数理解析研究所講究録
第 1892 巻 2014 年 1-12

1Global existence results for some double-diffusive convection system
based on the Brinkman-Forchheimer equation
with homogeneous Neumann boundary conditions \dag er
Shun Uchida
Graduate School of Advanced Science and Engineering,
Waseda University

1

Introduction
We consider the following system which describes double-diffusive convection phenomena of an incom-pressible viscous fluid in some porous medium:$(BF)[Matrix]$
where$\Omega\subset \mathbb{R}^{N}$is a bounded domain with smooth boundary$\partial\Omega,$$n$denotes the unit outward normal$:=\nabla T\cdot n$
vector onand. Unknown functions $u=(u_{1}, u_{2}, \cdots, u_{N})^{t},$and represent thefluid velocity, the temperature of the fluid, the concentration of a solute and the pressure of the fluid
respectively. Positive constants
are called the viscosity coefficient, Soret’s coefficient and Darcy’s
coefficient respectively. Constant vectors $g,$ are derived from gravity andare given external forces. Furthermore, we impose the solenoidal condition on the fluid velocity $u.$Double-diffusive convection is a model of convection in the fluid reflecting some interactions between$\partial\Omega$$T,$ $C$$\frac{\partial T}{\partial n}$$\nu,$$\rho,$

$p$

$a$

$h$

$f_{1}=(f_{1}^{1}, f_{1}^{2}, \cdots, f_{1}^{N})^{t},$$f_{2},$ $f_{3}$the temperature and the concentration of solute. The double-diffusive convection phenomena can be
represented by the second and the third equation of ( $BF$ ) which originate from a result of the irreversible

thermodynamics. The term
, which is called Soret’s effect term, describes a certain interaction
between the temperature of the fluid and the concentration of a solute. This interaction makes the$\rho\Delta T$

behavior of the fluid become more complicated than the simplified diffusion model and this Soret’s effectmainly characterizes the double-diffusive convection. Originally, the second equation also contains a, which is called Dufour’s effect term. However, Dufour’s effect is generally muchinteraction term
smaller than Soret’s effect, especially for the case where we deal with the liquid fluid. Therefore we here
consider only Soret’s effect term (for further details of physical background, see [1] and [8]).
The first equation of ( $BF$ ) comes from Brinkman-Forchheimer equation, which describes the behaviorof the fluid velocity in some porous medium with a relatively large porosity (the rate of void space in$\rho’\Delta C$\dag er Joint work with Professor Mitsuharu \^Otani, School of Science and Engineering, Waseda University.

2
a porous medium). Originally, Brinkman-Forchheimer equation has some nonlinear terms and a spacedependent function which stands for the porosity. However, under some physical assumptions, such as
the uniformity of the porosity, we can derive the linearized Brinkman-Forchheimer equation given in thefirst equation of ( $BF$ ). Here $gT,$ $hC$ are effects from gravity.
There are many studies for ( $BF$ ), for example, about the continuous dependence of the solutions on
Soret’s coefficient and so on. However, to the best of our knowledge, it seems that there are very few
studies for the solvability of ( $BF$ ). The first attempt in this direction is made in [13], where the initial$\rho$

boundary value problem for ( $BF$ ) with homogeneous Dirichlet boundary conditions is considered. In[13], they showed that this problem admits a unique global solution when $N\leq 3.$
In [11], the global solvability of the time periodic problem is shown for ( $BF$ ) with homogeneous Dirichletboundary conditions both for 2 and 3-dimensional cases.

Due to the convection terms$u\cdot\nabla T,$ $u\cdot\nabla C$, which are quite similar to that appearing in the Navier-Stokes equations, it apparently seems that it would be very difficult to obtain the global solvability” of( $BF$ ) in 3-dimensional case, i.e., the existence of the unique global solution of the initial boundary-value
problem for arbitrarily large initial data or the existence of time-periodic solutions for arbitrarily large
external forces. However, it is revealed that the global solvability holds true for these problems even forthe 3-dimensional case in [13] and [11].
The main purpose of this paper is to show that the global solvability results similar to [13] and[11] still hold true for ( $BF$ ) with homogeneous Neumann boundary conditions forand . In orderto carry out this purpose, we follow the basic strategy adopted in [13] and [11], i.e., we reduce ourproblem to some abstract equation in an appropriate Hilbert space and we rely on the abstract theory
developed in [9] and [10]. However, the lack of the coercivity of the Laplacian with homogeneous Neumannboundary conditions causes some difficulties in this procedure. Especially for the periodic problem, weneed to introduce some approximate system involving some dissipation terms and cut-off functions as in[11]. Unfortunately this hinders establishing desirable a priori estimates under the Neumann boundary
condition. In order to cope with this difficulty, we introduce another step of approximations for the
“

$T$

$C$

original system.

In section 2, we fix some notations for later use and we introduce abstract results. In section 3, ourmain results are stated. In section 4 and 5, we give proofs of main results for the initial boundary value
problem and the periodic problem respectively.

