DYNAMIC BIFURCATION THEORY OF RAYLEIGH-BÉNARD CONVECTION WITH INFINITE PRANDTL NUMBER

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1 DYNAMIC BIFURCATION THEORY OF RAYLEIGH-BÉNARD CONVECTION WITH INFINITE PRANDTL NUMBER JUNGHO PARK Abstract. We study in this paper the bifurcation and stability of the solutions of the Rayleigh-Bénard convection which has the infinite Prandtl number, using a notion of bifurcation called attractor bifurcation. We prove that the problem bifurcates from the trivial solution to an attractor A R when the Rayleigh number R crosses the first critical Rayleigh number R c. As a special case, we also prove another result which corresponds to the classical pitchfork bifurcation, that this bifurcated attractor A R consists of only two stable steady states when the first eigenvalue R 1 is simple. 1. Introduction We consider a Rayleigh-Bénard convection problem, i.e., a fluid motion induced by differential heating of a layer of fluids bounded by two horizontal parallel plates a distance h. After the non-dimensionalization using the units of the layer depth h as the typical length scale, the thermal diffusion time h 2 /κ as the typical time, the ratio of typical length over time, κ/h as the typical velocity, and the temperature on a scale where the top plane is kept at 0 and the bottom plane kept at 1, the Bénard experiment can be modeled by the Boussinesq equations; [ ] 1 u (1.1) + (u )u + p u RT k = 0, P r t (1.2) T + (u )T T = 0, t (1.3) u = 0, where u = (u 1, u 2, u 3 ) is the velocity field, p is the pressure function, T is the temperature function and k = (0, 0, 1) is the unit vector in the x 3 direction. Here R is a Rayleigh Number, ν and κ are the kinematic viscosity and thermal diffusive coefficient, respectively, and P r = ν κ is the Prandtl number. Bénard conducted the original experiments in this area. His fluid had a free upper surface and he found the cells to be in the shape of hexagons. Hot fluid at the bottom rose while cool fluid on top sunk by the force of gravity. In 1916, Lord Rayleigh developed a theory to interpret the phenomena of Bénard s experiment. He chose the Boussinesq equations with some boundary conditions to model Bénard s experiment and linearized these equations using normal modes Mathematics Subject Classification. 35Q, 35B, 37L. Key words and phrases. Rayleigh-Bénard Convection, bifurcation, infinite Prandtl number. The work was supported in part by the Office of Naval Research, and by the National Science Foundation. 1

2 2 JUNGHO PARK We will focus on a specific version of the problem modeled by the infinite Prandtl number limit of the Boussinesq equations, which is the standard model for mantle convection studied in terrestrial geophysics. In the limit of the infinite Prandtl number, the inertial terms in the momentum equation drop and we are left with a linear dependence of the velocity field on temperature; (1.4) p u RT k = 0, (1.5) T + (u )T T = 0, t (1.6) u = 0. Since we have normalized the time to the thermal diffusive time scale, we expect the velocity field to settle into some equilibrium state due to the viscosity effect of long time. Also, since the typical velocity is set to κ/h, the Reynolds number is expected to be small. Thus we anticipate creeping flow and hence the nonlinear advection term is negligible. Therefore the velocity filed should depend linearly on the temperature field. This is an advantage from the limit system, the infinite Prandtl number model that the velocity field has much more regular behavior. Extensive mathematical studies have been conducted for the Rayleigh-Bénard convection since Rayleigh s work. In particular the studies of Chandrasekhar [1], Drazin and Reid, who studied linear theories, Foias, Manley and Temam [3], who showed the existence and physical bounds of attractors and Rabinowitz [8], who studied the existence of Rectangular solutions and of the infinite Prandtl number limit the paper of X. Wang [10]. T. Ma and S. Wang [?] recently developed a new bifurcation theory based on a notion of bifurcation called attractor bif urcation. The authors interest in this paper is to apply the new bifurcation theory to the Rayleigh-Bénard convection with the infinite Prandtl number. The main theorem associated with attractor bifurcation states that when there are m eigenvalues crossing the imaginary axis, the system bifurcates from a trivial steady state solution to an attractor with dimension between m 1 and m, as the control parameter crosses a certain critical value, provided the critical state is asymptotically stable. For the Rayleigh-Bénard convection with the infinite Prandtl number, bifurcation is obtained with respect to the parameter R and the Rayleigh number. The main results obtained can be summarized as follows. First, we show that as the Rayleigh number R crosses the first critical value R c, the Boussinesq equations bifurcate from the trivial solution to an attractor A R, with dimension between m 1 and m. Here the first critical Rayleigh number R c is defined to be the first eigenvalue of the linear eigenvalue problem, and m is the multiplicity of R c. Second, as an attractor, the bifurcated attractor A R has asymptotic stability in the sense that it attracts all solutions with initial data in the phase space outside of the stable manifold of the trivial solution, with codimension m. Third, if, the first eigenvalue R c is simple, the system particularly bifurcates exactly two steady state solutions and each of them is a minimal attractor. Another important aspect of the nonlinear theory of the Benard Convection is to classify the structure of bifurcated solutions. Doing so can provide some insight on structure stability, and the pattern formation of the flow. Since it is technically

