Omega))\text{ for }1\leq\alpha,\beta<\infty\text{ and a.e. in }\Omega_{T}.

Growth condition on $M(u)$ and (101), (102) yield

	$\displaystyle M_{i}(u_{i})\rightarrow M(u)\text{ strongly in }C([0,T];L^{\beta}(\Omega))\text{ for }1\leq\beta<\infty,$
	$\displaystyle M_{i}(u_{i})\rightarrow M(u)\text{ strongly in }L^{\alpha}(0,T;L^{\beta}(\Omega))\text{ for }1\leq\alpha,\beta<\infty,$
	$\displaystyle\sqrt{M(_{i}(u_{i}))}\rightarrow\sqrt{M(u)}\text{ strongly in }C([0,T];L^{\gamma}(\Omega))\text{ for }1\leq\gamma<\infty.$

Hence $G_{i}(u_{i})$ converges to $G(u)$ a.e. in $\Omega_{T}$ and $\eta=G(u)$ . Passing to the limit in (88), we have

(103)

\int_{0}^{T}\left<g(u)\partial_{t}u,\phi\right>_{((W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega))}dt=-\int_{0}^{T}\int_{\Omega}\sqrt{M(u)}\xi\cdot\nabla\phi dxdt

for any $\phi\in L^{p}(0,T;W^{1,q}(\Omega))$ with $p,q>2$ .

Remark 4.1.

(Compactness in $L^{p}(0,T;B)$ Theorem 1 in [22]) Assume $B$ is a Banach space and $F\subset L^{p}(0,T;B)$ . $F$ is relatively compact in $L^{p}(0,T;B)$ for $1\leq p<\infty$ , or in $C([0,T],B)$ for $p=\infty$ if and only if

(104)

\left\{\int_{t_{1}}^{t_{2}}f(t)dt:f\in F\right\}\text{ is relatively compact in }B,\forall 0<t_{1}<t_{2}<T,

(105)

\left\Arrowvert\tau_{h}f-f\right\Arrowvert_{L^{p}(0,T;B)}\rightarrow 0\text{ as }h\rightarrow 0\text{ uniformly for }f\in F.

Here $\tau_{h}f(t)=f(t+h)$ for $h>0$ is defined on $[-h,T-h]$ .

4.2 Weak convergence of $\nabla\frac{\mu_{i}}{g_{i}(u_{i})}$

We now look for relation between $\xi$ and $u$ . Following the idea in [7], we decompose $\Omega_{T}$ as follows. Let $\delta_{j}$ be a positive sequence monotonically decreasing to $0$ . By (96) and Egorov’s theorem, for every $\delta_{j}>0$ , there exists $B_{j}\subset\Omega_{T}$ satisfying $|\Omega_{t}\backslash B_{j}|<\delta_{j}$ such that

(106)

u_{i}\rightarrow u\text{ uniformly in }B_{j}.

We can pick

(107)

B_{1}\subset B_{2}\subset\cdots\subset B_{j}\subset B_{j+1}\subset\cdots\subset\Omega_{T}.

Define

P_{j}:=\{(x,t)\in\Omega_{T}:|1-u^{2}|>\delta_{j}\}.

Then

(108)

P_{1}\subset P_{2}\subset\cdots\subset P_{j}\subset P_{j+1}\subset\cdots\subset\Omega_{T}.

Let $B=\cup_{j=1}^{\infty}B_{j}$ and $P=\cup_{j=1}^{\infty}P_{j}$ . Then $|\Omega_{T}\backslash B|=0$ and each $B_{j}$ can be split into two parts:

	$\displaystyle D_{j}=B_{j}\cap P_{j},\text{ where }\|1-u^{2}\|>\delta_{j},\text{ and }u_{i}\rightarrow u\text{ uniformly},$
	$\displaystyle\tilde{D}_{j}=B_{j}\backslash P_{j},\text{ where }\|1-u^{2}\|\leq\delta_{j},\text{ and }u_{i}\rightarrow u\text{ uniformly }.$

(107) and (108) imply

(109)

D_{1}\subset D_{2}\subset\cdots\subset D_{j}\subset D_{j+1}\subset\cdots\subset D:=B\cap P.

For any $\Psi\in L^{p}(0,T;L^{q}(\Omega,\mathbb{R}^{n}))$ with $p,q>2$ , we have

(110)		$\displaystyle\int_{\Omega_{T}}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\cdot\Psi dxdt$
	$\displaystyle=$	$\displaystyle\int_{\Omega_{T}\backslash B_{j}}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\cdot\Psi dxdt+\int_{D_{j}}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\cdot\Psi dxdt$
		$\displaystyle+\int_{\tilde{D}_{j}}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\cdot\Psi dxdt$

The left hand side of (110) converges to $\int_{\Omega_{T}}\sqrt{M(u)}\xi\cdot\Psi dxdt$ . We analyze the three terms on the right hand side separately. To estimate the first term on the right hand side of (110), noticing $|\Omega_{T}\backslash B_{j}|\rightarrow 0$ and

\displaystyle\lim_{i\rightarrow\infty}\int_{\Omega_{T}\backslash B_{j}}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\cdot\Psi dxdt=\ \int_{\Omega_{T}\backslash B_{j}}\sqrt{M(u)}\xi\cdot\Psi dxdt,

we have

\displaystyle\lim_{j\rightarrow\infty}\lim_{i\rightarrow\infty}\int_{\Omega_{T}\backslash B_{j}}M_{i}(u_{i})\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\cdot\Psi dxdt=0.

