ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang

Transcription

1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of Mathematics Sichuan University Chengdu, P. R. China and Department of Mathematics Indiana University Bloomington, IN Shouhong Wang Department of Mathematics Indiana university Bloomington, IN Abstract. We study in this article the large time asymptotic structural stability and structural evolution in the physical space for the solutions of the -D Navier- Stokes equations with the periodic boundary conditions. Both the Hamiltonian and block structural stabilities and structural evolutions are considered, and connections to the Lyapunov stability are also given. 1. Introduction. The main objective of this article is to study the large time structural evolution of the solutions of the -D Navier-Stokes equations with periodic boundary conditions. This is part of a research program initiated recently by the authors to study the structure and its evolution of fluid flows in the physical spaces; see [3, 4, 5, 6, 8, 7, 1, ]. The -D Navier-Stokes equations with periodic boundary conditions can be viewed as the fluid equations defined on the -D torus T = R /(πz). As we know, when the initial velocity field φ and external forcing f are all Hamiltonian, the solutions of the Navier-Stokes equations are Hamiltonian as well. Hence one objective of this article is to study the large time Hamiltonian structural stability and the Hamiltonian structural evolution of the solutions of the Navier-Stokes equations. The main results toward to this objective are two-fold. First, we show that for the Navier-Stokes equations with forcing with decay in time, the large time structure of the solutions is characterized by the unique solution of a Stokes problem. Second, we establish a natural connection between the notions of structural stability and the (dynamic) Lyapunov stability. However, when the initial velocity field φ and external forcing f are not Hamiltonian, the solutions of the Navier-Stokes equations are just divergence-free and are 1991 Mathematics Subject Classification. 34D, 35Q35, 58F, 76, 86A10. Key words and phrases. divergence-free vector fields, periodic boundary conditions, Navier- Stokes equations, structural stability, block stability, Hamiltonian stability, Lyapunov stability. The research work in this article was supported in part by the Office of Naval Research, by the National Science Foundation, and by the National Science Foundation of China. 189

2 190 T. MA AND S. WANG no longer Hamiltonian. In this case, the notion of structural stability and structural evolution is naturally replaced by the block stability introduced by the authors in [4, 7]. Hence another objective of this article is to study the block structural stability and the block structural evolution of the solutions of the Navier-Stokes equations as time goes to infinity. For this purpose, we obtain in this article a detailed characterization of the large time block structure of the solutions of the Navier-Stokes equations with forcing with decaying a Hamiltonian part and a (not necessarily decaying) harmonic part. The classification of the block structure are obtained in terms of the solutions of a suitable Stokes problem as well. This article is organized as follows. After some preliminaries in Section, we study the asymptotic Hamiltonian stability in Section 3, and block stability in Section 4.. Preliminaries. In this section, we recall some known results on Hamiltonian structural and block stabilities for incompressible flows on a -D torus, and on the -D Navier-Stokes equations with periodic boundary conditions..1. Hamiltonian Structural Stability. Let M = T be a two-dimensional smooth torus with a nature sympletic structure of the Riemannian area. Let C r (T M) be the space of all C r vector fields on M. If v C k (T M) and r = k + α with k 0 an integer and 0 < α < 1, then v C k (T M) and all derivatives of v up to order k are α Hölder continuous. We set D r (T M) = v C r (T M) div v = 0}, H r (T M) = v D r (T M) v is a Hamiltonian vector field}. where div is the divergence operator on M. In [4], we showed that a vector field v H r (T M) (r 1) is Hamiltonian structurally stable if and only if a) v is regular, and b) all saddle points are self connected. We now present more detailed structural analysis for Hamiltonian vector fields. Definition.1. Let v be a stable Hamiltonian vector field on M = T, and Q T be an invariant set of v. 1. Q is called a D block if Q is homeomorphic to an open disk such that Q is a saddle self connection of a saddle point p Q;. Q is called an S block if Q is homeomorphic to an open disk such that Q consists of two saddle self connections of a saddle point p Q. 3. Q is called a T block if Q is a closed domain such that the interior Q does not contain singular points of v, and T Q consists of only D and S blocks of v. In the above definition, the letters D, S and T stand for disk, sphere and torus respectively; see Figure.1 for schematic pictures of D, S and T blocks. We say that the D and S blocks Q have a right hand orientation (resp. a left hand orientation) if the interior of Q is on the left side of Q (resp. on the right side of Q), traveling in the direction of the orbits of v on Q. The Hamiltonian structural stability theorem tells us that a stable Hamiltonian vector field v on T has a block decomposition as follows I K T = Ω + D i + A k with D 0 =, (.1) i=0 k=1