3 RAYLEIGH-BÉNARD CONVECTION 3 more involved, we will give just an example concerning this and more details shall be reported elsewhere. In [?], Ma and Wang showed the existence of bifurcation of the Rayleigh-Bénard convection in the case of finite Prandtl number P r. In the infinite Prandtl number case, the following two features make this problem worthy of study. (i) The first equation of the Boussinesq equations does not involve the time derivative term, thus the solution u, the velocity field, becomes diagnostic and can be solved in terms of the temperature function T. This allows us to approach this problem from different point of view. That is, instead of considering the bifurcation problem of equations (1.4) (1.6) in (u, T ) L 2 () 3 L 2 (), we consider (1.7) T + (u(t ) )T T = 0, t (1.8) u = 0 in only T L 2 (). In Theorem 3.2, which will follow in Section 3, the Boussinesq equation bifurcates from T = 0 to an attractor A R. (ii) Since the velocity field depends on the temperature field T, we just investigate the structure of the attractor generated by only T. Moreover, since we can get the velocity field u(t ) associated with the temperature field T, the structure of the velocity field is reconstructed by T in terms of the eigenvectors. This tells us that we can reconstruct the attractor in terms of (u, T ), which has the same topological structure as that which is achieved in terms of only T. This paper is organized as follows. In Section 2, we will recall the attractor bifurcation theory and in Section 3, we will summarize the the Boussinesq equations and their mathematical setting, and study the bifurcation. An example of a topological structure that we got from the bifurcated solutions is addressed. 2. Abstract Bifurcation Theory In this section, we shall recall some results of dynamic bifurcation of abstract nonlinear evolution equations developed in [?]. Let H and H 1 be two Hilbert spaces, and H 1 H be a dense and compact inclusion. Consider the following nonlinear evolution equations (2.1) (2.2) du dt = L λu + G(u, λ), u(0) = u 0, where u : [0, ) H is the unknown function, λ R is the system parameter, and L λ : H 1 H are parameterized linear completely continuous fields continuously depending on λ R, which satisfy L λ = A + B λ a sectorial operator, (2.3) A : H 1 H a linear homeomorphism, B λ : H 1 H the parameterized linear compact operators. We can see that L λ generates an analytic semi-group {e tl λ } t 0 and then we can define fractional power operators L α λ for any 0 α 1 with domain H α = D(L α λ ) such that H α1 H α2 if α 2 < α 1, and H 0 = H.

