STATIONARY SOLUTIONS FOR THE NAVIER-STOKES EQUATIONS AND THE BOUSSINESQ EQUATIONS UNDER GENERAL OUTFLOW CONDITION(Nonlinear Evolution Equations and Applications)

$\mathrm{S}.\mathrm{T}_{1}.\mathrm{A}..\cdot \mathrm{T}.\mathrm{I}\mathrm{o}\mathrm{N}.\mathrm{A}.\mathrm{R}\mathrm{Y}\mathrm{s}\mathrm{o}\mathrm{L}.\mathrm{U}.\mathrm{T}\mathrm{I}\mathrm{o}\mathrm{N}_{\mathrm{t}}\mathrm{S}$

.

FOR

$\mathrm{T}\hat{\mathrm{H}}\mathrm{E}$

NAVIER-STOKES

EQUATIONS

$\cdot$

$\mathrm{A}\hat{\mathrm{N}}\mathrm{D}$

THE

BOUSSINESQ EQUATIONS

UNDER

GENERAL

OUTFLOW

CONDITION

. _{MORIMOTO, Hiroko}

Department of Mathematics

,

Meiji University

.. _森本浩子

明治大学理工学部数学科

Let $D$ bea bounded domain in$\mathrm{R}^{n}$ ( $\mathrm{n}=2$or3), $\partial D$itssmoothboundary. We consider

the existence of solutions to the stationary Navier-Stokes equations (1) $\{$

$-l\text{ノ}\triangle u+(u\cdot\nabla)u+\nabla p=$ $f$ in $D$,

$\mathrm{d}\mathrm{i}\mathrm{v}u$ _$=$ $0$ in $D$,

under the boundary condition (2). $u=\beta$ . on $\partial D$

where $u$ isthe velosity vector, $p$the pressure, $f$the external force, lノkinematic viscosity,

$\beta$ the velosity vector given on the boundary.

We consider also the similal$\cdot$

problem for the stationary

Boussines.q

equations

(3)

under

the

boundary condition

(4) $\{$

$u=\beta$ on $\partial D$

$T=$ $\theta$ on $\partial D$

where $T$ is the temperature, $\mathrm{C}1$ coefficient of volume expansion, $\chi$ thermal diffusivity, $\beta$

and $\theta$ prescribed velosity and temperature on the boundary, respectively.

Firstly, we mention some known results for these problem.

The existence of the stationary solutions to the Navier-Stokes equations (1), (2) and

the Boussinesq equations (3). (4) is known in general context if, for any $\epsilon>0$, there

exists an extension $b_{\text{\’{e}} i}$ of the boundary value $\beta$ to the domain $D$ such that $\mathrm{d}\mathrm{i}\mathrm{v}b_{\mathcal{E}}=0$ in

$D$ and the inequality

$(L)$ $|((u\cdot\nabla)b_{\epsilon}u))|\leq\epsilon||\nabla u||^{2}$, $\forall u\in C_{0,\sigma}^{\infty}(D)$

holds, where

$||u||=(u, u)^{1/2}$

$C_{0,\sigma}^{\infty}(D)=$

{

$u\in c_{\text{ノ}}0\infty(D)$ ; $\mathrm{d}\mathrm{i}\mathrm{v}u=0$ in $\mathrm{D}$

}.

Suppose that the boundary $()\prime D$ of _$D$ is multiply connected,

(5) $\partial D=\bigcup_{i=1i}^{k}\Gamma(k\geq 2)$ ($\Gamma_{i}$ : connected component of $(‘)D$) and $D$ is inside of $\Gamma_{k}$.

Since $\mathrm{d}\mathrm{i}\mathrm{v}u=0$, the integral _{$\int_{\partial D}u\cdot nd\sigma$} must vanish, where

$n$ denotes the outward

normal vector to the $\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{l}’ \mathrm{y}$. Let us call this

condition.

general outflow condition

(GOC).

$(GOC)$ $\int_{\partial D}\beta\cdot nd\sigma=\sum_{i=1}^{k}\int_{\Gamma_{i}}\beta\cdot nd\sigma=0$

Theorem 1. $(Lerayl\mathit{8}], Hopfl\mathit{6}]iFujita[\mathit{3}],$ $Lady\approx henSkayal7\mathit{1})$

Suppose the following condition is

_satisfied.

$(OC)$ $\int_{\Gamma_{i}}\beta\cdot nd\sigma=0(1\leq i\leq k)$

Then the inequality $(L)$ holds true,

Remark 1.

_If

the boundary is multiply $con\check{n}$ected, the condition _$(OC)$ is stronger than

the condition $(GOC)$. On $the\backslash \cdot$ other hand,

if

$\partial D$ is connected,

then.

$(GOC)$ and $(OC)$

are equivalent and

$\int_{\partial D}\beta\cdot nd\sigma=0\Rightarrow(L)$ holds true.

When $(OC)$ does not holds, we know the following fact due to the work of Takeshita.

