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

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Atttactor Bifurcation Theory and Its Applications to Rayleigh-Benard´ Convection 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-B¶enardConvection. 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-B¶enardconvection. In particular, we show that the problem bifurcates from the trivial solution an attractor AR when the Rayleigh number R crosses the ¯rst 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 B¶enardproblem. The proof of the main results will be published elsewhere. For this purpose, in this section, we introduce a new notion of dynamic bifur- cation called attractor bifurcation, and prove a general bifurcation theorem for nonlinear abstract evolution equations, depending on a parameter. In next sec- tion, applications to the Rayleigh-B¶enardconvection is given, providing a rigorous solution to the long standing open problem on the onset of the Rayleigh-B¶enard convection. 2000 Mathematics Subject Classi¯cation. 37L, 37G, 35Q, 58F, 76E, 76R. Key words and phrases. Attractor bifurcation, semigroup, central manifold theory, asymptotic stability, structural stability, Rayleigh-B¶enardconvection, roll structure . The work was supported in part by the O±ce 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; ¸); (1.1) dt ¸ u(0) = u0; (1.2) where u : [0; 1) ! H is the unknown function, ¸ 2 R is the system parameter, and L¸ : H1 ! H are parameterized linear completely continuous ¯elds continuously depending on ¸ 2 R1, which satisfy 8 <> L¸ = ¡A + B¸ is a sectorial operator; A : H1 ! H a linear homeomorphism; (1.3) :> B¸ : H1 ! H the parameterized linear compact operators. ¡tL It is easy to see [4] that L¸ generates an analytic semi-group fe ¸ gt¸0. Then ® we can de¯ne fractional power operators L¸ for any 0 · ® · 1 with domain ® H® = D(L¸ ) such that H®1 ½ H®2 if ®1 > ®2, and H0 = H. Furthermore, we assume that the nonlinear terms G(¢; ¸): H® ! H for some 1 > ® ¸ 0 are a family of parameterized Cr bounded operators (r ¸ 1) continuously depending on the parameter ¸ 2 R1, such that 1 G(u; ¸) = o(kukH® ); 8 ¸ 2 R : (1.4) For the linear operator L¸ = ¡A+B¸ we assume that there exist a real eigenvalue 1 sequence f½kg ½ R and and an eigenvector sequence fekg ½ H1 of A: 8 <> Aek = ½kek; 0 < ½1 · ½2 · ¢ ¢ ¢ ; (1.5) :> ½k ! 1 (k ! 1) such that fekg is an orthogonal basis of H. For the compact operator B¸ : H1 ! H, we also assume that there is a constant 0 < θ < 1 such that 1 B¸ : Hθ ¡! H bounded, 8 ¸ 2 R . (1.6) Let fS¸(t)gt¸0 be an operator semi-group generated by the equation (1.1) which enjoys the properties (i) For any t ¸ 0, S¸(t): H ! H is a linear continuous operator, (ii) S¸(0) = I : H ! H is the identity on H, and (iii) For any t; s ¸ 0, S¸(t + s) = S¸(t) ¢ S¸(s) Then the solution of (1.1) and (1.2) can be expressed as u(t) = S¸(t)u0; t ¸ 0: De¯nition 1.1. A set § ½ H is called an invariant set of (1.1) if S(t)§ = § for any t ¸ 0. An invariant set § ½ H of (1.1) is said to be an attractor if § is compact, and there exists a neighborhood U ½ H of § such that for any ' 2 U we have lim distH (u(t; '); §) = 0: (1.7) t!1 The largest open set U satisfying (1.7) is called the basin of attraction of §. ATTTACTOR BIFURCATION AND RAYLEIGH-BENARD¶ CONVECTION 593 De¯nition 1.2. 1. We say that the equation (1.1) bifurcates from (u; ¸) = (0; ¸0) an invariant set ­¸, if there exists a sequence of invariant sets f­¸n g of (1.1), 0 2= ­¸n such that lim ¸n = ¸0; n!1 lim max jxj = 0: n!1 x2­¸n 2. If the invariant sets ­¸ are attractors of (1.1), then the bifurcation is called attractor bifurcation. 3. If ­¸ are attractors and are homotopy equivalent to an m{dimensional sphere Sm, then the bifurcation is called Sm{attractor bifurcation. A complex number ¯ = ®1 + i®2 2 C is called an eigenvalue of L¸ if there are x; y 2 H1 such that L¸x = ®1x ¡ ®2y; L¸y = ®2x + ®1y: Now let the eigenvalues (counting the multiplicity) of L¸ be given by ¯1(¸); ¯2(¸); ¢ ¢ ¢ ; ¯k(¸) 2 C; where C is the complex space. Suppose that 8 <>< 0; ¸ < ¸0 Re¯i(¸) = = 0; ¸ = ¸0 (1 · i · m + 1) (1.8) :> > 0; ¸ > ¸0 Re¯j(¸0) < 0; 8 m + 2 · j: (1.9) Let the eigenspace of L¸ at ¸0 be © k ª E0 = u 2 H1 j L¸0 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 ¸ = ¸0. Then the following assertions hold true. 1. (1.1) bifurcates from (u; ¸) = (0; ¸0) an attractor A¸ for ¸ > ¸0, with m · dim A¸ · m + 1, which is connected as m > 0; 2. the attractor A¸ is a limit of a sequence of (m + 1){dimensional annulus Mk with Mk+1 ½ Mk; especially if A¸ is a ¯nite simplicial complex, then A¸ has the homotopy type of Sm; 3. For any u¸ 2 A¸, u¸ can be expressed as u¸ = v¸ + o(kv¸kH1 ); v¸ 2 E0; 4. If G : H1 ! H is compact, and the equilibrium points of (1.1) in A¸ are ¯nite, then we have the index formula ( X 2 if m = odd; ind [¡(L¸ + G); ui] = 0 if m = even. ui2A¸ 594 T. MA AND S. WANG 5. If u = 0 is globally stable for (1.1) at ¸ = ¸0, then for any bounded open set U ½ H with 0 2 U there is an " > 0 such that as ¸0 < ¸ < ¸0 + ", the attractor A¸ bifurcated from (0; ¸0) attracts U=¡ in H, where ¡ is the stable manifold of u = 0 with co-dimension m + 1. One important feature of this bifurcation theorem is that the bifurcation is ob- tained for the parameter ¸ crosses the critical value ¸c for any multipilicity m+1 of the eigenvalue ¸c 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 mathe- matical 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 di±cult 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 KirchgÄassnerindicated 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- B¶enardconvection problem was originated in the famous experiments conducted by H. B¶enardin 1900. B¶enardinvestigated 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 B¶enardexperiments. He chose the Boussinesq equations with some boundary conditions to model B¶enard'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, g®¯ R = h4 (2.1) ·º exceeds a certain critical value, where g is the acceleration due to gravity, ® the coe±cient of thermal expansion of the fluid, ¯ = jdT=dzj = T¹0 ¡ T¹1 the vertical temperature gradient with T¹0 the temperature on the lower surface and T¹1 on the upper surface, h the depth of the layer of the fluid, · the thermal di®usivity and º the kinematic viscosity.
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