4 4 JUNGHO PARK We now assume that the nonlinear terms G(, λ) : H α H for some 0 α < 1 are a family of parameterized C r bounded operators (r 1) continuously depending on the parameter λ R, such that (2.4) G(u, λ) = o( u Hα ), λ R. For the linear operator A we assume that there exists a real eigenvalue sequence {ρ k } R and an eigenvector sequence {e k } H 1, i.e., Ae k = ρ k e k, (2.5) 0 < ρ 1 ρ 2, ρ k (k ) where {e k } is an orthogonal basis of H. For the compact operator B λ : H 1 H, we also assume that there is a constant 0 < θ < 1 such that (2.6) B λ : H θ H bounded, λ R. Let {S λ (t)} t 0 be an operator semigroup generated by (2.1) then the solution of (2.1) and (2.2) can be expressed as u(t, u 0 ) = S λ (t)u 0, t 0. Definition 2.1. A set Σ H is called an invariant set of (2.1) if S(t)Σ = Σ for any t 0. An invariant set Σ H of (2.1) is said to be an attractor if Σ is compact, and there exists a neighborhood U H of Σ such that for any ϕ U we have (2.7) lim t dist H (u(t, ϕ), Σ) = 0. The largest open set U satisfying (2.7) is called the basin of attraction of Σ. Definition 2.2. (1) We say that the equation (2.1) bifurcates from (u, λ) = (0, λ 0 ) an invariant set λ, if there exists a sequence of invariant sets { λn } of (2.1), 0 / λn such that lim n λ n = λ 0, lim n max x λn x = 0. (2) If the invariant sets λ are attractors of (2.1), then the bifurcation is called attractor bifurcation. (3) If λ are attractors and are homotopy equivalent to an m-dimensional sphere S m, then the bifurcation is called S m -attractor bifurcation. Let the eigenvalues (counting multiplicity) of L λ be given by Suppose that β k (λ) C (k 1). < 0 if λ < λ 0, (2.8) Reβ i (λ) = 0 if λ = λ 0 > 0 if λ > λ 0, (2.9) Reβ j (λ 0 ) < 0 (m + 1 j). (1 i m),

5 RAYLEIGH-BÉNARD CONVECTION 5 Let the eigenspace of L λ at λ 0 be m { E 0 = u H1 (L λ0 β i (λ 0 )) k u = 0, k = 1, 2, }. i=1 The following dynamic bifurcation theorem for (2.1) was proved in [?]. Theorem 2.3 (Attractor Bifurcation). Assume that (2.3) (2.6) hold and u = 0 is a locally asymptotically stable equilibrium point of (2.1) at λ = λ 0. Then the following assertions hold. (1) The equation (2.1) bifurcates from (u, λ) = (0, λ 0 ) an attractor A λ for λ > λ 0, with m 1 dim A λ m, which is connected if m > 1. (2) The attractor A λ is a limit of a sequence of m-dimensional annulus M k with M k+1 M k ; in particular if A λ is a finite simplicial complex, then A λ has the homotopy type of S m 1. (3) For any u λ A λ, u λ can be expressed as u λ = v λ + o( v λ H1 ), v λ E 0. (4) If u = 0 is globally stable for (2.1) at λ = λ 0, then for any bounded open set U H with 0 U there is an ε > 0 such that as λ 0 < λ < λ 0 + ε, the attractor A λ bifurcated from (0, λ 0 ) attracts U/Γ in H, where Γ is the stable manifold of u = 0 with codimension m. The basic idea of the Theorem 2.3 can be demonstrated as follows, see [?, 5] for details. Near λ = λ 0, the flows of (2.1) in a small neighborhood of u = 0 is squeezed into an m dimensional center manifold which can be expressed by a h = h(x, λ), where x R m. The flow in the center manifold has the same topological structure as the flow in R m of the equations dx (2.10) dt = L λx + P G(x, h(x, λ)), where L λ = L λ E0, E 0 is the subspace of R m on which all eigenvalues of L λ possess nonnegative real part at λ = λ 0 and P is the projection. It suffices then to consider the dynamic bifurcation for (2.10). When λ < λ 0, near x = 0 the linear term L λ x determines the dynamical behavior of (2.10). Therefore the flow converges to x = 0. When λ = λ 0, assume that x = 0 is locally asymptotically stable for (2.10). Then when λ > λ 0 is sufficiently small, (2.10) can be rewritten as dx (2.11) dt = P G(x, h(x, 0)) + L λx + K(x, λ), where K(x, λ) = P G(x, h(x, λ)) P G(x, h(x, 0)) 0 as λ 0. Hence the flow of (2.11) is a superposition of the flow of dx (2.12) = P G(x, h(x, 0)) dt and the flow of dx (2.13) dt = L λx + K(x, λ) (λ > λ 0 ). The flow of (2.13) is outward from x = 0, and the flow of (2.12) is inward to x = 0. Near x = 0, the linear term dominates the flow and for x far away from x = 0, the

