数理解析研究所講究録 1436 巻 2005 年 22-44

22Quasi-variational inequalities for phase transitions
千葉大学大学院 自然科学研究科 阿曽 雅泰 (Masayasu Aso)
Graduate School of Science and Technology

Chiba University

Abstract. In this paper, we consider a quasi-variational problem for irreversible phasechange with temperature. We propose a mathematical model of a class of irreversible
phase change described by a system of PDEs including a quasi-variational inequality.
One of our interests is an existence of solutions of this system. The existence of solutions
is obtained as a limit of approximate solutions. Our approximate problems are formulated
by using the Moreau-Yosida approximation. The convergence of approximate solutions isbased on some uniform estimates and monotonicity techniques in the nonlinear operator
theory.

1. Introduction
We consider the following system of PDEs:

in $Q:=(0, T)\mathrm{x}$$\theta_{t}+w_{t}-\triangle\theta=h(t, x)$$w_{t}+\partial I_{\theta,N}(w_{t})-vlSw$ $\ni f(\theta, w)$in$\Omega$

(1.1)

,

(1.2)

$Q_{7}$

subject to the boundary conditions

on$\frac{\partial\theta}{\partial n}=\frac{\partial w}{\partial n}=0$$\Sigma:=(0, T)\mathrm{x}$$\Gamma$

(1.3)

,

(1.4)

and the initial conditions$\theta(0, \cdot)$$=\theta_{0}$,$w(0, \cdot)$$=w_{0}$in$\Omega$

is a finite positiveis a bounded domain inwith smooth boundary ,number, is a positive constant, andare the time derivatives of 0 and , A denotes
the Laplace operator in space variable anddenotes the outward normal derivative on
are the initial data ofandis a given function on , is a given function onand , respectively;
is the indicator function of the interval $[g(\theta), g(\theta)+N]$ witha non-negative bounded smooth function on and a sufficiently large positive number$N$ ;if
or $g(\theta)+N<w_{t}$ ,if $g(\theta)\leq w_{t}\leq g(\theta)+N$ ;
0

where

$\mathrm{R}^{3}$$\Omega$$\Gamma$$\theta_{t}$$\nu$$\Gamma;f$$\theta$$w$$w$$w_{t}$$x$$\mathrm{R}^{2}$$T$$\frac{\partial}{\partial n}$$h$$Q;\theta_{0}$$w_{0}$$I_{\theta,N}(\cdot)$$g$$+\infty$$\mathrm{R}$$w_{t}<g(\theta)$$I_{\theta,N}(w_{t}):=\{$

Keywords and phrases :irreversible phase change, quasi-variational inequality, subdifFerential,
AMS Subject Classification :$35\mathrm{K}45,35\mathrm{K}50$$35\mathrm{R}3$

23

is the subdifferential with respect to$\partial I_{\theta,N}(w_{t})$defined by$\emptyset(-\infty, 0]$$\partial I_{\theta,N}(w_{t}):=\{${0}$[0, +\infty)$$w_{f}$, namely, it is a set-valued mapping

if $w_{t}<g(\theta)$ or $g(\theta)+N<w_{t}$ ,if $w_{t}=g(\theta)$ ,if $g(\theta)<w_{t}<g(\theta)+N$,
if $w_{t}=g(\theta)+N$ .

For instance, in the context of solidification of multi-composite materials, the unknowns 0 and of the system (P) $:=\{(1.1)-(1.4)\}$ are explored, respectively, as the temperature and the irreversible solidification parameter, ( is often called a phase change( ) in (1.2) requires thatparameter or an order parameter.) Since the mapping
is within $(0\leq)g(\theta)\leq w_{t}\leq g(\theta)+N$ , our system possibly describes the irreversibility
effect. As for a mathematical treatment of irreversible phase change, there are some redoes
lated works [3,6,7,8,13,15,16] and so on, however, in any case the restriction of
dependsonthesetting,.
In
our
not depend on the unknown functions 0 andunknown function 0, which is one of new aspects of our work.
$w$

$w$

$\partial I_{\theta,N}$$w_{t}$$\cdot$$w_{t}$$w$$\partial I_{\theta,N}(\cdot)$In this paper, we give an existence result for the system (P) under some assumptions. Concerning the system (P) we have already discussed the caseon the data , , ,(cf.[3]). In the paper of [3], we used the abstract
is a bounded domain inwhenquasi-variational evolution inequality established in [2] to get approximate solutions:
$f$

$g$

$h$

$\theta_{07}w_{0}$$\mathrm{R}^{2}$$\Omega$$\partial\phi_{u(t)}(u’(t))+\partial\psi(u(t))\ni G(t, u(t))$

in $X$ for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ ,is a proper lower semi-continuous convex function
where $X$ is a real Hilbert space,on $X$ for each $u\in D(\psi):=\{z\in X; \psi(z)<+\infty\}$, is a proper lower semi-continuous
is a single-valued
are their subdifferentials in $X$ ,andconvex function on $X$ ,operator from $X$ into itself. Since we cannot apply such a procedure to (P), we shall
employ a fixed point argument to construct approximate solutions of (P) and obtain asolution of (P) by showing their convergence.
$\phi_{u}$$\psi$$\partial\phi_{u}$$G$$\partial\psi$with the usual inner
Throughout this paper, $H$ denotes the real Hilbert space
and it is a Hilbert space equipped
product ( , ) and denotes the Sobolev spacewith the following inner product:$L^{2}(\Omega)$$\cdot$$\cdot$$H^{1}(\Omega)$$V$$(z, v)_{V}:=(z, v)+a(z, v)$ ,$a(z, v):= \int_{\Omega}\nabla z(x)$.

$\nabla v(x)dx$,$\forall z$, $v\in V$

to indicate the operator A withand norm $|z|_{V}:=\sqrt{(z,z)_{V}}$ . We use the notation
is linear, closed, nonhomogeneous Neumann boundary condition; note here that$H$
negative and self-adjoint in ; in fact, we have$\triangle_{0}$$-\triangle_{0}$$D(- \triangle_{0})=\{z\in H^{2}(\Omega);\frac{\partial z}{\partial n}=0$and$-\triangle_{0}z=-\triangle z$

in $H$ ,in$H^{\frac{1}{2}}(\Gamma)\}$$\forall z\in D(-\triangle_{0})$.

24Notation$|\cdot$$|_{\infty}$

stands for various$L$“-norms, for instance,, “(0) and so on.
$L$$L^{\infty}(Q)$

Next we recall some basic properties on convex functions and their subdifferentials
in a real Hilbert space; precisely see [4,5,9,12]. Let $W$ be a real Hilbert space with inner$|w$ .
be a proper lower semi-continuous and convex
Letand norm
productfunction on $W$ . The subset $D(\varphi)=\{z\in W;\varphi(z)<+\infty\}$ of $W$ is called the effectiveof is a set-valued operator from $W$ into itself
domain of . The subdifferential
defined by$(z^{*}, v-z)_{W}\leq\varphi(v)-\varphi(z)$ ,(1.5)$(\cdot, \cdot)_{W}$$|\cdot$$\varphi$$\partial\varphi$$\varphi$$\varphi$$\forall v\in W$$z^{*}\in\partial\varphi(z)$$\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}$. For each $>0$ , we define
and its domain is defined bywhich is called the resolvent of , where I is the identity operator in$W$ . The Moreau-Yosida approximation
are defined byof and its subdifferential$D(\partial\varphi):=\{z\in W;\partial\varphi(z)\neq\emptyset\}$$\epsilon$$\partial\varphi$$J_{\epsilon}^{\varphi}:=(I+\epsilon\partial\varphi)^{-1}$$\partial\varphi_{\epsilon}$$\varphi$$\varphi_{\epsilon}$$\varphi_{\epsilon}(z)=\inf_{v\in W}\{\frac{1}{2\epsilon}|z-v|_{W}^{2}+\varphi(v)\}$,$\partial\varphi_{\epsilon}(z):=\frac{z-J_{\epsilon}^{\varphi}z}{\epsilon}$

Concerning the Moreau-Yosida approximation and the resolvent
are often used in this PaPer:$\varphi_{\epsilon}(z)=\varphi(J_{\epsilon}^{\varphi}z)+\frac{1}{2\epsilon}|z-J_{\epsilon}^{\varphi}z|_{W}^{2}$$\varphi(J_{\epsilon}^{\varphi}z)\leq\varphi_{\epsilon}(z)\leq\varphi(z)$,,$\forall\epsilon>0$$\lim_{\epsilon[searrow] 0}\varphi_{\epsilon}(z)=\varphi(z)$,,$| \partial\varphi_{\epsilon}(z)|_{W}\leq|\partial\varphi(z)|:=\inf\{|w|_{W}; w\in\partial\varphi(z)\}$Especially, in the case that$z\in H$ , define$\Omega$.

,$J_{\epsilon}^{\varphi}$, the following facts, Vz $\in W$,$(1.7_{\grave{J}}$

(1.8)

,

$\forall\epsilon>0_{7}\forall z\in W$$\forall\epsilon>0$,$\forall z\in D(\partial\varphi)$

.

(1.9)

is a bounded domain with smooth boundary, for every$\frac{1}{2}a(z, z)$

if $z\in V$,$+\infty$

otherwise.

$\varphi(z)=\{$(1.10)

is a proper lower semi-continuous and convex function on $H$ and Op$H$ (cf. [5]).

Then

(1.6)

$\forall z\in W$$\varphi$$=-\triangle_{0}$in

2. Main result
We make the following assumptions on the data:(1)

is a Lipschitz continuous function from
into and is a non-negative functionof -class from into itself such that the derivatives and $g’$ are bounded on R.
$\mathrm{R}^{2}$$f$$C^{2}$(2) $h\in L^{\infty}(Q)$ ,$\mathrm{R}$$\theta_{0}\in V\cap L^{\infty}(\Omega)$

$g$

$g’$

$\mathrm{R}$and$w_{0}\in D(-\triangle_{0})$.

Now we give the definition of a solution of (P).

Definition 2,1. A pair of functions
following conditions $(a1)-(a4)$ :$\{\theta,$w} is called a solution of (P) if it satisfies the25, $w\in W^{1,2}(0, T; H)\cap L^{\infty}(\mathrm{O}, T;V)\cap L^{2}(0,T; H^{2}(\Omega))$.

(01)$\theta$(a2)$\theta’(t)+w’(t)-\triangle_{0}\theta(t)=h(t)$in $H$ for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

(a3) There exists a function $\xi\in L^{2}(0, T;H)$ with$\xi\in\partial I_{\theta,N}(w’)\mathrm{a}.\mathrm{e}$$w’(t)+\xi(t)-l/\triangle_{0}w(t)=f(\theta(t), w(t))$(04)$\theta(0)=\theta_{0}$in $H$ for$\mathrm{a}.\mathrm{e}$

. on

$Q$

such that. $t\in(0, T)$ .

and $w(0)=w_{0}$ in $H$ .

We denote the time-derivatives of$\theta$

and

$w$

by

$\theta’$and $w’$ , respectively.

Theorem 2.1. Under the assumptions (1) and (2), problem (P) has at least one solution
and $w\in W^{1,2}(0, T; V)$in the sense of Definition 2.1 such that 0 6$L^{\infty}(Q)$$\{\theta, w\}$

$L^{\infty}(0, T;H^{2}(\Omega))$$\cap$.

The above existence result will be proved in the sections 3, 4 and 5.