Theorem 2. $(Takeshita[\mathit{1}\mathit{4}\mathit{1})$ Let $D$ be a bounded domain in $\mathrm{R}^{2}$ the boundary

of

which consists

_of

2

connected components $(‘)D=\Gamma_{1}\cup\Gamma_{2}$. Suppose that we can insert a circle

between $\Gamma_{1}$ and $\Gamma_{2}$.

If

the boundary integral

$\int_{\Gamma_{1}}\beta\cdot nd\sigma=-\int_{\Gamma_{2}}\beta\cdot nd\sigma\neq 0$,

then $(L)$ does not hold true.

Thereforewecan notusethe method of Theorem 1 to show the existence of stationary solutions to the Navier-Stokes equations (1), (2). Nevertherless this does not mean the non-existence of solutions. In fact, Amick showed the existence of solution under the assumption of “ _{symmetry”for 2-D case.}

Theorem 3. $(Amick[\mathit{1}\mathit{1})$ Let$D$ be a bounded domain in$\mathrm{R}^{2}$.

If

$D,$ $f$ , $\beta$ are symmetric

Motivated the work of Takeshita, we found the following exact solution for 2-D an-nular domain

$D=\{x\in \mathrm{R}^{2};R_{1}<|x|<R_{2}\},$ $\partial D=\Gamma_{1}\cup\Gamma_{2},$ $\Gamma_{i}=\{|x|=R_{\iota}\}(i=1,2)$.

Example 1. ($M_{\mathit{0}}\dot{n}motol^{g}\mathit{1}$, see also [10]) Suppose $f=0$ and the boundary value:

$\beta=\frac{\mu}{R}e,$ $+\omega_{i}R_{i}e_{\theta}$

on

$\Gamma_{i}(i=1,2)$,

where $\mu,\omega_{1},$$\omega_{2}$ are given constants. Then the boundary value problem (1) (2) has the

following solution. The velocity $u_{0}$ is given by

$u_{0}=u_{0}( \mu)=\frac{\mu}{r}e_{r}+b(\mu, r)e_{\theta}$.

(i)

_If

$\mu\neq-2_{l\text{ノ}},$ $b( \mu)r)=\frac{1}{r}(C_{1},+c2r^{2}\frac{\mu}{\nu})+$,

(ii)

_If

$\mu=-2|\text{ノ},$ $b(l^{l,r)(}= \frac{1}{r}C_{1}.+C_{2}\log r)$,

where $c_{1},,$$c_{2}$

are

$approp\dot{\eta}ate$ constants. The pressure$p_{0}=p_{0}(\mu)$ can be obtained

from

the

equation.

As for the perturbation of the above solution, we have

Theorem 4. $(Morimoto- Ukai[\mathit{1}\mathit{3}\mathit{1})$

Let $D=\{x\in \mathrm{R}^{2};R_{1}<|x|<R_{2}\}$ , $f=0$ and the boundary value:

$\beta=\{\frac{\mu}{R}+\varphi_{i}(\theta)\}e_{t}+\{\omega iR_{i}+^{\psi(\theta}i)\}e_{\theta}$ on $\Gamma_{i}(i=1,2)$,

where$\mu,$$\omega_{1},$$\omega_{2}$

are

given constants and$\varphi_{i}(\theta),$ $’\psi i(\theta)$

are

$2\pi$-periodicfunctions, the integral

of

which over the interval $[0,2\pi]$ vanishes. Suppose the inequality

$| \omega_{1}-\omega_{2}|.\frac{R_{1}^{2}R_{2}^{2}}{R_{2}^{2}-R_{1}^{2}}(\log\frac{R_{2}}{R_{1}})^{2}<2_{l}\text{ノ}$

hold. Then there exists at most discrete countable set$\mathcal{M}$ such that

for

each$\mu\in \mathrm{R}\backslash \mathcal{M}$ the

boundary valueproblem(1), (2) has asolution

_for

sufficiently small$\varphi_{i}(\theta),$ _{$\psi_{i}(\theta)(i=1,2)$}.

Remark 2. $\omega_{i}(i=1,2)$ can be large but the

diffenrence

$|\omega_{1}-\omega 2|$ should be small.

For the general domain $D$ in $\mathrm{R}^{2}$ or $\mathrm{R}^{3}$, the boundary of which is multiplyconnected,

we have

Theorem 5. $(Fujita-M_{\mathit{0}}rimoto[\mathit{4}])$

Suppose that $f\in V’$ is a potentialforce, that $\beta=\mu\beta_{0}+\beta_{1}$, where $\mu$ is a constant, $\beta_{0}$ is the boundary value

of

gradient

of

a harmonic

function

$\varphi\in H^{2}(D)$, and that $\beta_{1}$ is in $H^{1/2}(\partial D)$ with

$\int_{\partial D}\beta_{1}\cdot nCl\sigma=0$.