6 6 JUNGHO PARK nonlinear term P G(x, h(x, 0)) dominates the flow structure. Thus the inward flow and outward flow squeeze an attractor near x = 0. To apply the above dynamic bifurcation theorems it is crucial to verify the asymptotic stability of the critical states. Theorem 2.4. (T. Ma and S. Wang [?]) Let the linear operator L λ in (2.1) be symmetric and all the eigenvalues {β k } R of L λ at λ = λ 0 satisfy Set β i = 0 if 1 i m, β j < 0 if m + 1 j. E 0 = {u H 1 L λ0 u = 0}, E 1 = E 0 = {u H 1 u, v H = 0 v E 0 }. If G λ0 : H 1 H satisfies the following orthogonal condition: G λ0 u, u H = 0, u H 1. Then one and only one of the following two assertions holds: (1) There exists a sequence of invariant sets {Γ n } E 0 of (2.1) at λ = λ 0 such that 0 / Γ n, lim n dist(γ n, 0) = 0. (2) The trivial steady state solution u = 0 for (2.1) at λ = λ 0 is locally asymptotically stable under the H-norm. Furthermore, if (2.1) has no invariant sets in E 0 except the trivial one {0}, then u = 0 is globally asymptotically stable. 3. Attractor Bifurcation of the Bénard Problem 3.1. Mathematical setting. In this paper, we deal with the Boussinesq equations on the non-dimensional domain = D (0, 1) R 3, where D R 2 is an open set, and impose the boundary conditions at the plates which are u = 0, T = 0, at x 3 = 1, u = 0, T = 1, at x 3 = 0. (See the figure 3.1.) Note that (1.4) (1.6) have trivial solutions u = 0, T = 1 x 3 and p = p 0 + R(x 3 x2 3 2 )k. Let p be the difference between p and the trivial solutions that is, p = p 0 +R(x 3 x2 3 2 )k+ p and T be the difference between T and the trivial solutions, that is, T = (1 x 3 ) + T. Direct subtraction gives us and then R(1 x 3 )k + p R By the same way, we have [ (1 x 3 ) + T p R T k u = 0. T t + (u ) T u 3 T = 0. ] k u = 0,

7 RAYLEIGH-BÉNARD CONVECTION 7 u=t=0 x 3 =1 u=0,t=1 x 3 =0 So we have (3.1) (3.2) (3.3) Figure 3.1. Flow between two plates heated from below p R T k u = 0, T t + (u ) T u 3 T = 0, u = 0. and the boundary conditions changed into u = 0, T = T, at x3 = 1, u = 0, T = T 1, at x3 = 0. Let T := R T and p := p then (3.1) and (3.2) are changed into Hence we have (3.4) (3.5) (3.6) (3.7) p RT k u = 0, 1 T R t + (u ) 1 R T u 3 1 R T = 0. p u RT k = 0, T t + (u )T T Ru 3 = 0, u = 0, u = 0, T = 0 at x 3 = 0, 1. On the lateral boundary D [0, 1], we use the periodic condition: { u = 0, T = 0 at x3 = 0, 1, (3.8) (u, T )(x, t) = (u, T )(x 1 + k 1 L 1, x 2 + k 2 L 2, x 3, t), for x. Since (3.4) together with (3.6) is a Stokes equation, we have the solution u = u(t ) under the above boundary condition. By replacing u and its third component u 3 in (3.5) by solution which depends on T, we would consider only the following equation. T (3.9) t + (u(t ) )T T Ru 3 (T ) = 0,