By uniform convergence of $u_{i}$ to $u$ in $B_{j}$ , we introduce subsequence $u_{j,k}$ such that $u_{j,k}\rightarrow u$ uniformly in $B_{j}$ and there exists $N_{j}$ such that for all $k\geq N_{j}$ ,

(111)

|1-u^{2}_{j,k}|>\frac{\delta_{j}}{2}\text{ in }D_{j},\hskip 14.45377pt|1-u^{2}_{j,k}|\leq 2\delta_{j}\text{ in }\tilde{D}_{j}.

Thus the third term on the right hand side of (110) can be estimated by

			$\displaystyle\lim_{j\rightarrow\infty}\lim_{k\rightarrow\infty}\left\arrowvert\int_{\tilde{D}_{j}}M_{j,k}(u_{j,k})\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\cdot\Psi dxdt\right\arrowvert$
		$\displaystyle\leq$	$\displaystyle\lim_{j\rightarrow\infty}\lim_{k\rightarrow\infty}\left\{\left(\sup_{\tilde{D}_{j}}\sqrt{M_{j,k}(u_{j,k})}\right)\left\Arrowvert\Psi\right\Arrowvert_{L^{2}(\tilde{D}_{j})}\left\Arrowvert\sqrt{M_{j,k}(u_{j,k})}\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\right\Arrowvert_{L^{2}(\tilde{D}_{j})}\right\}$
		$\displaystyle\leq$	$\displaystyle\left(\sup_{\tilde{D}_{j}}\sqrt{M_{j,k}(u_{j,k})}\right)\|\Omega\|^{\frac{q-2}{2q}}\left\Arrowvert\Psi\right\Arrowvert_{L^{2}(0,T;L^{q}(\Omega)}\left\Arrowvert\sqrt{M_{j,k}(u_{j,k})}\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\right\Arrowvert_{L^{2}(\tilde{D}_{j})}$
			$\displaystyle\leq C\lim_{j\rightarrow\infty}\lim_{k\rightarrow\infty}\max\left\{(2\delta_{j})^{m/2},\theta_{j,k}^{m/2}\right\}$
		$\displaystyle=$	$\displaystyle 0.$

For the second term, we see that

			$\displaystyle\left(\frac{\delta_{j}}{2}\right)^{m}\int_{D_{j}}\|\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\|^{2}dxdt$
		$\displaystyle\leq$	$\displaystyle\int_{D_{j}}M_{j,k}(u_{j,k})\|\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\|^{2}dxdt$
		$\displaystyle\leq$	$\displaystyle\int_{\Omega_{T}}M_{j,k}(u_{j,k})\|\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\|^{2}dxdt\leq C.$

Therefore $\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}$ is bounded in $L^{2}(D_{j})$ and we can extract a further subsequence, not relabeled, which converges weakly to some $\xi_{j}\in L^{2}(D_{j})$ . Since $D_{j}$ is an increasig sequence of sets with $\lim_{j\rightarrow\infty}D_{j}=D$ , we have $\xi_{j}=\xi_{j-1}$ a.e. in $D_{j-1}$ . By setting $\xi_{j}=0$ outside $D_{j}$ , we can extend $\xi_{j}$ to a $L^{2}$ function $\tilde{\xi}_{j}$ defined in $D$ . Therefore for a.e. $x\in D$ , there exists a limit of $\tilde{\xi}_{j}(x)$ as $j\rightarrow\infty$ . Let $\xi(x)=\lim_{j\rightarrow\infty}\tilde{\xi}_{j}(x)$ , we see that $\xi(x)=\xi_{j}(x)$ for a.e $x\in D_{j}$ and for all $j$ .

By a standard diagonal argument, we can extract a subsequnce such that

(112)

\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}\rightharpoonup\zeta\text{ weakly in }L^{2}(D_{j})\text{ for all }j.

By strong convergence of $\sqrt{M_{i}(u_{i})}$ to $\sqrt{M(u)}$ in $C([0,T];L^{\beta}(\Omega))$ for $1\leq\beta<\infty$ , we obtain

\chi_{D_{j}}\sqrt{M_{k,N_{k}}(u_{k,N_{k}})}\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}\rightharpoonup\chi_{D_{j}}\sqrt{M(u)}\zeta

weakly in $L^{2}(0,T;L^{q}(\Omega))$ for $1\leq q<2$ and all $j$ . Recall $\sqrt{M_{i}(u_{i})}\nabla\frac{\mu_{i}}{g_{i}(u_{i})}\rightarrow\xi$ weakly in $L^{2}(\Omega_{T})$ , we have $\xi=\sqrt{M(u)}\zeta$ in $D_{j}$ for all $j$ . Hence $\xi=\sqrt{M(u)}\zeta$ in $D$ and consequently

\chi_{D}M_{k,N_{k}}(u_{k,N_{k}})\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}\rightharpoonup\chi_{D}M(u)\zeta

weakly in $L^{2}(0,T;L^{q}(\Omega))$ for $1\leq q<2$ .