32Preliminaries

2.1

Notation

In order to formulate our results, we fix the following notations.
$\mathbb{C}_{\sigma}^{\infty}(\Omega)=\{u=(u^{1}, u^{2}, \cdots, u^{N})^{t};u^{j}\in C_{0}^{\infty}(\Omega)^{\forall}j=1,2, \cdots, N, \nabla\cdot u=0\},$$\mathbb{L}^{2}(\Omega)=(L^{2}(\Omega))^{N}, \mathbb{H}^{1}(\Omega)=(H^{1}(\Omega))^{N}=(W^{1,2}(\Omega))^{N},$$\mathbb{L}_{\sigma}^{2}(\Omega)$: The closure of$\mathbb{H}_{\sigma}^{1}(\Omega)$: The closure of$\mathbb{C}_{\sigma}^{\infty}(\Omega)$$\mathbb{C}_{\sigma}^{\infty}(\Omega)$$H=\mathbb{L}_{\sigma}^{2}(\Omega)\cross L^{2}(\Omega)\cross L^{2}(\Omega)$-norm,

under the$\mathbb{L}^{2}(\Omega)$

under the$\mathbb{H}^{1}(\Omega)$-norm,: Hilbert space,$C_{n}([0, S];H)=\{U\in C([O, S];H);U(O)=U(S)\},$$\mathcal{P}_{\Omega}$: The orthogonal projection from$\mathbb{L}^{2}(\Omega)$

onto$\mathcal{A}=-\mathcal{P}_{l}\Delta$: The Stokes operator with domain$A_{N}=-\Delta$

with domain$\mathcal{A}^{\alpha},$$A_{N}^{\alpha}$$\mathbb{L}_{\sigma}^{2}(\Omega)$,$D(\mathcal{A})=\mathbb{H}^{2}(\Omega)\cap \mathbb{H}_{\sigma}^{1}(\Omega)$,$D(A_{N})= \{u\in H^{2}(\Omega);\frac{\partial’u}{\partial n}=0 on \partial\Omega\},$denote the fractional powers of$\mathcal{A},$$A_{N}$of order$\alpha.$2.2 Subdifferential Operator and Nonlinear Interpolation Class
Let be a proper lower semi-continuous convex function from $H$ into $(-\infty, +\infty]$ . Define the effective
domain of by $D(\varphi)=\{U\in H;\varphi(U)<+\infty\}$ and the subdifferential of by$\varphi$$\varphi$$\varphi$$\partial\varphi(U)=\{f\in H;\varphi(V)-\varphi(U)\leq(f, V-U)_{H}$with domain

for all $V\in H\}$$D(\partial\varphi)=\{U\in H;\partial\varphi(U)\neq\emptyset\}.$Generally, subdifferential operators are multivalued maximal monotone operators. However, since
Subdifferential operators used in this paper are always single-valued, we restrict ourselves to the single-

valued subdifferential operators. It will be shown that the leading terms of the system ( $BF$ ) can be given

as the subdifferential of some lower semi-continuous convex function in the next subsection.
It is well known that for any maximal monotone operator
in $H$ , the resolvent of $A;J_{\lambda}=(I+$, is well defined on $H$ andasfor all. Then for $\alpha\in(0,1),$, by measuring how fast
converges to , we can define a nonlinear interpolation class$A$$\lambda A)^{-1}(\lambda>0)$$J_{\lambda}Uarrow U$$[1, \infty]$

associated with

$U\in\overline{D(A)}$$p\in$$U$$J_{\lambda}U$$A$$\lambdaarrow 0$$\mathcal{B}_{\alpha,p}(A)$by$\mathcal{B}_{\alpha,p}(A)=\{U\in\overline{D(A)};t^{-\alpha}|U-J_{t}U|_{H}\in L_{*}^{p}(0,1)\},$

where $L_{*}^{p}=L^{p}(dt/t)$ , i.e., $|f|_{L^{p}(0,S)}=( \int_{0}^{s}|f(t)|^{p}t^{-1}dt)^{1/p}$ for $1\leq p<\infty$ and$L_{*}^{\infty}=L^{\infty}$. We often use

the notation$|U|_{\mathcal{B}_{\alpha p}(A)}=|t^{-\alpha}|U-J_{t}U|_{H}|_{L_{*}^{p}(0,1)}.$This nonlinear interpolation class$\mathcal{B}_{\alpha,p}(A)$such as Besov spaces. In particular, if$A$covers a very wide class of interpolation spaces already known
is non-negative self-adjoint operator, then the domain of the4(see [2], [3] and [4]). In what follows, we
fractional power of of order is given by
use this nonlinear interpolation theory for the special case where$A$$D(A^{\alpha})=\mathcal{B}_{\alpha,2}(A)$$\alpha$$A=\partial\varphi.$

2.3 Reduction to an Abstract Problem
In this subsection, we reduce our problem to an abstract problem in some Hilbert space. Operating the

projection
equations:$\mathcal{P}_{t\}}$to the first equation of ( $BF$ ) to erase the pressure term$\nabla p$. Then we obtain the following(2.1)$\{\begin{array}{l}\partial_{t}u+\nu \mathcal{A}u=-au+\mathcal{P}_{\Omega}gT+\mathcal{P}_{\Omega}hC+\mathcal{P}_{\Omega}f_{1},\partial_{t}T+A_{N}T+u\cdot\nabla T=f_{2},\partial_{t}’C+A_{N}C+u\cdot\nabla C=-\rho A_{N}T+f_{3}.\end{array}$

We introduce the Hilbert spacefor each parameter $\eta\in(0,1]$ , which designates the Hilbert space$H$ endowed withthe following inner product:$H_{\eta}$$(U_{1}, U_{2})_{H_{1}},=(u_{1}, u_{2})_{L_{\sigma}^{2}}+(T_{1}, T_{2})_{L^{2}}+\frac{\eta^{2}}{9\rho^{2}}(C_{1}, C_{2})_{L^{2}}$

(2.2)

for

$U_{i}=(u_{i}, T_{i}, C_{i})^{t},$$(i=1,2)$ .