8 8 JUNGHO PARK where u(t ) and u 3 (T ) satisfy (3.4) together. The Boussinesq equations (3.9) is a crucial equation to study the Rayleigh- Bénard problem in this paper. For an initial value problem we also provide the initial value of T as (3.10) T = T 0 at t = 0. In addition, we can consider boundary condition such as { T = 0 at x3 = 0, 1, (3.11) T (x, t) = T (x 1 + k 1 L 1, x 2 + k 2 L 2, x 3, t), for x instead of boundary condition (3.8). We recall here the functional setting of equation (3.9) with initial and boundary conditions (3.10) and (3.11). Let H = L 2 () and H 1 = H 1 0 () H 2 () where H 1 0 () is the space of functions in H 1 (), which vanish at x 3 = 0, 1 and are periodic in the x i -directions (i = 1, 2). Thanks to existence results, we can define a semi-group which enjoys the semi-group properties. S(t) : T 0 T (t), Remark 3.1. We can use other boundary conditions as following; (1) Dirichlet boundary condition: u = 0, (2) Free boundary condition: T = 0 (or T n = 0), u τ T = 0, u n = 0, n = 0, where n and τ are the unit normal and tangent vectors on D [0, 1] respectively, and u n = u n, u τ = u τ Bifurcation Theorems. First, we want to find the critical Rayleigh number which plays the role of the critical value in the abstract theory. To do so, we consider the linearized equations of (3.9) with the boundary conditions (3.11): (3.12) T Ru 3 (T ) = 0 where u 3 (T ) satisfies equation (3.4) Since this eigenvalue problem for the Rayleigh number R is symmetric, all eigenvalues R k with multiplicities m k of (3.12) with (3.11) are real numbers, and (3.13) 0 < R 1 < < R k < R k+1 < in the sense that is equivalent to u + p λt k = 0, T λu 3 = 0, u = 0 [ u ũ + T T ] [ ] dx = λ T ũ 3 + u 3 T dx,

9 RAYLEIGH-BÉNARD CONVECTION 9 where λ = R. The first eigenvalue R 1, denoted by R 1 = R c, is called the critical Rayleigh number. Let the multiplicity of R c be m 1 = m (m 1) and the first eigenvectors T 1, T m of (3.12) be orthonormal: T i, T j H = T i T j dx = δ ij. Then E 0, the first eigenspace of (3.12) with (3.11) is { m } (3.14) E 0 = α k T k α k R, 1 k m. k=1 Now we are ready to introduce the main result. Theorem 3.2. For the Bénard problem (3.9) with (3.11), we have the following assertions: (1) If R R c, the steady state T = 0 is a globally asymptotically stable equilibrium point of the equation. (2) The equation bifurcates from (T, R) = (0, R c ) to an attractor A R for R > R c, with m 1 dim A R m, which is connected when m > 0. (3) For any T A R, the associated velocity field u = u(t ) which is achieved from a given T, can be expressed as ( m m ) (3.15) u = α k e k + o α k e k, k=1 where e k are eigenvectors of (3.9) corresponding to each T k. (4) The attractor A R has the homotopy type of an (m 1)-dimensional sphere S m 1 provided A R is a finite simplicial complex. (5) For any bounded open set U L 2 () with 0 U there is an ε > 0 such that as R c < R < R c + ε, the attractor A R attracts U/Γ in L 2 (), where Γ is the stable manifold of T = 0 with co-dimension m. Proof. Step 1. In the first step we will prove that T = 0 is a globally asymptotically stable equilibrium point of (3.9) at λ 1 = R c. We know that the Boussinesq equation (3.9) has a bounded absorbing set in H. Assume that (3.9) has an invariant set B E 0, with B {0} at λ 1 = R c. If restricted in B, (3.9) can be rewritten as T (3.16) + (u(t ) )T = 0. t Let T B be the solutions of (3.16). Then αt (αt) αb E 0 are also solutions of (3.16), since T (s) (αt (αt)) + [u(αt (αt) ](αt (αt)) = α2 + [u(αt (s)) ] (αt (s)) t [ s ] T (s) = α 2 + (u(t (s)) )T (s) = 0, s where s := αt, that is, for any α R, the set αb E 0 is an invariant set of (3.16). But it is an unbounded invariant set, which contradicts the existence of a bounded absorbing set since all invariant sets have the same bound in H as the absorbing k=1