Replacing $u_{i}$ by subsequence $u_{k,N_{k}}$ in (110) and letting $k\rightarrow\infty$ then $j\rightarrow\infty$ , we have

(113)		$\displaystyle\int_{\Omega_{T}}\sqrt{M(u)}\xi\cdot\Psi dxdt$	$\displaystyle=$	$\displaystyle\lim_{j\rightarrow\infty}\int_{D_{j}}M(u)\zeta\cdot\Psi dxdt$
(113)			$\displaystyle=$	$\displaystyle\int_{D}M(u)\zeta\cdot\Psi dxdt.$

It follows from (103) and (113) that

(114)

\int_{0}^{T}\left<g(u)\partial_{t}u,\phi\right>_{((W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega))}dt=-\int_{D}M(u)\zeta\cdot\nabla\phi dxdt

for all $\phi\in L^{p}(0,T;W^{1,q}(\Omega))$ where $p,q>2$ .

4.3 Relation between $\zeta$ and $u$

The desired relation between $\zeta$ and $u$ is

(115)		$\displaystyle\zeta$	$\displaystyle=$	$\displaystyle\frac{1}{g}\nabla\mu-\mu\frac{g^{\prime}(u)}{g^{2}(u)}\nabla u$
(115)			$\displaystyle=$	$\displaystyle\frac{1}{g}\left(-\nabla\Delta u+q^{\prime\prime}(u)\nabla u\right)-\frac{g^{\prime}(u)}{g^{2}(u)}\nabla u\left(-\Delta u+q^{\prime}(u)\right).$

Given the known regularity $u\in L^{\infty}(0,T;H^{1}(\Omega))$ and degeneracy of $g(u)$ , the right hand side of (115) might not be defined as a function. We can, however, under the additional assumption $u\in L^{2}(0,T;H^{2}(\Omega))$ and suitable assumptions on integrability of $\nabla\Delta u$ , find an explicit expression of $\zeta$ in terms of (115) in suitable subset of $\Omega_{T}$ .

Claim I: If $u\in L^{2}(0,T:H^{2}(\Omega))$ and for some $j$ , the interior of $D_{j}$ , denoted by $(D_{j})^{\circ}$ , is not empty, then

\nabla\Delta u\in L^{1}((D_{j})^{\circ})

and

\zeta=\frac{-\nabla\Delta u+q^{\prime\prime}(u)\nabla u}{g(u)}-\frac{g^{\prime}(u)}{g^{2}(u)}\left(-\Delta u+q^{\prime}(u)\right)\nabla u\text{ a.e. in }(D_{j})^{\circ}.

Proof of the claim I. Since $u\in L^{2}(0,T;H^{2}(\Omega))$ , we can have a subsequence, not relabeled such that, $u_{k,N_{k}}$ converges weakly to $u$ in $L^{2}(0,T;H^{2}(\Omega))$ . Since

(116)

\mu_{k,N_{k}}=-\Delta u_{k,N_{k}}+q^{\prime}(u_{k,N_{k}})\text{ in }\Omega_{T},

The right hand side of (116) weakly converges to $-\Delta u+q^{\prime}(u)$ in $L^{2}(\Omega_{T})$ . Hence

\mu_{k,N_{k}}\rightharpoonup\mu=-\Delta u+q^{\prime}(u)\text{ weakly in }L^{2}(\Omega_{T}).

On the other hand, using $u_{k,N_{k}}$ and $u$ as test functions in (82) yield

	$\displaystyle\int_{\Omega_{T}}\mu_{k,N_{k}}u_{k,N_{k}}dxdt=\int_{\Omega_{T}}\left(\left\arrowvert\nabla u_{k,N_{k}}\right\arrowvert^{2}+q^{\prime}(u_{k,N_{k}})u_{k,N_{k}}\right)dxdt$
	$\displaystyle\int_{\Omega_{T}}\mu_{k,N_{k}}udxdt=\int_{\Omega_{T}}\left(\nabla u_{k,N_{k}}\cdot\nabla u+q^{\prime}(u_{k,N_{k}})u\right)dxdt.$

Passing to the limit, by (102), growth assumptions on $q^{\prime}$ and (116), we have

\lim_{k\rightarrow\infty}\int_{\Omega_{T}}\left\arrowvert\nabla u_{k,N_{k}}\right\arrowvert^{2}=\int_{\Omega_{T}}\left\arrowvert\nabla u\right\arrowvert^{2}.

Therefore

\nabla u_{k,N_{k}}\rightarrow\nabla u\text{ strongly in }L^{2}(\Omega_{T}).

