$N=2$ $Q=$ - Kobe Universityfe.math.kobe-u.ac.jp/.../fe40-019-039/fe40-019-039.pdfFunkcialaj...

Funkcialaj Ekvacioj, 40 (1997) 19-39

Periodic Problems for Heat Convection Equationsin Noncylindrical Domains

By

Hiroshi INOUE and Mitsuharu OTANI(Ashikaga Institute of Technology and Waseda University, Japan)

Abstract. We discuss the existence and regularity of strong solutions of the periodicproblems for heat convection equations in regions with periodically moving bound-aries in $R^{2}$ or $R^{3}$ . Our main tool is a perturbation theory for time dependentsubdifferential operators.

1. Introduction

We consider the periodic problem for the so-called Oberbeck-Boussinesqequations in noncylindrical domains, which describe the heat convection inthe viscous incompressible flow in bounded regions whose boundaries vary astime $t$ goes on.

For each $t¥in[0, T]$ , $T$ be any positive number, let $Q(t)$ be a boundeddomain in $R^{N}$ ($N=2$ or 3) with smooth boundary $¥Gamma(t)$ , and let $Q=$$¥bigcup_{0¥leq t¥leq T}Q(t)¥times¥{t¥}$ and $¥Gamma=¥bigcup_{0¥leq t¥leq T}¥Gamma(t)¥times¥{t¥}$ . Then the equation to be con-sidered in this paper is given as follows:

(1.1) $¥frac{¥partial}{¥partial t}u-v¥Delta u+(u¥cdot¥nabla)u=-¥frac{1}{¥rho}¥nabla p+¥{1-¥eta(¥theta-d)¥}g+f_{1}$ , $(x, t)¥in Q$,

(1.2) $¥mathrm{d}¥mathrm{i}¥mathrm{v}$ $u=0$, $(x, t)¥in Q$ ,

(1.3) $¥frac{¥partial}{¥partial t}¥theta-¥kappa¥Delta¥theta+(u¥cdot¥nabla)¥theta=f_{2}$ , $(¥mathrm{x}, t)¥in Q$ ,

(1.4) $u(x, t)=a(x, t)$, $¥theta(x, t)=b(x, t)$, $(x, t)¥in¥Gamma$,

(1.5) $u(x, 0)=u(¥mathrm{x}, T)$, $¥theta(x, 0)=¥theta(x, T)$, $x$ $¥in Q(0)=Q(T)$,

where $(u. ¥nabla)=¥sum_{j=1}^{N}u^{j}(¥partial/¥partial ¥mathrm{x}_{j})$. Unknown functions $u=(u^{1}, u^{2}, ¥ldots, u^{N})$ , $¥rho$ and $¥theta$are the solenoidal velocity, pressure and temperature of the fluid which occupies$Q$ respectively; $a$, $b$ , $u_{0}$ , $¥theta_{0}$ are given data and $g$ is the body force field (saygravity); $f_{1}$ and $f_{2}$ are external forces; constants $v$ , $¥rho$ , $¥kappa$ , $¥eta$ , $d$ represent kinematicviscosity, density, thermal conductivity, volume expansion coefficient and somedatum point of the temperature of the fluid respectively (see Joseph [8]).

20 Hiroshi INOUE and Mitsuharu OTANI

The initial boundary value problems for this type of equations have beeninvestigated by many peoples, e.g., we refer to Kirchgassner-Kielhofer [10],Ghidaglia [5], ?eda [13], Morimoto [12], Hishida [6], and Inoue-Otani [7].

As for the periodic problems, however, it seems that the intensive studyis not yet fully pursued. When $Q$ is a cylindrical domain, i.e., $Q(t)¥equiv Q_{0}$ ,Morimoto [12] constructed a weak solution of (1.1)?(1.5) with the boundarycondition for $¥theta$ replaced by a discontinuous Neumann-Dirichlet condition.

The problem in noncylindrical domain was studied by ?eda [13] for thecase where $N=3$, $f_{1}=0$, $f_{2}=0$, and $b$ is a constant in $t$ and constant in $¥mathrm{x}$on each connected component of $¥Gamma$. He applied the method of Otani-Yamada[17] to show the existence of strong periodic solution of (1.1)?(1.5) by assuming$v$ is sufficiently large and given data $a$ and $b$ are sufficiently small.

The main purpose of this paper is to show that the general case ofperiodic problem can be solved under the conditions analogous to those forthe initial boundary value problems in our previous work [7]. Our argumentdoes not require the largeness of $v$ but only some smallness conditions on $a$,$b$ , $f_{1}$ , and $¥eta gf_{2}$ , and furthermore for the case of $N=2$, we do not need anycondition on the size of the external forces $f_{1}$ and $f_{2}$ .

Moreover their smallness conditions are given explicitly in terms of $v$, $¥kappa$and the first eigenvalues of the Stokes operator and $-¥Delta$ with domain $H^{2}¥cap H_{0}^{1}$ .

Our basic tool to investigate these problems is the perturbation theoryfor time dependent subdifferential operators developed in (Otani [15], [7]).

Furthermore we need a delicate approximation procedure for $N=2$ orreduction to the fixed point argument for $N=3$, and at the same time carefula priori estimates. The basic lemmas for establishing a priori estimates aregiven in Lemma 3.3 and 3.4.

Main results are stated in §2 and their proofs are given in §3.

2. Notations and main results

2.1. Notations and some function spaces

To formulate our results we need the same function spaces and use thesame notations as in [7].

Let $¥Omega$ be a bounded domain in $R^{N}$ and $X(¥Omega)$ be a function space definedon $¥Omega$ . We denote the norm of $X(¥Omega)$ by $|¥cdot|_{X(¥Omega)}$ or simply by $|¥cdot|_{X}$ if no ambiguityarises. The norm of $L^{2}(¥Omega)$ is designated by $|¥cdot|$ . We prepare the followingfunction spaces:

$H^{s}(¥Omega)=$ the Sobolev space of order $s$ in $L^{2}(¥Omega)$ with norm $|¥cdot|_{H^{s}}$ ,$H_{0}^{1}(¥Omega)=$ the completion of $C_{0}^{¥infty}(¥Omega)$ under the $H^{1}(¥Omega)-$norm,

Heat Convection Equations 21

$L^{2}(¥Omega)=(L^{2}(¥Omega))^{N}$ with norm $||¥cdot||$ ,$H^{s}(¥Omega)=(H^{s}(¥Omega))^{N}$ with norm $||¥cdot||_{H^{s}}$ ,$H_{0}^{1}(¥Omega)=(H_{0}^{1}(¥Omega))^{N}$ ,$C_{¥sigma}^{¥infty}(¥Omega)=¥{u =(¥mathcal{U}^{1}, ¥mathcal{U}_{ }^{2},¥ldots, u^{N});u^{j}¥in C_{0}^{¥infty}(¥Omega),j=1, ¥ldots, N, ¥mathrm{d}¥mathrm{i}¥mathrm{v} u=0¥}$ ,$H_{¥sigma}^{1}(¥Omega)=H^{1}(¥Omega)¥cap L_{¥sigma}^{2}(¥Omega)$,$L_{¥sigma}^{2}(¥Omega)=$ the completion of $C_{¥sigma}^{¥infty}(¥Omega)$ under the $L^{2}(¥Omega)-$norm.

The Stokes operator is defined as follows: $ A(¥Omega)=-P_{¥Omega}¥Delta$ with domain$D(A(¥Omega))=H^{2}(¥Omega)¥cap H_{¥sigma}^{1}(¥Omega)$ , where $P_{¥Omega}$ is the orthogonal projection from $L^{2}(¥Omega)$onto $L_{¥sigma}^{2}(¥Omega)$ .

In what follows, we often use the notations:

$¥left¥{¥begin{array}{l}||u||_{p}=||u||_{(L^{p})^{N}},¥¥||u||_{¥infty,T}=¥sup_{0¥leq t¥leq T}||u(t)||,¥¥||u||_{H^{2},2,T}^{2}=¥{¥¥||u||_{2,T}^{2}=||u||_{H^{0},2,T}^{2}=||u||_{L^{2},2,T}^{2},¥end{array}¥right.$

$¥mathrm{f}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{o}¥mathrm{r}0T¥geq 1


2.2. Main results

Throughout this paper, we always assume the following conditions on $Q$ ,$a$, $b$ , $f_{1}$ , $f_{2}$ , $g$ :

(A.Q)$¥pi$ There exists a level preserving $C^{3_{-}}$diffeomorphism $¥ovalbox{¥tt¥small REJECT}$ from $Q$ onto$Q_{0}¥times[0, T]$ for some bounded domain in $Q_{0}$ in $R^{N}$ and $Q(0)=Q(T)$.