10 10 JUNGHO PARK set. Hence the invariant set B can only consist of T = 0. By Theorem 2.4, T = 0 at λ = R c is globally asymptotically stable. Step 2. Let H and H 1 be defined as in section 3.1. Define L λ = A + B λ : H 1 H and G : H 1 H by A(T ) = T, (3.17) B λ (T ) = λu 3 (T ), G(T ) = (u(t ) )T, where λ = R. Then A(T ), T H = T T dx = T ( T )dx = T, A( T ) H. For arbitrary T and T, we have two equations: and p u(t ) λt k = 0 p u( T ) λ T k = 0, which are true for T and T, respectively. Moreover, these equations become p u( T )dx u(t ) u( T )dx λ T u 3 ( T )dx = 0 and p u(t )dx u( T ) u(t )dx λ T u 3 (T )dx = 0. Due to the incompressibility of u, we have B λ (T ), T H = λ u 3 (T ) T dx = λ T u 3 ( T )dx = T, B λ ( T ) H. Thus, A, B λ are symmetric operators and so is L λ = A + B λ. Moreover, G is orthogonal since G(T ), T H = (u(t ) )T T dx = div u(t ) T 2 dx = 0. By (3.13) we have (2.5). Moreover (2.4) and (2.6) also hold. Due to the above arguments, we have an operator equation which is equivalent to the Boussinesq equation (3.9): (3.18) dt dt = L λt + G(T ).

11 RAYLEIGH-BÉNARD CONVECTION 11 The eigenvalue problem of the linear equation (3.19) L λ T = β(λ)t is equivalent to Because u(t ) satisfies T λu 3 (T ) + β(λ)t = 0. u(t ) + p λt k = 0, we can put two equations together as follows: u(t ) + p λt k = 0, (3.20) T λu 3 (T ) + β(λ)t = 0, u = 0. Since L λ is a symmetric completely continuous field, the eigenvalue β k (λ) (k = 1, 2, ) of (3.20) are real numbers satisfying (3.21) { β1 (λ) β 2 (λ) β k (λ), β k (λ) as k. The first eigenvalue β 1 (λ) of (3.20) has the Rayleigh s formula: [ T 2 λt u 3 (T ) ] dx β 1 (λ) = min T H 1() T. 2 dx For the critical Rayleigh number λ 1 = R c, (3.12) and (3.13) lead us to [ u(t ) 2 + T 2] > 2 λt u 3(T )dx if λ < λ 1, dx = 2 λt u 3(T )dx if λ = λ 1, < 2 λt u 3(T )dx if λ > λ 1. From the first equation of (3.20), we have u(t ) 2 dx = which gives us [ T 2 λt e 3 (T ) ] dx = λt u 3 (T )dx, [ T 2 e(t ) 2] > 0 if λ < λ 1, dx = 0 if λ = λ 1, < 0 if λ > λ 1 for the first eigenvectors (e(t ), T ) H0 1 () 3 H 1. Therefore, < 0 if λ < λ 1, β 1 (λ) = 0 if λ = λ 1, > 0 if λ > λ 1, and β j (λ 1 ) < 0 (2 j < ). Thus the condition (2.8) and (2.9) are obtained. By Theorem 3.2, we complete the proof.