Since $u_{k,N_{k}}\in L^{2}(0,T;W^{3,s}(\Omega))$ , we can differentiate (116) and get

(117)

\nabla\mu_{k,N_{k}}=-\nabla\Delta u_{k,N_{k}}+q^{\prime\prime}(u_{k,N_{k}})\nabla u_{k,N_{k}},

and

(118)

\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}=\frac{1}{g_{k,N_{k}}(u_{k,N_{k}})}\nabla\mu_{k,N_{k}}-\mu_{k,N_{k}}\frac{g^{\prime}_{k,N_{k}}(u_{k,N_{k}})}{g^{2}_{k,N_{k}}(u_{k,N_{k}})}\nabla u_{k,N_{k}}

on $D_{j}^{\circ}$ . Thus

(119)

\nabla\mu_{k,N_{k}}=g_{k,N_{k}}(u_{k,N_{k}})\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}+\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}g^{\prime}_{k,N_{k}}(u_{k,N_{k}})\nabla u_{k,N_{k}}.

Since

	$\displaystyle g_{k,N_{k}}(u_{k,N_{k}})\rightarrow g(u)\text{ uniformly in }D_{j}^{\circ},$
	$\displaystyle\frac{g^{\prime}_{k,N_{k}}(u_{k,N_{k}})}{g_{k,N_{k}}(u_{k,N_{k}})}\rightarrow\frac{g^{\prime}(u)}{g(u)}\text{ uniformly in }D_{j}^{\circ},$
	$\displaystyle\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}\rightharpoonup\zeta\text{ weakly in }L^{2}(D_{j}^{\circ}),$
	$\displaystyle\mu_{k,N_{k}}\rightharpoonup\mu\text{ weakly in }L^{2}(\Omega_{T}),$
	$\displaystyle\nabla u_{k,N_{k}}\rightarrow\nabla u\text{ strongly in }L^{2}(\Omega_{T}),$

we have, for any $\phi\in L^{\infty}(D_{j}^{\circ})$ ,

	$\displaystyle\int_{D_{j}^{\circ}}\phi\left(g_{k,N_{k}}(u_{k,N_{k}})\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}+\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}g^{\prime}_{k,N_{k}}(u_{k,N_{k}})\nabla u_{k,N_{k}}\right)dxdt$
	$\displaystyle\rightarrow\int_{D_{j}^{\circ}}\phi\left(g(u)\zeta+\frac{g^{\prime}(u)}{g(u)}\mu\nabla u\right)dxdt,$

i.e.

\nabla\mu_{k,N_{k}}\rightharpoonup\eta\coloneq g(u)\zeta+\frac{g^{\prime}(u)}{g(u)}\mu\nabla u\ \text{ weakly in }L^{1}(D_{j}^{\circ}).

Passing to the limit in (117), we obtain, in the sense of distribution, that

\eta=-\nabla\Delta u+q^{\prime\prime}(u)\nabla u.

Since $q^{\prime\prime}(u)\nabla u\in L^{2}(\Omega_{T})$ , we have $-\nabla\Delta u\in L^{1}(D_{j}^{\circ})$ , hence

(120)

\eta=-\nabla\Delta u+q^{\prime\prime}(u)\nabla u\text{ a.e. in }D_{j}^{\circ}

Since $\frac{1}{g_{k,N_{k}}(u_{k,N_{k}})}\rightarrow\frac{1}{g(u)}$ uniformly in $D_{j}$ , we have

\frac{1}{g_{k,N_{k}}}\nabla\mu_{k,N_{k}}\rightharpoonup\frac{1}{g(u)}\eta\text{ weakly in }L^{1}(D_{j}^{\circ}).

Since $\frac{g^{\prime}_{k,N_{k}}(u_{k,N_{k}})}{g^{2}_{k,N_{k}}(u_{k,N_{k}})}\rightarrow\frac{g^{\prime}(u)}{g^{2}(u)}$ uniformly in $D_{j}$ , we have

\frac{g^{\prime}_{k,N_{k}}(u_{k,N_{k}})}{g^{2}_{k,N_{k}}(u_{k,N_{k}})}\mu_{k,N_{k}}\nabla u_{k,N_{k}}\rightharpoonup\frac{g^{\prime}(u)}{g^{2}(u)}\mu\nabla u\text{ weakly in }L^{1}(D_{j}^{\circ}).

Passing to the limit in (118), we have

\displaystyle\zeta

\displaystyle=

\displaystyle\frac{1}{g(u)}\eta-\mu\frac{g^{\prime}(u)}{g^{2}(u)}\nabla u=\frac{-\nabla\Delta u+q^{\prime\prime}(u)\nabla u}{g(u)}-\frac{g^{\prime}(u)}{g^{2}(u)}\left(-\Delta u+q^{\prime}(u)\right)\nabla u

on $(D_{j})^{\circ}$ . Noticing the value of $\zeta$ on $\Omega_{T}\backslash D$ doesn’t matter since it does not appear on the right hand side of (113).

Claim II: For any open set $U\in\Omega_{T}$ in which $\nabla\Delta u\in L^{p}(U)$ for some $p>1$ and $g(u)>0$ , we have

(121)

\zeta=\frac{-\nabla\Delta u+q^{\prime\prime}(u)\nabla u}{g(u)}-\frac{g^{\prime}(u)}{g^{2}(u)}\left(-\Delta u+q^{\prime}(u)\right)\nabla u.

in $U$ .