(A.a)$¥pi$ There exists a vector function $¥overline{u}$ in $C^{1}(Q)$ such that $¥overline{u}¥in C([0, T]$ ;$H^{1}(Q(t)))¥cap L^{2}(0, T;H^{2}(Q(t))),¥overline{u}_{t}¥in L^{2}(0, T;L^{2}(Q(t)))$, $¥mathrm{d}¥mathrm{i}¥mathrm{v}¥overline{¥mathrm{u}}=0$ in $Q,¥overline{u}=a$on $¥Gamma$ and $¥overline{u}(x, 0)=¥overline{u}(x, T)$ .

$(¥mathrm{A}.b)_{¥pi}$ There exists a function $¥overline{¥theta}$ in $C^{1}(Q)$ such that $¥overline{¥theta}¥in C([0, T];H^{1}(Q(t)))¥cap$$L^{2}(0, T;H^{2}(Q(t))),¥overline{¥theta}_{t}¥in L^{2}(Q),¥overline{¥theta}=b$ on $¥Gamma$ and $¥overline{¥theta}(x, 0)=¥overline{¥theta}(x, T)$ .

$(¥mathrm{A}./)$ $f_{1}¥in L^{2}(0, T;L^{2}(Q(t)))$ and $f_{2}¥in L^{2}(Q)$.(A.g) $g$ has the potential $G$, $i.e.$ , $g$ $=¥nabla G$ (this is always satisfied, when $g$ is

the gravity) such that $G¥in L^{¥infty}(0, T;W^{1,¥infty}(Q(t)))$ .

To formulate our results, we put

$F_{1}(t)=P_{B}[-¥overline{u}_{t}+v¥Delta¥overline{u}-(¥overline{u}¥cdot¥nabla)¥overline{u}-¥eta¥overline{¥theta}g+f_{1}]^{¥wedge}$ ,

$F_{2}(t)=[-¥overline{¥theta}_{t}+¥kappa¥Delta¥overline{¥theta}-(¥overline{u}¥cdot¥nabla)¥overline{¥theta}+f_{2}]^{¥wedge}$ ,

and

$¥lambda=¥min_{0¥leq t¥leq T}¥lambda_{1}(t)$ , $¥mu=¥min_{0¥leq t¥leq T}¥mu_{1}(t)$ ,

where $¥lambda_{1}(t)$ (resp. $¥mu_{1}(t)$ ) is the first eigenvalue of the Stokes operator $A(Q(t))$(resp. $-¥Delta$ with domain $H^{2}(Q(t))¥cap H_{0}^{1}(Q(t))$). Then our results are stated asfollows:

Theorem I. Let $N=2$ and let conditions $(¥mathrm{A}.Q)_{¥pi}$ , (A.a)$¥pi’(¥mathrm{A}.b)_{¥pi}$ , $(¥mathrm{A}./)$ and(A.g) be satisfied. Assume that $||¥nabla¥overline{u}||_{¥infty,T}+|¥nabla¥overline{¥theta}|_{¥infty,T}/2


Theorem $¥mathrm{I}¥mathrm{I}$. Let $N=3$ and let conditions $(¥mathrm{A}.Q)_{¥pi}$ , $(¥mathrm{A}.a)_{¥pi}$ , $(¥mathrm{A}.b)_{¥pi}$ , $(¥mathrm{A}./)$ and(A.g) be satisfied. Then there exists a positive function $¥beta=¥beta(v, ¥lambda)$ depending$a¥leq¥beta¥sqrt{v},|F_{2}|_{2,T}^{2}||¥eta g||_{L^{¥infty}(Q)}^{2}¥leq¥{2¥sqrt{3}¥kappa^{2}¥mu^{2}lsoonQsuchthatif||F_{1}||_{2,T}^{2}¥leq 80¥sqrt{3}¥mathcal{F}_{¥mathcal{V}/(2¥kappa¥mu+3)¥}¥beta^{2}and|¥nabla^{¥frac{T}{¥theta}}|_{¥infty,T}^{2}||¥eta g||_{L^{¥infty}(Q)}^{2}¥leq}v¥beta^{2},||¥nabla¥overline{u}||_{¥infty,T}¥cdot||¥nabla^{2}¥overline{u}||_{2},¥leq¥beta,||¥overline{u}||_{H^{1},¥infty,T}^{2}$

$¥{¥sqrt{3}¥kappa^{2}¥mu(¥lambda¥mu)^{1/4}/2(2¥kappa¥mu+3)¥}¥beta$, then (1.1)?(1.5) has a periodic solution $(u, ¥theta)$ sat-isfying $(¥neq)_{¥pi}$ , where $||¥nabla^{2}¥overline{u}||^{2}=¥sum_{i,j,k=1}^{N}||¥partial¥overline{u}^{i}/¥partial x_{j}¥partial x_{k}||^{2}$ .

Remark, (i) If $b$ is a constant, then we can take $¥overline{¥theta}(x, t)¥equiv b$ . Therefore,in Theorem $¥mathrm{I}$ , we have only to assume that $||¥nabla¥overline{u}||_{¥infty,T}


multivalued) maximal monotone operator. The unique element of the leastnorm in $¥partial¥varphi(u)$ is designated by $¥partial^{¥mathrm{o}}¥varphi(u)$ .

Consider the following abstract periodic problem in a real Hilbert space $¥mathrm{H}$ ,

(P.P) $¥left¥{¥begin{array}{l}¥frac{du}{dt}(t)+¥partial¥varphi^{t}(u(t))+B(t,u(t))¥ni f(t),¥¥u(0)=u(T),¥end{array}¥right.$$01$ such that

$k_{0}|u|_{H}^{p}¥leq¥varphi^{t}(u)$ , for all $u¥in D(¥varphi^{t})$ .

(iv) For each $t¥in[0, T]$ , $¥partial¥varphi^{t}$ is strictly monotone, $i.e.$ , $(w_{1}-w_{2}, u_{1}-u_{2})_{H}$$=0$ with $u_{i}¥in D(¥partial¥varphi^{t})$ and $w_{i}¥in¥partial¥varphi^{t}(u_{i})(i=1,2)$ implies $u_{1}=u_{2}$ .

Then, for the case where $B(t, ¥cdot)¥equiv 0$, the following result holds (see [20]).

Theorem 3.1. Let $(¥mathrm{A}.¥varphi^{t})_{¥pi}$ be satisfied and $f(t)¥in L^{2}(0, T;H)$. Then (P.P)with $B(t, ¥cdot)¥equiv 0$ has a unique periodic solution $u$ such that $du/dt$ $¥in L^{2}(0, T;H)$and $¥varphi^{t}(u(t))$ is absolutely continuous on $[0, T]$ .


As for the perturbed equation, we impose the following conditions on$¥partial¥varphi^{t}$ and $B(t, ¥cdot)$ .

(A.1) For each $t¥in[0, T]$ and $L¥in(0, ¥infty)$, the set $¥{u ¥in H;¥varphi^{t}(u(t))+|u|_{H}^{2}¥leq L¥}$is compact in $H$.

(A.2) $B(t, ¥cdot)$ is measurable and demiclosed in the following sense:(i) For each function $u¥in C([0, T];H)$ such that $du/dt$ $¥in L^{2}(0, T;H)$ , and

there exists a function $g(t)¥in L^{2}(0, T;H)$ with $g(t)¥in¥partial¥varphi^{t}(u(t))$ for $a.e$.$t¥in[0, T]$ , $B(t, u(t))$ is measurable with respect to $t¥in[0, T]$ ,

(ii) If $u_{n}¥rightarrow u$ in $C([0, T];H)$, $g_{n}¥rightarrow g$ weakly in $L^{2}(0, T;H)$ with $ g_{n}(t)¥in$$¥partial¥varphi^{t}(u_{n}(t))$ , $g(t)¥in¥partial¥varphi^{t}(u(t))$ for $a.e$ . $t¥in[0, T]$ and if $B(¥cdot, u_{n}(¥cdot))¥rightarrow b(¥cdot)$weakly in $L^{2}(0, T;H)$, then $b(t)=B(t, u(t))$ for $a.e$ . $t¥in[0, T]$ .