12 12 JUNGHO PARK Remark 3.3. In Theorem 3.2 and its proof, we can say that all statements are true for (u, T ) instead of T alone, i.e., (u, T ) = 0 is a globally asymptotically stable solution of (3.4) (3.6) and the equations (3.4) (3.6) bifurcate from (u, T ) = (0, R c ) to an attractor A R L 2 () 3 L 2 (), since we have u(t ) associated with a given T which satisfies equation (3.4), and u = 0 when T = 0. Now we have another result which corresponds to the classical pitchfork bifurcation. But the below theorem gives us the stability of bifurcated steady states and that is another advantage of the attractor bifurcation theory. Theorem 3.4. Let A R be the bifurcated attractor of (3.9) with (3.11) achieved in Theorem 3.2. If the first eigenvalue R 1 is simple, then A R consists of exactly two points, φ 1, φ 2 H 1 given by φ 1 = αt 1 + o( α ), φ 2 = αt 1 + o( α ), for some α 0. Moreover, for any bounded open set U H with 0 U, if R > R c and R R c is small enough, then U can be decomposed into two open sets U 1 and U 2 such that (1) Ū = Ū1 + Ū2, U 1 U 2 = and 0 U 1 U 2, (2) φ i U i (i = 1, 2), and (3) for any φ 0 U i (i = 1, 2), lim t S λ (t)φ 0 = φ i, where S λ (t)φ 0 is the solution of (3.9) with (3.11) and initial data φ 0 = T 0. Proof. Since L λ + G : H 1 H defined in the proof of Theorem 3.2 is a symmetric completely continuous field, H 1 can be decomposed into H 1 = E λ 1 E λ 2, E λ 1 = {xt 1 (λ) : x R}, E λ 2 = {y H 1 : y, T 1 H = 0}. Let P i : H 1 Ei λ be the canonical projection. For φ H, φ can be expressed by φ = xt 1 + y, for some x R, y E2 λ. Then 0 = P 1 (L λ (φ) + G(φ)) = P 1 (xl λ (T 1 ) + L λ (y)) + P 1 (G(φ)) and = xl λ (T 1 ) + G(φ), T 1 H = xβ 1 (λ) + G(φ), T 1 H, 0 = (I P 1 )(L λ (φ) + G(φ)) = (I P 1 )(xl λ (T 1 ) + L λ (y)) + (I P 1 )(G(φ)) = L λ (y) + (I P 1 )G(φ) = L λ (y) + P 2 G(φ). So, the equation L λ (φ) + G(φ) = 0 can be decomposed into (3.22) (3.23) β 1 (λ)x + G(φ), T 1 (λ) H = 0, L λ (y) + P 2 G(φ) = 0.

