Atttactor Bifurcation Theory and Its Applications to Rayleigh-Benard´ Convection

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COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 2, Number 4, December 2003 pp. 591–599

ATTTACTOR BIFURCATION THEORY AND ITS APPLICATIONS TO RAYLEIGH-BENARD´ CONVECTION

Tian Ma Department of Mathematics Sichuan University Chengdu, P. R. China and Department of Mathematics Indiana University Bloomington, IN 47405

Shouhong Wang Department of Mathematics Indiana university Bloomington, IN 47405

Abstract. In this note, we present a fast communication of a new bifurcation theory for nonlinear evolution equations, and its application to Rayleigh-BénardConvection. The proofs of the main theorems presented will appear elsewhere. The bifurcation theory is based on a new notion of bifurcation, called attractor bifurcation. We show that as the parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m + 1, where m + 1 is the number of eigenvalues crosses the imaginary axis. Based on this new bifurcation theory, we obtain a nonlinear theory for bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Bénardconvection. In particular, we show that the problem bifurcates from the trivial solution an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number Rc for all physically sound boundary conditions.

1. Attractor Bifurcation Theory of Nonlinear Evolution Equations. As we mentioned in the abstract, we present in this paper a fast communication on a new bifurcation theory, and its applications to the classical Bénardproblem. The proof of the main results will be published elsewhere. For this purpose, in this section, we introduce a new notion of dynamic bifurcation called attractor bifurcation, and prove a general bifurcation theorem for nonlinear abstract evolution equations, depending on a parameter. In next section, applications to the Rayleigh-Bénardconvection is given, providing a rigorous solution to the long standing open problem on the onset of the Rayleigh-Bénard convection.

2000 Mathematics Subject Classification. 37L, 37G, 35Q, 58F, 76E, 76R. Key words and phrases. Attractor bifurcation, semigroup, central manifold theory, asymptotic stability, structural stability, Rayleigh-Bénardconvection, roll structure . The work was supported in part by the Office of Naval Research, by the National Science Foundation, and by the National Science Foundation of China.

591 592 T. MA AND S. WANG

Let H and H1 be two Hilbert spaces, and H1 ,→ H be a dense and compact inclusion. We consider the following nonlinear evolution equations du = L u + G(u, λ), (1.1) dt λ u(0) = u0, (1.2) where u : [0, ∞) → H is the unknown function, λ ∈ R is the system parameter, and Lλ : H1 → H are parameterized linear completely continuous ﬁelds continuously depending on λ ∈ R1, which satisfy   Lλ = −A + Bλ is a sectorial operator, A : H1 → H a linear homeomorphism, (1.3)  Bλ : H1 → H the parameterized linear compact operators.

−tL It is easy to see [4] that Lλ generates an analytic semi-group {e λ }t≥0. Then α we can deﬁne fractional power operators Lλ for any 0 ≤ α ≤ 1 with domain α Hα = D(Lλ ) such that Hα1 ⊂ Hα2 if α1 > α2, and H0 = H. Furthermore, we assume that the nonlinear terms G(·, λ): Hα → H for some 1 > α ≥ 0 are a family of parameterized Cr bounded operators (r ≥ 1) continuously depending on the parameter λ ∈ R1, such that 1 G(u, λ) = o(kukHα ), ∀ λ ∈ R . (1.4)

For the linear operator Lλ = −A+Bλ we assume that there exist a real eigenvalue 1 sequence {ρk} ⊂ R and and an eigenvector sequence {ek} ⊂ H1 of A:   Aek = ρkek, 0 < ρ1 ≤ ρ2 ≤ · · · , (1.5)  ρk → ∞ (k → ∞) such that {ek} is an orthogonal basis of H. For the compact operator Bλ : H1 → H, we also assume that there is a constant 0 < θ < 1 such that 1 Bλ : Hθ −→ H bounded, ∀ λ ∈ R . (1.6)

Let {Sλ(t)}t≥0 be an operator semi-group generated by the equation (1.1) which enjoys the properties

(i) For any t ≥ 0, Sλ(t): H → H is a linear continuous operator, (ii) Sλ(0) = I : H → H is the identity on H, and (iii) For any t, s ≥ 0, Sλ(t + s) = Sλ(t) · Sλ(s) Then the solution of (1.1) and (1.2) can be expressed as

u(t) = Sλ(t)u0, t ≥ 0. Deﬁnition 1.1. A set Σ ⊂ H is called an invariant set of (1.1) if S(t)Σ = Σ for any t ≥ 0. An invariant set Σ ⊂ H of (1.1) 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

lim distH (u(t, ϕ), Σ) = 0. (1.7) t→∞ The largest open set U satisfying (1.7) is called the basin of attraction of Σ. ATTTACTOR BIFURCATION AND RAYLEIGH-BENARD´ CONVECTION 593