(A.3) There exist positive constants $k$, $¥delta$ , $c_{1}$ and a monotone increasing func-tion $¥parallel(¥cdot)$ on [0, $¥infty$ ) such that(i) $|B(t, u)|_{H}^{2}¥leq k|¥partial^{¥mathrm{o}}¥varphi^{t}(u)|_{H}^{2}+l(|u|_{H})(¥varphi^{t}(u)+1)^{2},0¥leq k


Then we have

(3.3) $¥sup_{0¥leq t¥leq T}|y(t)|¥leq(2+¥frac{3}{¥alpha_{0}})|f(t)|_{1,T}$ ,

where

$|f(t)|_{1,T}=¥left¥{¥begin{array}{l}¥sup_{1¥leq t¥leq T}¥int_{t-1}^{t}|f(s)|ds¥¥¥frac{1}{T}¥int_{¥mathrm{o}}^{T}|f(s)|ds¥end{array}¥right.$ $ffoorr01¥leq T


$K_{1}(t)=¥{u ¥in L_{¥sigma}^{2}(B);u=0 ¥mathrm{a}.¥mathrm{e}. x¥in B¥backslash Q(t)¥}$ ,

$¥varphi_{2}^{t}(¥theta)=¥{^{¥frac{¥kappa}{2}¥int_{+¥infty}|¥nabla¥theta|^{2}dx}B,$

’

$¥theta¥theta¥in¥in H_{0}^{1}(B)¥cap K_{2}(t)L^{2}(B)¥backslash (H_{0}^{1}(B)’¥cap K_{2}(t))$

,

$K_{2}(t)=$ $¥{¥theta¥in L^{2}(B);¥theta=0 ¥mathrm{a}.¥mathrm{e}. x¥in B¥backslash Q(t)¥}$ .

Then $¥varphi_{1}^{t}¥in¥Phi(L_{¥sigma}^{2}(B))$ , $¥varphi_{2}^{t}¥in¥Phi(L^{2}(B))$, and their subdifferentials are characterizedas follows:

$¥partial¥varphi_{1}^{t}(u)=¥{f¥in L_{¥sigma}^{2}(B);P_{Q(t)}f|_{Q(t)}=vA(Q(t))f|_{Q(t)}¥}$,

$D(¥partial¥varphi_{1}^{t})=¥{u¥in L_{¥sigma}^{2}(B);u|_{Q(t)}¥in H^{2}(Q(t))¥cap H_{¥sigma}^{1}(Q(t)), u|_{B¥backslash Q(t)}=0¥}$ ,

$||¥partial^{¥mathrm{o}}¥varphi_{1}^{t}(u)||=||vA(Q(t))u|_{Q(t)}||$ ,

$¥partial¥varphi_{2}^{t}(¥theta)=¥{h¥in L^{2}(B);h|_{Q(t)}=-¥kappa¥Delta¥theta|_{Q(t)}¥}$ ,

$D(¥partial¥varphi_{2}^{t})=¥{¥theta¥in L^{2}(B);¥theta|_{Q(t)}¥in H^{2}(Q(t))¥cap H_{0}^{1}(Q(t)), ¥theta|_{B¥backslash Q(t)}=0¥}$ ,

$|¥partial^{¥mathrm{o}}¥varphi_{2}^{t}(¥theta)|=|¥kappa¥Delta¥theta|_{Q(t)}|$ .

Furthermore put

$B_{1}(u)=P_{B}¥{(u¥cdot¥nabla)u+([¥overline{u}]^{¥wedge}¥cdot¥nabla)u+[(u|_{Q(t)}¥cdot¥nabla)¥overline{u}]^{¥wedge}¥}$ ,

$ B_{2}(u, ¥theta)=(u¥cdot¥nabla)¥theta+([¥overline{u}]^{¥wedge}¥cdot¥nabla)¥theta$ ,

$F_{2}(u, t)=F_{2}(t)-[(u|_{Q(t)}¥cdot¥nabla)¥overline{¥theta}]^{¥wedge}$ ,

and consider the following abstract periodic problems in $L_{¥sigma}^{2}(B)$ and $L^{2}(B)$:

(3.4) $¥left¥{¥begin{array}{l}¥frac{d}{dt}v+¥partial¥varphi_{1}^{t}(v)+B_{1}(v)+P_{B}(¥eta¥phi¥hat{g})¥ni F_{1}(t),¥¥v(0)=v(T),¥end{array}¥right.$

(3.5) $¥left¥{¥begin{array}{l}¥frac{d}{dt}¥phi+¥partial¥varphi_{2}^{t}(¥phi)+B_{2}(v,¥phi)¥ni F_{2}(v,t),¥¥¥phi(0)=¥phi(T).¥end{array}¥right.$

Then our original problem is reduced to (3.4)?(3.5) in the following sense:

Lemma 3.5. Suppose that $(v, ¥phi)$ is a solution of (3.4)?(3.5) satisfying

(3.6) $¥left¥{¥begin{array}{l}v¥in C_{¥pi}([0,T],.L_{¥sigma}^{2}(Q(t))),v_{t},¥Delta v,(v¥cdot¥nabla)v¥in L^{2}(0,T..L^{2}(Q(t))),¥¥||¥nabla v||^{2}isabsolutelycontinuouson[0,T],¥¥v(t)¥in D(¥partial¥varphi_{1}^{t})fora.e.t¥in[0,T],¥end{array}¥right.$


(3.7) $¥left¥{¥begin{array}{l}¥phi¥in C_{¥pi}([0,T],.L^{2}(Q(t))),¥phi_{t},¥Delta¥phi,(v¥cdot¥nabla)¥phi¥in L^{2}(0,T..L^{2}(Q(t))),¥¥|¥nabla¥phi|^{2}isabsolutelycontinuouson[0,T],¥¥¥phi(t)¥in D(¥partial¥varphi_{2}^{t})fora.e.t¥in[0,T].¥end{array}¥right.$

Then $(u, ¥theta)=(v|_{Q(t)}+¥overline{u}, ¥phi|_{Q(t)}+¥overline{¥theta})$ gives a solution of (1.1)?(1.5) satisfying$(¥neq)_{¥pi}$ .

Proof. In view of (A.a)$¥pi$ and $(¥mathrm{A}.b)_{¥pi}$ , we easily see that $(u, ¥theta)$ satisfies(1.2)?(1.5). By virtue of the boundedness of $||¥nabla u(t)||$ , $|¥nabla¥theta(t)|$ and reflexivityof $H_{¥sigma}^{1}$ , $H_{0}^{1}$ , we can easily derive the weak continuity of $[u(t)-¥overline{u}(t)]^{¥wedge}$ and$[¥theta(t)-¥overline{¥theta}(t)]^{¥wedge}$ in $H_{¥sigma}^{1}(B)$ and $H_{0}^{1}(B)$ . Hence the continuity in the strong topol-ogy is also assured by the continuity of $||¥nabla u(t)||$ and $|¥nabla¥theta(t)|$ . Other propertiesin $(¥neq)_{¥pi}$ are direct consequences of (3.6)?(3.7).

In order to check (1.1), it suffices to use (A.g), Helmholtz’s decomposition,(3.4) operated by $P_{Q(t)}$ and the fact that $P_{Q(t)}(P_{B}h)|_{Q(t)}=P_{Q(t)}h|_{Q(t)}$ for all $h¥in L^{2}(B)$ .

Q.E.D.

3.3. Some lemmas

We here collect several inequalities without their proofs, which will beused in establishing a priori estimates in later arguments. For their proofs,see (Ladyzhenskaya [11], Temam [19], [7]).

Lemma 3.6. The following inequalities hold.

(3.8) $||u||¥leq¥frac{1}{¥sqrt{¥lambda}}||¥nabla u||$ for all $t¥in[0, T]$ and $u¥in H_{¥sigma}^{1}(Q(t))$,

(3.9) $||¥nabla u||¥leq¥frac{1}{¥sqrt{¥lambda}}||¥Delta u||$ for all $t¥in[0, T]$ and $u¥in H^{2}(Q(t))¥cap H_{¥sigma}^{1}(Q(t))$ ,

(3.10) $|¥theta|¥leq¥frac{1}{¥sqrt{¥mu}}|¥nabla¥theta|$ for all $t¥in[0, T]$ and $¥theta¥in H_{0}^{1}(Q(t))$.

Lemma 3.7. There exist constants $K_{1}$ and $K_{2}$ such that

(3.11) $||¥nabla^{2}u||¥leq K_{1}||¥Delta u||$ for all $t¥in[0, T]$ and $u¥in H^{2}(Q(t))¥cap H_{¥sigma}^{1}(Q(t))$,

(3.12) $|¥nabla^{2}¥theta|¥leq K_{2}|¥Delta¥theta|$ for all $t¥in[0, T]$ and $¥theta¥in H^{2}(Q(t))¥cap H_{0}^{1}(Q(t))$ .

Lemma 3.8. The following inequalities hold.