13 RAYLEIGH-BÉNARD CONVECTION 13 Since the eigenvalues β j (λ) in (3.21) satisfy that β j (λ 1 ) 0 for j 2, and λ 1 = R c, the restriction L λ E λ 2 : E λ 2 E λ 2 is invertible. The decomposed equation (3.23) and the implicit function theorem together imply that y is a function of x: (3.24) y = y(x, λ), which satisfies (3.23). Since G(φ) = G(xT 1 + y) is an analytic function of φ, y is also analytic in x and so is the function f(x, λ) := G(xT 1 + y(x, λ)), T 1 (λ) H. Hence there exist α(λ) 0 in R and k 2 such that β 1 (λ)x + f(x, λ) has the Taylor expansion as follows: (3.25) β 1 (λ)x + f(x, λ) = β 1 (λ)x + α(λ)x k + o( x k ). By Theorem 3.2, if λ λ 1 and λ 1 λ is small, (3.22) and (3.23) have no non-zero solutions, which implies that α(λ 1 ) < 0 and k=odd. Thus, (3.25) has exactly two solutions ( ) ( 1/(k 1) (β(λ) ) ) 1/(k 1) β(λ) x ± = ± + o, α α for λ > λ 1 and λ λ 1 small. By Theorem 3.2 again, (3.9) bifurcates from (0, λ 1 ) an attractor A R with dim A R 1. It is clear that A R consists of exactly two singular points φ λ := x ± T 1 +o( x ± ) if λ > λ 1 and λ λ 1 is small. By the stable manifold theorem, there is a stable manifold with codimension 1 of (3.9) at φ = 0 dividing the open set U into two parts: U 1 and U 2, such that φ i U i and φ i attracts U i. This completes the proof. We have the following remarks. Remark 3.5. By the same arguments in Remark 3.3, Theorem 3.4 hold true for A R L 2 () 3 L 2 (). That is, the bifurcated attractor A R consists of exactly two points φ 1, φ 2 H 1 0 () 3 H 1 () given by φ 1 = α(e 1 (T 1 ), T 1 ) + o( α ), φ 2 = α(e 1 (T 1 ), T 1 ) + o( α ), where e 1 (T 1 ) is an eigenvector of (3.9) corresponding to T 1. Remark 3.6. Both theorems hold for Boussinesq equations (3.9) with different combinations of boundary conditions as described in Remark Two dimensional Benard problem. In this subsection, we introduce one example of the critical eigenvalue for some simple geometry and given boundary condition. Let us consider 2-D Boussinesq equations in = [0.L] [0, 1] with the infinite Prandtl number: p u RT k = 0, T + (u )T T = 0, t u = 0, where u = (u 1, u 2 ), x = (x 1, x 2 ) and k = (0, 1). For simplicity we consider the periodic boundary condition in the x 1 direction and the free-free boundary condition at the top and the bottom as follows: (u, T )(x, t) = (u, T )(x 1 + kl, x 2, t), for any k Z,

14 14 JUNGHO PARK u 1 T = 0, u 2 = 0, = 0 at x 2 = 0, 1. x 2 Then the critical Rayleigh number can be easily achieved as R c = 27π4 = We can get the first eigenvectors as follows: T 1 = a k cos 2π L x 1 sin πx 2, T 2 = a k sin 2π L x 1 sin πx 2, where a k = π 2 + ( 2π L )2, and induced velocity fields are u 1 = u(t 1 ) = ( π 2 sin 2πL x 1 cos πx 2, 2πL cos 2πL ) x 1 sin πx 2, u 2 = u(t 2 ) = (π 2 cos 2πL x 1 cos πx 2, 2πL sin 2πL ) x 1 sin πx 2. By Theorem 3.2, we can see when R > R c that the structure of the bifurcated solutions as shown in Figure 3.2. Since the analysis is more involved, details will be reported elsewhere. X 2 Figure 3.2. The structure of solutions of Benard convection with infinite Prandtl number X 1 References [1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, Inc [2] Y. Benoist, P. Foulon and F. Labourie, Flots d Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5 (1992), [3] C. Foias, O. Manley, and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal., 11 (1987), pp [4] C. Doering, P. Constantin,On upper bounds for infinite Prandtl number convection with or without rotation, J. mathematical physics, vol.42, No.2, (2001), pp [5] T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, [6] T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, vol. 119 of Mathematical Surveys and Monographs, Americal Mathematical Society, Providence, RI, 2005.

15 RAYLEIGH-BÉNARD CONVECTION 15 [7] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, [8] P.H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. Rational Mech. Anal., 29 (1968), pp [9] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, vo. 68 of Applied Mathematical Sciences, Springer Verlag, New York, second ed., [10] X. Wang, Infinite prandtl number limit of Rayleigh-Bénard convection, Comm. Pure Appl. Math.,vol.57, (2004) pp Institute for Scientific Computing and Applied mathematics & Department of Mathematics, Indiana University, Bloomington, IN address: