MORIMOTO, Hiroko

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TitleSTATIONARY SOLUTIONS FOR THE NAVIER-STOKESEQUATIONS AND THE BOUSSINESQ EQUATIONSUNDER GENERAL OUTFLOW CONDITION(NonlinearEvolution Equations and Applications)Author(s)MORIMOTO, HirokoCitation数理解析研究所講究録 (1997), 1009: 100-104Issue Date1997-08URL http://hdl.handle.net/2433/61502RightType Departmental Bulletin PaperTextversionpublisherKyoto University$\mathrm{S}.\mathrm{T}_{1}.\mathrm{A}..\cdot \mathrm{T}.\mathrm{I}\mathrm{o}\mathrm{N}.\mathrm{A}.\mathrm{R}\mathrm{Y}\mathrm{s}\mathrm{o}\mathrm{L}.\mathrm{U}.\mathrm{T}\mathrm{I}\mathrm{o}\mathrm{N}_{\mathrm{t}}\mathrm{S}$ .FOR $\mathrm{T}\hat{\mathrm{H}}\mathrm{E}$ NAVIER-STOKES EQUATIONS$\cdot$ $\mathrm{A}\hat{\mathrm{N}}\mathrm{D}$ THEBOUSSINESQ EQUATIONSUNDER GENERAL OUTFLOW CONDITION. MORIMOTO, HirokoDepartment of Mathematics , Meiji University..Let $D$ be a bounded domain in $\mathrm{R}^{n}$ ( $\mathrm{n}=2$ or 3), $\partial D$ its smooth boundary. We considerthe existence of solutions to the stationary Navier-Stokes equations(1) $\{$$-l\text{ }\triangle u+(u\cdot\nabla)u+\nabla p=$ $f$ in $D$ ,$\mathrm{d}\mathrm{i}\mathrm{v}u$ $=$ $0$ in $D$ ,under the boundary condition(2). $u=\beta$ . on $\partial D$where $u$ is the velosity vector, $p$ the pressure, $f$ the external force, l kinematic viscosity,$\beta$ the velosity vector given on the boundary.We consider also the similal$\cdot$ problem for the stationary Boussines.q equations(3)under the boundary condition(4) $\{$$u=\beta$ on $\partial D$$T=$ $\theta$ on $\partial D$where $T$ is the temperature, $\mathrm{C}1$ coefficient of volume expansion, $\chi$ thermal diffusivity, $\beta$and $\theta$ prescribed velosity and temperature on the boundary, respectively.Firstly, we mention some known results for these problem.The existence of the stationary solutions to the Navier-Stokes equations (1), (2) andthe Boussinesq equations (3). (4) is known in general context if, for any $\epsilon>0$ , thereexists an extension $b_{\text{\’{e}} i}$ of the boundary value $\beta$ to the domain $D$ such that $\mathrm{d}\mathrm{i}\mathrm{v}b_{\mathcal{E}}=0$ in$D$ and the inequality$(L)$ $|((u\cdot\nabla)b_{\epsilon}u))|\leq\epsilon||\nabla u||^{2}$, $\forall u\in C_{0,\sigma}^{\infty}(D)$holds, where$(u, v)= \sum_{i}\int_{D}u_{i}(x)v_{i}(X)‘ l_{X}$1009 1997 100-104 100$||u||=(u, u)^{1/2}$$C_{0,\sigma}^{\infty}(D)=$ { $u\in c_{\text{ }}0\infty(D)$ ; $\mathrm{d}\mathrm{i}\mathrm{v}u=0$ 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 outwardnormal 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 thanthe 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 whichconsists of 2 connected components $(‘)D=\Gamma_{1}\cup\Gamma_{2}$ . Suppose that we can insert a circlebetween $\Gamma_{1}$ 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.Therefore we can not use the method of Theorem 1 to show the existence of stationarysolutions to the Navier-Stokes equations (1), (2). Nevertherless this does not mean thenon-existence of solutions. In fact, Amick showed the existence of solution under theassumption 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 symmetricwith respect to a line $p$ , and every $\Gamma_{i}$ intersects with $p$ , then a solution exists.101Motivated 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 thefollowing 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 $\mu\neq-2_{l\text{ }},$ $b( \mu)r)=\frac{1}{r}(C_{1},+c2r^{2}\frac{\mu}{\nu})+$ ,(ii) If $\mu=-2|\text{ },$ $b(l^{l,r)(}= \frac{1}{r}C_{1}.+C_{2}\log r)$ ,where $c_{1},,$ $c_{2}$ are $approp\dot{\eta}ate$ constants. The pressure $p_{0}=p_{0}(\mu)$ can be obtained from theequation.As for the perturbation of the above solution, we haveTheorem 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 $2\pi$ -periodic functions, the integralof which over the interval $[0,2\pi]$ vanishes. Suppose the inequality$| \omega_{1}-\omega_{2}|.\frac{R_{1}^{2}R_{2}^{2}}{R_{2}^{2}-R_{1}^{2}}(\log\frac{R_{2}}{R_{1}})^{2}<2_{l}\text{ }$hold. Then there exists at most discrete countable set $\mathcal{M}$ such that for each $\mu\in \mathrm{R}\backslash \mathcal{M}$ theboundary value problem (1), (2) has a solution 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 multiply connected,we haveTheorem 5. $(Fujita-M_{\mathit{0}}rimoto[\mathit{4}])$Suppose that $f\in V’$ is a potential force, 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 $\beta_{1}$ isin $H^{1/2}(\partial D)$ 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 $\mu\in \mathrm{R}\backslash \mathcal{M}$ , thereexists a weak solution to (1), (2) if $\beta_{1}$ satisfies the inequality $||\beta_{1}||_{H}1/2.(\partial D)<C^{*}for$ somepositive constant $c_{\text{ }}*=C^{*}$ ( $\nu,$ $l^{\iota}$ , D. $\beta_{0}$ ).102Remark 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 forsufficiently 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 potential force, 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 $\alpha,$ $||\beta_{1}||H^{1}/2(\partial D)$ ’$||\theta_{0}||_{H^{1}(D)}/2\partial<C_{\text{ }^{}*}$ 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$\beta_{0\mathrm{g}r}=\nabla \mathrm{l}\mathrm{o}|_{\partial D}$the set of exceptional values $\mathcal{M}$ in Theorem 7 is void.103References[1] Amick, C. J., Existence of $\mathrm{s}\mathrm{o}^{\underline{\rceil}}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$ to the nonhomogeneous steady Navier-Stokesequations, Indiana Univ. Math. J. 33(1984), pp.817-830.[2] Cattabriga, L., Su un problema al contorno relativo 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 theNavier-Stokes equation, J. Fac. Sci., Unuiv.Tokyo, Sec.I. 9(1961), pp.59-102.[4] Fujita, H., Morimoto, H.. A $\perp\cdot \mathrm{e}\mathrm{m}_{\mathrm{c}\mathrm{u}\cdot \mathrm{k}}$‘ on the existence of the Navie.$\mathrm{r}$-Stokes flow withnon-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 of the Naviel$\cdot$-Stokes flowsin 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,Gordon and Breach, New York, 1969.[8] Leray, J., Etude de diverses \’equations int\’egrales nonlin\’eaires et de quelquesprobl\‘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 theboundary conditon with non-vanishing outflow, Memoirs of the Institute of Scienceand 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 equationsunder general outflow $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{i}6\lambda \mathrm{o}\mathrm{n}$ , preprint[12] Morimoto, H., Note on the boundary value problem for the Navier-Stokes equationsin 2-D annular domain with general outflow condition(in Japanese), to appear inMemoirs of the Institute of Science and Technology, Meiji Univ.,[13] Morimoto, H., Ukai, S., Pel.tul.bation of the Navier-Stokes flow in an annular domainwith the non-vanishing outHoW condition, J. Math. Sci., Univ. Tokyo, 3(1996),pp.73-82.[14] Takeshita, A.. A remark $\mathrm{o}\mathrm{n}^{1}-\lrcorner \mathrm{e}1^{\cdot}\mathrm{a}\mathrm{y}\mathrm{s}$ inequality, Pacific J.Math., 157(1993), pp.151-158.[15] Temam, R., Navier-Stokes Equations, $\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}- \mathrm{H}_{0}11\vee \mathrm{a}\mathrm{n}\mathrm{d}$ . Amsterdam, 1977.104

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

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

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