(3.13) $|h|_{4}¥leq¥sqrt{42}|¥nabla h||h|$ for all $h¥in H^{1}(R^{2})$ ,for all $h¥in H^{1}(R^{2})$ ,


(3.14) $¥int_{Q(t)}|(u¥cdot¥nabla)v¥cdot w|dx¥leq¥sqrt{2}||u||^{1/2}||¥nabla u||^{1/2}||¥nabla v||||w||^{1/2}||¥nabla w||^{1/2}$ if $N=2$,

for all $u$, $w$ $¥in H_{0}^{1}(Q(t))$ and $v¥in H^{1}(Q(t))$,

(3.15) $¥int_{Q(t)}|(u¥cdot¥nabla)¥phi¥theta|dx¥leq¥{¥sqrt{2}2||u||1/4||¥nabla u||3/4|¥nabla¥phi||¥theta|1/4|¥nabla¥theta|||u||^{1/2}||¥nabla u||^{1/2}|¥nabla¥phi||¥theta|^{1/2}|¥nabla¥theta|^{1/2}3/4,$’

$iiff$ $NN=3=2,$

,

for all $u¥in H_{0}^{1}(Q(t))$, $¥theta¥in H_{0}^{1}(Q(t))$ and $¥emptyset¥in H^{1}(Q(t))$ .Lemma 3.9. Let $(¥mathrm{A}.Q)_{¥pi}$ be satisfied, then the following inequalities hold.

(3.16) $||(v¥cdot ¥nabla)w||^{2}¥leq¥left¥{¥sqrt{2}4¥sqrt{3}c_{0}^{3}||v||_{H^{1}}^{2}||¥nabla w||||¥nabla^{2}w||c_{0}^{4}||v||||v||_{H^{1}}||¥nabla w||||¥nabla^{2}w,||,¥right.$ $iiff$ $NN=2=3,$’

for all $v¥in H^{1}(Q(t))$ , and $w¥in H^{2}(Q(t))$,

(3.17) $|(v¥cdot¥nabla)¥phi|^{2}¥leq¥left¥{¥sqrt{2}4¥sqrt{3}c_{0}^{3}||v||_{H^{1}}^{2}|¥nabla¥phi||¥nabla^{2}¥phi|c_{0}^{4}||v||||v||_{H^{1}}|¥nabla¥phi||¥nabla^{2},¥phi|,¥right.$ $iiff$ $NN=2=3,$’

for all $v¥in H^{1}(Q(t))$ , and $w¥in H^{2}(Q(t))$,

Here $||¥nabla^{2}w||=|(¥sum_{i,j,k=1}^{N}|¥frac{¥partial^{2}}{¥partial_{X_{i}}¥partial x_{j}}w^{k}|^{2})^{1/2}|_{L^{2}}$, $|¥nabla^{2}¥phi|=|(¥sum_{i,j=1}^{N}|¥frac{¥partial^{2}}{¥partial_{X_{i}}¥partial x_{j}}¥phi|^{2})^{1/2}|_{L^{2}}$

and $c_{0}$ is a constant satisfying$|¥tilde{h}|_{L^{2}(R^{N})}¥leq c_{0}|h|_{L^{2}(Q(t))}$ and $|¥nabla¥tilde{h}|_{L^{2}(R^{N})}¥leq c_{0}|h|_{H^{1}(Q(t))}$ ,

where $¥tilde{h}$ is an extension of $h$ to $H^{1}(R^{N})$. Furthermore, if $v¥in H_{0}^{1}(Q(t))$, thenthe term $||u||_{H^{1}}$ in (3.16)?(3.17) can be replaced by $||¥nabla u||$ .

3.4. The proof of Theorem I

We are going to solve (3.4)?(3.5) in the Hilbert space $H=L_{¥sigma}^{2}(B)¥times L^{2}(B)$with the standard inner product $(U_{1}, U_{2})_{H}=(u_{1}, u_{2})_{L_{¥sigma}^{2}}+$ $(¥theta_{1}, ¥theta_{2})_{L^{2}}$ for $U_{i}=$

$¥left(¥begin{array}{l}u_{i}¥¥¥theta_{i}¥end{array}¥right)=^{t}(u_{i}, ¥theta_{i})¥in H(i =1,2)$ . To this end, we put

$¥Psi^{t}(U(t))=¥varphi_{1}^{t}(u(t))+¥varphi_{2}^{t}(¥theta(t))$, $U(t)=¥left(¥begin{array}{l}u(t)¥¥¥theta(t)¥end{array}¥right)$ ,

$A^{t}(U(t))=¥left(¥begin{array}{ll}¥partial¥varphi_{1}^{t}(u(t)) & 0¥¥0 & ¥partial¥varphi_{2}^{t}(¥theta(t))¥end{array}¥right)$ , $B(U)=¥left(¥begin{array}{ll}B_{1}(u)+ & P_{B}(¥eta g¥theta)¥¥B_{2}(u,¥theta)+(u¥cdot¥nabla)¥overline{¥theta} & ¥end{array}¥right)$,

$F(t)=¥left(¥begin{array}{l}F_{1}(t)¥¥F_{2}(t)¥end{array}¥right)$ .

Here and in what follows, we omit the notations. $|_{Q(t)}$ and $(¥cdot)^{¥wedge}$ for the sakeof simplicity.


Then it is easy to see that $¥partial¥Psi=A^{t}$ with domain $D(¥partial¥Psi^{t})=D(¥partial¥varphi_{1}^{t})¥times D(¥partial¥varphi_{2}^{t})$and that (3.4)?(3.5) is reduced to the equation:

(3.18) $¥frac{d}{dt}U(t)+A^{t}U(t)+B(U(t))¥ni F(t)$ , $U(0)=U(T)$.

Because of the presence of the coupling term $¥eta g¥theta$ , the direct application ofabstract results such as in [15] requires more restrictive assumptions thanthose in Theorem I. In order to avoid this difficulty, we introduce the follow-ing modified equation:

$(3.18)_{M}$ $¥frac{d}{dt}U(t)+A^{t}U(t)+B_{M}(U(t))¥ni F(t)$, $U(0)=U(T)$,

where $M$ is a parameter to be fixed later and we put

$B_{M}(U)=¥left(¥begin{array}{l}B_{1}(u)+P_{B}(¥eta g¥theta_{M})¥¥B_{2}(u,¥theta)+(u¥cdot¥nabla)¥overline{¥theta}¥end{array}¥right)$, $¥theta_{M}=¥left¥{¥begin{array}{l}¥theta ¥mathrm{i}¥mathrm{f}|¥theta|_{L^{2}}¥leq M,¥¥¥frac{M}{|¥theta|_{L^{2}}}¥theta ¥mathrm{i}¥mathrm{f}|¥theta|_{L^{2}}>M.¥end{array}¥right.$

The existence of a solution of this equation is assured by the following lemma.

Lemma 3.10. Let $||¥nabla¥overline{u}||_{¥infty,T}+|¥nabla¥overline{¥theta}|_{¥infty,T}/2


Hence (i) of (A.3) is assured by (A.a)$¥pi’(¥mathrm{A}.b)_{¥pi}$ and (A.g). Noticing that$((u¥cdot¥nabla)v, v)_{L_{¥sigma}^{2}}=((u¥cdot¥nabla)¥theta, ¥theta)_{L^{2}}=0$ if $¥mathrm{d}¥mathrm{i}¥mathrm{v}$ $u=0$, we have

$(-g-B_{M}(U), U)_{H}=-v||¥nabla u||^{2}-¥kappa|¥nabla¥theta|^{2}+¥int_{Q(t)}(u¥cdot¥nabla)¥overline{u}¥cdot ud¥mathrm{x}$

$+¥int_{Q(t)}(u¥cdot¥nabla)¥overline{¥theta}¥theta dx+¥int_{Q(t)}¥eta g¥theta_{M}¥cdot udx$,

for all $g¥in¥partial¥Psi(U)$ . Furthermore, (3.8), (3.10), (3.13), (3.14) and (3.15) give

(3. 19) $|¥int_{Q(t)}(u¥cdot¥nabla)¥overline{u}¥cdot udx|¥leq||u||_{4}^{2}||¥nabla¥overline{u}||¥leq¥frac{¥sqrt{2}}{¥sqrt{¥lambda}}||¥nabla u||^{2}||¥nabla¥overline{u}||$ ,

(3.20) $|¥int_{Q(t)}(u¥cdot¥nabla)¥overline{¥theta}¥theta dx|¥leq||u||_{4}|¥theta|_{4}|¥nabla¥overline{¥theta}|¥leq¥frac{1}{¥sqrt{2}}(¥frac{1}{¥sqrt{¥lambda}}||¥nabla u||^{2}+¥frac{1}{¥sqrt{¥mu}}|¥nabla¥theta|^{2})||¥nabla¥overline{¥theta}|$,

(3.21)

$|¥int_{Q(t)}¥eta g¥theta_{M}¥cdot udx|¥leq||¥eta g||_{4}|¥theta_{M}|||u||_{4}¥leq¥frac{r_{0}}{4¥sqrt{¥lambda}}||¥nabla u||^{2}+¥frac{¥sqrt{2}}{r_{0}}M^{2}||¥eta g||_{L^{4}}^{2}$ , $¥forall r_{0}>0$ ,

whence follows

$(-g-B_{M}(U), U)_{H}+(v-¥frac{¥sqrt{2}}{¥sqrt{¥lambda}}||¥nabla¥overline{u}||-¥frac{1}{¥sqrt{2¥lambda}}|¥nabla¥overline{¥theta}|-¥frac{r_{0}}{4¥sqrt{¥lambda}})||¥nabla u||^{2}$

$+(¥kappa-¥frac{1}{¥sqrt{2¥mu}}|¥nabla¥overline{¥theta}|)|¥nabla¥theta|^{2}¥leq¥frac{¥sqrt{2}}{r_{0}}M^{2}||¥eta g||_{4}^{2}$.

Then, for sufficiently small $r_{0}$ , (ii) of (A.3) is verified. Q.E.D.

Proof Theorem I. We are going to show below that$¥underline{¥mathrm{c}¥mathrm{l}¥mathrm{a}¥mathrm{i}¥mathrm{m}}$ : There exists a sufficiently large $M$ such that $¥sup_{0¥leq t¥leq T}|¥theta_{M}(t)|


together with the definition of $r_{1}$ yield

(3.22) $¥frac{d}{dt}||u(t)||^{2}+¥frac{r_{1}}{2¥sqrt{¥lambda}}||¥nabla u(t)||^{2}¥leq¥frac{4}{r_{1}}¥{¥frac{1}{¥sqrt{¥lambda}}||F_{1}||^{2}+¥sqrt{2}M^{2}||¥eta g||_{4}^{2}¥}$.

Consequently, from (3.8) and Lemma 3.4, we can derive the a priori estimate

(3.23) $||u||_{¥infty,T}^{2}¥leq(1+¥frac{3}{r_{1}¥sqrt{¥lambda}})¥frac{8}{r_{1}}¥{¥frac{1}{¥sqrt{¥lambda}}||F_{1}||_{2,T}^{2}+¥sqrt{2}||¥eta g||_{L^{¥infty}(0,T,L^{4})}^{2}.M^{2}¥}$ .

Moreover, the integration of (3.22) over $[¥mathrm{f} -1, t]$ (or [0, $T]$ ) and (3.23) give

(3.24) $||¥nabla u||_{2,T}^{2}¥leq(1+¥frac{2}{r_{1}¥sqrt{¥lambda}})¥frac{24¥sqrt{¥lambda}}{r_{1}^{2}}¥{¥frac{1}{¥sqrt{¥lambda}}||F_{1}||_{2,T}^{2}+¥sqrt{2}||¥eta g||_{L^{¥infty}(0,T;L^{4})}^{2}M^{2}¥}$ .

Next we multiply (3.18)$M$ by $t(0, ¥theta)$ and integrate over $Q(t)$ to get

$¥frac{1}{2}¥frac{d}{dt}|¥theta(t)|^{2}+¥kappa|¥nabla¥theta(t)|^{2}¥leq|¥int_{Q(t)}(u¥cdot¥nabla)¥overline{¥theta}¥theta dx|+|F_{2}||¥theta|$

By virtue of (3.20) and the estimate

$|F_{2}||¥theta|¥leq¥frac{1}{¥sqrt{¥mu}}|F_{2}||¥nabla¥theta|¥leq¥frac{r_{2}}{4¥sqrt{¥mu}}|¥nabla¥theta|^{2}+¥frac{1}{r_{2}¥sqrt{¥mu}}|F_{2}|^{2}$

together with the definition of $r_{2}$ , we have

$¥frac{d}{dt}|¥theta(t)|^{2}+¥frac{r_{2}}{2¥sqrt{¥mu}}|¥nabla¥theta|^{2}¥leq¥frac{¥sqrt{2}}{¥sqrt{¥lambda}}|¥nabla¥overline{¥theta}|||¥nabla u||^{2}+¥frac{2}{r_{2}¥sqrt{¥mu}}|F_{2}|^{2}$.

Hence, by (3.10), Lemma 3.4 and (3.24), we obtain

$|¥theta|_{¥infty,T}^{2}¥leq 2(1+¥frac{3}{r_{2}¥sqrt{¥mu}})¥{¥frac{¥sqrt{2}}{¥sqrt{¥lambda}}|¥nabla¥overline{¥theta}|_{¥infty,T}||¥nabla u||_{2,T}^{2}+¥frac{2}{r_{2}¥sqrt{¥mu}}|F_{2}|_{2,T}^{2}¥}$

$¥leq 2(1+¥frac{3}{r_{2}¥sqrt{¥mu}})¥{¥frac{24¥sqrt{2}}{r_{1}^{2}}(1+¥frac{2}{r_{1}¥sqrt{¥lambda}})|¥nabla¥overline{¥theta}|_{¥infty,T}(¥frac{1}{¥sqrt{¥lambda}}||F_{1}||_{2,T}^{2}$

$+¥sqrt{2}||¥eta g||_{L^{¥infty}(0,T;L^{4})}^{2}M^{2})+¥frac{2}{r_{2}¥sqrt{¥mu}}|F_{2}|_{2,T}^{2}¥}$ .

Since the coefficient of $M^{2}$ in the right hand side is $|¥nabla¥overline{¥theta}|_{¥infty,T}||¥eta g||_{L^{¥infty}(0,T;L^{4})}^{2}a^{-1}$ ,if $|¥nabla¥overline{¥theta}|_{¥infty,T}||¥eta g||_{L^{¥infty}(0,T,L^{4})}^{2}.


3.5. The proof of Theorem II

To prove Theorem $¥mathrm{I}¥mathrm{I}$, we rely on the Schauder’s fixed point argumentin $L^{2}(0, T;L_{¥sigma}^{2}(B))$ . The closed convex subset where we work is defined asfollows

$K_{R}=¥{h ¥in L^{2}(0, T;L_{¥sigma}^{2}(B));||h||_{2,T}¥leq R¥}$.

Given $b$ in $K_{R}$ , we first solve

(3.25) $¥left¥{¥begin{array}{l}¥frac{d}{dt}v_{b}+¥partial¥varphi_{1}^{t}(v_{b}(t))+b¥ni F_{1}(t),¥¥v_{b}(0)=v_{b}(T).¥end{array}¥right.$

Next, for each $v_{b}$ , we construct the solution $¥theta_{b}=¥theta v_{b}$ of the equation:

(3.26) $¥left¥{¥begin{array}{l}¥frac{d}{dt}¥theta_{b}(t)+¥partial¥varphi_{2}^{t}(¥theta_{b}(t))+B_{2}(v_{b}(t),¥theta_{b}(t))¥ni F_{2}(t)-(v_{b}(t)¥cdot¥nabla)¥overline{¥theta}(t),¥¥¥theta_{b}(0)=¥theta_{b}(T).¥end{array}¥right.$

Define the operator $¥ovalbox{¥tt¥small REJECT}$ by$¥ovalbox{¥tt¥small REJECT}:b$ $¥mapsto B_{1}(v_{b})+P_{B}¥eta g¥theta_{b}$ .

In what follows, we are going to show that $¥ovalbox{¥tt¥small REJECT}$ has a fixed point $¥overline{b}$ in $K_{R}$ fora suitably chosen $R$, which completes the proof, since $(v_{¥overline{b}}, ¥theta_{¥overline{b}})$ gives a solutionfor (3.4)?(3.5). This existence of solutions for (3.25) and (3.26) is assured bythe following lemmas.

Lemma 3.11. Let (A.Q)$¥pi’(¥mathrm{A}.a)_{¥pi}$ , $(¥mathrm{A}.b)_{¥pi}$ , $(¥mathrm{A}./)$ and (A.g) be satisfied andlet $b$ $¥in K_{R}$ . Then (3.25) has a unique solution $v_{b}$ satisfying (3.6).