Deﬁnition 1.2. 1. We say that the equation (1.1) bifurcates from (u, λ) =

(0, λ0) an invariant set Ωλ, if there exists a sequence of invariant sets {Ωλn }

of (1.1), 0 ∈/ Ωλn such that

lim λn = λ0, n→∞ lim max |x| = 0. n→∞ x∈Ωλn

2. If the invariant sets Ωλ are attractors of (1.1), then the bifurcation is called attractor bifurcation. 3. If Ωλ are attractors and are homotopy equivalent to an m–dimensional sphere Sm, then the bifurcation is called Sm–attractor bifurcation.

A complex number β = α1 + iα2 ∈ C is called an eigenvalue of Lλ if there are x, y ∈ H1 such that

Lλx = α1x − α2y,

Lλy = α2x + α1y.

Now let the eigenvalues (counting the multiplicity) of Lλ be given by

β1(λ), β2(λ), ··· , βk(λ) ∈ C, where C is the complex space. Suppose that  < 0, λ < λ0 Reβi(λ) = = 0, λ = λ0 (1 ≤ i ≤ m + 1) (1.8)  > 0, λ > λ0

Reβj(λ0) < 0, ∀ m + 2 ≤ j. (1.9)

Let the eigenspace of Lλ at λ0 be © k ª E0 = u ∈ H1 | Lλ0 u = 0, k = 1, 2, ··· . .

It is known that dim E0 = m + 1. Theorem 1.3 (Attractor Bifurcation). Assume that the conditions (1.3)-(1.6), (1.8) and (1.9) hold true, and u = 0 is a locally asymptotically stable equilibrium point of (1.1) at λ = λ0. Then the following assertions hold true.

1. (1.1) bifurcates from (u, λ) = (0, λ0) an attractor Aλ for λ > λ0, with m ≤ dim Aλ ≤ m + 1, which is connected as m > 0; 2. the attractor Aλ is a limit of a sequence of (m + 1)–dimensional annulus Mk with Mk+1 ⊂ Mk; especially if Aλ is a ﬁnite simplicial complex, then Aλ has the homotopy type of Sm; 3. For any uλ ∈ Aλ, uλ can be expressed as

uλ = vλ + o(kvλkH1 ), vλ ∈ E0;

4. If G : H1 → H is compact, and the equilibrium points of (1.1) in Aλ are ﬁnite, then we have the index formula ( X 2 if m = odd, ind [−(Lλ + G), ui] = 0 if m = even. ui∈Aλ 594 T. MA AND S. WANG

5. If u = 0 is globally stable for (1.1) at λ = λ0, then for any bounded open set U ⊂ H with 0 ∈ U there is an ε > 0 such that as λ0 < λ < λ0 + ε, the attractor Aλ bifurcated from (0, λ0) attracts U/Γ in H, where Γ is the stable manifold of u = 0 with co-dimension m + 1. One important feature of this bifurcation theorem is that the bifurcation is obtained for the parameter λ crosses the critical value λc for any multipilicity m+1 of the eigenvalue λc of the linear problem. As we know in the classical Krasnoselskii- Rabinowitz bifurcation theorem for steady state solutions, one requires that m + 1 be an odd number, which is normally not the case for many problems in mathematical physics. Another important feature is that the bifurcation attractor does not include the trivial steady state, and is asymptotically stable; hence it is physically important. In classical general, it is usually a diﬃcult problem to derive the asymptotical stability of the bifurcated solution. Many known stability results are obtained in some class of functions with certain symmetry, which are less physically relevant. As Kirchg¨assnerindicated in [5], an ideal stability theorem would include all physically meaningful perturbations, and today we are still far from this goal. We believe that the stability of bifurcated attractor provides a right notion of stability toward to this problem, as the local attractor is the right object to better describe the system instead of a single steady state solution. The bifurcated attractor contains a collection of solutions of the evolution equation, including possibly steady states, periodic orbits, as well as homoclinic and heteoclinic orbits.