3 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NSE 191 where Ω is a unique T block, D i (0 i I) are D blocks and A k (1 k K) are S blocks, which are topologically equivalent to a block structure as shown in Figure.1 below. D 1 Ω A 1 A D Γ 1 Γ Figure.1 In Figure.1, A 1 and A are two S blocks with left-hand and right-hand orientations respectively, D 1 and D are D blocks with left-hand and right-hand orientations respectively, and Ω is a T block. More precisely, the following theorem characterizes the block structure of a stable Hamiltonian vector field. Theorem.1. (Ma and Wang [4]) A vector field v H r (T M) (r 1) is structurally stable in H r (T M) if and only if v has a block decomposition as in (.1), and 1. v has exactly one T block;. v has a least two S blocks and the number K of the S blocks of v is even, i.e. K = m (m 1), half of which has right-hand orientation and the other half have left-hand orientation; 3. all saddles points in D and S blocks are self-connected. Moreover, the set H r 1 (T M) of all stable Hamiltonian vector fields is open and dense in H r (T M)... Block Stability. As we know, Hamiltonian vector fields on a torus do not exhaust all divergence free vector fields, i.e. H r (T M) D r (T M). In fact, no vector field on T is structurally stable in D r (T M), and the right notion of stability is the so called block stability in D r (T M) introduced in [4]. Definition.. A regular vector field v D r (T M) is called a basic vector field if M = T can be decomposed into invariant blocks as M = Ω K k=0 A k (A 0 = ) with empty intersections between them, such that 1. each A k is an open flow invariant sub manifold which is homeomorphic to an open disk, and Ω is a compact invariant sub manifold with genus one, and. all saddle points of v in A k (1 k K) are self connected, 3. v has exactly one saddle point in each A k (1 k K), and has no singular points in the interior Ω. We denote by D r B (T M) the set of all Cr basic vector fields. Then we have the following block stability theorem. Theorem.. (Block Stability Theorem [4, 7]) Let v D r B (T M) be a basic vector field with block decomposition M = Ω k k=0 A k (A 0 = φ). Then there exists a neighborhood O D r (T M) of v such that

4 19 T. MA AND S. WANG 1. each v 1 O has block decomposition M = Ω (1) K k=0 A(1) k which is isomorphic to that of v, i.e. there exists a homeomorphism ϕ : M M such that ϕ takes the blocks Ω (1) and A (1) k of v 1 the blocks Ω and A k (1 k K) respectively, preserving the flow orientation;. v Ai is topologically equivalent to v 1 (1) A for v 1 O; i 3. there is a dense set Õ O with O Õ φ such that for any v 1 Õ, Ω(1) is an ergodic set of v 1, and for v 1 O Õ, Ω(1) consists of periodic orbits of v 1 ; 4. DB r (T M) is open and dense in Dr (T M)..3. Navier Stokes Equations on the Torus. We consider the Navier Stokes equations defined on the torus M = T = R /(πz) : u t + (u )u = µ u p + f(x, t), div u = 0, (.) u(x 1 + k 1 π, x + k π) = u(x 1, x ), u(x, 0) = ϕ(x), where k 1, k Z are integers. For any u D r (T M), we have the following Fourier expansion: u = (u 1, u ), u 1 = a nm e i(nx1+mx), u = n,m= n,m= b nm e i(nx1+mx), a n m = a nm, b n m = b nm, na nm + mb nm = 0. By the Hodge decomposition, we have C r (T M) = D r (T M) G r (T M), G r (T M) = φ φ C r+1 (M)}, D r (T M) = H r (T M) H. Here H contains all harmonic fields and has dimension, which is the first Betti number of M. It is easy to see that n,m= n 0,m 0 H = u = (a, b) a, b, R} = R. In view of the Fourier expansion, if a 00 = b 00 = 0, then u given by (.) is a Hamiltonian vector field, whose Hamiltonian function is given by 1 H = i m a nme i(nx 1+mx ) 1 [ i a0n e inx + b n0 e inx ] 1. n n= The following theorem is useful for the discussion of asymptotic structural stability of solutions of the Navier Stokes equations. Theorem.3. [4] If ϕ H r (T M), f H r (T M) G r (T M), (r ), then the solution u(x, t) of (.1) is a one parameter family of Hamiltonian vector fields, i.e. u(, t) H r (T M) for any t 0.

5 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NSE Asymptotic Hamiltonian Structural Stability. We study in this section the large time Hamiltonian structural stability for the solutions of the Navier-Stokes equations. Two cases will be addressed. The first case is on the solutions of the Navier-Stokes equations with forcing with decay. The second case deals with the structural stability of Lyapunov stable solutions Decay of the solutions and Hamiltonian structural stability. We consider problem (.) with forcing decaying in time t as t goes to infinity. More precisely, let f C 1 ([0, 1], H r (T M)) (r 3) such that for some n > 1/ f(x, t) = t n (f 0 (x) + f 1 (x, t)), (3.1) lim f 1(, t) C r = 0, t 0 t f 1 dt <. (3.) L In order to study the asymptotic structural stability of (.), it is necessary to consider the Stokes problem µ u + p = f 0 (x), x M = T, (3.3) div u = 0. Since the Laplace operator : H r+ (T M) H r (T M) (r 0) is an isomorphism, we infer from Theorem.1 that there exists an open and dense F H r (T M) F = ( µ )(H r 1 (T M)), (3.4) such that for any f 0 F, the solution u(x, t) of (3.3) is Hamiltonian structurally stable, where H r 1 (T M) is the set of Hamiltonian structurally stable fields. Obviously the set F is independent of the kinematic viscosity µ > 0. The main result in this section is as follows: Theorem 3.1. Let f C ([0, ), H r (T M)) (r 3) have the expansion (3.1 3.) near t =, and ϕ H r (T M). If f 0 F, then there is a time t 0 > 0 depending on ϕ and f such that the solution u(, t) of (.) is Hamiltonian structurally stable for all t > t 0. Moreover, u is topologically equivalent to the unique solution of (3.3) for any t > t 0. Proof. Let the solution u(x, t) and p(x, t) be expressed near t = by u(x, t) = t n (u 1 (x) + v(x, t)), p(x, t) = t n (p 1 (x) + q(x, t)), where (u 1, p 1 ) is the unique solution of the Stokes problem (3.3), and (v, q) satisfies v t µ v + q = nt 1 (u 1 + v) t n [(u 1 + v) ](u 1 + v) + f 1, (3.5) div v = 0. Here f 1 H r (T M) (r 3) satisfies (3.). For any T 0 > 0 sufficiently large, the initial value for v in (3.5) is given by v(x, T 0 ) = T n 0 u(x, T 0 ) u 1 (x). (3.6) To prove the theorem, it suffices then to prove that lim v(, t) C1 = 0. (3.7) t

6 194 T. MA AND S. WANG For this purpose, first it follows from (3.5) and (3.6) that d v dx = µ v + f 1 v + nt 1 (u 1 + v) v (3.8) dt T T + t 1 [(u 1 + v) ](u 1 + v) v]}dx, with initial condition v (x, T 0 )dx = [T0 n u(x, T 0 ) u 1 (x)] dx. (3.9) T T Then we infer from (3.8) that for t > T 0 sufficiently large, d dt v L C v L + g(t), g(t) = 0, lim t for some constant C > 0. By the Gronwall inequality and using (3.9), we deduce that lim v L = 0. (3.10) t By Sobolev s embedding theorems, it suffices for us to prove [ ] 3 v W 3, = D k v dx C, C > 0 a constant, (3.11) T k=0 which implies lim v W,p = 0, 1 p <. t Hence (3.7) follows. To prove (3.11), we infer from (3.5) that d dt v L + µ v L (3.1) = nt n 1 ((u 1 + v) )(u 1 + v) v T [nt (u 1 + v) + nt 1 v t n (v )v]v [t n (v )v + f 1]v }dx. Here v 0 = v(x, T 0 ), q 0 = q(x, T 0 ), f 0 1 = f 1 (x, T 0 ), and v = v/ t is given by v (x, T 0 ) =µ v(x, T 0 ) q 0 + nt 1 0 (u 1 + v 0 ) (3.13) t n ((u 1 + v 0 ) )(u 1 + v 0 ) + f 0 1. For t T 0 > 0 large enough, we obtain from (3.13) and (3.13) that d dt v L + µ v L C f 1 L + C v L v L + Ct 1 n v L v L. Then by the Gronwall inequality, we have d [ v L e ] t 0 v L dt dt C f 1 L + Ct n 1 v L v L. Hence we obtain v L C (3.14) where C > 0 is a constant independent of t.

7 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NSE 195 Now (3.5) can be rewritten as µ v + q = F (x, t) div v = 0 (3.15) where F = nt 1 (u 1 + v) t n [(u 1 + v) ](u 1 + v) + f 1 v t. Since v, v L ([0, ), L (T M)), we deduce from (3.5) that v L ([0, ), W 1, (T M)). Using the same method as in the proof of the regularity theorem, Theorem in [9], we obtain from(3.15) that v L ([0, ), W, (T M)). Moreover, let z = D 3 v. Then by (3.5), z is governed by the equation z r µ z + p = nt 1 (D 3 u 1 + z) t n D 3 [((u 1 + v) )(r 1 + v)] + D 3 f 1, div z = 0. (3.16) Using v L ([0, ), W, (T M)) and (3.16), it is easy to show z L ([0, ), L (T M)), and the estimate (3.11) holds true. The proof is complete. 3.. Structural stability vs Lyapunov stability. In this section, we are mainly concerned with the relationship between the Lyapunov stability and the asymptotic Hamiltonian structural stability. Let f C ([0, ), H r (T M)) (r 3) such that lim f(x, t) = λf 0(x) in H r (T M), (3.17) t f t dxdt <, (3.18) 0 T where λ > 0 is a parameter. The Lyapunov Stability Theorem amounts to saying that for given f 0, if λ > 0 is small enough then the solution u(x, t) of (.) converges to the steady state of (.), namely lim t u(, t) u 0 C 1 = 0, (3.19) lim t p(, t) p 0 C 0 = 0, where (u 0, p 0 ) is a solution of the stationary equations of (.). For convenience, we restate the Lyapunov Stability Theorem as follows. Theorem 3.. (Lyapunov Stability Theorem) Let f C ([0, ), H r (T M)) (r 3) satisfying (3.17) and (3.18). If λ = µ with µ being the kinematic viscosity, then there exists a constant α > 0 independent of µ such that as f 0 H r < α, for any ϕ H r (T M) the solution u(x, t) of (.) satisfies (3.19). Moreover, the stationary solution u 0 of (.) has the form as u 0 = µv 0 (x), where v 0 (x) solves the equation v + (v )v + p = f 0 (x), (3.0) div u = 0.