Here, in order to deal with the perturbation term $\rho\Delta T=-\rho A_{N}T$ as a small perturbation to our

problem, we put the weight depending on$\eta$

and$\rho$

to the last term.

Next, as a lower semi-continuous convex function from

$H_{\eta}$

to $[0, +\infty]$ , we define$\varphi$

by
(2.3)

$\varphi(U)=\{\begin{array}{ll}\frac{\nu}{2}\Vert|\nabla u|\Vert_{L^{2}}^{2}+\frac{1}{2}\Vert\nabla T\Vert_{L^{2}}^{2}+\frac{\eta^{2}}{18\rho^{2}}\Vert\nabla C\Vert_{L^{2}}^{2} if U\in D(\varphi) ,+\infty if U\in H_{\eta}\backslash D(\varphi) ,\end{array}$where$D(\varphi)=\mathbb{H}_{\sigma}^{1}(\Omega)\cross H^{1}(\Omega)\cross H^{1}(\Omega)$is the effective domain of$\varphi$. Then the subdifferential of$\varphi$isgiven by$\partial\varphi(U)=(\begin{array}{l}-\nu \mathcal{P}_{tl}\Delta u-\Delta T-\Delta C\end{array})$

with domain$D(\partial\varphi)=(\mathbb{H}^{2}\cap \mathbb{H}_{\sigma}^{1})\cross D(A_{N})\cross D(A_{N})$

(2.4)

.

Furthermore, we put$U=(\begin{array}{l}uTC\end{array}),$$\frac{dU}{dt}=(\begin{array}{l}\partial_{t}u\partial_{t}T\partial_{t}C\end{array}),$$B(U)=(\begin{array}{l}au-\mathcal{P}_{(\}}gT-\mathcal{P}_{l\}}hCu\cdot\nabla Tu\cdot\nabla C-\rho\Delta T\end{array}),$$F=(\begin{array}{l}\mathcal{P}_{t1}f_{1}f_{2}f_{3}\end{array})$

.

(2.5)

Then the initial boundary value problem for (2.1) is reduced to the following abstract Cauchy problem:
in$H_{\eta}$( $CP$ ) $\{\begin{array}{l}\frac{dU}{dt}(t)+\partial\varphi(U(t))+B(U(t))=F(t) t\in[0, S],U(0)=U_{0},\end{array}$and the periodic problem for (2.1) is reduced to the following abstract periodic problem in( $AP$ ) $\{\begin{array}{l}\frac{dU}{dt}(t)+\partial\varphi(U(t))+B(U(t))=F(t) t\in[0, S],U(0)=U(S) .\end{array}$(2.6)$H_{\eta}$

:
(2.7)

5
2.4

Known Abstract Theorem

In order to assure the existence of the solutions, we rely on abstract results given in [9] and [10]. To

formulate these results, we introduce the following conditions.

Assumptions(Al) For any $L\in(O, +\infty)$ , the set $\{U\in H;\varphi(U)+\Vert U\Vert_{H}^{2}\leq L\}$ is compact in $H.$
(A2)

is -demiclosed in the following sense:$U_{n}arrow U$ strongly in $C([O, S];H),$
weakly in $L^{2}(0, S;H),$$L^{2}(0, S;H)$ , then $b(t)=B(U(t))$ holds for a.e. $t\in[0, S].$
$B(\cdot)$$\varphi$$B(U_{n})arrow b$$\partial\varphi(U_{n})arrow\partial\varphi(U)$(A3) For a given exponent $\alpha\in(0,1/2)$ , there exists a monotone increasing function$\ell(\cdot)$

weakly insuch that$\Vert B(U)\Vert_{H}\leq\ell(\Vert U\Vert_{H})\{\epsilon\Vert\partial\varphi(U)\Vert_{H}+\frac{1}{\epsilon}|\varphi(U)|^{\frac{1-\alpha}{1-2\alpha}}+1\}\forall_{U\in D(\partial\varphi)},$is a positive constant determined by the initial data
precisely, is a monotone decreasing function ofwhere$\epsilon$$U_{0}$and the external force $F(t)$ , more$|U_{0}|_{H}+|U_{0}|_{\mathcal{B}_{\alpha,p}(\partial\varphi)}+|F|_{L^{2}(0,S;H)}.$$\epsilon$(A4) There exists a monotone increasing function$\ell(\cdot)$

and $k\in(O, 1)$ such that$\Vert B(U)\Vert_{H}^{2}\leq k\Vert\partial\varphi(U)\Vert_{H}^{2}+\ell(\varphi(U)+\Vert U\Vert_{H}^{2}) \forall_{U}\in D(\partial\varphi)$(A5) There exists a monotone increasing function$\ell(\cdot)$.

and a constant $k\in[0,1)$ such that.

$\Vert B(U)\Vert_{H}^{2}\leq k\Vert\partial\varphi(U)\Vert_{H}^{2}+P(\Vert U\Vert_{H})(\varphi(U)+1)^{2} \forall_{U}\in D(\partial\varphi)$(A6) There exist positive constants$\alpha,$$K$such that$(-\partial\varphi(U)-B(U), U)_{H}+\alpha\varphi(U)\leq K\forall_{U}\in D(\partial\varphi)$.