2. Boussinesq Equations and their Mathematical Setting. The Rayleigh- Bénardconvection problem was originated in the famous experiments conducted by H. Bénardin 1900. Bénardinvestigated a fluid, with a free surface, heated from below in a dish, and noticed a rather regular cellular pattern of hexagonal convection cells. In 1916, Lord Rayleigh [8] developed a theory to interpret the phenomena of Bénardexperiments. He chose the Boussinesq equations with some boundary conditions to model Bénard’sexperiments, and linearized these equations using normal modes. He then showed that the convection would occur only when the non-dimensional parameter, called the Rayleigh number, gαβ R = h4 (2.1) κν exceeds a certain critical value, where g is the acceleration due to gravity, α the coefficient of thermal expansion of the fluid, β = |dT/dz| = T¯0 − T¯1 the vertical temperature gradient with T¯0 the temperature on the lower surface and T¯1 on the upper surface, h the depth of the layer of the fluid, κ the thermal diffusivity and ν the kinematic viscosity. Since Rayleigh’s pioneering work, there have been intensive studies for this problem; see among others Chandrasekhar [1] and Drazin and Reid [2] for linear theories, and Kirchgässner[5], Rabinowitz [7], and Yudovich [9, 10], and the references therein for nonlinear theories. Most, if not all, known results on bifurcation and stability analysis of the Rayleigh-Benard problem are restricted to the bifurcation and stability analysis when the Rayleigh number crosses a simple eigenvalue in certain subspaces of the entire phase space obtained by imposing certain symmetry. It is clear that a complete nonlinear bifurcation and stability theory for this problem should at least include ATTTACTOR BIFURCATION AND RAYLEIGH-BENARD´ CONVECTION 595

1) bifurcation theorem when the Rayleigh number crosses the first critical number for all physically sound boundary conditions, 2) asymptotic stability of bifurcated solutions, and 3) the structure/patterns and their stability and transitions in the physical space. The main difficulties for such a complete theory are two-fold. The first is due to the high nonlinearity of the problem as in other fluid problems, and the second is due to the lack of a theory to handle bifurcation and stability when the eigenvalue of the linear problem has even multiplicity. The main objective of this article is to try to establish such a nonlinear theory for the Rayleigh-Bénardconvection using a new notion of bifurcation, called attractor bifurcation, and the corresponding theory. We now address each aspects of our results in this article following the three aspects of a complete theory for the problem just mentioned along with the main idea and methods used.

2.1. Rayleigh-Bénardconvection and the Boussines equations. The Bénard experiment can be modeled by the Boussinesq equations; see among others Rayleigh [8], Drazin and Reid [2] and Chandrasekhar [1]. They read ∂u + (u · ∇)u − ν∆u + ∇p = −gk[1 − α(T − T¯ )], (2.2) ∂t 0 ∂T + (u · ∇)T − κ∆T = 0, (2.3) ∂t div u = 0, (2.4) where ν, κ, α, g are the constants defined as in (2.1), u = (u1, u2, u3) the velocity field, p the pressure function, T the temperature function, T¯0 a constant represent- ing the lower surface temperature at x3 = 0, and k = (0, 0, 1) the unit vector in x3-direction. To make the equations non-dimensional, let x = hx0, t = h2t0/κ, u = κu0/h, √ 0 ¯ 0 T = βh(T / R) + T0 − βhx3, 2 0 2 0 2 0 2 p = ρ0κ Prp /h + p0 − gρ0(hx3 + αβh (x3) /2),

Pr = ν/κ.

Here the Rayleigh number R is deﬁned by (2.2), and Pr = ν/κ is the Prandtl number. Omitting the primes, the equations (2.3)-(2.5) can be rewritten as follows · ¸ 1 ∂u √ + (u · ∇)u + ∇p − ∆u − RT k = 0, (2.5) Pr ∂t ∂T √ + (u · ∇)T − Ru − ∆T = 0, (2.6) ∂t 3 div u = 0. (2.7)

The non-dimensional domain is Ω = D × (0, 1) ⊂ R3, where D ⊂ R2 is an open 3 set. The coordinate system is given by x = (x1, x2, x3) ∈ R . 596 T. MA AND S. WANG

The Boussinesq equations (2.5)—(2.7) are supplemented with the following initial value conditions (u, T ) = (u0,T0) at t = 0. (2.8) Boundary conditions are needed at the top and bottom and at the lateral boundary ∂D × (0, 1). At the top and bottom boundary (x3 = 0, 1), either the so-called rigid or free boundary conditions are given On the lateral boundary ∂D × [0, 1], either the periodic, or the Dirichlet or the free boundary conditions ccan be used. For simplicity, we proceed in this article with the following set of boundary conditions, and all results hold true as well for other combinations of boundary conditions. ½ T = 0, u = 0 at x3 = 0, 1, (2.9) u(x, t) = u(x + L, t),T (x, t) = T (x + L, t) for x ∈ Ω. The results concerning the existence of a solution for (2.5)-(2.7) with initial and boundary conditions (2.8) and (2.9) are classical, we refer the interested readers to Foias, Manley and Temam [3] for details. Here we recall only the functions spaces needed for stating the main results. To this end, let 2 3 2 H = {(u, T ) ∈ L (Ω) × L (Ω) | divu = 0, u3|x3=0,1 = 0, (2.10)

ui is periodic in the xi direction (i = 1, 2)}, 1 4 V = {(u, T ) ∈ H0 (Ω) | divu = 0, (2.11)

ui is periodic in the xi direction (i = 1, 2)}, 1 1 where H0 (Ω) is the space of functions in H (Ω), which vanish at x3 = 0, 1 and are 1 periodic in the xi-directions (i = 1, 2). Here H (Ω) is the usual Sobolev space. 2.2. Attractor Bifurcation of the B´enardProblem. The linearized equations of (2.5)-(2.7) are given by  √  − ∆u + ∇p − RT k = 0,  √ − ∆T − Ru3 = 0, (2.12)   div u = 0, where R is the Rayleigh number. These equations are supplemented with the same boundary conditions (2.9) as the nonlinear Boussinesq system. This eigenvalue problem for the Rayleigh number R is symmetric. Hence, we know that all eigenvalues Rk with multiplicities mk of (2.12) with (2.9) are real numbers, and

0 < R1 < ··· < Rk < Rk+1 < ··· . (2.13)

The ﬁrst eigenvalue R1, also denoted by Rc = R1, is called the critical Rayleigh number. Let the multiplicity of Rc be m1 = m+1 (m ≥ 0), and the ﬁrst eigenvectors Ψ1 = (e1(x),T1), ··· , Ψm+1 = (em+1,Tm+1) of (2.12) be orthonormal: Z

hΨi, ΨjiH = [ei · ej + TiTj]dx = δij . Ω

For simplicity, let E0 be the ﬁrst eigenspace of (2.12) with with (2.9) ( ) mX+1 E0 = αkΨk | αk ∈ R, 1 ≤ k ≤ m + 1 . (2.14) k=1 The main results in this section are the following theorems. ATTTACTOR BIFURCATION AND RAYLEIGH-BENARD´ CONVECTION 597

Theorem 2.1. For the B´enard problem (2.5-2.7) with (2.9), the following assertions hold true. 1. When the Rayleigh number is less than or equal to the critical Rayleigh number: R ≤ Rc, the steady state (u, T ) = 0 is a globally asymptotically stable equilibrium point of the equations. 2. The equations bifurcate from ((u, T ),R) = (0,Rc) an attractor AR for R > Rc, with m ≤ dim AR ≤ m + 1, which is connected when m > 0. 3. For any (u, T ) ∈ AR, the velocity ﬁeld u can be expressed as Ã ! mX+1 mX+1 u = αkek + o αkek (2.15) k=1 k=1

where ek are the velocity fields of the first eigenvectors in E0. m 4. The attractor AR has the homotopy type of an m-dimensional sphere S provided AR is a finite simplicial complex. 5. For any bounded open set U ⊂ H with 0 ∈ U there is an ε > 0 such that as Rc < R < Rc + ε, the attractor AR attracts U/Γ in H, where Γ is the stable manifold of φ = 0 with co-dimension m + 1.