8 196 T. MA AND S. WANG Proof. Let f(x, t) = µ f 0 (x) + h(x, t). By (3.17) and (3.18), we have lim h(, t) Cr = 0 (r 3), (3.1) t h t dxdt <. (3.) 0 T Let u 0 = µv 0 and v(x, t) = u(x, t) µv 0 (x). It is easy to see that v 0 satisfies the equation (3.0), and v(x, t) is governed by the following equation v t + (v )v = µ[ v (v 0 )v (v )v 0 ] p + h(x, t), (3.3) div u = 0, v(x, 0) = ϕ(x) µv 0 (x). Then the following estimate can be obtained in the same fashion as in Ma and Wang [3]: v 0 C +r C [ f 0 C r + f 0 p Cr], for some p > 8 (3.4) where C > 0 is independent of µ. Hence, there exists a constant α > 0 independent of µ such that when f 0 C r < α, we have [ v + (v 0 )v + (v )v 0 ] vdx T [ = v + (v )v 0 v ] dx T ρ v dx, T for some 0 < ρ < 1, which yields from (3.3) that lim v(, t) L t = 0. Moreover, by (3.1) (3.3) it is easy to show that v(, t) W 3, C, C > 0 a constant, which implies lim t v(, t) C 1 = 0. The proof is complete. We are now in position to state the main result on the relationship between the Lyapunov stability and the asymptotic Hamiltonian structural stability for solutions of the Navier-Stokes equations (.). Theorem 3.3. Let f C ([0, ), H r (T M)) (r 3) satisfying (3.17) and (3.18). Let F H r (T M) be the structurally stable solution set of the Stokes equation (3.3) as given by (3.4). Let λ = µ k (k > ) and f 0 C r = 1. Then for µ > 0 sufficiently small, there exists a time t 0 > 0 such that for all t > t 0 the solution u(x, t) of (.) is Hamiltonian structurally stable, and is topologically equivalent to the solution w(x) of the Stokes problem w + p = f0 (x), div w = 0.

9 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NSE 197 Proof. By Theorem 3., u(x, t) u 0 (x) (t ) in C 1 norm, where u 0 satisfies the equation µ u 0 + (u 0 )u 0 + p = µ k f 0 (x) (k > ) (3.5) div u 0 = 0. If u 0 is Hamiltonian structurally stable, then it is clear that u(x, t) are topologically equivalent to u 0 for all t > 0 sufficiently large. Let u 1 be a solution of the Stokes problem µ u 1 + p = µ k f 0, div u 1 = 0. Let u 1 = µ k 1 w, then the above Stokes problem is equivalent to w + p = f 0, div w = 0. (3.6) Let the solution u 0 (x) of (3.5) have the form u 0 = µu 1 + µv(x, µ) = µ k w + µv(x, µ). Then v(x, µ) solves the equation v + (v )v + µ k 1 [(w )v + (v )w] + p = µ k (w )w, div u = 0. From (3.6) we get (3.7) w C r+ C f 0 C r, (3.8) where C > 0 is a constant independent of µ. The equation (3.7) can be written as v + µ k 1 [(w )v + (v )w] + p = F (x), (3.9) div v = 0, where F (x) = µ k (w )w (v )v. For the equation(3.9), the L p estimates are valid (see [9]): v W,p C [ F L p + v L p] (3.30) C [ v L p + v L p v L p + µ k w C 1 ] where p 1 + q 1 = 1. From (3.30) we have v W,p C [ v L p + µ k f 0 C 0 ]. (3.31) On the other hand, we derive from (3.9) that v L T = v [ dx µ k (w )w + µ k 1 (v )w ] vdx. T Then for µ > 0 sufficiently small, by (3.8) we have ( ) 1 µ 1 C v L Cµk w 4 C1. (3.3) It follows from (3.31) and (3.3) that v W,p Cµ k [ f 0 C 0 + f 0 4 C 0 ], p > 1.