Then the following results hold (see [9] and [10]).
Theorem 2.1. Let$U_{0}\in \mathcal{B}_{\alpha,p}(\partial\varphi)$satisfied. Then there exists$U(t)$

in$[0, S_{0}]$with $p\in[1,2]$ and $F\in L^{2}(0, S;H)$ , and let (Al), (A2) and$S_{0}\in(0, S]$

depending$on |U_{0}|_{H}and|U_{0}|_{\mathcal{B}_{\alpha p}(\partial\varphi)} such that (CP)$$(A3)_{\alpha}^{0}$be

has a solution

satisfying$t^{1/2-\alpha}dU/dt, t^{1/2-\alpha}\partial\varphi(U(t)), t^{1/2-\alpha}B(U(t))\in L^{2}(0, S_{0};H)$,$t^{-\alpha}\Vert U(t)-U_{0}\Vert_{H}, t^{1/2-\alpha}|\varphi(U(t))|^{1/2}\in L_{*}^{q}(0, S_{0})\forall_{q}\in[2, \infty].$

Theorem 2.2. Let (Al), (A2) and (A4) be satisfied and let
and $F\in L^{2}(0, S;H)$ . Then there
exists $S_{0}\in(0, S]$ depending $on |U_{0}|_{H} and \varphi(U_{0})$ such that ( $CP$ ) has a solution $U(t)$ in
satisfying$U_{0}\in D(\varphi)$$[0, S_{0}]$,$dU/dt, \partial\varphi(U(t)), B(U(t))\in L^{2}(0, S_{0};H)$$\varphi(U(t))$

is absolutely continuous on $[0, S_{0}].$

Theorem 2.3. Let (Al), (A2), (A5) and (A6) be satisfied. Then for every $F\in L^{2}(0, S;H)$ , ( $AP$ ) has
a strong solution $U\in C_{\pi}([O, S];H)$ such that$dU/dt, \partial\varphi(U), B(U)\in L^{2}(0, S;H)$$\varphi(U)$is absolutely continuous on,$[0, S]$

and $\varphi(U(O))=\varphi(U(S))$ .

6Remarkwhich is slightly stronger thanIn [9], Theorem 2.3 is actually proved under a different assumption
replaced by(A3). However it is easy to see that the proof of Theorem 2.3 holds true with(see the proof of Theorem I in [9]).
$(A3)_{\alpha}$$(A3)_{\alpha}$$(A3)_{\alpha}^{0}$

3 Main Results
Theorem 3.1. $([12J$ : Initial Boundary Value Problem)

Let $N\leq 3$ and let$f_{1}\in L^{2}(0, S;\mathbb{L}^{2}(\Omega)),$$f_{2},$$f_{3}\in L^{2}(0, S;L^{2}(\Omega))$. Then for each initial data $U_{0}=$with $\alpha\in[1/4,1/2]$ , ( $BF$ ) admits a unique solution $U=$$(u, T, C)^{t}\in C([0, S];H)$ satisfying $U(0)=U_{0}$ and$(u_{0}, T_{0}, C_{0})^{t}\in D(\mathcal{A}^{\alpha})\cross D(A_{N}^{\alpha})\cross D(A_{N}^{\alpha})$$(\#)_{\alpha}$$\{\begin{array}{l}t^{1/2-\alpha}\partial_{t}u, t^{1/2-\alpha}\mathcal{A}u\in L^{2}(0, S;\mathbb{L}_{\sigma}^{2}(\Omega)) ,t^{1/2-\prime}||\nabla u||_{V(\Omega)}\in L_{*}^{p}(0, S) for all p\in[2, \infty],t^{1/2-\alpha}\partial_{t}T, t^{1/2-\alpha}\partial_{t}C, t^{1/2-\alpha}\Delta T, t^{1/2-\alpha}\Delta C\in L^{2}(0, S;L^{2}(\Omega)) ,t^{1/2-\alpha}||\nabla T||_{L^{2}(\Omega)}, t^{1/2-\alpha}||\nabla C||_{L^{2}(\Omega)}\in L_{*}^{p}(0, S) for all p\in[2, \infty],\end{array}$

Theorem 3.2. $([12J$ : Time Periodic Problem)$f_{3}\in L^{2}(0, S;L^{2}(\Omega))$ . Furtherrnooe, we assume thatLet$f_{1}\in L^{2}(0, S;\mathbb{L}^{2}(\Omega)),$$f_{2},$$\int_{0}^{s}\int_{1l}f_{2}(x, t)dxdt=\int_{0}^{s}\int_{l}f_{3}(x, t)dxdt=0$$f_{2},$.

$f_{3}$

satisfy(3.1)

Then ( $BF$ ) admits a solution $U=(u, T, C)^{t}\in C_{\pi}([O, S];H)$ satisfying$(\neq)_{1/2}$

Remarks(1) Ifbelongs to$U_{0}$

the solution$U$$\{\begin{array}{l}\partial_{t}u, \mathcal{A}u\in L^{2}(0, S;\mathbb{L}_{\sigma}^{2}(\Omega)) ,u\in C([0, S];\mathbb{H}_{\sigma}^{1}(\Omega)) ,\partial_{t}T, \partial_{t}C, \triangle T, \triangle C\in L^{2}(0, S;L^{2}(\Omega)) ,T, C\in\dot{C}([0, S];H^{1}(\Omega)) .\end{array}$$D(\mathcal{A}^{1/2})\cross D(A_{N}^{1/2})\cross D(A_{N}^{1/2})=\mathbb{H}_{\sigma}^{1}(\Omega)\cross H^{1}(\Omega)\cross H^{1}(\Omega)$satisfies property$(\#)_{1/2}$

in Theorem 3.1, thengiven in Theorem 3.2.

(2) It can be shown that the required Condition (3.1) is also the necessary condition for the existenceof the periodic solution of ( $BF$ ) satisfying the homogeneous Neumann boundary condition. In fact,
integrating the second and the third equations over$\Omega\cross[0, S]$, we can derive (3.1).

(3) In [13], the same result as in Theorem 3.1 is given for ( $BF$ ) with the homogeneous Neumann boundary

condition replaced by the homogeneous Dirichlet boundary condition only for the case $\alpha=1/2$ . However,with obvious modifications, we can show that if$U_{0}\in D(\mathcal{A}^{\alpha})\cross D(A_{D}^{\alpha})\cross D(A_{D}^{\alpha})$then the Dirichlet problem for ( $BF$ ) admits a unique solution$U$satisfying(4) The characterizations for the domains of the fractional powers of[6].

$A_{N}$

with $\alpha\in[1/4,1/2],$$(\#)_{\alpha}.$and$\mathcal{A}$can be found in [7] and7
4

Initial Boundary Value problem

In this section, we give an outline of our proof of Theorem 3.1. This argument is divided into threeparts, i.e., the local existence, the global existence and the uniqueness.

4.1

Local Existence

In order to prove the local existence, we are going to check conditions required in Theorems 2.1 and2.2. For this purpose, we choose, where is an exponent appearing inFirst, it is easy to see that the compactness of -level set and the -demiclosedness of $B(U)$ (therequired condition (Al) and (A2) can be satisfied.
Furthermore, in spite of the lack of coercivity of the leading term, we can take almost the sameprocedure as that in [13] and we can derive the following estimate of the perturbation term
with
$N\leq 3$ :$\eta=\epsilon$$(A3)_{\alpha}^{0}.$$\epsilon$$\varphi$$\varphi$

$)$

$B$

$\Vert B(U)\Vert_{H}\leq\epsilon\Vert\partial\varphi(U)\Vert_{H_{\eta}}+\frac{\gamma}{\epsilon}(\varphi^{3/2}(U)+\Vert U\Vert_{H_{\eta}}+1)$

(4.1)

,

where is a constant which depends on some Sobolev’s embedding constants, the positive coefficientsand the constant vectors in ( $BF$ ). This estimate (4.1) ensures the condition
with $\alpha\in[1/4,1/2)$
and the condition (A4).
Hence, Theorem 2.1 assures the existence of local solutions $U(t)$ on
satisfying
with$\gamma$$(A3)_{\alpha}^{0}$$[0, S_{0}]$replaced by

when
Theorem 2.2, ( $BF$ ) has local solutions satisfying $(\#)_{1/2}$ with$(\#)_{\alpha}$$S$

for $\alpha\in[1/4,1/2)$ . Moreover, from
replaced bywhen $U_{0}=(u_{0}, T_{0}, C_{0})^{t}\in$$U_{0}=(u_{0}, T_{0}, C_{0})^{t}\in D(\mathcal{A}^{\alpha})\cross D(A_{N}^{\alpha})\cross D(A_{N}^{\alpha})$$S_{0}$$S$$S_{0}$.

$D(\mathcal{A}^{1/2})\cross D(A_{N}^{1/2})\cross D(A_{N}^{1/2})=\mathbb{H}_{\sigma}^{1}(\Omega)\cross H^{1}(\Omega)\cross H^{1}(\Omega)$

4.2 Global Existence
In this subsection, we show that every local solutions can be continued globally to $[0, S]$ by establishing some a priori estimates.
Althoughloses the coercivity because of the Neumann boundary condition, we can obtain theboundedness of the solution. Indeed, establishing the following a priori estimate, we can derive theboundedness of(boundedness of-norm) of the solutions:$\partial\varphi(U)$$H^{1}$$\sup_{0\leq t\leq S}\varphi(U(t))$(1)

2nd equation(2)

3rd equation$\cross C,$(3)

lst equation$\cross\partial_{t}u$

(4)

(5)

$\cross T,$

,

(4.2)

2nd equation$\cross-\Delta T,$$\partial_{t}’T,$

3rd equation$\cross-\Delta C,$$\partial_{t}C.$

Therefore, we can assure that the local solutions can be globally extended when $\alpha=1/2.$
In the cases where $\alpha\in[1/4,1/2)$ , the local solutions also can be extended from the regularity property$(\#)_{\alpha}$, i.e., from the fact that there exist $0<t_{0}\leq S_{0}$ where the solutions satisfy $(u(t_{0}), T(t_{0}), C(t_{0}))^{t}\in$$D(\varphi)=\mathbb{H}_{\sigma}^{1}(\Omega)\cross H^{1}(\Omega)xH^{1}(\Omega)$.

8
4.3 UniquenessIn this subsection, we are going to prove the uniqueness of the solution of the initial boundary value
problem for ( $BF$ ).

Let$U^{1}$

and$U^{2}$

be solutions of ( $BF$ ) for the same initial data:$U^{i}=(\begin{array}{l}u\dot{.}T^{i}C^{t}\end{array})(i=1,2)$

and let$(\begin{array}{l}w\tau\theta\end{array})=U^{1}-U^{2}.$Then we can obtain the following inequality of$(w, \tau, \theta)^{t}$as in [13].

$\frac{1}{2}\frac{d}{dt}y(t)\leq\gamma y(t)+\frac{\gamma^{2}}{\nu}\Vert\nabla T^{2}(t)\Vert_{L^{2}}^{4}\Vert w(t)\Vert_{L_{\sigma}^{2}}^{2}+\frac{\gamma^{2}}{2\rho^{4}\nu^{2}}\Vert\nabla C^{2}\Vert_{L^{2}}^{4}\Vert w(t)\Vert_{L_{\sigma}^{2}}^{2}$$\leq\gamma(\Vert\nabla T^{2}\Vert_{L^{2}}^{4}+\Vert\nabla C^{2}\Vert_{L^{2}}^{4}+1)y(t)$where$y(t)=\Vert w(t)\Vert_{L_{\sigma}^{2}}^{2}+\Vert\tau(t)\Vert_{L^{2}}^{2}+=^{1}2\rho\Vert\theta(t)\Vert_{L^{2}}^{2}$. Here we note that$(\#)_{\alpha}$with $\alpha\in[1/4,1/2]$ implies

that

.

$t^{1/2-\alpha}\Vert\nabla T^{2}\Vert_{L^{2}}, t^{1/2-\alpha}\Vert\nabla C^{2}\Vert_{L^{2}}\in L_{*}^{4}(0, S) \Rightarrow \Vert\nabla T^{2}\Vert_{L^{2}}, \Vert\nabla C^{2}\Vert_{L^{2}}\in L^{4}(0, S)$

Hence, the uniqueness follows from Gronwall’s inequality.

5 Periodic Problem
In this section, we give an outline of our proof of Theorem 3.2. When we try to apply the abstract result
Theorem 2.3 directly to ( $BF$ ), some difficulties arise. Therefore, we first introduce some approximate
systems with two parameters
and we show the existence of periodic solutions for these approximate
systems. We next consider the convergence of solutions of approximate equations as. In the, we face some difficulty which does not appear in the case where we impose Dirichletargument asboundary condition.
$\lambda$$\epsilon,$$\epsilon,$$\lambdaarrow 0$$\lambdaarrow 0$

5.1 Approximate Equations
When one tries to apply Theorem 2.3 to ( $AP$ ), one faces some difficulties. The most serious one

arises in checking (A5). In fact, from (4.1), whose growth order for
is cubic, it is difficult to showthat $B(U)$ satisfies the required growth order in (A5). Moreover, when the constant vectors $g,$arevery large, it is difficult to examine whether (A6) is satisfied. From these reasons, we are led to introduce$\varphi(U)$$h$the same type of relaxed approximate problems as in [11].
However, approximate problems introduced in [11] prevents establishing desirable a priori estimates
under the homogeneous Neumann boundary conditions. In order to manage with this difficulty, we
introduce another approximation procedure.

9More precisely, we add linear coercive terms to the second and the third equations and also replace. We consider the following approximate equations.
and $C$ by their cut-off functions
and$[C]_{\epsilon}$$[T]_{\epsilon}$(5.1)$(BF)_{e,\lambda}\{\begin{array}{l}\partial_{t}u=\nu \mathcal{P}_{\Omega}\Delta u-au+\mathcal{P}_{\Omega}g[T]_{\epsilon}+\mathcal{P}_{\Omega}h[C]_{\epsilon}+\mathcal{P}_{\Omega}f_{1},\partial_{t}T+u\cdot\nabla T=\Delta T-\epsilon|T|^{p-2}T-\lambda T+f_{2},\partial_{t}’C+u\cdot\nabla C=\triangle C+p\Delta T-\epsilon|C|^{p-2}C-\lambda C+f_{3},\end{array}$where$\epsilon,$$\lambda\in(0,1)$$T$

are approximation parameters and the cut-off function$[T]_{\epsilon}$is defined by(5.2)$[T]_{\epsilon}=\{\begin{array}{ll}T if |T|\leq 1/\epsilon,(Sgn T) 1/\epsilon if |T|\geq 1/\epsilon, \epsilon\in(0,1) ,\end{array}$and is some sufficiently large exponent.
We are going to reduce these approximate equations (5.1) to an abstract problem ( $AP$ ). For the
perturbation term, we replace it by$p$$B_{\epsilon}(U)=(\begin{array}{llll}au-\mathcal{P}_{\Omega} g[T]_{\Xi}-\mathcal{P}_{\Omega} h[C]_{\epsilon} u\cdot\nabla T \Delta Tu\cdot\nabla C-\rho \end{array})$

We also need to replace the lower semi-continuous convex function$\varphi_{\epsilon,\lambda}(U)=\varphi(U)+\psi_{\epsilon,\lambda}(U)$$\varphi$

.
by

(5.3)

$\varphi_{\epsilon,\lambda}$

which is given by,$\psi_{\epsilon,\lambda}(U)=\{\begin{array}{ll}\frac{\epsilon}{p}\Vert T\Vert_{L^{p}}^{p}+\frac{\epsilon}{9\rho^{2}p}\Vert C\Vert_{Lp}^{p}+\frac{\lambda}{2}\Vert T\Vert_{L^{2}}^{2}+\frac{\lambda}{18\rho^{2}}||C\Vert_{L^{2}}^{2} if U\in D(\psi_{\epsilon,\lambda})+\infty if U\in H_{\eta}\backslash D(\psi_{\epsilon,\lambda}) ,\end{array}$where$D(\psi_{\epsilon,\lambda})=\mathbb{L}_{\sigma}^{2}(\Omega)\cross L^{p}(\Omega)\cross Iy(\Omega)$.

Here and henceforth, we choose $\eta=1$ in (2.2), definition of the inner product of. Then it is clear
thatis a lower semi-continuous convex function on
and Fr\’echet differentiable on
andthat the subdifferential ofcoincides with the sum of dissipation terms and coercive terms, i.e.,$H_{\eta}$$\psi_{\epsilon,\lambda}$$D(\psi_{\epsilon,\lambda})$$H_{\eta}$$\psi_{\epsilon,\lambda}$$\partial\psi_{\epsilon,\lambda}(U)=(O, \epsilon|T|^{p-2}T+\lambda T, \epsilon|C|^{p-2}C+\lambda C)^{t}.$In general, the sum of two subdifferentials is not always maximal monotone. However, for this case,

we have the following good property:$(\partial’\varphi(U), \partial’\psi_{\epsilon,\lambda}(U))_{H}=(-\Delta T, \epsilon|T|^{p-2}T+\lambda T)_{L^{2}}+(-\Delta C, \epsilon|C|^{p-2}C+\lambda C)_{L^{2}}$$= \lambda\Vert\nabla T\Vert_{L^{2}}^{2}+\epsilon(p-1)\int_{\Omega}|T|^{p-2}|\nabla T|^{2}dx$

(5.4)

.

$+ \lambda\Vert\nabla C\Vert_{L^{2}}^{2}+\epsilon(p-1)\int_{\Omega}|C|^{p-2}|\nabla C|^{2}dx\geq 0$

By virtue of (5.4), together with Proposition 2.17, Theorem 4.4 and Proposition 4.6 in Br\’ezis [5], we candeduce that$\partial\varphi+\partial\psi_{\epsilon,\lambda}$

becomes maximal monotone, and hence we get.

$D(\partial(\varphi+\psi_{\epsilon,\lambda}))=D(\partial\varphi)\cap D(\partial\psi_{\epsilon,\lambda})$$\partial(\varphi+\psi_{\epsilon,\lambda})=\partial\varphi+\partial\psi_{\epsilon,\lambda}$

with

10
Thus, we have another abstract problem associated with approximate problems:$(AP)_{\epsilon,\lambda}\{\begin{array}{l}\frac{dU(t)}{dt}+\partial\varphi_{\epsilon,\lambda}(U(t))+B_{\epsilon}(U(t))=F(t) t\in[O, S],U(0)=U(S) .\end{array}$

By almost the same arguments as that of [11], we can assure that

(5.5)

satisfies required conditions(A5) and (A6) by virtue of replacing by cut-off functions and adding dissipation terms $-\epsilon|T|^{p-2}T,$ $-$where $p\geq 12$ . Hence, applying the abstract result Theorem 2.3 to our approximate problems,$(BF)_{\epsilon,\lambda}$$\epsilon|C|^{p-2}C$

we can show that$(BF)_{\epsilon,\lambda}$

5.2 Convergence as

has time periodic solutions$U_{\epsilon,\lambda}=(u_{\epsilon},{}_{\lambda}T_{\epsilon},{}_{\lambda}C_{\epsilon,\lambda})^{t}$$\epsilonarrow 0$

In this subsection, we discuss the convergence of the approximate solutions of approximate equations.
First, we consider the convergence as$\epsilonarrow 0.$

In spite of the lack of coercivity of$\partial\varphi$, due to relaxation terms$-\lambda T,$ $-\lambda C$, we can use the same

convergence argument as in [11]. Indeed, we can derive the following boundedness$\sup_{0\leq t\leq S}\Vert U_{\epsilon,\lambda}(t)\Vert_{H},\sup_{0\leq t\leq S}\varphi_{\epsilon,\lambda}(U_{\epsilon,\lambda}(t)), \Vert\partial\varphi(U_{\vee}.,)\Vert_{L^{2}(0,S,H)}, \Vert\frac{dU_{\epsilon,\lambda}}{dt}\Vert_{L^{2}(0,S;H)}\leq\gamma_{\lambda}$(5.6)

from the following a priori estimates:(1)(2)

2nd equation
3rd equation(3)

lst equation(4)

lst equation
2nd equation
3rd equation

(5)
(6)

$\cross T_{\epsilon,\lambda},$$\cross C_{\epsilon,\lambda},$$\cross u_{\epsilon,\lambda},$(5.7)$\cross\partial_{t}u_{\epsilon,\lambda},$$\cross-\Delta T_{\epsilon,\lambda},$$\cross-\Delta C_{\epsilon,\lambda},$$\partial_{t}T_{\epsilon,\lambda},$$\partial_{t}C_{\epsilon,\lambda},$where
denotes the general constant depending on the external forces, positive constants, constantvectors in the systemand but not on
Hence, by the standard argument, letting, we can assure that the following equations$\gamma_{\lambda}$$(BF)_{\epsilon,\lambda}$$\lambda$$\epsilon.$$\epsilonarrow 0$

have time periodic solutions$U_{\lambda}=(u_{\lambda}, T_{\lambda}, C_{\lambda})^{t}$$(BF)_{\lambda}$, for each parameter :$\lambda$$(BF)_{\lambda}\{\begin{array}{l}\partial_{t}u=\nu \mathcal{P}_{\Omega}\Delta u-au+\mathcal{P}_{\Omega}gT+\mathcal{P}_{\Omega}hC+\mathcal{P}_{\Omega}f_{1},\partial_{t}T+u\cdot\nabla T=\Delta T-\lambda T+f_{2},\partial_{t}C+u\cdot\nabla C=\triangle C+\rho\Delta T-\lambda C+f_{3}.\end{array}$

5.3 Convergence as(5.8)$\lambdaarrow 0$

As the last step, we discuss the convergence of solutions when
. To do this, we need to establish
appropriate a priori estimates. However, because of the lack of coercivity of the leading term, it isdifficult to establish appropriate a priori estimates. To cope with this difficulty, we use the assumption
for external forces (3.1).
$\lambdaarrow 0$

11Integrating the second equation of$(BF)_{\lambda}$

over$\Omega$, we get(5.9).

$\frac{d}{dt}\int_{\Omega}T_{\lambda}(x, t)dx+\lambda\int_{\Omega}T_{\lambda}(x, t)dx=\int_{\Omega}f_{2}(x, t)dx \forall_{t\in}[0, S]$

Here we used the following facts:$\int_{\Omega}\Delta T_{\lambda}dx=\int_{\^{o}\Omega}\frac{\partial T_{\lambda}}{\partial n}dS=0, \int_{\Omega}u_{\lambda}\cdot\nabla T_{\lambda}dx=\int_{\Omega}div(u_{\lambda}T_{\lambda})dx=\int_{\partial\Omega}u_{\lambda}T_{\lambda}dS=0.$

Integrating (5.9) over $(0, S)$ and using the periodic condition and (3.1), we find that$\lambda\int_{0}^{s}\int_{\Omega}T_{\lambda}(x, t)dxdt=0.$

Therefore, from the continuity of the solutions$T_{\lambda}$, there exist $t_{0}\in[0, S]$ such that$\int_{\Omega}T_{\lambda}(x, t_{0})dx=0.$

Hence by (5.9), we obtain(5.10).

$\int_{1}T_{\lambda}(x, t)dx=\int_{t_{0}}^{t}e^{-\lambda(t-s)}\int_{l}f_{2}(x, t)dxdt \forall_{t}\in[t_{0}, t_{0}+S]$

Then applying Poincar\’e-Wirtinger’s inequality$\Vert v-\overline{\tau,}\Vert_{L^{2}}\leq C_{W}\Vert\nabla e\Vert_{L^{2}} \forall_{8,\in H^{1}(\Omega)}, \overline{v}=\frac{1}{|\Omega|}\int_{\Omega}v(x)dx,$

we obtain$\Vert T_{\lambda}\Vert_{L^{2}(0,S;L^{2}(\Omega))}\leq C_{W}\Vert\nabla T_{\lambda}\Vert_{L^{2}(0,S,L^{2}(\Omega))}+S\Vert f_{2}\Vert_{L^{2}(0,S;L^{2}(\Omega))}$where
is a suitable constant which depends only on
inequahty from the third equation of:$C_{W}$$\Omega$.

,(5.11)

Similarly, we can derive the following$(BF)_{\lambda}$(5.12).

$\Vert C_{\lambda}\Vert_{L^{2}(0,S,L^{2}(\Omega))}\leq C_{W}\Vert\nabla C_{\lambda}\Vert_{L^{2}(0,S;L^{2}(\Omega))}+S\Vert f_{3}\Vert_{L^{2}(0,S;L^{2}(\Omega))}$Then, using these inequalities and repeating exactly the same arguments as in section 5.2, we canassure the existence of periodic solutions of the original system ( $BF$ ). Indeed, (5.11) and (5.12) together
with the following manipulation:
(1)
(2)
(3)
(4)
(5)

2nd equation
3rd equation
lst equation
2nd equation
3rd equation$\cross T_{\lambda},$$\cross C_{\lambda},$$\cross\partial_{t}’u_{\lambda}$,$\cross-\Delta T_{\lambda},$$\cross-\Delta C_{\lambda},$(5.13)$\partial_{t}T_{\lambda},$$\partial_{t}C_{\lambda},$lead us to appropriate a priori estimates. By using this uniform boundedness of$U_{\lambda}$

derived from the, we can assureasthe existence of periodic solution of the original system ( $BF$ ), whence follows our result.

above and considering the convergence of the solutions and the equations$(BF)_{\lambda}$$\lambdaarrow 0$

12Reference[1] A. Brandt and H. J. S. Fernando, Double-Diffusive Convection (Geophysical Monograph), Amer.

Geophysical Union, 1995.
[2] D. Br\’ezis, Classes d’interpolation associ\’es \‘a un op\’erateur monotone, C.R. Acad. Sci. Paris 276 s\’erie

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[3] D. Br\’ezis, Perturbations singuli\‘eres et probl\‘emes d’\’evolution avec d\’efaut d’adjustement, C.R. Acad.

Sci. Paris 276, s\’erie A (1973) 1597-1600.
[4] D. Br\’ezis, “Interpolation et op\’erateurs non lin\’eaires,’’ Th\‘eses de Universit\’e Paris VI (1974).
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[8] D. A. Nield and A. Bejan, Convection in Porous Medium, Third Edition, New York: Springer, 2006.
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$\hat{O}$tani, Global solvability of double-diffusive convection systems based upon

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Graduate School of Advanced Science and Engineering,
Waseda University
Tokyo 169-8555
JAPANE–mail address: u-shun@suou.waseda.jp$FTgfflx_{\neq^{p}}^{\succ^{\backslash }}\pi_{J}^{\backslash }\Subset\not\in 1^{\mapsto\backslash }\neq\varpi Fae\ovalbox{\tt\small REJECT}^{\backslash }\}\backslash$$|\triangleleft ffl$ce