Then, there exists a discrete countable set$\mathcal{M}\subset \mathrm{R}$ such that

for

each $\mu\in \mathrm{R}\backslash \mathcal{M}$, there

exists

a

weak solution to (1), (2)

_if

$\beta_{1}$

satisfies

the inequality

$||\beta_{1}||_{H}1/2.(\partial D)<C^{*}for$

some

positive constant $c_{\text{ノ}}*=C^{*}$(

Remark 3. The boundary value $\beta_{0}$ may not satisfy the vanishing

outflow

condition. $A$

non-trivial example

_of

such $\beta_{0}$ in 3-dimensinal case is

$\sum_{i=1}^{k-1}\nabla(\frac{q_{i}}{4\pi|x-ai|})$

where $q_{i}’ s$ are constants and₍$\iota_{i}s$ are points outside $D$, each $a_{i}$ being enclosed by $\Gamma_{i}$

.

In the following case, the set $\mathcal{M}$ is void, that is, for every

$\mu$, solutions exist for

sufficiently small $\beta_{1}$.

Theorem 6. $(Fujita- M_{\mathit{0}\dot{n}}mot_{ook}-amotol\mathit{5}], M_{\mathit{0}}\dot{n}motol^{\mathit{1}\mathit{2}}])$

In case

_of

2-D annular domain and

$\beta_{0}=\nabla\log r|_{\partial D}$

the set

_of

exceptional values$\mathcal{M}$ in Theorem 5 is void.

As for the Boussinesq equations, we obtain the following results.

Theorem

7.

$(Morimoto[\mathit{1}\mathit{1}])$ Suppose that _{$f\in V’$} is a potentialforce, that $\beta=\mu\beta_{0}+$

$\beta_{1}$, where

$\mu$ is a constant, $\beta_{0}$ is ifie boundary value

of

gradient

of

a harmonic

function

$\varphi\in H^{2}(D)$, and that $\beta_{1}$ is in $H^{1/2}((‘)D)$ with

$\int_{\partial D}\beta_{1}\cdot nd\sigma=0$.

Suppose that $\theta_{0}$ is in $H^{1/2}(\partial D)$. Then, there exists a discrete countable set $\mathcal{M}\subset \mathrm{R}$

such that

_for

each $\mu\in \mathrm{R}\backslash \mathcal{M}$, there exists a solution to (3) (4),

if

$\alpha,$ $||\beta_{1}||H^{1}/2(\partial D)$

’

$||\theta_{0}||_{H^{1}(D)}/2\partial<C_{\text{ノ}^{}*}$ holds

for

some positive constant $c*=C^{*}(\nu, \chi)\mu,$ $D,$$\beta_{0})$.

Remark 4. The set

_of

exceptional value $\mathcal{M}$ is the same as in the Navier-Stokes equa-tions case.

Theorem 8. $(Morimoto[\mathit{1}\mathit{1}])$ In case

of

2-D annular domain and

$\beta_{0\mathrm{g}r}=\nabla \mathrm{l}\mathrm{o}|_{\partial D}$

References

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nonhomogeneous steady Navier-Stokes

equations, Indiana Univ. Math. J. 33(1984), pp.817-830.

[2] Cattabriga, L., Su unproblema al contornorelativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ.Padova 31(1961) pp.308-340.

[3] Fujita, H., On the existence and $1^{\backslash }\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}1’ \mathrm{i}\mathrm{t}\mathrm{y}$ of the steady-state solutions of the

Navier-Stokes equation, J. Fac. Sci., Unuiv.Tokyo,

Sec.I.

9(1961), pp.59-102.

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Navie.

$\mathrm{r}$-Stokes flow with

non-vanishing outflow condition, to appear in Proceeding of Nonlinear Wave 1995

[5] Fujita,H., Morimoto, H.. $\mathrm{O}\mathrm{k}_{\mathfrak{c}}^{C}\mathrm{m}\mathrm{o}\mathrm{t}\mathrm{o}$. H., Satbility analysis ofthe Naviel$\cdot$-Stokes flows

in annuli, to appear in Mathematical Methods in the Applied Sciences

[6] Hopf,E., Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann.

117(1941) pp.764-775

[7] Ladyzhenskaya. O. A.. The $-|\mathrm{v}^{f}\neg r1\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{a}1$ Theory of Viscous Incompressible Flow,

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[8] Leray, J., Etude de diverses \’equations int\’egrales nonlin\’eaires et de quelques

probl\‘emes que pose

1

hydrodynamique, J. Math. Pure Appl. 12(1933) pp.1-82.

[9] Morimoto, H., _{A solution to the stationary Navier-Stokes equations under the}

boundary conditon with non-vanishingoutflow, Memoirs of the Institute of Science

and Technology, Meiji Univ.. 31(1992). pp.7-12.

[10]

_Morimoto:

H., Stationary Navier-Stokes equations under general outflow conditon,

Hokkaido Math.J. 24 (1995), pp.641-648.

[11] Morimoto, H., On the existence of solutions to stationary Boussinesq equations under general outflow $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{i}6\lambda \mathrm{o}\mathrm{n}$, preprint

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Memoirs ofthe Institute ofScience and Technology, Meiji Univ.,

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pp.73-82.

[14] Takeshita,

A..

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