To prove this, since

(122)

\nabla\mu_{k,N_{k}}=-\nabla\Delta u_{k,N_{k}}+q^{\prime\prime}(u_{k,N_{k}})\nabla u_{k,N_{k}}\text{ in }\Omega_{T}

and

(123)

\nabla\frac{\mu_{k,N_{k}}}{g_{k,N_{k}}(u_{k,N_{k}})}=\frac{1}{g_{k,N_{k}}(u_{k,N_{k}})}\nabla\mu_{k,N_{k}}+\mu_{k,N_{k}}\cdot\nabla\frac{1}{g_{k,N_{k}}(u_{k,N_{k}})}\text{ on }D_{j}.

The right hand side of (122) converges weakly to $-\nabla\Delta u+q^{\prime\prime}(u)\nabla u$ in $L^{q}(U)$ for $q=\min\{p,2\}>1$ . Hence

\nabla\mu_{k,N_{k}}\rightharpoonup\eta=-\nabla\Delta u+q^{\prime\prime}(u)\nabla u\text{ weakly in }L^{q}(U).

The right hand side of (123) converges weakly to

\frac{\eta}{g(u)}-\frac{g^{\prime}(u)}{g^{2}(u)}\mu\cdot\nabla u

in $L^{1}(U\cap D_{j})$ for each $j$ and therefore

\zeta=\frac{-\nabla\Delta u+q^{\prime\prime}(u)\nabla u}{g(u)}-\frac{g^{\prime}(u)}{g^{2}(u)}\left(-\Delta u+q^{\prime}(u)\right)\nabla u

in $U\cap D$ . The definition of $\zeta$ can be extended to $U\backslash D$ by our integrability assumption on $u$ . Define

\tilde{\Omega}_{T}=\{U\subset\Omega_{T}:g(u)>0\text{ on }U\text{ and }\nabla\Delta u\in L^{p}(U)\text{ for some }p>1\text{ depending on }U\}.

Then $\tilde{\Omega}_{T}$ is open and $\zeta$ is defined by (121) on $\tilde{\Omega}_{T}$ . Since $|\Omega_{T}\backslash B|=0$ , $M(u)=0$ on $\Omega_{T}\backslash P$ and

\Omega_{T}\backslash\{D\cup\tilde{\Omega}_{T}\}\subset\{\Omega_{T}\backslash B\}\cup\{\Omega_{T}\backslash P\},

we can take the value of $\zeta$ to be zero outside $D\cup\Omega_{T}$ , and it won’t affect the integral on the right side of (13).

Lastly the energy inequality (15) follows by taking limit in the energy inequality for $u_{k,N_{k}}$ .

Remark 4.2.

In Cahn-Hilliard case, there is convergence of $\nabla\mu_{k}$ on $L^{2}(D_{j})$ , and relation between $\xi$ and $u$ can be derived directly. Here we only have convergence of $\nabla\frac{\mu_{k}}{g_{k}(u_{k})}$ on $L^{2}(D_{j})$ . In order to obtain convergence of $\nabla\mu_{k}$ , we need convergence $\mu_{k}$ on $L^{p}(\Omega_{T})$ for suitable $p$ , this is where we used the additional assumption $u\in L^{2}(0,T;H^{2}(\Omega))$ .

5 A Modified phase field model for self-climb of prismatic dislocation loops

Dislocations are line defects in crystals [12, 23]. A phase field model [20] was derived based on the pipe diffusion model for self-climb of prismatic dislocation loops [18, 19] that describes the conservative climb of dislocation loops observed in experiments of irradiated materials [13, 12, 8]. In this section, we study the wellposedness of the following modified phase field model for self-climb of prismatic dislocation loops:

(124)		$\displaystyle g(u)\partial_{t}u$	$\displaystyle=$	$\displaystyle\nabla\cdot(M(u)\nabla\frac{\mu}{g(u)})\text{ for }x\in\Omega\subset\mathbb{R}^{2},t\in[0,\infty)$
(125)		$\displaystyle\mu$	$\displaystyle=$	$\displaystyle-\Delta u+\frac{1}{\varepsilon^{2}}q^{\prime}(u)+\frac{1}{\varepsilon}f_{cl}$

Where $M(u)=M_{0}g(u)$ , $g(u)=|1-u^{2}|^{m}$ for $2\leq m<\infty$ , $q(u)$ satisfy same assumptions (6)-(7) as those for Eqs. (1)-(2). Here $f_{cl}$ is the total climb force with

f_{cl}=f_{cl}^{d}+f_{cl}^{app}

where $f_{cl}^{app}$ is the applied climb force, and

(126)

f_{cl}^{d}(x,y,u)=\frac{Gb^{2}}{4\pi(1-\nu)}\int_{\Omega}\left(\frac{x-\overline{x}}{R^{3}}u_{\overline{x}}+\frac{y-\overline{y}}{R^{3}}u_{\overline{y}}\right)d\overline{x}d\overline{y}

represents the climb force generated by all the dislocations. Here $\Omega\subset\mathbb{R}^{2}$ is a bounded domain, $G$ is the shear modulus, $\nu$ is the Poisson ratio, and $R=\sqrt{(x-\overline{x})^{2}+(y-\overline{y})^{2}}$ . In this model, we assume that the prismatic dislocation loops lie and evlove by self-climb in the $xy$ plane and all dislocation loops have the same Burgers vector $\mathbf{b}=(0,0,b)$ .

The chemical potential $\mu$ comes from variations of the classical Cahn-Hilliard energy and the elastic energy due to dislocations, i.e.

(127)

\mu=\frac{\delta E_{CH}}{\delta u}+\frac{\delta E_{el}}{\delta u},

where

(128)		$\displaystyle E_{CH}(u)=\int_{\Omega}\left(\frac{1}{2}\|\nabla u\|^{2}+q(u)\right)dx,$
(129)		$\displaystyle E_{el}=\int_{\Omega}\left(\frac{1}{2}uf^{d}_{cl}+uf_{cl}^{app}\right)dx$

are classical Cahn-Hilliard energy and elastic energy, respectively. Under periodic boundary conditions, the climb force generated by the dislocations can be expressed as

(130)

\displaystyle f^{d}_{cl}(x,y,u)=\frac{Gb^{2}}{2(1-\nu)}(-\Delta)^{\frac{1}{2}}u.

Here $(-\Delta)^{s}u$ is a fractional operator defined by

\mathcal{F}((-\Delta)^{s}f)=(\xi_{1}^{2}+\xi_{2}^{2})^{\frac{s}{2}}\mathcal{F}(f)(\xi)

for $\xi\in\mathbb{Z}^{2}$ . In the analysis below, without loss of generality, we set the coefficient of the climb force $\frac{Gb^{2}}{2(1-\nu)}=1$ .

System (124)-(125) is a modified version of the phase field model introduced in [20], which does not have the $g(u)$ term on the left side of (124). Putting an extra factor $h=H_{0}g$ in front of the nonlocal climb force $f^{d}_{cl}$ , the asymptotic analysis in [20] showed that the proposed phase field model yields accurate dislocation self-climb velocity in the sharp interface limit. Moreover, numerical simulations in [20] showed excellent agreement with experimental observations and discrete dislocation dynamics simulation results. Now we prove the wellposedness of the modified model (124)-(125). There is an extra nonlocal term $f^{d}_{cl}$ in this model compared with the model considered in previous sections.

Define

u^{N}(x,t)=\sum_{j=1}^{N}c_{j}^{N}(t)\phi_{j}(x),\hskip 36.135pt\mu^{N}(x,t)=\sum_{j=1}^{N}d^{N}_{j}(t)\phi_{j}(x),

where $\{c_{j}^{N},d^{N}_{j}\}$ satisfy

(131)	$\displaystyle\int_{\Omega}\partial_{t}u^{N}\phi_{j}dx$	$\displaystyle=$	$\displaystyle-\int_{\Omega}M_{\theta}(u^{N})\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\cdot\nabla\frac{\phi_{j}}{g_{\theta}(u^{N})}dx,$
(132)	$\displaystyle\int_{\Omega}\mu^{N}\phi_{j}dx$	$\displaystyle=$	$\displaystyle\int_{\Omega}\left(\nabla u^{N}\cdot\nabla\phi_{j}+q^{\prime}(u^{N})\phi_{j}+\phi_{j}(-\Delta)^{\frac{1}{2}}u^{N}\right)dx,$
(133)	$\displaystyle u^{N}(x,0)$	$\displaystyle=$	$\displaystyle\sum_{j=1}^{N}\left(\int_{\Omega}u_{0}\phi_{j}dx\right)\phi_{j}(x).$

(131)-(133) is an initial value problem for a system of ordinary equations for $\{c_{j}^{N}(t)\}$ . Since right hand side of (131) is continuous in $c_{j}^{N}$ , the system has a local solution.

Define energy functional

F(u)=\int_{\Omega}\left\{\frac{1}{2}|\nabla u|^{2}+q(u)+|(-\Delta)^{\frac{1}{4}}u|^{2}\right\}dx.

Direct calculation yields

\frac{d}{dt}F(u^{N}(x,t))=-\int_{\Omega}M_{\theta}(u^{N})\left|\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right|^{2}dx,

integration over $t$ gives the following energy identity for any $t>0$

			$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u^{N}(x,t)\|^{2}+q(u^{N}(x,t))+u^{N}(-\Delta)^{\frac{1}{2}}u^{N}\right)dx$
			$\displaystyle+\int_{0}^{t}\int_{\Omega}M_{\theta}(u^{N}(x,\tau))\left\|\nabla\frac{\mu^{N}(x,\tau)}{g_{\theta}(u^{N}(x,\tau))}\right\|^{2}dxd\tau$
		$\displaystyle=$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u^{N}(x,0)\|^{2}+q(u^{N}(x,0))+u^{N}(x,0)(-\Delta)^{\frac{1}{2}}u^{N}(x,0)\right)dx$
		$\displaystyle\leq$	$\displaystyle\int_{\Omega}\left\Arrowvert\nabla u_{0}\right\Arrowvert_{L^{2}(\Omega)}^{2}+C\left(\left\Arrowvert u_{0}\right\Arrowvert^{r+1}_{H^{1}{\Omega}}+\|\Omega\|\right)+\frac{1}{2}\left\Arrowvert u_{0}\right\Arrowvert_{L^{2}(\Omega)}^{2}\leq C<\infty,$

where $C$ represents a generic constant possibly depending only on $T$ , $\Omega$ , $u_{0}$ but not on $\theta$ . Since $\Omega$ is bounded region, by growth assumption assumption (6) and Poincare’s inequality, the energy identity (5) implies $u^{N}\in L^{\infty}(0,T;H^{1}(\Omega))$ with

(135)

\left\Arrowvert u^{N}\right\Arrowvert_{L^{\infty}(0,T;H^{1}(\Omega))}\leq C\text{ for all }N,

and

(136)

\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega_{T})}\leq C\text{ for all }N.

Repeat the argument in Section 3 and Section 4, replacing energy functional $F(u)$ by $E(u)$ when necessary, we can prove the following existence theorem for (124)-(125) with nondegenerate and degenerate mobilities respectively.

Theorem 5.1.

Let $M_{\theta},g_{\theta}$ be defined by (9) and (10), under the assumptions (6)-(8), for any $u_{0}\in H^{1}(\Omega)$ and any $T>0$ , there exists a function $u_{\theta}$ such that

a)

$u_{\theta}\in L^{\infty}(0,T;H^{1}(\Omega))\cap C([0,T];L^{p}(\Omega))\cap L^{2}(0,T;W^{3,s}(\Omega))$ , where $1\leq p<\infty$ , $1\leq s<2$ ,
b)

$\partial_{t}u_{\theta}\in L^{2}(0,T;(W^{1,q}(\Omega))^{\prime})$ for $q>2$ ,
c)

$u_{\theta}(x,0)=u_{0}(x)$ for all $x\in\Omega$ ,

which satisfies (4)-(5) in the following weak sense

	$\displaystyle\int_{0}^{T}<\partial_{t}u_{\theta},\phi>_{(W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega)}dt$
(137)		$\displaystyle=-\int_{0}^{T}\int_{\Omega}M_{\theta}(u_{\theta})\nabla\frac{-\Delta u_{\theta}+q^{\prime}(u_{\theta})+(-\Delta)^{\frac{1}{2}}u_{\theta}}{g_{\theta}(u_{\theta})}\cdot\nabla\frac{\phi}{g_{\theta}(u_{\theta})}dxdt$

for all $\phi\in L^{2}(0,T;W^{1,q}(\Omega))$ with $q>2$ . In addition, the following energy inequality holds for all $t>0$ .

(138)		$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{\theta}(x,t)\|^{2}+q(u_{\theta}(x,t))+u_{\theta}(x,t)(-\Delta)^{\frac{1}{2}}u_{\theta}\right)dx$
		$\displaystyle+\int_{0}^{t}\int_{\Omega}M_{\theta}(u_{\theta}(x,\tau)\left\|\nabla\frac{-\Delta u_{\theta}(x,\tau)+q^{\prime}(u_{\theta}(x,\tau)+(-\Delta)^{\frac{1}{2}}u_{\theta}}{g_{\theta}(u_{\theta}(x,\tau))}\right\|^{2}dxd\tau$
	$\displaystyle\leq$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{0}(x)\|^{2}+q(u_{0}(x))+u_{0}(x)(-\Delta)^{\frac{1}{2}}u_{0}\right)dx.$

Theorem 5.2.

For any $u_{0}\in H^{1}(\Omega)$ and $T>0$ , there exists a function $u:\Omega_{T}=\Omega\times[0,T]\rightarrow\mathbb{R}$ satisfying

i)

$u\in L^{\infty}(0,T;H^{1}(\Omega))\cap C([0,T];L^{s}(\Omega))$ , where $1\leq s<\infty$ ,
ii)

$g(u)\partial_{t}u\in L^{p}(0,T;(W^{1,q}(\Omega))^{\prime})$ for $1\leq p<2$ and $q>2$ .
iii)

$u(x,0)=u_{0}(x)$ for all $x\in\Omega$ ,

which solves (4)-(5) in the following weak sense

(139)

\int_{0}^{T}<g(u)\partial_{t}u,\phi>_{(W^{1,q}(\Omega))^{\prime},W^{1,q}(\Omega)}dt=-\int_{B\cap P}M(u)\zeta\cdot\nabla\phi dxdt

Assume $u\in L^{2}(0,T;H^{2}(\Omega)).$ For any open set $U\in\Omega_{T}$ on which $g(u)>0$ and $\nabla\Delta u\in L^{p}(U)$ for some $p>1$ , we have

(140)

\zeta=\frac{-\nabla\Delta u+q^{\prime\prime}(u)\nabla u+\nabla(-\Delta)^{\frac{1}{2}}u}{g(u)}-\frac{g^{\prime}(u)}{g^{2}(u)}\left(-\Delta u+q^{\prime}(u)+(-\Delta)^{\frac{1}{2}}u\right)\nabla u.

a.e in $U$ .

Moreover, the following energy inequality holds for all $t>0$

(141)				$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u(x,t)\|^{2}+q(u(x,t))\right)dz+\int_{\Omega_{r}\cap B\cap P}M(u(x,\tau))\|\zeta(x,\tau)\|^{2}dxd\tau$
(141)			$\displaystyle\leq$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{0}(x)\|^{2}+q(u_{0}(x))\right)dx.$

ACKNOWLEDGEMENTS X.H. Niu’s research is supported by National Natural Science Foundation of China under the grant number 11801214 and the Natural Science Foundation of Fujian Province of China under the grant number 2021J011193. Y. Xiang’s research is supported by the Hong Kong Research Grants Council General Research Fund 16307319. X. Yan’s research is supported by a Research Excellence Grant and CLAS Dean’s Summer Research Grant from University of Connecticut.

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(12)		$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{\theta}(x,t)\|^{2}+q(u_{\theta}(x,t))\right)dx$
		$\displaystyle+\int_{0}^{t}\int_{\Omega}M_{\theta}(u_{\theta}(x,\tau)\left\|\nabla\frac{-\Delta u_{\theta}(x,\tau)+q^{\prime}(u_{\theta}(x,\tau)}{g_{\theta}(u_{\theta}(x,\tau))}\right\|^{2}dxd\tau$
	$\displaystyle\leq$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u_{0}(x)\|^{2}+q(u_{0}(x))\right)dx.$

			$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u^{N}(x,t)\|^{2}+q(u^{N}(x,t))\right)dx$
			$\displaystyle+\int_{0}^{t}\int_{\Omega}M_{\theta}(u^{N}(x,\tau))\left\|\nabla\frac{\mu^{N}(x,\tau)}{g_{\theta}(u^{N}(x,\tau))}\right\|^{2}dxd\tau$
		$\displaystyle=$	$\displaystyle\int_{\Omega}\left(\frac{1}{2}\|\nabla u^{N}(x,0)\|^{2}+q(u^{N}(x,0))\right)dx$
		$\displaystyle\leq$	$\displaystyle\left\Arrowvert\nabla u_{0}\right\Arrowvert_{L^{2}(\Omega)}^{2}+C\left(\left\Arrowvert u_{0}\right\Arrowvert^{r+1}_{H^{1}{\Omega}}+\|\Omega\|\right)\leq C<\infty$

	$\displaystyle\|II_{12}\|$	$\displaystyle\leq$	$\displaystyle\int_{[0,T]\backslash T_{\delta}}\|\alpha(t)\|\left\Arrowvert\frac{g^{\prime}_{\theta}(u^{N})}{g^{\frac{3}{2}}_{\theta}(u^{N})}\nabla u^{N}-\frac{g^{\prime}_{\theta}(u_{\theta})}{g^{\frac{3}{2}}_{\theta}(u_{\theta})}\nabla u_{\theta}\right\Arrowvert_{L^{2}(\Omega)}\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega)}dt$
		$\displaystyle=$	$\displaystyle\int_{[0,T]\backslash T_{\delta}}\|\alpha(t)\|f^{N}(t)\|\left\Arrowvert\sqrt{M_{\theta}(u^{N})}\nabla\frac{\mu^{N}}{g_{\theta}(u^{N})}\right\Arrowvert_{L^{2}(\Omega)}dt.$

	$\displaystyle\left\|G_{i}(u_{i}(x,t+h))-G_{i}(u_{i}(x,t))\right\|$	$\displaystyle\geq$	$\displaystyle\left\|\int_{J_{i}(x,t;h)\cap I_{\delta}^{c}}g_{i}(s)ds\right\|$
		$\displaystyle\geq$	$\displaystyle\frac{\delta^{2m}}{3}\|u_{i}(x,t+h)-u_{i}(x,t)\|.$

			$\displaystyle\left(\frac{\delta_{j}}{2}\right)^{m}\int_{D_{j}}\|\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\|^{2}dxdt$
		$\displaystyle\leq$	$\displaystyle\int_{D_{j}}M_{j,k}(u_{j,k})\|\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\|^{2}dxdt$
		$\displaystyle\leq$	$\displaystyle\int_{\Omega_{T}}M_{j,k}(u_{j,k})\|\nabla\frac{\mu_{j,k}}{g_{j,k}(u_{j,k})}\|^{2}dxdt\leq C.$

Well-posedness of a modified degenerate Cahn-Hilliard model for surface diffusion

Abstract

keywords:

1 Introduction

Theorem 1.1.

Theorem 1.2.

2 Sharp interface limit via asymptotic expansions

2.1 Outer expansions

2.2 Inner expansions

Remark 2.1.

3 Weak solution for phase field model with positive mobilities

3.1 Galerkin approximations

3.2 Convergence of uNu^{N}

Remark 3.1.

3.3 Weak solution

Remark 3.2.

3.4 Regularity of uθu_{\theta}

Remark 3.3.

3.5 Energy Inequality

4 Phase field model with degenerate mobility

4.1 Convergence of uiu_{i} and equation for the limit function

Remark 4.1.

4.2 Weak convergence of ∇μigi​(ui)\nabla\frac{\mu_{i}}{g_{i}(u_{i})}

4.3 Relation between ζ\zeta and uu

Remark 4.2.

5 A Modified phase field model for self-climb of prismatic dislocation loops

Theorem 5.1.

Theorem 5.2.

References

3.2 Convergence of $u^{N}$

3.4 Regularity of $u_{\theta}$

4.1 Convergence of $u_{i}$ and equation for the limit function

4.2 Weak convergence of $\nabla\frac{\mu_{i}}{g_{i}(u_{i})}$

4.3 Relation between $\zeta$ and $u$