Theorem 2.2. If the ﬁrst eigenvalue of is simple, i.e. dim E0 = 1, then the bifurcated attractor AR of the B´enard problem (2.5-2.7) with (2.9) consists of exactly ¯ ¯ 2 4 two points, φ1, φ2 ∈ H1 = V ∩ H (Ω) given by

¯ ¯ φ1 = αΨ1 + o(|α|), φ2 = −αΨ1 + o(|α|), for some α 6= 0, where Ψ1 is the ﬁrst eigenvector generating E0 in (2.14). Moreover, for any bounded open set U ∈ H with 0 ∈ U, there is an ε > 0, as Rc < R < Rc +ε, U can be decomposed into two open sets U1 and U2 such that

1. U¯ = U¯1 + U¯2, U1 ∩ U2 = ∅ and 0 ∈ ∂U1 ∩ ∂U2, ¯ 2. φi ⊂ Ui (i = 1, 2), and ¯ 3. for any φ0 ∈ Ui (i = 1, 2), limt→∞ Sλ(t)φ0 = φi, where Sλ(t)φ0 is the solution of the B´enard problem (2.5-2.7) with (2.9) with initial data φ0 = (u0,T0).

2.3. Two-Dimensional Rayleigh-BénardConvection: Asymptotic and Struc- tural Stabilities of Bifurcated Solution. The main objective of this section is to study the dynamic bifurcation and the structural stability of the bifurcated solutions of the 2-D Boussinesq equations related to the Rayleigh-Bénardconvection. It is easy to see that both Theorems 2.1 and 2.1 hold true for the 2D Boussinesq equations with any combination of boundary conditions as discussed in Section 2. Hence we focus in this section on structural stability in the physical space of the bifurcated solutions, justifying the roll pattern formation in the Rayleigh-Bénard convection. For consistency, we always assume that the domain Ω = [0,L] × [0, 1] with coordinate system x = (x1, x3). The 2-D Boussinesq equations for the 2-D Bénard convection take the same form as the 3-D Boussinesq equations (2.5-2.7). with the velocity field being replaced by u = (u1, u3), and the operators by the corresponding 2-D operators in the x = (x1, x3) coordinate system. For simplicity, we consider 598 T. MA AND S. WANG here only the free-free boundary conditions as follows:   ∂uτ  u · n = 0, = 0, on ∂Ω, ∂n (2.16)  ∂T  T = 0 at x3 = 0, 1, = 0, at x1 = 0, L. ∂x1 In this case, the function space H defined by (2.10) is replaced here by

2 3 H = {(u, T ) ∈ L (Ω) | divu = 0, u3|x3=0,1 = 0, u1|x1=0,L = 0}.

It is then easy to see that in this case, the ﬁrst eigenspace E0 is one-dimensional, and is given by   E0 = Span {Ψ1 = (e1,T1)},  µ ¶  L kπx kπx e = − sin 1 cos πx , cos 1 sin πx , 1 k L 3 L 3 (2.17)   1 p kπx  T = L2 + k2 cos 1 sin πx . k L 3

The topological structure of e1 in (2.17) consists of k vortices as shown in Fig- ure 2.1(a) and (b)

(a) (b)

Figure 2.1. Rolls with reverse orientations

By the structural bifurcation theorem in [6], the ﬁrst eigenvectors (2.17) are structurally stable; therefore, from Theorem 2.2 we immediately obtain the following result. Theorem 2.3. For any bounded open set U ⊂ H with 0 ∈ U, there is an ε > 0, as the Rayleigh number Rc < R < Rc + ε, U can be decomposed into two open sets U1 and U2 depending on R such that

1. U¯ = U¯1 + U¯2,U1 ∩ U2 = ∅, 0 ∈ ∂U1 ∩ ∂U2; 2. for any initial value φ0 ∈ Ui (i = 1, 2) there exists a time t0 > 0 such that the solution SR(t)φ0 of the two-dimensional Boussinesq equations with (2.16) is topologically equivalent to either the structure as shown in Figure 2.1(a) or that as shown in (b) for all t > t0. ATTTACTOR BIFURCATION AND RAYLEIGH-BENARD´ CONVECTION 599

REFERENCES [1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, Inc.1981. [2] P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge University Press, 1981. [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. 939–967. [4] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981. [5] K. Kirchgassner¨ , Bifurcation in nonlinear hydrodynamic stability, SIAM Rev., 17 (1975), pp. 652–683. [6] T. Ma and S. Wang, Structural classification and stability of incompressible vector fields, Physica D, 171 (2002), pp. 107–126. [7] P. H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. Rational Mech. Anal., 29 (1968), pp. 32–57. [8] L. Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag., 32 (1916), pp. 529–46. [9] V. I. Yudovich, Free convection and bifurcation, J. Appl. Math. Mech., 31 (1967), pp. 103– 114. [10] , Stability of convection flows, J. Appl. Math. Mech., 31 (1967), pp. 272–281. Received for publication October 2003. E-mail address: [email protected]