3. Approximate problem
In this section, we consider the following approximate problem $(P_{\epsilon}):=\{(3.1)-(3.3)\}$ :in$\theta_{t}+w_{t}-\triangle_{0}\theta=h$

$Q$

(3.1)

,

in

$w_{t}+\partial I_{\theta,N}(w_{t})+\nu\partial\varphi_{\epsilon}(w)\ni f(\theta, J_{\epsilon}^{\varphi}w)$$\theta(0, \cdot)=\theta_{0}$,in$w(0, \cdot)=w_{0}$whereis the Moreau-Yosida approximation of.
in $H$ andferential of$\varphi_{\Xi}$$\varphi$$\Omega$

$Q$

(3.2)

,

(3.3)

,

defined by $(1,10)$ ,$\partial\varphi_{\epsilon \mathrm{i}}$is the subdif-$J_{\epsilon}^{\varphi}=(I+\epsilon\partial\varphi)^{-1}$$\varphi_{\epsilon}$, a pair of functions
Definition 3.1. For every fixed
if it satisfies the following conditions $(b1)-(b4)$ :$\epsilon>0$$\{\theta_{\epsilon}, w_{\epsilon}\}$is called a solution of$(P_{\epsilon})$

(01)
(02)

$\theta_{\epsilon}$,$J_{\epsilon}^{\varphi}w_{\xi}\in W^{1,2}(0,T;H)$$\cap L^{\infty}(0, T;V)$$\theta_{\epsilon}’(t)+w_{\epsilon}’(t)-\triangle_{0}\theta_{\epsilon}(t)=h(t)$(b3) There exists a function, $w_{\epsilon}\in W^{1,2}(0, T;H)$ .

in $H$ for$\xi_{\mathcal{E}}\in L^{2}(0,T;H)$$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

with$\xi_{\epsilon}\in\partial I_{\theta_{\epsilon},N}(w_{\epsilon}’)\mathrm{a}.\mathrm{e}$$w_{\xi \mathrm{j}}’(t)+\xi_{\epsilon}(t)+\iota/\partial\varphi_{\epsilon}(w_{\epsilon}(t))=f(\theta_{\epsilon}(t), J_{\epsilon}^{\varphi}w_{\epsilon}(t))$(b4) $0(0)=\theta_{0}$ and$w_{\epsilon}(0)=w_{0}$for$\mathrm{a}.\mathrm{e}$.

$Q$such that$t\in(\mathrm{O}, T)$

.

in H.

Theorem 3.1. Under the assumption (1), $/or$ any
each $>0_{2}$ There exists at least one solution
3. 1.
$\epsilon$

1n $H$. on$\theta_{0}$$\{\theta_{\epsilon}, w_{\epsilon}\}$, $w_{0}\in V$, $h\in L^{2}(0, T;H)$ and forof$(P_{\epsilon})$in the sense of Definition

26

First, we construct a local in time solution ofdefined by
To do so, prepare a set

by using the fixed point argument.

$(P_{\epsilon})$$X_{T}^{\epsilon}(M_{0})$,$\overline{\theta}\in W^{1,2}(0, T; H)\cap L^{\infty}(0, T; V)$$\overline{\theta}(0)=\theta_{0}$,$J^{\varphi}\overline{w}\in W^{1,2}(0, T;H)\cap L^{\infty}(0, T;V),\overline{w}(0)=w0$$X_{T}^{\epsilon}(M_{0}):=\{(\overline{\theta},\overline{w})$$|^{\frac{\epsilon}{\theta}}’|_{L^{2}(0,T;H)}^{2}\leq 1+M_{0}$,$|\overline{w}’|_{L^{2}(0,T;H)}^{2}\leq 1+M_{0}$,$\sup_{t\in[0,T]}|\nabla J_{\epsilon}^{\varphi}\overline{w}(t)|_{H}^{2}\leq 1+M_{0}\}$$\sup_{t\in[0,T]}|\nabla\overline{\theta}(t)|_{H}^{2}\leq 1+M_{0}$,, (3.4),is a positive constant dependent on the norm of initial data and a fixed number
whereis the convex and, more precisely, $M_{0}:=|\theta_{0}|_{V}^{2}+2(1+\nu)|w_{0}|_{V}^{2}$ . We see thatcompact subset of the product space $C([0, T];H)\mathrm{x}$ $C_{w}([0, T];H)$ . We fix $>0$ and takefor , in the right side of (3.2), namely, and substituteinan element$M_{0}$$X_{T}^{\epsilon}(M_{0})$$\nu$$\epsilon$$\overline{\theta},\overline{w}$$X_{T}^{\epsilon}(M_{0})$$(\overline{\theta},\overline{w})$$w$$\theta$in $H$ for$w’(t)+\partial I_{\overline{\theta}(t),N}(w’(t))+\nu\partial\varphi_{\epsilon}(w(t))\ni f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))$

We denote byevery$(P_{\epsilon})_{(\overline{\theta},\overline{w})}$$(\overline{\theta},\overline{w})\in X_{T}^{\epsilon}(M_{0})$the system (3.1), (3.5) for each
, (3.5) can be written in the form$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

$(\overline{\theta},\overline{w})\in X_{T}^{\epsilon}(M_{0})$$w’(t)=(I+\partial I_{\overline{\theta}(t),N})^{-1}(f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))-\iota/\partial\varphi_{\epsilon}(w(t)))$(3.5)

and (3.3). For,

are Lipschitz continuous
andwhere I is the identity in $H$ . Noting that$H$ofuniquesolutioninand the equation (3.1) is linear, we can find awith $0<T_{0}\leq T$ , (determined later), we define a mappingNow, taking a number
into $W^{1,2}(0, T;H)\cap L^{\infty}(0, T;V)$ by the formulafrom$(I+\partial I_{\overline{\theta},N})^{-1}$$\partial\varphi_{\epsilon}$$\{\theta, w\}$$(P_{\mathcal{E}})(\overline{\theta},\overline{w}\rangle\cdot$$S$$T_{0}$$X_{T}^{\epsilon}(M_{0})$$(\theta(t), w(t))$$S(\overline{\theta},\overline{w})(t)=\{$$(\theta(T_{0}), w(T_{0}))$wherelemma:$\{\theta, w\}$is the solution of$(P_{\epsilon})_{(\overline{\theta},\overline{w})}$.

if $0\leq t\leq T_{0}$ ,if $T_{0}\leq t\leq T$ ,As to this mapping$S$, we see the following

Lemma 3.1. There exists T0 with $0<T_{0}\leq T$ such that
$S(X_{T}^{\epsilon}(M_{0}))\subseteq X_{T}^{\epsilon}(M_{0})$.

Proof. Letandbe the solution of(1), without loss of generality, we may assume that andR. Multiplying (3.5) byin $H$ and noting that$(\overline{\theta},\overline{w})\in X_{T}^{\epsilon}(M_{0})$$\{\theta, w\}$$g$

. From the assumption
are Lipschitz continuous on$(P_{\epsilon})_{(\overline{\theta},\overline{w})}$$g’$$w’-g(\overline{\theta})$$\bullet$$\bullet$$\bullet$($w’(t)$ ,$w’(t)-g( \overline{\theta}(t)))\geq\frac{3}{4}|w’(t)|_{H}^{2}-|g(\overline{\theta}(t))|_{H}^{2}$$(\xi(t),$ $w’(t)-g(\overline{\theta}(t)))\geq 0$($\partial\varphi_{\epsilon}(w(t))$,, V4(t),$\in\partial I_{\overline{\theta}(t),N}(w’(t))$,$w’(t)-g( \overline{\theta}(t)))=\frac{d}{dt}\varphi_{\epsilon}(w(t))-(\partial\varphi_{\epsilon}(w(t)), g(\overline{\theta}(t)))$,27$\bullet(f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t)),$$w’(t)-g(\overline{\theta}(t)))$$=(f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))$,$w’(t))-(f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t)),$ $g(\overline{\theta}(t)))$$\leq\frac{1}{4}|w’(t)|_{H}^{2}+|f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))|_{H}^{2}+\frac{1}{2}|f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))|_{H}^{2}+\frac{1}{2}|g(\overline{\theta}(t))|_{H}^{2}$$\leq\frac{1}{4}|w’(t)|_{H}^{2}+K_{1}(|\overline{\theta}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}\overline{w}(t)|_{H}^{2}+1)$,

we have$\frac{1}{2}|w’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(w(t))\leq K_{1}(|\overline{\theta}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}\overline{w}(t)|_{H}^{2}+1)+\nu(\partial\varphi_{\epsilon}(w(t)), g(\overline{\theta}(t)))$$(0, T)$ , wherefor .
of and , and norms
inequalities:$t\in$$\mathrm{a}.\mathrm{e}$$f$$K_{1}$$|f|_{\infty}$$g$$(\partial\varphi_{\epsilon}(w(t)), g(\overline{\theta}(t)))$is a positive constant depending only on Lipschitz constants. Combining the above inequality with the following
and$|g|_{\infty}$$=$$(-\triangle_{0}J_{\epsilon}^{\varphi}w(t),$ $g(\overline{\theta}(t)))$$=$$(\nabla J_{\epsilon}^{\varphi}w(t),$$\leq$$\leq$$\nabla g(\overline{\theta}(t)))$$\frac{1}{2}|\nabla J_{\epsilon}^{\varphi}w(t)|_{H}^{2}+\frac{1}{2}|\nabla g(\overline{\theta}(t))|_{H}^{2}$$\varphi_{\epsilon}(w(t))+\frac{|g’|_{\infty}^{2}(1+M_{0})}{2}$,$|\overline{\theta}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}\overline{w}(t)|_{H}^{2}\leq|\overline{\theta}(t)|_{H}^{2}+|\overline{w}(t)|_{H}^{2}$$=| \overline{\theta}(0)+\oint_{0}^{t}\overline{\theta}’(s)ds|2H +|w-(0)+ \int_{0}^{t}\overline{w}’(s)ds|_{H}^{2}$$\leq 2|\theta_{0}|_{H}^{2}+2\int_{0}^{t}|\overline{\theta}’(s)|_{H}^{2}ds+2|w_{0}|_{H}^{2}+2\int_{0}^{t}|\overline{w}’(s)|_{H}^{2}ds$$\leq 2|\theta_{0}|_{H}^{2}+2t|\overline{\theta}’|_{L^{2}(0,T_{j}H)}^{2}+2|w_{0}|_{H}^{2}+2t|\overline{w}’|_{L^{2}(0,T;H)}^{2}$$\leq 2(|\theta_{0}|_{H}^{2}+|w_{0}|_{H}^{2})+4T(1+M_{0})$,we see that

(3.6)$\frac{1}{2}|w’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(w(t))\leq\nu\varphi_{\epsilon}(w(t))+K_{2}$is a positive constant depending only on the Lipschitz. $t\in(0, T)$ , wherefor. Applying theand constants and,,constants of , , norms
Gronwall’s lemma to (3.6), we obtain that
$K_{2}$$\mathrm{a}.\mathrm{e}$

$f$

$g$

$|f|_{\infty}$$|g|_{\infty},$$|\theta_{0}|_{H}$$|w_{0}|_{H}$$M_{0}$$\nu$$\nu\varphi_{\epsilon}(w(t))\leq(\nu\varphi_{\epsilon}(w_{0})+K_{2}T)e^{T}\leq(\nu|\nabla w_{0}|_{H}^{2}+K_{2}T)e^{T}=:K_{3}$,$\forall t\in[0, T]$

Hence by (3.6), the following holds:$\frac{1}{2}|w’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(w(t))\leq K_{2}+K_{3}$for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

.

28

Integrating the above in over $[0, T’]$ with $(0<T’\leq T)$ , we obtain that$t$$\frac{1}{2}\oint_{0}^{T’}|w’(t)|_{H}^{2}dt+\mathit{1}/\varphi_{\epsilon}(w(T’))\leq\nu\varphi_{\epsilon}(w_{0})+T’(K_{2}+K_{3})$,$\forall T’\in(0, T]$

.

(3.7)

Next multiplying 0’ by (3.1) in $H$ , we get that$\frac{1}{2}|\theta’(t)|_{H}^{2}+\frac{1}{2}\frac{d}{dt}|\nabla\theta(t)|_{H}^{2}\leq|w’(t)|_{H}^{2}+|h(t)|_{H}^{2}$Integrating the above in

$t$

over

$[0, \tilde{T}]$$\frac{1}{2}\int_{0}^{\tilde{T}}|\theta’(t)|_{H}^{2}dt+\frac{1}{2}|\nabla\theta(\tilde{T})|_{H}^{2}$$(0<\tilde{T}\leq T’)$for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

and using (3.7), we see that$\leq$$\frac{1}{2}|\nabla\theta_{0}|_{H}^{2}+\int_{0}^{T’}|w’(t)|_{H}^{2}dt+\int_{0}^{T’}|h(t)|_{H}^{2}dt$$\leq$$K_{0}+2T’(K_{2}+K_{3})+ \int_{0}^{T’}|h(t)|_{H}^{2}dt$

,

where
$K_{0}:= \frac{1}{2}|\nabla\theta_{0}|_{H}^{2}+\nu|\nabla w_{0}|_{H}^{2}\leq\frac{M_{0}}{2}$Taking$T_{0}$.

with $T_{0}\leq T$ such that
$2T_{0}(1+ \frac{1}{\nu})(K_{2}+K_{3})+\oint_{0}^{T_{0}}|h(t)|_{H}^{2}dt\leq\frac{1}{2}$we have the conclusion.

,$\phi$$w_{0}\in V$,, there exists a solution
Lemma 3.2. For any $>0$ and any
41,
such
that
on
the
time-intervalof
and $w\in W^{1,2}(0, T_{0;}H)$ , whereis (small) positive number determined in Lemma 3. 1.
$\theta_{0},$$\epsilon$$\{\theta, w\}$$h\in L^{2}(\mathrm{O}, T;H)$$[0, T_{0}]$$(P_{\Xi})$$T_{0}$$J_{\epsilon}^{\varphi}w\in W^{1,2}(\mathrm{O}, T_{0;}H)\cap L^{\infty}(0, T_{0;}V)$$a$

Proof. In order to get the conclusion of this lemma, we shall use the Schauder’s fixed
point theorem for the mapping . First we show
is continuous in the topology of$C([0, T];H)\mathrm{x}$ $C_{w}([0, T];H)$ . We take a sequence
converging to some$C([0,T];H)\mathrm{x}$$C_{w}([0, T];H)$ . Letelement
in the topology ofbethe solution ofThencoupleofeach forN.
thefunctions satisfies$S$$\mathrm{S}$$\{(\overline{\theta}_{n},\overline{w}_{n})\}\subset X_{T}^{\epsilon}(M_{0})$$(\overline{\theta},\overline{w})\in X_{T}^{\epsilon}(M_{0})$$\{\theta_{i}, w_{i}\}$$\mathrm{i}\in$$(P_{\epsilon})_{(\overline{\theta}_{i},\overline{w}_{i})}$$\{\theta_{i}, w_{i}\}$$\theta_{i}’(t)+w_{i}’(t)-\triangle_{0}\theta_{i}(t)=h(t)$in $H$ for$\mathrm{a}.\mathrm{e}$. $t\in(0, T_{0})$ ,(3.8)

in $H$ for. $t\in(0, T_{0})$ ,(3.8)(3.10)
and $w_{i}(0)=w_{0}$ in $H$,
in $H$ for. $t\in(0, T_{0})$ . By (3.8) and (3.9), two solutions, , satisfy that$w_{i}’(t)+\xi_{i}(t)+\nu\partial\varphi_{\epsilon}(w:(t))=f(\overline{\theta}_{i}(t), J_{\mathcal{E}}^{\varphi}\overline{w}_{\mathrm{z}}(t))$$\mathrm{a}.\mathrm{e}$$\theta_{i}(0)=\theta_{0}$where$\xi_{i}(t)\in\partial I_{\overline{\theta}_{i}(t),N}(w_{i}’(t))$$\{\theta_{i}, w_{i}\}$,$\mathrm{i}=m$$\mathrm{a}.\mathrm{e}$$n$$\theta_{m}’(t)-\theta_{n}’(t)+w_{m}’(t)-w_{n}’(t)-\triangle_{0}(\theta_{m}(t)-\theta_{n}(t))=0$

in $H$(3.11)

and$w_{m}’(t)-w_{n}’(t)+\xi_{m}(t)-\xi_{n}(t)+\nu\partial\varphi_{\epsilon}(w_{m}(t))-\nu\partial\varphi_{\epsilon}(w_{n}(t))$

in $H$. $t\in(0, T_{0})$ . For the sake of simplicity, we denote $w_{m}-w_{n}$ ,, and , respectively. Multiplying (3.12) by
by
$H$
in
and noting that

(3.12)$=f(\overline{\theta}_{m}(t), J_{\epsilon}^{\varphi}\overline{w}_{m}(t))-f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t))$

for
and

$\mathrm{a}.\mathrm{e}$$\theta_{m}-\theta_{n}$$\theta\wedge,\frac{\hat}{\theta}\hat{\xi}$$\overline{w}_{m}-\overline{w}_{n}$$g(\overline{\theta}_{n}))$$\hat{w},$$\frac{\hat}{w}$,$\xi_{m}-\xi_{n},\overline{\theta}_{m}-\overline{\theta}_{n}$$w_{m}’-g(\overline{\theta}_{m})-(w_{n}’$-29$\bullet(w_{m}’(t)-w_{n}’(t),$ $w_{m}’(t)-g(\overline{\theta}_{m}(t))-(w_{n}’(t)-g(\overline{\theta}_{n}(t))))$$=|w_{m}’(t)-w_{n}’(t)|_{H}^{2}-(w_{m}’(t)-w_{n}’(t),$ $g(\overline{\theta}_{m}(t))-g(\overline{\theta}_{n}(t)))$$\geq\frac{3}{4}|w_{m}’(t)-w_{n}’(t)|_{H}^{2}-|g(\overline{\theta}_{m}(t))-g(\overline{\theta}_{n}(t))|_{H}^{2}$$\geq\frac{3}{4}|\hat{w}’(t)|_{H}^{2}-L_{g}^{2}|^{\frac{\hat}{\theta}}(t)|_{H}^{2}$$\bullet$($\xi_{rn}(t)-\xi_{n}(t)$,,,$w_{m}’(t)-g(\overline{\theta}_{m}(t))-(w_{n}’(t)-g(\overline{\theta}_{n}(t))))\geq 0$$\bullet(\partial\varphi_{\epsilon}(w_{m}(t))-\partial\varphi_{\epsilon}(w_{n}(t)))w_{m}’(t)-g(\overline{\theta}_{m}(t))-(w_{n}’(t)-g(\overline{\theta}_{n}(t))))$$=(\partial\varphi_{\epsilon}(w_{m}(t)-w_{n}(t)), w_{m}’(t)-w_{n}’(t))-(\partial\varphi_{\epsilon}(w_{m}(t)-w_{n}(t)), g(\overline{\theta}_{m}(t))-g(\overline{\theta}_{n}(t)))$$\geq\frac{d}{dt}\varphi_{\epsilon}(w_{m}(t)-w_{n}(t))-\frac{1}{2}|\partial\varphi_{\epsilon}(w_{m}(t)-w_{n}(t))|_{H}^{2}-\frac{1}{2}|g(\overline{\theta}_{m}(t))-g(\overline{\theta}_{n}(t))|_{H}^{2}$,$\geq\frac{d}{dt}\varphi_{\epsilon}(\hat{w}(t))-\frac{1}{2\epsilon^{2}}|\hat{w}(t)|_{H}^{2}-\frac{L_{g}^{2}}{2}|^{\frac{\hat}{\theta}}(t)|_{H}^{2}$$\bullet(f(\overline{\theta}_{m}(t), J_{\epsilon}^{\varphi}\overline{w}_{m}(t))-f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t)),$$w_{m}’(t)-g(\overline{\theta}_{m}(t))-(w_{n}’(t)-g(\overline{\theta}_{n}(t))))$$=(f(\overline{\theta}_{m}(t), J_{\epsilon}^{\varphi}\overline{w}_{m}(t))-f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}w_{n}(t)),$$w_{m}’(t)-w_{n}’(t))$$-(f(\overline{\theta}_{m}(t), J_{\epsilon}^{\varphi}\tilde{w}_{m}(t))-f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t)),$$g(\overline{\theta}_{m}(t))-g(\overline{\theta}_{n}(t)))$$\leq\frac{1}{4}|w_{m}’(t)-w_{n}’(t)|_{H}^{2}+|f(\overline{\theta}_{m}(t), J_{\epsilon}^{\varphi}\overline{w}_{m}(t))-f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t))|_{H}^{2}$$+ \frac{1}{2}|f(\overline{\theta}_{m}(t), J_{\epsilon}^{\varphi}\overline{w}_{m}(t))-f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t))|_{H}^{2}+\frac{1}{2}|g(\overline{\theta}_{m}(t))-g(\overline{\theta}_{n}(t))|_{H}^{2}$,$\leq\frac{1}{4}|\hat{w}’(t)|_{H}^{2}+3L_{f}^{2}(|^{\frac{\hat}{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi\frac{\hat}{w}}(t)|_{H}^{2})+\frac{L_{g}^{2}}{2}|\hat{\overline{\theta}}(t)|_{H}^{2}$

we have that
(3.13)$\frac{1}{2}|\hat{w}’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(\hat{w}(t))\leq K_{4}(|^{\frac{\hat}{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi\frac{\hat}{w}}(t)|_{H}^{2}+|\hat{w}(t)|_{H}^{2})$andis a positive constant dependent on $>0$ and. $t\in(0, T_{0})$ , where
are the Lipschitz constants of and , respectively. By the simple calculation, we havefor$\mathrm{a}.\mathrm{e}$$\epsilon$$K_{4}$$f$$L_{g}$$L_{f}$$g$$\frac{d}{dt}|\hat{w}(t)|_{H}^{2}=2(\hat{w}’(t),\hat{w}(t))\leq|\hat{w}’(t)|_{H}^{2}+|\hat{w}(t)|_{H}^{2}$.

(3.14)

It follows from (3.13) and (3.14) that$\frac{d}{dt}\{\frac{1}{2}|\hat{w}(t)|_{H}^{2}+\nu\varphi_{\epsilon}(\hat{w}(t))\}\leq K_{5}(|^{\frac{\hat}{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi\frac{\hat}{w}}(t)|_{H}^{2})+K_{6}(\frac{1}{2}|\hat{w}(t)|_{H}^{2}+\nu\varphi_{\epsilon}(\hat{w}(t)))$30, whereandare positive constants. Applying the Gronwall’s.
forlemma to the above inequality, we have$\mathrm{a}.\mathrm{e}$$K_{6}$$K_{5}$$t\in(\mathrm{O}, T_{0})$$\frac{1}{2}|\hat{w}(t)|_{H}^{2}+\nu\varphi_{\epsilon}(\hat{w}(t))\leq e^{K_{6}T_{0}}\{K_{5}\int_{0}^{T_{0}}(|\hat{\overline{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}\hat{\overline{w}}(t)|_{H}^{2})dt\}$,$\forall t\in[0, T_{0}]$.

This implies that$|\hat{w}(t)|_{H}^{2}\leq 2e^{K_{6}T_{0}}K_{5}f_{0}^{T_{0}}$$(|^{\frac{\hat}{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi\frac{\hat}{w}}(t)|_{H}^{2})dt$,$\forall t\in[0, T_{0}]$.

Then (3. 13) gives$\frac{1}{2}|\hat{w}’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(\hat{w}(t))\leq K_{7}\{|^{\frac{\hat}{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}\hat{\overline{w}}(t)|_{H}^{2}+\int_{0}^{T_{0}}(|^{\frac{\hat}{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}\hat{\overline{w}}(t)|_{H}^{2})dt\}$

for . $t\in(0, T_{0})$ , wherethen$\mathrm{a}.\mathrm{e}$$K_{7}$

is a positive constant. Integrating the above in over$t$$\frac{1}{2}\oint_{0}^{T_{0}}|\hat{w}’(t)|_{H}^{2}dt+\nu\varphi_{\epsilon}(\hat{w}(T_{0}))\leq K_{7}(1+T_{0})\int_{0}^{T_{0}}(|^{\frac{\hat}{\theta}}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi\frac{\hat}{w}}(t)|_{H}^{2})dt$$[0, T_{0}]$,.

Taking , $marrow+\infty$ , we see that$n$$K_{7}(1+T_{0}) \int_{0}^{T_{0}}(|\hat{\overline{\theta}}(t)|_{H}^{2}+|J_{5}^{\varphi\frac{\hat}{w}}(t)|_{H}^{2})dtarrow \mathrm{O}$.

is a Cauchy sequence in $L^{2}(0, T_{0;}H)$ . Hence there exists a function$w\in W^{1,2}(0, T_{0;}H)$ such that
This shows that$w_{n}arrow w$

For every fixed$\{w_{n}’\}$

in $C([0, T_{0}];H)$ and$\epsilon>0$$w_{n}’arrow w’$$\leq$$\{\partial\varphi_{\epsilon}(w_{n})\}_{n=1}^{\infty}$

is bounded in$\int_{0}^{T_{0}}|\partial\varphi_{\epsilon}(w_{n}(t))-\partial\varphi_{\epsilon}(0)|_{H}^{2}dt$$\frac{1}{\epsilon^{2}}\oint_{0}^{T_{0}}|w_{n}(t)|_{H}^{2}dt$$L^{2}$( , To), $H)$ . Putting$0$$\xi_{n}(t):=-w_{n}’(t)-\nu\partial\varphi_{\epsilon}(w_{n}(t))+f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t))$

we see that

(3.15), from (3.15) and the following inequality$\oint_{0}^{T_{0}}|\partial\varphi_{\epsilon}(w_{n}(t))|_{H}^{2}dt=$

we see thatin $L^{2}(0, T_{0;}H)$ as $narrow+\infty$ .

in $H$ for$\mathrm{a}.\mathrm{e}$.

$t\in(0, T_{0})$,is bounded in
in $H$ for. $t\in(0, T_{0})$ and$L^{2}(0, T_{0;}H)$ . We may assume that for a subsequence,
converges weakly to 4
in $L^{2}(0,T_{0;}H)$ as $karrow+\infty$ and, because
in $C([0, T_{0}];H)$ as $narrow+\infty$ . For simplicity, we use again instead of . Moreover, we
can easily show that$\xi_{n}(t)\in\partial I_{\overline{\theta}_{n}(t),N}(w_{n}’(t))$$\{\xi_{n}\}$$\mathrm{a}.\mathrm{e}$$\{n_{k}\}$$\{\xi_{n_{k}}\}$$\xi=-w’-\nu\partial\varphi_{\epsilon}(w)+f(\overline{\theta}, J_{\epsilon}^{\varphi}\overline{w})$$Jfwnarrow J_{\epsilon}^{\varphi}\overline{w}$$n$$\lim\sup\int_{0}^{T_{0}}narrow+\infty(\xi_{n}(t), w_{n}’(t))dt\leq\int_{0}^{T_{0}}(\xi(t), w’(t))dt$$n_{k}$,(3.16)

31

because we see that$\int_{0}^{T_{0}}(\xi_{n}(t), w_{n}’(t))dt=f_{0}^{T_{\mathit{0}}}(-w_{n}’(t)$$=$-, $w_{n}’(t))dt$$\nu\partial\varphi_{\epsilon}(w_{n}(t))+f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t))$$- \oint_{0}^{T_{0}}|w_{n}’(t)|_{H}^{2}dt-\nu\varphi_{\epsilon}(w_{n}(T_{0}))+\nu\varphi_{\epsilon}(w_{0})$$+ \int_{0}^{T_{0}}$($f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t))$,$w_{n}’(t)$

)

$dt$

,

then
$\lim_{narrow+}\sup_{\infty}\mathfrak{l}\oint_{0}^{T_{0}}(\xi_{n}(t), w_{n}’(t))dt$$\leq$$- \lim_{narrow+}\inf_{\infty}|w_{n}’|_{L^{2}(0,T_{0j}H)}^{2}-\nu\lim_{n\prec+}\inf_{\infty}\varphi_{\epsilon}(w_{n}(T_{0}))+\nu\varphi_{\epsilon}(w_{0})$$+ \lim_{narrow+\infty}\oint_{0}^{T_{0}}$$\leq$$=$$f(\overline{\theta}_{n}(t), J_{\epsilon}^{\varphi}\overline{w}_{n}(t))$,$w_{n}’(t)$

)

$dt$

$-|w’|_{L^{2}(0,T_{0;}H)}^{2}-\nu\varphi_{\epsilon}(w(T_{0}))+\nu\varphi_{\epsilon}(w_{0})$$+ \oint_{0}^{T_{0}}$$=$($\int_{0}^{T_{0}}$($f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))$, $w’(t)$ )$dt$, $w’(t)$ ) $dt$$(-w’(t)-\nu\partial\varphi_{\xi}(w(t))+f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))$$\int_{0}^{T_{0}}(\xi(t), w’(t))dt$.

on $H$ in the sense of Mosco (cf.[4,12,17]) as $narrow+\infty$ , by the usual
monotonicity technique with the Mosco convergence and (3.16), we have the inclusion. $t\in(0, T_{0})$ . Finally, we have the following:in $H$ for
Since$I_{\tilde{\theta}_{n},N}(\cdot)arrow I_{\overline{\theta},N}(\cdot)$$\xi(t)\in\partial I_{\overline{\theta}(t),N}(w’(t))$$\mathrm{a}.\mathrm{e}$$\uparrow v’(t)+\xi(t)+\nu\partial\varphi_{\epsilon}(w(t))=f(\overline{\theta}(t), J_{\epsilon}^{\varphi}\overline{w}(t))7$

for$\mathrm{a}.\mathrm{e}$. $t\in(0, T_{0})$ . Multiplying (3.11) by$\hat{\theta}’$$\xi(t)\in\partial I_{\overline{\theta}(t),N}(w’(t))$

in $H$(3.17)

in $H$ with the following calculations,$\bullet(\theta_{m}’(t)-\theta_{n}’(t), \theta_{m}’(t)-\theta_{n}’(t))=(\hat{\theta}’(t),\hat{\theta}’(t))=|\hat{\theta}’(t)|_{H}^{2}$,$\bullet(w_{m}’(t)-w_{n}’(t), \theta_{\tau n}’(t)-\theta_{n}’(t))=(\hat{w}’(t),\hat{\theta}’(t))\geq-\frac{1}{2}|\hat{\theta}’(t)|_{H}^{2}-\frac{1}{2}|\hat{w}’(t)|_{H}^{2}$\bullet$(- \triangle_{0}(\theta_{m}(t)-\theta_{n}(t)),\theta_{m}’(t)-\theta_{\acute{n}}(t))=(-\triangle_{0}\hat{\theta}(t),\hat{\theta}’(t))=\frac{1}{2}\frac{d}{dt}|\nabla\hat{\theta}(t)|_{H}^{2}$,

we have that$| \hat{\theta}’(t)|_{H}^{2}+\frac{d}{dt}|\nabla\hat{\theta}(t)|_{H}^{2}\leq|\hat{w}’(t)|_{H}^{2}$for$\mathrm{a}.\mathrm{e}$. $t\in(0, T_{0})$ .

(3.18)

is a Cauchy sequence
Then on account of (3.15), the above inequality implies that
in $W^{1,2}(0, T_{0};H)\cap L^{\infty}(0, T_{0};V)$ . Therefore we may assume that there exists a function$\theta\in W^{1,2}(0, T_{0;}H)\cap L^{\infty}(0,T_{0;}V)$ such that$\{\theta_{n}\}$$\theta_{n}$& in $C([0, T_{0}];H)$ and$\theta_{n}’arrow\theta’$

in $L^{2}(0, T_{0;}H)$ as $narrow+\infty$ .

It follows from (3.15) and (3.19) that$-\triangle_{0}\theta_{n}arrow-\triangle_{0}\theta$

weakly in $L^{2}(0, T_{0;}H)$ as $narrow+\infty$ .

(3.19)

32

Hence the limit functions$\theta$

and

$w$

enjoy

in $H$ for$\theta’(t)+\mathrm{W}’ \mathrm{m}(\mathrm{t})-\triangle_{0}\theta(t)=h(t)$$\mathrm{a}.\mathrm{e}$. $t\in(0, T_{0})$ .

(3.20)

is the solution to
Therefore from (3.17) and (3.20), the pair of limit functions
and $w\in$on $(0, T_{0})$ with the regularities ,onto the time interval $[0, T]$ by$W^{1,2}(0, T_{0;}H)$ .
on
Here extend andand is continuous in the topology of $C([0, T];H)\mathrm{x}$and $w(T_{0})$ . Then
$C_{w}([0, T];H)$ . Hence we can apply the Schauder’s fixed point theorem with respect toonof which is a solution toto find a fixed point
the mapping in$\{\theta, w\}$$\theta$$J_{\epsilon}^{\varphi}w\in W^{1,2}(0, T_{0;}H)\cap L^{\infty}(0, T_{0;}V)$$(P_{\epsilon})_{(\overline{\theta},\overline{w})}$$S(^{\frac{7}{\theta}},\overline{w})=(\theta, w)$$S$$[0, T_{0}]$.

$\theta(T_{0})$$[0, T_{0}]$$w$$\theta$$S$$S$$(\theta, w)$$X_{T}^{\epsilon}(M_{0})$$(P_{\epsilon})$$\theta$

Lemma 3.3. For every fixedinterval [0,$T’]$ $(0<T’\leq T)$ .

$\epsilon>0$, the solution$\{\theta,$w} ofis unique on any time$(P_{\epsilon})$

on $[0, T’]$ $(0<T’\leq T)$ withbe the solutions toandProof. Let
the same initial data, namely, they satisfy the following equations:$\{\theta_{m}, w_{m}\}$$(P_{\epsilon})$$\{\theta_{n}, w_{n}\}$in $H$ for$\theta_{i}’(t)+w_{i}’(t)-\triangle_{0}\theta_{i}(t)=h(t)$$w_{i}’(t)+\mathrm{h}\mathrm{i}\mathrm{t})+\nu\partial\varphi_{\epsilon}(w_{i}(t))=f(\theta_{i}(t), J_{\epsilon}^{\varphi}w_{i}(t))$$\theta_{i}(0)=\theta_{0}$wheretwo solutions$\{\theta_{i}, w_{i}\}$. $t\in(0, T’)$ ,in $H$ for$\mathrm{a}.\mathrm{e}$. $t\in(0, T’)$ ,$\mathrm{a}.\mathrm{e}$$\mathrm{i}=m$(3.21)(3.22)(3.23)

and $w_{i}(0)=w_{0}$ in $H$ ,. $t\in(0, T’)$ ,in $H$ for, $i=m$ , , satisfy that$\xi_{i}(t)\in\partial I_{\theta_{i}(t),N}(w_{i}’(t))$$\mathrm{a}.\mathrm{e}$, . By the above equations,
$n$

$n$

$\theta_{m}’(t)-\theta_{n}’(t)+w_{m}’(t)-w_{n}’(t)-\triangle_{0}(\theta_{m}(t)-\theta_{n}(t))=0$

in $H$(3.24)

and$w_{m}’(t)-w_{n}’(t)+\xi_{m}(t)-\xi_{n}(t)+\nu\partial\varphi_{\epsilon}(w_{m}(t))-\nu\partial\varphi_{\epsilon}(w_{n}(t))$$=f(\theta_{m}(t), J_{\epsilon}^{\varphi}w_{m}(t))-f(\theta_{n}(t), J_{\epsilon}^{\varphi}w_{n}(t))$

in $H$(3.25). $t\in(0, T’)$ . Then by the same calculations to get (3.13) and (3.18) as in Lemma
3.2; (3.24),
and (3.25)

for$\mathrm{a}.\mathrm{e}$$\mathrm{x}\hat{\theta}’$$\mathrm{x}\{w_{m}’-g(\theta_{m})-(w_{n}’-g(\theta_{n}))\}$$\bullet(w_{m}’(t)-w_{n}’(t), w_{m}’(t)-g(\theta_{m}(t))-(w_{n}’(t)-g(\theta_{n}(t))))$$=|w_{m}’(t)-w_{n}’(t)|_{H}^{2}-(w_{m}’(t)-w_{n}’(t), g(\theta_{m}(t))-g(\theta_{n}(t)))$$\geq\frac{3}{4}|w_{m}’(t)-w_{n}’(t)|_{H}^{2}-|g(\theta_{m}(t))-g(\theta_{n}(t))|_{H}^{2}$$\geq\frac{3}{4}|\hat{w}’(t)|_{H}^{2}-L_{g}^{2}|\hat{\theta}(t)|_{H}^{2}$$\bullet$,$(\xi_{m}(t)-\xi_{n}(t), w_{m}’(t)-g(\theta_{m}(t))-(w_{n}’(t)-g(\theta_{n}(t))))\geq 0_{7}$$\bullet\frac{1}{2}|\nabla J_{\epsilon}^{\varphi}(\hat{w}(t))|_{H}^{2}=\varphi(J_{\epsilon}^{\varphi}\hat{w}(t))\leq\varphi_{\epsilon}(\hat{w}(t))$,

33$\bullet\frac{1}{2}|\nabla g(\theta_{m}(t))-\nabla g(\theta_{n}(t))|_{H}^{2}=\frac{1}{2}|g’(\theta_{m}(t))\nabla\theta_{m}(t)-g’(\theta_{n}(t))\nabla\theta_{n}(t)|_{H}^{2}$$\leq|g’(\theta_{m}(t))\nabla\theta_{m}(t)-g’(\theta_{m}(t))\nabla\theta_{n}(t)|_{H}^{2}+|g’(\theta_{m}(t))\nabla\theta_{n}(t)-g’(\theta_{n}(t))\nabla\theta_{n}(t)|_{H}^{2}$$\leq|g’(\theta_{m}(t))|_{H}^{2}|\nabla\theta_{m}(t)-\nabla\theta_{n}(t)|_{H}^{2}+|\nabla\theta_{n}(t)|_{H}^{2}|g’(\theta_{m}(t))-g’(\theta_{n}(t))|_{H}^{2}$$\leq|g’(\theta_{m}(t))|_{H}^{2}|\nabla\hat{\theta}(t)|_{H}^{2}+L_{\mathit{9}}^{2},|\nabla\theta_{n}(t)|_{H}^{2}|\hat{\theta}(t)|_{H}^{2}$$\leq K_{8}(|\nabla\hat{\theta}(t)|_{H}^{2}+|\hat{\theta}(t)|_{H}^{2})$,$\bullet(\partial\varphi_{\epsilon}(w_{m}(t))-\partial\varphi_{\epsilon}(w_{n}(t)), w_{m}’(t)-g(\theta_{m}(t))-(w_{n}’(t)-g(\theta_{n}(t))))$$=(\partial\varphi_{\epsilon}(w_{m}(t)-w_{n}(t)), w_{m}’(t)-w_{n}’(t))-(\partial\varphi_{\epsilon}(w_{m}(t)-w_{n}(t)), g(\theta_{m}(t))-g(\theta_{n}(t)))$$\geq\frac{d}{dt}\varphi_{\epsilon}(w_{m}(t)-w_{n}(t))-\frac{1}{2}|\nabla J_{\epsilon}^{\varphi}(w_{m}(t)-w_{n}(t))|_{H}^{2}-\frac{1}{2}|\nabla g(\theta_{m}(t))-\nabla g(\theta_{n}(t))|_{H}^{2}$$\geq\frac{d}{dt}\varphi_{\epsilon}(\hat{w}(t))-\varphi_{\epsilon}(\hat{w}(t))-K_{8}(|\nabla\hat{\theta}(t)|_{H}^{2}+|\hat{\theta}(t)|_{H}^{2})$$\bullet(f(\theta_{m}(t), J_{\epsilon}^{\varphi}w_{m}(t))-f(\theta_{n}(t), J_{\epsilon}^{\varphi}w_{n}(t)),$,$w_{m}’(t)-g(\theta_{m}(t))-(w_{n}’(t)-g(\theta_{n}(t))))$$=(f(\theta_{m}(t\}, J_{\epsilon}^{\varphi}w_{m}(t))-f(\theta_{n}(t), J_{\epsilon}^{\varphi}w_{n}(t)),$$w_{m}’(t)-w_{n}’(t))$$-(f(\theta_{m}(t), J_{\epsilon}^{\varphi}w_{m}(t))-f(\theta_{n}(t), J_{\epsilon}^{\varphi}w_{n}(t)),$$g(\theta_{m}(t))-g(\theta_{n}(t)))$$\leq\frac{1}{4}|w_{m}’(t)-w_{n}’(t)|_{H}^{2}+|f(\theta_{m}(t), J_{\epsilon}^{\varphi}w_{m}(t))-f(\theta_{n}(t), J_{\epsilon}^{\varphi}w_{n}(t))|_{H}^{2}$$+ \frac{1}{2}|f(\theta_{m}(t), J_{\epsilon}^{\varphi}w_{m}(t))-f(\theta_{n}(t), J_{\epsilon}^{\varphi}w_{n}(t))|_{H}^{2}$$ $\frac{1}{2}|g(\theta_{m}(t))-g(\theta_{n}(t))|_{H}^{2}$$\leq\frac{1}{4}|\hat{w}’(t)|_{H}^{2}+3L_{f}^{2}(|\hat{\theta}(t)|_{H}^{2}+|J_{\xi j}^{\varphi}\hat{w}(t)|_{H}^{2})+\frac{L_{g}^{2}}{2}|\hat{\theta}(t)|_{H}^{2}$$\leq\frac{1}{4}|\hat{w}’(t)|_{H}^{2}+$K9$(|\hat{w}(t)|_{H}^{2}+|\hat{\theta}(t)|_{H}^{2})$,we deduce that$\frac{1}{2}|\hat{w}’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(\hat{w}(t))\leq\nu\varphi_{\epsilon}(\hat{w}(t))+K_{10}(|\hat{\theta}(t)|_{H}^{2}+|\nabla\hat{\theta}(t)|_{H}^{2}+|w^{\mathrm{A}}(t)|_{H}^{2})$and(3.26)(3.27)$| \hat{\theta}’(t)|_{H}^{2}+\frac{d}{dt}|\nabla\hat{\theta}(t)|_{H}^{2}\leq|\hat{w}’(t)|_{H}^{2}$are positiveand,for . $t\in(0, T’)$ , where $\theta=\theta_{m}-\theta_{n},\hat{w}=w_{m}-w_{n}$ andis a Lipschitz constant of . Computing (3.26)+
constants independent of $>0$ and, we have that
(3.27)$K_{8}$$K_{9}$$K_{10}$$\mathrm{a}.\mathrm{e}$$\epsilon$$\mathrm{x}$$L_{g’}$$g’$$\frac{1}{4}$$\frac{1}{4}|\hat{w}’(t)|_{H}^{2}+\frac{1}{4}|\hat{\theta}’(t)|_{H}^{2}+\frac{d}{dt}\{\nu\varphi_{\epsilon}(\hat{w}(t))+\frac{1}{4}|\nabla\hat{\theta}(t)|_{H}^{2}\}\leq\nu\varphi_{\epsilon}(\hat{w}(t))+K_{10}(|\hat{\theta}(t)|_{V}^{2}+|\hat{w}(t)|_{H}^{2})$

34

for

$\mathrm{a}.\mathrm{e}$. $t\in(0, T’)$ . Making use of (3.14) for both$w$

and 0, we have the following$\frac{d}{dt}\{\frac{1}{4}|\hat{\theta}(t)|_{V}^{2}+\frac{1}{4}|\hat{w}(t)|_{H}^{2}+\nu\varphi_{\epsilon}(\hat{w}(t))\}\leq K_{11}(\frac{1}{4}|\hat{\theta}(t)|_{V}^{2}+\frac{1}{4}|\hat{w}(t)|_{H}^{2}+\nu\varphi_{\epsilon}(\hat{w}(t)))$is a positive constant. Applying the Gronwall’s lemma to. $t\in(0, T’)$ , whereforthe above inequality, the uniqueness follows at once.
$K_{11}$$\mathrm{a}.\mathrm{e}$$\theta$

Lemma 3.4. For every fixed
the interval [0, T].

Let$T_{0}$$>0_{1}$the solution$\{\theta,$w} of$(P_{\epsilon})$can be extended in time tohas a (unique) solution
be the supremum of all $T_{0}\in[0, T]$ such that.
is uniquely determined on the interval. By Lemma 3.3, &,onsatisfies that:be any number such that $0<T_{0}<T^{*}$ . The solution

Proof. Let$\{\theta, w\}$$\epsilon$$T^{*}$$(P_{\epsilon})$$[0, T_{0}]$$[0, T^{*})$

$w\}$

$\{$

$\{\theta, w\}$$w’(t)+\xi(t)+\iota/\partial\varphi_{\epsilon}(w(t))=f(\theta(t), J_{\epsilon}^{\varphi}w(t))$$\theta(0)=\theta_{0}$(3.28)

in $H$,$\theta’(t)+w’(t)-\triangle_{0}\theta(t)=h(t)$in $H$,(3.29)(3.30)

and $w(0)=w_{0}$ in $H$ ,in $H$ . Multiplying (3.29) by $w’-g(\theta)$ and (3.28) by 0’ in $H$
wherewith the following calculations:$\xi(t)\in\partial I_{\theta(t),N}(w’(t))$$\bullet(w’(t), w’(t)-g(\theta(t)))\geq\frac{3}{4}|w’(t)|_{H}^{2}-|g(\theta(t))|_{H}^{2}$$\bullet$$(\xi(t), w’(t)-g(\theta(t)))\geq 0$

,

V4

,

$(t)\in\partial I_{\theta\langle t),N}(w’(t))$,$\bullet(\partial\varphi_{\epsilon}(w(t)), w’(t)-g(\theta(t)))=\frac{d}{dt}\varphi_{\epsilon}(w(t))-(\partial\varphi_{\epsilon}(w(t)), g(\theta(t)))$$\bullet(\partial\varphi_{\epsilon}(w(t)), g(\theta(t)))\leq\varphi_{\epsilon}(w(t))+\frac{|g’|_{\infty}^{2}}{2}|\nabla\theta(t)|_{H}^{2}$$\bullet(f(\theta(t), J_{\epsilon}^{\varphi}w(t)),$

$=$

,

,

$w’(t)-g(\theta(t)))$$(f(\theta(t), J_{\epsilon}^{\varphi}w(t))$, $w’(t))-(f(\theta(t), J_{\epsilon}^{\varphi}w(t)),$$g(\theta(t)))$$\leq\frac{1}{4}|w’(t)|_{H}^{2}+|f(\theta(t),J_{\epsilon}^{\varphi}w(t))|_{H}^{2}+\frac{1}{2}|f(\theta(t),J_{\epsilon}^{\varphi}w(t))|_{H}^{2}+\frac{1}{2}|g(\theta(t))|_{H}^{2}$$\leq\frac{1}{4}|w’(t)|_{H}^{2}+K_{12}(|\theta(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}w(t)|_{H}^{2}+1)$,

we have that$\frac{1}{2}|w’(t)|_{H}^{2}$$+ \nu\frac{d}{dt}\varphi_{\epsilon}(w(t))\leq K_{13}(|\theta(t)|_{V}^{2}+|w(t)|_{H}^{2}+\nu\varphi_{\epsilon}(w(t))+1)$.

(3.31)

and$\frac{1}{2}|\theta’(t)|_{H}^{2}+\frac{1}{2}\frac{d}{dt}|\nabla\theta(t)|_{H}^{2}\leq|w’(t)|_{H}^{2}+|h(t)|_{H}^{2}$(3.30)

35. $t\in(0, T_{0})$ , respectively, where K12 andfor(3.31) $+(3.32)\mathrm{x}$ , we have that$\mathrm{a}.\mathrm{e}$$K_{13}$are positive constants. Computing$\frac{1}{4}$$\frac{1}{4}|w’(t)|_{H}^{2}+\frac{1}{8}|\theta’(t)|_{H}^{2}+\frac{d}{dt}\{\frac{1}{8}|\nabla\theta(t)|_{H}^{2}+\nu\varphi_{\epsilon}(w(t))\}$(3.33)$\leq K_{14}(|\theta(t)|_{V}^{2}+|w(t)|_{H}^{2}+\nu\varphi_{\epsilon}(w(t))+|h(t)|_{H}^{2}+1)$

is a positive constant. Using (3.14) for both 0 and. $t\in(0, T_{0})$ , whereforthe suitable arrangement, we have the following:$K_{14}$$\mathrm{a}.\mathrm{e}$$\frac{d}{dt}E(t)\leq K_{15}(E(t)+|h(t)|_{H}^{2}+1)$

for$\mathrm{a}.\mathrm{e}$$w$. $t\in(0, T_{0})$ ,with(3.34)

where$E(t):= \frac{1}{8}|\theta(t)|_{V}^{2}+\frac{1}{4}|w(t)|_{H}^{2}+\iota/\varphi_{\epsilon}(w(t))$

and$K_{15}$for$\mathrm{a}.\mathrm{e}$. $t\in(0, T_{0})$is a positive constant. APPlying the Gronwall’s lemma to (3.34), we obtain that$\mathrm{E}(\mathrm{t})\leq(E(0)+K_{15}\oint_{0}^{T_{0}}|h(t)|_{H}^{2}dt+K_{15}T_{0})e^{K_{15}T_{0}}$Integrating (3.33) in over$t$$[0, T_{0}]$,$\forall t\in[0, T_{0}]$.

(3.35), then by (3.35) we have that$\frac{1}{4}\int_{0}^{T_{0}}|w’(t)|_{H}^{2}dt+\frac{1}{8}\oint_{0}^{T_{0}}|\theta’(t)|_{H}^{2}dt+\frac{1}{8}|\nabla\theta(T_{0})|_{H}^{2}+\nu\varphi_{\epsilon}(w(T_{0}))$(3.36)$\leq K_{16}(\varphi_{\epsilon}(w_{0})+|\theta_{0}|_{V}^{2}+\int_{0}^{T}|h(t)|_{H}^{2}dt+1)1$

is a positive constant. Noting that (3.36) is valid for any $T_{0}\in[0, T’)$ because the value of right hand side of (3.36) is independent of , and, we obtain that

where

$K_{16}$$T_{0}$$|(J_{\epsilon}^{\varphi}w)’|_{L^{2}\{0,T_{0j}H)}\leq$$|w’|_{L^{2}(0,T_{0_{1}}\cdot H)}$0,

$J_{\epsilon}^{\varphi}w\in W^{1,2}(0, T^{*}; H)\cap L^{\infty}(0_{7}T^{*}; V)$

and $w\in W^{1,2}(0, T_{1}^{*}.H)$ .

Therefore the following limits exist:$\lim_{t^{\mathrm{r}}}\theta(t)=:\theta^{*}$

and$\lim_{t^{*}}w(t)=:w^{*}$

in $H$ .

can be extended to the time
Hence by the local existence result again we see thatbeyond . It contradicts the hypothesis of . Finally, we obtain that $T=T^{*}$ .
$\{\theta w\}\rangle$$T^{*}$$\theta$$T^{*}$

Proof of Theorem 3.1: It follows immediately from Lemmas 3.2-3.4.

$\theta$

38

4. Convergence of approximate solutions
In this section we discuss the convergence of approximate solutions. Let
the solution ofobtained in Theorem 3.1, namely, it satisfies that$\{\theta_{\epsilon}, w_{\epsilon}\}$be$(P_{e})$$\theta_{\epsilon}’(t)+w_{\epsilon}’(t)-\triangle_{0}\theta_{\epsilon}(t)=h(t)$in$H\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ ,$w_{\epsilon}’(t)+\partial I_{\theta_{\epsilon}(t),N}(w_{\epsilon}’(t))+\nu\partial\varphi_{\epsilon}(w_{\epsilon}(t))\ni f(\theta_{\epsilon}(t), J_{\epsilon}^{\varphi}w_{\epsilon}(t))$$\theta_{\epsilon}(0)=\theta_{0}$in $H$ andin$H\mathrm{a}.\mathrm{e}$(4.1). $t\in(0, T)$ ,

(4.3)

in H.

$w_{\epsilon}(0)=w_{0}$We need some uniform estimates of approximate solutions(4.2)$\{\theta_{\epsilon}, w_{\epsilon}\}$to discuss theconvergence.

Lemma 4.1. Any approximate solution$|\theta_{\epsilon}|_{\infty}$,$\{\theta_{\epsilon}, w_{\epsilon}\}$satisfies$|w_{\rho}’.|_{\infty}\leq M_{1}+M_{1}T$,

where $M_{1}=|\theta_{0}|_{\infty}+|h|_{\infty}+|g|_{\infty}+N$ .

Proof. Define a function$p$

on $[0, T]$ by$p(t):=M_{1}+M_{1}t$ .

holds for any

Noting that
we observe that$|w_{\epsilon}’|_{\infty}\leq|g|_{\infty}+N$$\epsilon$$>0$ by thedefinition of a solution of$(\theta_{\epsilon}-p)’-\triangle_{0}(\theta_{\epsilon}-p)=h-w_{\epsilon}’-M_{1}\leq 0$

Multiplying (4.4) by$[\theta_{\epsilon}-p]^{+}$

in Q.

$(P_{\epsilon})$(4.4)

in $H$ , we have thatfor$\frac{1}{2}\frac{d}{dt}|[\theta_{\epsilon}(t)-p(t)]^{+}|_{H}^{2}+|\nabla[\theta_{\epsilon}(t)-p(t)]^{+}|_{H}^{2}\leq 0$$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

Integrating the above inequality in , we see that$t$,$|[\theta_{\epsilon}(t)-p(t)]^{+}|_{H}^{2}\leq|[\theta_{0}-p(0)]^{+}|_{H}^{2}=0$This implies that$\theta_{\epsilon}\leq p\leq M_{1}+M_{1}T$$\forall t\in[0, T]$

Multiplying (4.5) by$[-\theta_{\epsilon}-p]^{+}$.

. On the other hand,$(-\theta_{\epsilon}-p)’-\triangle_{0}(-\theta_{\epsilon}-p)=-h+w_{\epsilon}’-M_{1}\leq 0$

in Q.

in $H$ , we have that$\frac{1}{2}\frac{d}{dt}|[-\theta_{\epsilon}(t)-p(t)]^{+}|_{H}^{2}+|\nabla[-\theta_{\epsilon}(t)-p(t)]^{+}|_{H}^{2}\leq 0$

for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

Integrating the above in , we see that$t$$|[-\theta_{\epsilon}(t)-p(t)]^{+}|_{H}^{2}\leq|[-\theta_{0}-p(0)]^{+}|_{H}^{2}=0$,,

V# $\in[0,T]$ ,

(4.5)

37

which gives$\theta_{\epsilon}(t)\geq-p(t)\geq-M_{1}$-$M_{1}T$. Hence we complete the proof.

Lemma 4.2. There exists a positive constant$R_{1}$

independent of$\epsilon$$\theta$$>0$ such that$|w_{\epsilon}’|_{L^{2}(0,T_{1}H)}^{2}.+|\theta_{\epsilon}’|_{L^{2}(0,T_{j}H)}^{2}+|\triangle_{0}\theta_{\epsilon}|_{L^{2}(0,T;H)}^{2}\leq R_{1}$and$\sup_{t\in[0,T]}|\nabla\theta_{\epsilon}(t)|_{H}^{2}+\sup_{t\in[0,T]}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}(t)|_{H}^{2}\leq R_{1}$Proof.

Multiplying$w_{\epsilon}’-g(\theta_{\epsilon})$$(\xi_{\epsilon}(t), w_{\epsilon}’(t)-g(\theta_{\epsilon}))\geq 0$,.

by (4.2) in $H$ and noting that$\forall\xi_{\epsilon}(t)\in\partial I_{\theta_{\epsilon}(t),N}(w_{\epsilon}’(t))$

in $H$ for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ ,we have with the similar calculation to obtain (3.31)$\frac{1}{2}|w_{\epsilon}’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(w_{\epsilon}(t))\leq N_{1}(|\theta_{\epsilon}(t)|_{H}^{2}+|w_{\epsilon}(t)|_{H}^{2}+1)+\nu(\partial\varphi_{\epsilon}(w_{\epsilon}(t)), g(\theta_{\epsilon}(t)))(4.6)$

for$\mathrm{a}.\mathrm{e}$.

$t\in(\mathrm{O}, T)$, where$\partial\varphi_{\epsilon}(w_{\epsilon}(t))=-\triangle_{0}J_{\epsilon}^{\varphi}w_{\epsilon}(t)$is a positive constant independent of$H$inand the boundedness of we have$N_{1}$$\epsilon$$>0$ .

Noting that$g_{7}$$( \partial\varphi_{\epsilon}(w_{\epsilon}(t)),g(\theta_{\epsilon}(t)))=\leq\leq\varphi_{\epsilon}(w_{\epsilon}(t))+\frac{\nabla g(\theta_{\epsilon}|^{2}+|g’|_{\infty}^{2}}{2}|\nabla\theta_{\epsilon}(t)|_{H}^{2}\frac{1}{2}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}(t)_{H}\frac{1}{2}|\nabla g(\theta_{\epsilon}(t))|_{H}^{2}(\nabla J_{\epsilon}^{\varphi}w_{\epsilon}(t),(t)))$for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ . From the above inequality and (4.6) we observe that$\frac{1}{2}|u_{\Xi}’’(t)|_{H}^{2}+\nu\frac{d}{dt}\varphi_{\epsilon}(w_{\epsilon}(t))\leq N_{2}(|\theta_{\epsilon}(t)|_{H}^{2}+|\nabla\theta_{\epsilon}(t)|_{H}^{2}+|w_{\epsilon}(t)|_{H}^{2}+1)+\nu\varphi_{\xi}(w_{\epsilon}(t))(4.7)$

is a positive constant independent offor . $t\in(0, T)$ , where$H$ , we havein(4.1) by and$N_{2}$$\mathrm{a}.\mathrm{e}$$\theta_{\epsilon}’$$\epsilon>0$. Next, multiplying$-\triangle_{0}\theta_{\epsilon}$(4.8)$\frac{1}{2}|\theta_{\epsilon}’(t)|_{H}^{2}+\frac{1}{2}\frac{\mathrm{t}l^{\tau}}{dt}|\nabla\theta_{\epsilon}(t)|_{H}^{2}\leq|w_{\epsilon}’(t)|_{H}^{2}+|h(t)|_{H}^{2}$

and

(4.9)

$\frac{1}{2}\frac{d}{dt}|\nabla\theta_{\epsilon}(t)|_{H}^{2}+\frac{1}{2}|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}\leq|w_{\epsilon}’(t)|_{H}^{2}+|h(t)|_{H}^{2}$for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ , respectively. Computing (4.7)$+(4.8)\mathrm{x}$$\frac{1}{8}+(4.9)\mathrm{x}\frac{1}{8}$, we infer that$\frac{1}{16}|\theta_{\epsilon}’(t)|_{H}^{2}+\frac{1}{4}|w_{\epsilon}’(t)|_{H}^{2}+\frac{1}{16}|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+\frac{d}{dt}\{\frac{1}{8}|\nabla\theta_{\epsilon}(t)|_{H}^{2}+v\varphi_{\epsilon}(w_{\epsilon}(t))\}$$\leq N_{3}(|\theta_{\epsilon}(t)|_{H}^{2}+|\nabla\theta_{\epsilon}(t)|_{H}^{2}+|w_{\epsilon}(t)|_{H}^{2}+1)+\nu\varphi_{\epsilon}(w_{\zeta}(t))$

is a positive constant independent of, where.
some suitable arrangements in (4.10), we deduce thatfor$\mathrm{a}.\mathrm{e}$$t\in(\mathrm{O}, T)$$N_{3}$

Oe(t) $\leq N_{4}(E(t)+1)$

for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ ,$\epsilon>0$(4.10). By (3.14) with

(4. 1)

38

where$N_{4}$

is a positive constant independent of$>0$$\epsilon$

and,$\forall t\in[0, T]$$E(t):= \frac{1}{16}|\theta_{\epsilon}(t)|_{H}^{2}+\frac{1}{8}|\nabla\theta_{\epsilon}(t)|_{H}^{2}+\frac{1}{4}|w_{\epsilon}(t)|_{H}^{2}+\nu\varphi_{\epsilon}(w_{\epsilon}(t))$.

(4.12)

Applying the Gronwall’s lemma to (4.11), we have the following:$E(t)\leq(E(0)+N_{4}T)e^{N_{4}T}$ ,$\in[0, T]$$\forall t$

Combining (4.12) with (4.13), we can find a positive constantsuch that.

(4.13)$N_{5}$$\sup_{t\in[0,T]}|\nabla\theta_{\epsilon}(t)|_{H}^{2}+\sup_{t\in[0,T]}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}(t)|_{H}^{2}\leq N_{5}$By (4.10) and (4.13), we can find a positive constant

independent of$|w_{\epsilon}’|_{L^{2}(0,T;H)}^{2}+|\theta_{\epsilon}’|_{L^{2}(0,T;H)}^{2}+|\triangle_{0}\theta_{\epsilon}|_{L^{2}(0,T;H)}^{2}\leq N_{6}$By (4.14) and (4.15), put $R_{1}:=N_{5}+N_{6}$ to get the conclusion.
Lemma 4.3. There exists a positive constant$R_{2}$

for

$a.e$

$\epsilon$$>0$such that.

(4.15)$\theta$

independent of$|\triangle_{0}g(\theta_{\epsilon}(t))|_{H}^{2}\leq R_{2}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+1)$(4.14).

independent of$N_{6}$$\epsilon>0$$\epsilon>0$such that. $t\in(0, T)$ .

Proof. From the fact that,$\triangle_{0}g(\theta_{\epsilon}(t))=g’(\theta_{\epsilon}(t))\triangle_{0}\theta_{\epsilon}(t)+g^{l\prime}(\theta_{\epsilon}(t))|\nabla\theta_{\epsilon}(t)|^{2}$

it follows that

(4.16)$|\triangle_{0}g(\theta_{\epsilon}(t))|_{H}^{2}\leq N_{7}(|\nabla\theta_{\epsilon}(t)|_{L^{4}(\Omega)}^{4}+|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2})$for . $t\in(0, T)$ , wherelation inequality (cf. [18] ):$N_{7}:=2 \max\{|g’|_{\infty}, |g’’|_{\infty}\}$$\mathrm{a}.\mathrm{e}$$|\nabla z|_{L^{4}(\Omega)}\leq C|z|_{2}^{\frac{1}{H2}}(\Omega)|z|^{\frac{1}{\infty 2}}$,. By the Gagliardo-Nirenberg interpo-$\forall z\in L^{\infty}(\Omega)\cap H^{2}(\Omega)$and Lemma 4.1 and 4.2, the following inequalities hold:$|\nabla\theta_{\epsilon}(t)\}_{L^{4}(\Omega)}^{4}\leq C^{4}|\theta_{\epsilon}(t)|_{H^{2}(\Omega)}^{2}|\theta_{\epsilon}(t)|_{\infty}^{2}$$\leq c^{4}(|\theta_{\epsilon}(t)|_{H}^{2}+|\nabla\theta_{\epsilon}(t)|_{H}^{2}+|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2})|\theta_{\epsilon}(t)|_{\infty}^{2}$$\leq N_{6}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+1)$

for

where andare positive constants independent offind the desired constant .
$C$$N_{8}$$R_{2}$$\mathrm{a}.\mathrm{e}.t\in(0,T)$$\epsilon$

$>0$

,

. In virtue of (4.16) we can$\theta$, we see that the above constantis independent
Remark 4.1. In the case that St$N$of both parameters 6 and . By the Gagliardo-Nirenberg interpolation inequality for
2-dimensional case,,$\subset \mathrm{R}^{2}$$|z|_{L^{4}(\Omega)}\leq\tilde{C}|z|^{\frac{1}{H2}}|z|^{\frac{1}{V\mathrm{z}}}$$R_{2}$$\forall z\in V$

39

and Lemma 4.2 we can get the following inequalities without thetained in Lemma 4.1:$|\triangle_{0}g(\theta_{\epsilon}(t))|_{H}^{2}$where$\tilde{C}$and$N_{9}$$L$“-estimate of$\leq$$2|g’(\theta_{\epsilon}(t))\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+2|g’(\theta_{\epsilon}(t))|\nabla\theta_{\epsilon}(t)|^{2}|_{H}^{2}$$\leq$$N_{7}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+|\nabla\theta_{\epsilon}(t)|_{L^{4}(\Omega)}^{4})$$\leq$$N_{7}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+\tilde{C}^{4}|\nabla\theta_{\epsilon}(t)$$\leq$$\tilde{C}^{4}N_{7}R_{1}^{4}+N_{7}(1+\tilde{C}^{4}R_{1}^{2})|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}$$\leq$$N_{9}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+1)$

for$|\begin{array}{l}2H\end{array}|\nabla\theta_{\epsilon}(t)|_{V}^{2})$$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ ,are positive constants independent of and N.
$\epsilon$

Lemma 4.4. There exists a positive constant$R_{3}$

independent of$>0$ such that$\epsilon$.

$| \nabla J_{\epsilon}^{\varphi}w_{\epsilon}’|_{L^{2}(0,T_{j}H)}^{2}+\sup_{t\in[0,T]}|\triangle_{0}J_{\epsilon}^{\varphi}w_{\epsilon}(t)|_{H}^{2}\leq R_{3}$Proof. Multiplying (4.2) by$\partial\varphi_{\epsilon}(w_{\epsilon}’-g(\theta_{\epsilon}))$in $H$ and note that$\bullet(w_{\epsilon}’(t), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)-g(\theta_{\epsilon}(t))))=(w_{\epsilon}’(t), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)))-(w_{\epsilon}’(t), \partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t))))$,$\bullet(w_{\epsilon}’(t), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)))=(w_{\epsilon}’(t)-J_{\epsilon}^{\varphi}w_{\epsilon}’(t), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)))+(J_{\epsilon}^{\varphi}w_{\epsilon}’(t), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)))$$=\epsilon|\partial\varphi_{\epsilon}(w_{\epsilon}’(t))|_{H}^{2}+|\nabla J_{\epsilon}^{\epsilon}w_{\epsilon}’(t)|_{H}^{2}$$\geq|\nabla J_{\epsilon}^{\epsilon}w_{\epsilon}’(t)|_{H}^{2}$,$\bullet(w_{\epsilon}’(t), \partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t))))=(\nabla J_{\epsilon}^{\varphi}w_{\epsilon}’(t), \nabla g(\theta_{\epsilon}(t)))$$\leq\frac{1}{4}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}’(t)|_{H}^{2}+|\nabla g(\theta_{\epsilon}(t))|_{H}^{2}$$\leq\frac{1}{4}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}’(t)|_{H}^{2}+|g’|_{\infty}^{2}|\nabla\theta_{\epsilon}(t)|_{H}^{2}$$\leq\frac{1}{4}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}’(t)|_{H}^{2}+N_{10}$,$\bullet(\xi_{\epsilon}(t), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)-g(\theta_{\epsilon}(t))))\geq 0$,$\forall\xi_{\epsilon}(t)\in\partial I_{\theta_{\epsilon}(t),N}(w_{\epsilon}’(t))$$\bullet(\partial\varphi_{\epsilon}(w_{\epsilon}(t)), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)-g(\theta_{\epsilon}(t))))$$=(\partial\varphi_{\epsilon}(w_{\epsilon}(t)), \partial\varphi_{\epsilon}(w_{\epsilon}’(t)))-(\partial\varphi_{\epsilon}(w_{\epsilon}(t)), \partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t))))$$= \frac{1}{2}\frac{d}{dt}|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}-(\partial\varphi_{\epsilon}(w_{\epsilon}(t)), \partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t))))$$\bullet(\partial\varphi_{\epsilon}(w_{\epsilon}(t)), \partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t))))$$\leq\frac{1}{2}|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}+\frac{1}{2}|\partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t)))|_{H}^{2}$,,$\theta_{\epsilon}$

ob-

40

$\leq\frac{1}{2}|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}+\frac{1}{2}|\partial\varphi(g(\theta_{\epsilon}(t)))|_{H}^{2}$$= \frac{1}{2}|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}+\frac{1}{2}|-\triangle_{0}g(\theta_{\epsilon}(t)))|_{H}^{2}$,$\bullet(f(\theta_{\epsilon}(t),J_{\epsilon}^{\varphi}w_{\epsilon}(t)),\partial\varphi_{\epsilon}(w_{\acute{\epsilon}}(t)-g(\theta_{\epsilon}(t))))\leq\frac{1}{2}|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}+\frac{R_{2}}{2}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+1)$$=(f(\theta_{\epsilon}(t),J_{\epsilon}^{\varphi}w_{\epsilon}(t)),\partial\varphi_{\epsilon}(w_{\epsilon}’(t)))-(f(\theta_{\epsilon}(t),J_{\epsilon}^{\varphi}w_{\epsilon}(t)),\partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t))))$$\bullet(f(\theta_{\epsilon}(t), J_{\epsilon}^{\varphi}w_{\epsilon}(t)),$,$\partial\varphi_{\epsilon}(w_{\epsilon}’(t)))$$\leq\frac{1}{4}|\nabla J_{\epsilon}^{\varphi}w’(t)|_{H}^{2}+|\nabla f(\theta_{\epsilon}(t), J_{\epsilon}^{\varphi}w_{\epsilon}(t))|_{H}^{2}$$\leq\frac{1}{4}|\nabla J_{\epsilon}^{\varphi}w’(t)|_{H}^{2}+N_{11}(|\nabla\theta_{\epsilon}(t)|_{H}^{2}+|\nabla J_{\epsilon}^{\varphi}(w_{\epsilon}(t))|_{H}^{2})$$\leq\frac{1}{4}|\nabla J_{\epsilon}^{\varphi}w’(t)|_{H}^{2}+N_{11}(|\nabla\theta_{\epsilon}(t)|_{H}^{2}+\varphi_{\epsilon}(w_{\epsilon}(t)))$$\leq\frac{1}{4}|\nabla J_{\epsilon}^{\varphi}w’(t)|_{H}^{2}+N_{12}$$\bullet(f(\theta_{\epsilon}(t), J_{\epsilon}^{\varphi}w_{\epsilon}(t)),$,$\partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t))))$$\leq\frac{1}{2}|f(\theta_{\epsilon}(t), J_{\epsilon}^{\varphi}w_{\epsilon}(t))|_{H}^{2}+\frac{1}{2}|\partial\varphi_{\epsilon}(g(\theta_{\epsilon}(t)))|_{H}^{2}$$\leq N_{13}(|\theta_{\epsilon}(t)|_{H}^{2}+|J_{\epsilon}^{\varphi}w_{\epsilon}(t)|_{H}^{2}+|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+1)$$\leq N_{14}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+1).)$

we have with the help of Lemmas 4.2 and 4.3 that
(4.17)$\frac{1}{2}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}’(t)|_{H}^{2}+\frac{\nu}{2}\frac{d}{dt}|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}\leq N_{15}(|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}+|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}+1)$

forof$\mathrm{a}.\mathrm{e}$$\epsilon$. $t\in(0, T)$ , where $N_{i}(\mathrm{i}=10, 11,12,13_{2}14,15)$ are positive constants independent. Moreover, multiply (4.1) by
to have$>0$$-\triangle_{0}\theta_{\epsilon}$(4.18)$\frac{1}{2}\frac{d}{dt}|\nabla\theta_{\epsilon}(t)|_{H}^{2}+\frac{1}{2}|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}\leq|w_{5}’(t)|_{H}^{2}+|h(t)|_{H}^{2}$for$\mathrm{a}.\mathrm{e}$.

$t\in(\mathrm{O}, T)$. Integrating (4.18) in over $[0, T]$ and using Lemma 4.2, we have that$t$$\int_{0}^{T}|\triangle_{0}\theta_{\epsilon}(t)|_{H}^{2}dt\leq|\theta_{0}|_{V}^{2}+2R_{1}^{2}+2\int_{0}^{T}|h(t)|_{H}^{2}dt=$: N16.

(4.19)

Applying the Gronwall’s lemma to (4.17) and using (4.19), we obtain that.

$\sup_{t\in[0,T]}|\partial\varphi_{\epsilon}(w_{\epsilon}(t))|_{H}^{2}\leq(\frac{2(N_{15}N_{16}+N_{15}T)}{\mathrm{z}\prime}+|w_{0}|_{H^{2}(\Omega)}^{2})\exp\{\frac{2N_{15}T}{\nu}\}=:N_{17}$(4.20)

41

Again by (4.17), we have with the above estimate (4.20) that$\int_{0}^{T}|\nabla J_{\epsilon}^{\varphi}w_{\epsilon}’(t)|_{H}^{2}dt\leq\nu|w_{0}|_{H^{2}(\Omega)}^{2}+2N_{15}(N_{16}+N_{17}T+T)=:N_{18}$Putting $R_{3}:=N_{17}+N_{18}$ , we complete the proof.

.

$\theta$

5. Proof of Theorem 2.1
We are now in a position to give a proof of Theorem 2.1.

. By the uniform estimates
be the solution of
Proof of Theorem 2.1 Let
as $narrow+\infty$ with functions
totending0
sequenceain Lemmas 4.1-4.4 we can choose$w\inW^{1,2}(0,T;V)\cap L^{\infty}(0, T;H^{2}(\Omega))$
06 $W^{1,2}(0, T;H)\cap L^{\infty}(0, T;V)\cap L^{2}(0, T;H^{2}(\Omega))$ andsuch that(5.1)
in $C([0, T];H)$ as $narrow+\infty$$(P_{\epsilon})$$\{\theta_{\epsilon 2}w_{\epsilon}\}$$\{\epsilon_{n}\}$$\theta_{n}:=\theta_{\epsilon_{n}}arrow\theta$

and$J_{\epsilon_{n}}^{\varphi}w_{n}:=J_{\epsilon_{n}}^{\varphi}w_{\epsilon_{\mathfrak{n}}}arrow w$(5.2)

in $C([0, T];V)$ as $narrow+\infty$ .

It follows from Lemma 4.4 and (1.6) that$|w_{n}(t)-J_{\epsilon_{n}}^{\varphi}w_{n}(t)|_{\overline{H}}\leq\epsilon_{n}|\partial\varphi_{\epsilon_{n}}(w_{n}(t))|_{H}\leq\epsilon_{n}R_{3}$This implies that,$\forall t\in[0, T]$.

in $C([0, T];H)$ as $narrow+\infty$ .

$w_{n}arrow w$(5.3)

Now let $narrow+\infty$ . Then it follows that$\partial\varphi_{5_{n}}(w_{n})arrow\partial\varphi(w)=-\mathrm{A}\mathrm{q}\mathrm{w}$weakly in $L^{2}(0, T,\cdot H)$ as $narrow+\infty$ .

Here, we observe that the function$\xi_{n}(t):=-w_{n}’(t)-\nu\partial\varphi_{\epsilon_{n}}(w_{n}(t))+f(\theta_{n}(t), J_{\epsilon_{n}}^{\varphi}w_{n}(t))$

in $H$ forsatisfies that$L^{2}(0, T;H)$ . Therefore there exist a subsequencesuch that$\xi_{n}(t)\in\partial I_{\theta_{n}(t),N}(w_{n}’(t))$$\xi_{n_{k}}arrow\xi$$\mathrm{a}.\mathrm{e}$in $H$ for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

is bounded in. $t\in(0, T)$ andof and a function $\xi\in L^{2}(0, T;H)$$\{\xi_{n}\}$$\{n_{k}\}$weakly in $L^{2}(0, T;H)$ as $karrow+\infty$ .

. Now let us show the inclusion $\xi(t)\in$
Clearly $\xi=-w’+\nu\triangle_{0}w+f(\theta, w)$ in
in $H$ for . $t\in(0, T)$ . In order to do so, we employ the usual monotonicity( ) on $H$ in the sense of Mosco, we have only to showtechnique. Since$L^{2}(0, T,\cdot H)$$\partial I_{\theta(t),N}(w’(t))$$\mathrm{a}.\mathrm{e}$$I_{\theta_{n},N}(\cdot)arrow I_{\theta,N}$$\cdot$.

$\lim\sup\int_{0}^{T}k\prec+\infty(\xi_{n_{k}}(t), w_{n_{k}}’(t))dt\leq\int_{0}^{T}(\xi(t), w’(t))dt$

(5.4)

42

This can be proved as follows:$\lim\sup\int_{0}^{T}karrow+\infty(\xi_{n_{k}}(t), w_{n_{k}}’(t))dt$$\leq-\lim_{karrow+}\inf_{\infty}|w_{n_{k}}’|_{L^{2}(0_{2}T;H)}^{2}+\nu\lim_{karrow+\infty}\varphi_{\epsilon_{n_{k}}}(w_{0})-\nu$ $\lim_{karrow+}\inf_{\infty}\varphi_{\epsilon_{n_{k}}}(w_{n_{k}}(T))$$+ \lim_{karrow+\infty}\oint_{0}^{T}(f(\theta_{n_{k}}(t), J_{\epsilon_{n_{k}}}^{\varphi}w_{n_{k}}(t))$,$w_{n_{k}}’(t))dt$$\leq-|w’|_{L^{2}(0,T;H)}^{2}+\frac{\nu}{2}|\nabla w_{0}|_{H}^{2}-\frac{\nu}{2}|\nabla w(T)|_{H}^{2}+\int_{0}^{T}(f(\theta(t), w(t)),$$=I_{0}^{T}(\xi(t), w’(t))dt$

Therefore we obtain.

$\xi(t)\in\partial I_{\theta(t),N}(w’(t))$$w’(t)+\ \{\mathrm{t}$$w’(t))dt$

in $H$ for$\mathrm{a}.\mathrm{e}$. $t\in(0,T)$ and) $-\nu\triangle_{0}w(t)=f(\theta(t), w(t))$ in $H$ for$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

By using regularity $w\in L^{\infty}(0, T;H^{2}(\Omega))$ and Lemma 4.1, we obtain $w\in W^{1,\infty}(Q)$ . Since
the equation $(4,1)$ is linear, it is easy to discuss the convergence for $(4,1)$ and obtain thatin $H$ for$\theta’(t)+w’(t)-\triangle_{0}\theta(t)=h(t)$This completes the proof of Theorem 2.1.

$\mathrm{a}.\mathrm{e}$. $t\in(0, T)$ .

$\theta$, from Remark 44 it is possible to take N $-++\infty$ in
Remark 5.1. In the case of(.) and hence the solutions obtained in the above satisfy the following system (P)’:$\Omega\subset \mathrm{R}^{2}$$I_{\theta(t),N}$$\theta_{t}+w_{8}-\triangle_{0}\theta=h(t, x)$

(P)’

$\{$

$w_{t}+\partial I_{\theta}(w_{t})-\nu\triangle_{0}w\ni f(\theta, w)$$\theta(0, \cdot)=\theta_{0}$,$w(0, \cdot)=w_{0}$where$+\infty$$I_{\theta}(w_{t}):=\{$

and$\partial I_{\theta}(w_{t})$

0

if
if

$w_{t}<g(\theta)$$w_{t}\geq g(\theta)$

in
in
in

$Q$
$Q$
$\Omega$

,
,
,

,
;

is its subdifferential:$\emptyset(-\infty, 0]$$\partial I_{\theta}(w_{t}):=\{${0}

if $w_{t}<g(\theta)$ ,if $w_{t}=g(\theta)$ ,if.
$w_{t}\geq g(\theta)$(P)’ is the same system that we have already discussed in [3]. We note here that if $N$ is a
. Therefore we can consider thatsufficiently large positive number,is close to
the problem (P) is one of approximate problems of (P)’, because the indicator function( ) can be regarded as an approximation of.
$I_{\theta}(\cdot)$$I_{\theta,N}(\cdot)$$I_{\theta,N}$$I_{\theta}(\cdot)$$\cdot$Remark 5.2. In the case of, we are very interested in the situation when the
fixed large number $N$ goes to -f-oo. In that case we can not obtain the uniform estimate
in Lemma 4.1. This unables us to discuss the convergence of approximate solutions$\Omega\subset \mathrm{R}^{3}$

43

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F. Luterotti, G. Schimperna and U. Stefanelli, Global solution to a phase field modelwith irreversible and constrained phase evolution, Qurt Appl. Math., 60 (2002),

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E. Zeidler, Nonlinear Functional Analysis and Its Applications. $11/A$ . Linear Mono, 1990.
tone Operators, Springer-Verlag, New$\mathrm{Y}\mathrm{o}\mathrm{r}\mathrm{k}/\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{n}$Masayasu ASO

Department of Mathematics
Graduate School of Science and Technology
Chiba University
1-33 Yayoi-ch\={o}, Inage-ku, Chiba, 263-8522, Japan-mail: masayasu.aso@graduate.chiba-u.jp$\mathrm{E}$