Lemma 3.12. Let $(¥mathrm{A}.Q)_{¥pi}$ , $(¥mathrm{A}.a)_{¥pi}$ , $(¥mathrm{A}.b)_{¥pi}$ , $(¥mathrm{A}./)$ and (A.g) be satisfied andlet $v_{b}$ be a solution of (3.25) satisfying (3.6). Then (3.26) has a unique solution$¥theta_{b}$ satisfying (3.7).

Proof of Lemma 3.11. Assumption $(¥mathrm{A}.Q)_{¥pi}$ assures $(¥mathrm{i})-(¥mathrm{i}¥mathrm{i})$ of $(¥mathrm{A}.¥varphi^{t})_{¥pi}$ with$¥varphi^{t}=¥varphi_{1}^{t}$ and (3.8) implies $(¥mathrm{i}¥mathrm{i}¥mathrm{i})-(¥mathrm{i}¥mathrm{v})$ of $(¥mathrm{A}.¥varphi)_{¥pi}$ . It follows from (A.a)$¥pi’(¥mathrm{A}.b)_{¥pi}$ ,$(¥mathrm{A}./)$ and (A.$g$) that $F_{1}-b$ belongs to $L^{2}(0, T;L_{¥sigma}^{2}(B))$. Therefore we can applyTheorem 3.1. Q.E.D.

Proof of Lemma 3.12. $¥underline{¥mathrm{E}¥mathrm{x}¥mathrm{i}¥mathrm{s}¥mathrm{t}¥mathrm{e}¥mathrm{n}¥mathrm{c}¥mathrm{e}}$: It easily follows from (A.a)$¥pi’(¥mathrm{A}.b)_{¥pi}$ , $(¥mathrm{A}./)$and (3.6) that $F_{2}(t)$ and $(v_{b}¥cdot¥nabla)¥overline{¥theta}$ belong to $L^{2}(0, T;L^{2}(B))$. Conditions $(¥mathrm{A}.¥varphi^{t})_{¥pi}$ ,(A.I) and (A.2) with $¥varphi^{t}=¥varphi_{2}^{t}$ and $B(t, ¥cdot)=B_{2}(v_{b}(t), ¥cdot)$ are verified as in the proofof Lemma 3.10. Since $((v. ¥nabla)¥theta, ¥theta)_{L^{2}}=0$ for all $¥theta¥in H_{0}^{1}(B)$ and $v¥in H^{1}(B)$ with$¥mathrm{d}¥mathrm{i}¥mathrm{v}$ $v=0$, we obtain $(B_{2}(v_{b}, ¥theta), ¥theta)_{L^{2}}=0$, hence $(-g_{2}-B_{2}(v_{b}, ¥theta), ¥theta)+2¥varphi_{2}^{t}(¥theta)=0$for all $g_{2}¥in¥partial¥varphi_{2}^{t}(¥theta)$ , whence follows (ii) of (A.3).


Furthermore, by virtue of (3.17) and (3.12), we get

$|B_{2}(v_{b}, ¥theta)|^{2}¥leq 8¥sqrt{3}c_{0}^{3}K_{2}(||¥nabla u||^{2}+||¥overline{u}||_{H^{1}}^{2})|¥nabla¥theta||¥Delta¥theta|$

$¥leq¥frac{1}{2}|¥Delta¥theta|^{2}+384¥kappa^{-1}c_{0}^{6}K_{2}^{2}(||¥nabla u||^{4}+||¥overline{u}||_{H^{1}}^{4})¥varphi_{2}^{t}(¥theta)$.

We here note that $||¥nabla u||+||¥overline{u}||_{H_{1}}¥in L^{¥infty}(0, T)$ and so (i) of (A.3) is satisfied.Thus the existence of solution $¥theta_{b}$ of (3.26) satisfying (3.7) is verified.

Uniqueness: Let $¥theta_{1}$ and $¥theta_{2}$ be solutions of (3.26) satisfying (3.7) and put$¥phi=¥overline{¥theta_{1}-¥theta_{2}.}$Then $¥emptyset$ satisfies

(3.27) $¥phi_{t}-¥kappa¥Delta¥phi+B(v_{b}, ¥phi)=0$, $¥phi(0)=¥phi(T)=0$.

Multiplication of (3.27) by $¥emptyset$ yield

$¥frac{1}{2}¥frac{d}{dt}|¥phi(t)|^{2}+¥kappa|¥nabla¥phi(t)|^{2}=0$ for $¥mathrm{a}.¥mathrm{e}$ . $t¥in[0, T]$ .

Integrating this over $[0, T]$ , we get $¥int_{¥mathrm{o}}^{T}|¥nabla¥phi(t)|^{2}dt=0$, which together with

(3.10) implies $¥phi(t)=0¥mathrm{a}.¥mathrm{e}$. $t¥in[0, T]$ . Q.E.D.

We now proceed to the proof of Theorem $¥mathrm{I}¥mathrm{I}$ . By the preceding arguments,the operator $¥ovalbox{¥tt¥small REJECT}$ is proved to be a well-defined single-valued operator from$¥ovalbox{¥tt¥small REJECT}=L^{2}(0, T;L_{¥sigma}^{2}(B))$ into itself. Furthermore, we are going to show that $¥ovalbox{¥tt¥small REJECT}$maps $K_{R}$ into itself for an appropriately chosen $R$ . For this purpose we needthe following a priori estimates.

Lemma 3.13. Let $b$ $¥in K_{R}$ and let $v_{b}$ and $¥theta_{b}$ be the solutions of (3.25) and(3.26). Then we have

(3.28) $||v_{b}||_{¥infty,T}^{2}¥leq¥frac{2}{v¥lambda}(2+¥frac{3}{v¥lambda})(||F_{1}||_{2,T}^{2}+||b||_{2,T}^{2})$ ,

(3.29) $||¥nabla v_{b}||_{2,T}^{2}¥leq¥frac{6}{v^{2}¥lambda}(1+¥frac{1}{v¥lambda})(||F_{1}||_{2,T}^{2}+||b||_{2,T}^{2})$ ,

(3.30) $||¥nabla v_{b}||_{¥infty,T}^{2}¥leq¥frac{2}{v}(2+¥frac{3}{v¥lambda})¥{2+¥frac{3(m^{2}+m)}{v¥lambda}(1+¥frac{1}{v¥lambda})¥}(||F_{1}||_{2,T}^{2}+||b||_{2,T}^{2})$ ,

(3.31) $||¥Delta v_{b}||_{2,T}^{2}¥leq¥frac{6}{v^{2}}(1+¥frac{1}{v¥lambda})¥{2+¥frac{3(m^{2}+m)}{v¥lambda}(1+¥frac{1}{v¥lambda})¥}(||F_{1}||_{2,T}^{2}+||b||_{2,T}^{2})$,

(3.32) $|¥theta_{b}|_{¥infty,T}^{2}¥leq¥frac{2}{¥kappa}(2+¥frac{3}{¥kappa¥mu})(¥frac{1}{¥mu}|F_{2}|_{2,T}^{2}+4(¥lambda¥mu)^{-1/4}||¥nabla v_{b}||_{¥infty,T}^{2}|¥nabla¥overline{¥theta}|_{2,T}^{2})$.


Proof. For the sake of brevity, we here denote $v_{b}$ and $¥theta_{b}$ by $v$ and $¥theta$ .Multiply (3.25) by $v$ and integrate over $B$, then we get

$¥frac{1}{2}¥frac{d}{dt}||v(t)||^{2}+v||¥nabla v(t)||^{2}¥leq$ $(||F_{1}(t)||+||b(t)||)||v(t)||$

$¥leq¥frac{v¥lambda}{2}||v||^{2}+¥frac{1}{v¥lambda}(||F_{1}||^{2}+||b||^{2})$,

hence, by (3.8),

(3.33) $¥frac{d}{dt}||v(t)||^{2}+v||¥nabla v(t)||^{2}¥leq¥frac{2}{v¥lambda}(||F_{1}||^{2}+||b||^{2})$.

Therefore Lemma 3.4 with the use of (3.8) gives (3.28), and the integration of(3.33) over $[t -1, t]$ (or [0, $T]$ ) together with (3.28) gives (3.29).

Next multiply (3.25) by $g_{1}(t)=F_{1}(t)-dv(t)/dt-b(t)¥in¥partial¥varphi_{1}^{t}(v(t))$ and inte-grate over $B$ . Then recalling the fact that $¥varphi_{1}^{t}$ satisfies (ii) of $(¥mathrm{A}.¥varphi^{t})_{¥pi}$ with$¥gamma=1/2$ and $m_{2}=0$ (see [17], [14]) and using Lemma 3.3, we obtain

$¥frac{d}{dt}¥varphi_{1}^{t}(v(t))+||g_{1}(t)||^{2}¥leq(||F_{1}(t)||+||b(t)||)||g_{1}(t)||+m(¥varphi_{1}^{t}(v(t))+(¥varphi_{1}^{t}(v(t))^{1/2})||g_{1}(t)||)$

$¥leq¥frac{1}{2}||g_{1}(t)||^{2}+2(||F_{1}||^{2}+||b||^{2})+(m^{2}+m)¥varphi_{1}^{t}(v(t))$.

Hence, in view of the fact that $||g_{1}(t)||¥geq||¥partial^{¥mathrm{o}}¥varphi_{1}^{t}(v(t))||¥geq v||¥Delta v(t)||_{L^{2}(Q(t))}$, we get

(3.34) $¥frac{d}{dt}||¥nabla v(t)||^{2}+v||¥Delta v(t)||^{2}¥leq¥frac{4}{v}(||F_{1}(t)||^{2}+||b(t)||^{2})+(m^{2}+m)||¥nabla v(t)||^{2}$.

Now (3.30) is derived from (3.9), (3.29) and Lemma 3.4, and the integrationof (3.34) over $[t -1, t]$ (or [0, $T]$ ) together with (3.30) gives (3.31).

To see (3.32), multiply (3.26) by $¥theta$ and integrate over $B$, then by using(3.8), (3.10) and (3.15), we have

$¥frac{1}{2}¥frac{d}{dt}|¥theta(t)|^{2}+¥kappa|¥nabla¥theta(t)|^{2}¥leq|F_{2}(t)||¥theta(t)|+|¥int_{Q(t)}(v¥cdot¥nabla)¥overline{¥theta}¥theta(t)dx|$

$¥leq|F_{2}|¥frac{1}{¥sqrt{¥mu}}|¥nabla¥theta|+2(¥lambda¥mu)^{-1/8}||¥nabla v|||¥nabla¥overline{¥theta}||¥nabla¥theta|$

$¥leq¥frac{1}{2}¥kappa|¥nabla¥theta|^{2}+¥frac{1}{¥kappa¥mu}|F_{2}|^{2}+¥frac{4}{¥kappa}(¥lambda¥mu)^{-1/4}||¥nabla v||^{2}|¥nabla¥overline{¥theta}|^{2}$ ,

which implies


$¥frac{d}{dt}|¥theta(t)|^{2}+¥kappa¥mu|¥theta(t)|^{2}¥leq¥frac{2}{¥kappa¥mu}|F_{2}|^{2}+¥frac{8}{¥kappa}(¥lambda¥mu)^{-1/4}||¥nabla v||^{2}|¥nabla¥overline{¥theta}|^{2}$ .

Thus Lemma 3.4 assures (3.32). Q.E.D.

We are now ready to prove Theorem $¥mathrm{I}¥mathrm{I}$ .

Proof of Theorem $¥mathrm{I}¥mathrm{I}$ . $¥underline{¥mathrm{S}¥mathrm{t}¥mathrm{e}¥mathrm{p}}$I: We first fix two numbers $d_{0}$ and $¥beta$ asfollows.

(3.36) $¥{_{¥beta=(160¥sqrt{3}c_{0}^{3}K_{1}d_{0})^{-1}}^{d_{0}=6v^{-1}(1+¥frac{1}{v¥lambda})¥{2+¥frac{3(m^{2}+m)}{v¥lambda}(1+¥frac{1}{v¥lambda})¥}}.$,

Define $R$ by

(3.36) $R^{2}=80¥sqrt{3}¥sqrt{v}¥beta^{2}$ ,

and take $F_{1}$ sufficiently small so that

(3.37) $||F_{1}||_{2,T}¥leq R$.

Then it follows from (3.30) maps (3.31) that

(3.38) $||¥nabla v_{b}||_{¥infty,T}^{2}¥leq 2d_{0}R^{2}$,

(3.39) $||¥Delta v_{b}||_{2,T}^{2}¥leq 2d_{0}v^{-1}R^{2}$ .

We here claim that $¥ovalbox{¥tt¥small REJECT}$ and $K_{R}$ into itself, provided that $¥overline{u},¥overline{¥theta}$, $F_{2}$ are takensufficiently small in the following sense.

(3.40) $||¥nabla¥overline{u}||_{¥infty,T}||¥nabla^{2}¥overline{u}||_{2,T}¥leq¥beta$,

(3.41) $||¥overline{u}||_{H^{1},¥infty,T}^{2}¥leq¥sqrt{v}¥beta$,

(3.42) $||¥eta g||_{L^{¥infty}(Q)}^{2}|F_{2}|_{2,T}^{2}¥leq¥frac{2¥sqrt{3}¥kappa^{2}¥mu^{2}¥sqrt{v}}{(2¥kappa¥mu+3)}¥beta^{2}$,

(3.38) $||¥eta g||_{L^{¥infty}(Q)}^{2}|¥nabla¥overline{¥theta}|_{2,T}^{2}¥leq¥frac{¥sqrt{3}¥kappa^{2}¥mu(¥lambda¥mu)^{1/4}}{2(2¥kappa¥mu+3)}¥beta$.

To show this, we first note that

$||¥ovalbox{¥tt¥small REJECT}(b)||_{2,T}^{2}¥leq 4(I(v_{b}, v_{b})+I(¥overline{u}, v_{b})+I(v_{b},¥overline{u})+||¥eta g¥theta_{b}||_{2,T}^{2})$,

where $I(v, w)=||(v¥cdot¥nabla)w||_{2,T}^{2}$ . Recalling (3.16) and (3.11), and using (3.38)?(3.39),we get

$I(v_{b}, v_{b})¥leq 4¥sqrt{3}c_{0}^{3}K_{1}||¥nabla v_{b}||_{¥infty,T}^{3}¥cdot||¥Delta v_{b}||_{2,T}¥leq 4¥sqrt{3}c_{0}^{3}K_{1}(2d_{0})^{2}R^{4}v^{-1/2}$.


Hence, (3.36), the definition of $R$ implies that

(3.44) $I(v_{b}, v_{b})¥leq¥frac{1}{20}R^{2}$ ,

where we used the fact that $c_{0}¥geq 1$ and $K_{1}¥geq 1$ .The same argument as above together with (3.40) and (3.41) assures

respectively

(3.45) $I(¥overline{u}, v_{b})¥leq¥frac{1}{20}R^{2}$ ,

(3.46) $I(v_{b},¥overline{u})¥leq¥frac{1}{20}R^{2}$ .

Furthermore, (3.30) and (3.38) yield

$||¥eta g¥theta_{b}||_{2,T}^{2}¥leq||¥eta g||_{¥infty,T}^{2}¥{¥frac{2}{¥kappa¥mu}(2+¥frac{3}{¥kappa¥mu})|F_{2}|_{2,T}^{2}+¥frac{16}{¥kappa}(¥lambda¥mu)^{-1/4}(2+¥frac{3}{¥kappa¥mu})|¥nabla¥overline{¥theta}|_{2,T}^{2}d_{0}R^{2}¥}$ .

Then, from (3.42) and (3.43), we can easily derive the estimate $||¥eta g¥theta_{b}||_{2,T}^{2}¥leq$$R^{2}/20+R^{2}/20$ . Thus it is shown that $¥ovalbox{¥tt¥small REJECT}$ maps $K_{R}$ into itself.

$¥underline{¥mathrm{S}¥mathrm{t}¥mathrm{e}¥mathrm{p}¥mathrm{I}¥mathrm{I}}$ : Let $¥ovalbox{¥tt¥small REJECT}_{W}$ be $L^{2}(0, T;L_{¥sigma}^{2}(B))$ endowed with the weak topology.Since $K_{R}$ is a closed convex compact subset in $¥ovalbox{¥tt¥small REJECT}_{W}$ and $¥ovalbox{¥tt¥small REJECT}$ is already shownto be a mapping from $K_{R}$ into itself, Schauder-Tychonoff’s fixed point theoremsays that $¥ovalbox{¥tt¥small REJECT}$ has a fixed point in $K_{R}$ , provided that $¥ovalbox{¥tt¥small REJECT}$ is continuous from$¥ovalbox{¥tt¥small REJECT}_{W}$ into $¥ovalbox{¥tt¥small REJECT}_{W}$ . To show the continuity of $¥ovalbox{¥tt¥small REJECT}$ , we can apply much the sameargument as given in Lemma 3.15 in [14].

Let $b_{n}¥in K_{R}$ and $b_{n}-b$ weakly in $L^{2}(0, T;L_{¥sigma}^{2}(B))$ , and let $v_{n}=v_{b_{n}}$ and$¥theta_{n}=¥theta_{v_{bn}}$ be the solutions of (3.25) and (3.26) with $b$ $=b_{n}$ and $v_{b}=v_{b_{n}}$ . Apriori estimate (3.31) assures that $g_{l}^{n}=F_{l}-dv_{n}/dt-b_{n}¥in¥partial¥varphi^{t}(v_{n})$ and dvjdt arebounded in $L^{2}(0, T;L_{¥sigma}^{2}(B))$ , so in particular, $¥{v_{n}¥}_{n¥in N}$ forms an equicontinuousfamily in $C([0, T];L_{¥sigma}^{2}(B))$. Furthermore it follows from (3.30) that $¥{v_{n}(t)¥}_{n¥in N}$is a precompact set in $L_{¥sigma}^{2}(B)$ for all $t¥in[0, T]$ . Hence, by Ascoli’s theorem,we can extract a subsequence $¥{v_{n_{¥mathrm{k}}}¥}$ of $¥{v_{n}¥}$ such that

(3.47) $¥left¥{¥begin{array}{l}v_{n_{k}}¥rightarrow v¥mathrm{s}¥mathrm{t}¥mathrm{r}¥mathrm{o}¥mathrm{n}¥mathrm{g}¥mathrm{l}¥mathrm{y}¥mathrm{i}¥mathrm{n}C([0,T],.L_{¥sigma}^{2}(B)),¥¥dv_{n_{k}}/dt¥rightarrow dv/dt¥mathrm{w}¥mathrm{e}¥mathrm{a}¥mathrm{k}¥mathrm{l}¥mathrm{y}¥mathrm{i}¥mathrm{n}L^{2}(¥prime 0,T,.L_{¥sigma}^{2}(B)),¥¥g_{1}^{n_{k}}¥rightarrow g_{1}¥mathrm{w}¥mathrm{e}¥mathrm{a}¥mathrm{k}¥mathrm{l}¥mathrm{y}¥mathrm{i}¥mathrm{n}L^{2}(0,T,.L_{¥sigma}^{2}(B)).¥end{array}¥right.$

Since $¥partial¥varphi_{1}^{t}(¥cdot)$ is demiclosed in $L^{2}(0, T;L_{¥sigma}^{2}(B))$ , we can easily see that $ g_{1}(t)¥in$$¥partial¥varphi_{1}^{t}(v(t))$ for $a.e$ . $t¥in¥llcorner¥ulcorner 0$, $T$], which means that $v$ is a solution of (3.25). More-over, the above argument does not depend on the choice of subsequences,therefore we find that (3.47) holds true with $v_{n_{k}}=v_{n}$ and $g^{n_{k}}=g^{n}$ .


Multiplying (3.26) by $ g_{2}^{n}(t)=F_{2}(t)-(v_{n}¥cdot¥nabla)¥overline{¥theta}(t)-d¥theta_{n}/dt-B_{2}(v_{n}(t), ¥theta_{n}(t))¥in$$¥partial¥varphi_{2}^{t}(¥theta_{n}(t))$ and using (3.17), we can obtain in parallel with (3.34),

$¥frac{d}{dt}|¥nabla¥theta_{n}(t)|^{2}+¥kappa|¥Delta¥theta_{n}(t)|^{2}¥leq¥frac{8}{¥kappa}(|F_{2}(t)|^{2}+|(v_{n}¥cdot¥nabla)¥theta_{n}|^{2}+|(¥overline{u}¥cdot¥nabla)¥theta_{n}|^{2}+|(v_{n}¥cdot¥nabla)¥overline{¥theta}|^{2})$

$+(m^{2}+m)|¥nabla¥theta_{n}(t)|^{2}$

$¥leq¥epsilon|¥Delta¥theta_{n}(t)|^{2}+|¥nabla¥theta_{n}(t)|^{2}¥{C_{¥epsilon}(||¥nabla v_{n}||^{4}+||¥nabla u||_{H^{1}}^{4})+m^{2}+m¥}$

$+¥frac{8}{¥kappa}(|F_{2}|^{2}+4¥sqrt{3}c_{0}^{3}||¥nabla v_{n}||^{2}|¥nabla¥overline{¥theta}||¥nabla^{2}¥overline{¥theta}|)$,

where $¥epsilon$ is an arbitrary positive number and $C_{¥epsilon}$ is a number depending on $¥kappa$ ,$¥epsilon$ , $c_{0}$ , $K_{2}$ . Hence we can derive a priori bound for $|¥nabla¥theta_{n}|_{¥infty,T}$ , $|¥Delta¥theta_{n}|_{2,T}$ and$|d¥theta_{n}/dt|_{2,T}$ , and therefore by the same argument as above, there exists a sub-sequence $¥{¥theta_{n_{k}}¥}$ of $¥{¥theta_{n}¥}$ such that

(3.48) $¥left¥{¥begin{array}{l}¥theta_{n_{k}}¥rightarrow¥theta ¥mathrm{s}¥mathrm{t}¥mathrm{r}¥mathrm{o}¥mathrm{n}¥mathrm{g}¥mathrm{l}¥mathrm{y}¥mathrm{i}¥mathrm{n}C([0,T],.L^{2}(B)),¥¥d¥theta_{n_{k}}/dt¥rightarrow d¥theta/dt¥mathrm{w}¥mathrm{e}¥mathrm{a}¥mathrm{k}¥mathrm{l}¥mathrm{y}¥mathrm{i}¥mathrm{n}L^{2}(0,T,.L^{2}(B)),¥¥g_{2^{k}}^{n}-g_{2}¥in¥partial¥varphi_{2}^{t}(¥theta)¥mathrm{w}¥mathrm{e}¥mathrm{a}¥mathrm{k}1¥mathrm{y}¥mathrm{i}¥mathrm{n}L^{2}(0,T,.L^{2}(B)),¥¥B_{2}(v_{n_{k}},¥theta_{n_{k}})-b_{2}¥mathrm{w}¥mathrm{e}¥mathrm{a}¥mathrm{k}¥mathrm{l}¥mathrm{y}¥mathrm{i}¥mathrm{n}L^{2}(0,T,.L^{2}(B)).¥end{array}¥right.$

Recalling the fact that $((w¥cdot¥nabla)¥theta, ¥phi)=-((w¥cdot¥nabla)¥phi, ¥theta)$ holds for all $w¥in L_{¥sigma}^{2}(B)$, $¥theta¥in$$H_{0}^{1}(B)$ and $¥emptyset¥in C_{0}^{¥infty}(B)$ , we can see that

(3.49) $B_{2}(v_{n_{k}}, ¥theta_{n_{k}})¥rightarrow B_{2}(v, ¥theta)$ in $¥ovalbox{¥tt¥small REJECT}^{¥prime}([0, T] ¥times B)$,

whence follows $b_{2}=B_{2}(v, ¥theta)$. Thus $¥theta$ is shown to be a solution of (3.26).Since the above argument does not depend on the choice of subsequences,the original sequence $¥theta_{n}$ converges to $¥theta$ in $C([0, T];L^{2}(B))$. Furthermore,repeating the same verification as for (3.49), we can show that $B_{1}(v_{n})-B_{1}(v)$weakly in $L^{2}(0, T;L_{¥sigma}^{2}(B))$. Now we conclude that $¥ovalbox{¥tt¥small REJECT}$ is continuous from $¥ovalbox{¥tt¥small REJECT}_{W}$into $¥ovalbox{¥tt¥small REJECT}_{W}$ . Q.E.D.

References

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M.J.M. 3 (1973), 257-318.[11] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow,

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Sci. Appl. 3 (1993/1994), 383-399.[17] Otani, M. and Yamada, Y., On the Navier-Stokes equations in noncylindrical domains:

An approach by the subdifferential operator theory, J. Fac. Sci. Univ. Tokyo, 25 (1978),185-204.

[18] Serrin, J., The initial value problem for the Navier-Stokes equations, Nonlinear Problems.Univ. Wisconsin Press, 69-98 (1963).

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nuna adreso:Hiroshi InoueAshikaga Institute of Technology268-1 Omae Ashikaga, Tochigi, 326Japan

Mitsuharu otaniDepartment of Applied PhysicsSchool of Science and EngineeringWaseda University3-4-1, Okubo, Shinjuku, Tokyo 169Japan

(Ricevita la 19-an de julio, 1995)

1. Introduction2. Notations and main results2.1. Notations and some function spaces2.2. Main results

3. Proofs of theorems3.1. Abstract preliminaries3.2. Abstract formulation3.3. Some lemmas3.4. The proof of Theorem I3.5. The proof of Theorem II

References

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