10 198 T. MA AND S. WANG Thus, for the solution u 0 of (3.5) we get u 0 = µ k [w(x) + O(µ k )] in H 1 (T M) (k > ). Thus, for all µ > 0 sufficiently small, u 0 is topologically equivalent to the solution w(x) of (3.6). By assumption, w(x) is Hamiltonian structurally stable, hence the proof of the theorem is complete. Remark 3.1. The same assertions as in Theorems are valid for the Navier Stokes equations with the homogeneous Dirichlet boundary condition and the free boundary condition Some Examples. In the above stability theorems, it is important to know the structurally stable solution set F of the Stokes problem defined by (3.4). For this purpose, we give some examples of the f F. First, when f = (sin x, 1 cos x 1) the solution of (3.6) is as follows u(x 1, x ) = (sin x, 1 cos x 1). Obviously, the solution u(x) has exactly four singular points ( π ) ( ) ( π ) ( ), 0, 3 π, 0,, π, 3 π, π which are nondegenerate. By Theorem.1, the block structure of u must be as T = T A 1 A, i.e. u has exactly two S blocks. Hence u(x) is Hamiltonian structurally stable. In general, let f(x 1, x ) = (α 1 sin x, α cos kx 1 ), k = 1,,..., where α 1, α 0 and α 1 α. Then the solution of (3.6) is Hamiltonian structurally stable, and is given by u(x 1, x ) = (α 1 sin x, α k cos kx 1 ). (3.33) For the case where α 1 = α k = 1, the structure of (3.33) is schematically shown by Figure 3.3 below. Similarly, for the functions given by f(x) = α 1 sin(x +θ 1 ), α sin(kx 1 +θ )} or f(x) = α 1 sin(kx + θ 1 ), α sin(x 1 + θ )}, the solutions of (3.6) are Hamiltonian structurally stable. π x π x 1 o π k 3 k π 4k-3 π k 4k-1 π k π Figure 3.1

11 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NSE Asymptotic Block Stability Main Theorems. We know that when the harmonic vector fields appear in the data (ϕ, f), the solution u of (.) is not Hamiltonian. Thanks to the block stability theorem-theorem., we shall consider the asymptotic block stability instead of the Hamiltonian structural stability. Assume that ϕ = ϕ H + a (4.1) f = f H (x, t) + b(t), b(t) dt < where ϕ H and f H are Hamiltonian vector fields, a = (a 1, a ) is a constant vector, b = b 1 (t), b (t)} is continuous harmonic field on t. Denote by k(t) = a + t 0 b(τ)dτ, 0 (k(t) = (k 1 (t), k (t))). We recall that a T block of a stable Hamiltonian vector field u H r (T M) consists of closed orbits of u in its interior. All closed orbits in the T block as the elements in homology H 1 (T, Z) are equal to each other, which are nonzero. This will characterize the stable Hamiltonian vector fields as follows. We know that a torus is homeomorphic to a rectangle with its two pairs of edges identified as shown in Figure 4.1. We take two closed curves e 1 and e with orientations (see Figure 4.1) in M as two basic elements in H 1 (T, Z). γ e e 1 Then for any γ H 1 (T, Z), we have Figure 4.1 γ = ne 1 + me, n, m Z. If γ M is a closed curve with orientation, then n = the intersection number of γ with e 1 m = the intersection number of γ with e. Let u H r (T M) be a stable Hamiltonian vector field, and Ω M = T be the T block of u. We say that u is of (n, m) type if for any closed orbit γ Ω of u we have γ = ±(ne 1 + me ). Now we return to consider the asymptotic block stability of the Navier Stokes equations (.). The main theorems in this section are given by the following, which are the counterparts of Theorem 3.1 and Theorem 3.3 respectively.

12 00 T. MA AND S. WANG Theorem 4.1. Let (f H, ϕ H ) satisfy the hypotheses of Theorem 3.1, and k(t) = 0 t > 0. Then the following assertions hold true: 1. If lim t k(t) = C = (C 1, C ) 0, then there is a t 0 > 0 such that the solution u(x, t) of (.) is block stable for any t > t 0, whose block structure only consists of a T block Ω(t), i.e. T = Ω(t), and u(x, t) are ergodic on T for almost all t > t 0 provided b(t) 0, or C 1 /C is irrational.. Let the solution u 0 of the Stokes equation (3.3) be of (n, m) type ( n + m = 0). If k(t) = o(t n ) (t ) and k 1 (t)/k (t) n m for all t > 0 sufficiently great, then there is a t 0 > 0 such that for all t > t 0 the solutions of (.) are block stable, whose block structures are isomorphic to the unique stable block structure of the solution of (3.3) with a harmonic perturbation. Theorem 4.. Let (f H, ϕ H ) satisfy the hypotheses of Theorem 3.3, and k(t) = 0 t > 0. Then we have 1. If lim t k(t) = C µ, then the assertion 1 in Theorem 4.1 holds.. Assume the solution of (3.3) is of (n, m) type, if k(t) 0 (t ) and k 1 (t)/k (t) n m for all t > 0 sufficiently great, then the assertion in Theorem 4.1 holds. Remark 4.1. In assertion of Theorems , the phrase: the unique stable block structure of the solution of (3.3) with a harmonic perturbation means that for a stable Hamiltonian solution u 0 (x) of (3.3) and almost all harmonic vector fields a = (a 1, a ) sufficiently small the vector fields u 0 + a are block stable, and the block structure in the sense of isomorphism is unique. For example, if u 0 = (sin x, cos kx 1 ) (k 1), then u 0 +a have the block structure as shown in Figure 4.1 for all a = (a 1, a ) with a 1, a 0 sufficiently small. A 1 A k 1 A A k Figure Connection lemmas. In order to prove Theorems we need to introduce a connection lemma. Let M be an m-dimensional manifold and u C r (T M) be a vector field. A curve L M is called an orbit line of u if L is a union of curves L = i=1 γ i

13 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NSE 01 such that either γ i is an orbit of u, or γ i consists of singular points of u, and if γ i and γ i+1 are orbits of u, then the ω limit set of γ i is the α limit set of γ i+1 ω(γ i ) = α(γ i+1 ), i.e., the end points of γ i are singular points of u, and the starting end point of γ i+1 is the finishing end point of γ i ; see Figure 4.. γ i+1 p p i+1 L γ i γ i p i p 1 Figure 4.3 Let M be a two-dimensional manifold with boundary. We set D r (T M) = u C r (T M) div u = 0, u n M = 0} i.e. D r (T M) consists of all divergence free vector fields on M with no normal flow boundary condition. Lemma 4.1. (Connection Lemma). Let M R be a C r manifold, u, v D r (T M) with v sufficiently small, and L M be an orbit line of u starting at p. Then the orbit line γ of u + v starting at p will pass through a point q L if and only if v dl = 0, (4.) L[p,q] where L[p, q] is the curve segment on L from p to q. Proof. We divide the proof into two steps. Step 1. We first discuss the geometrical meaning of the condition (4.). By the Green formula, we know that for a curve Γ with end points p and q, v D r (T M) satisfies v dl = v n ds = 0 Γ if and only if there is an orbit line Γ of v passing through p and q, i.e. Γ[p, q] Γ[p, q] consists of simple closed curves, see Figure 4. below Γ Γ Γ 1 p Γ 1 Γ q Figure 4.4

14 0 T. MA AND S. WANG Hence, to achieve this lemma it is equivalent to prove that the orbit line γ of u + v starting at p will pass thorough q L if and only if the orbit line L of v starting at p will connect to q. Step. Because M R, there exist the Hamiltonian functions H 1, H C 1 (M) such that ( H1 u = J H 1 =, H ) 1 x x 1 ( H v = J H =, H ). x x 1 Namely, the equi-value lines of H 1 and H are respectively the orbit lines of u and v. Hence, the orbit line v of u + v starting at p will pass through q L if and only if p and q are on the same equi-value line of H 1 + H, i.e. L and L intersect at p and q. Thus, this lemma is proven. Likewise, for the Hamiltonian vector fields on general orientable two-dimensional manifold M, the connection lemma is also valid. Lemma 4.. Let M be an orientable two-dimensional manifold. Let u, v H r (T M) with v sufficiently small, and L be the orbit line of u starting at p. Then the orbit line γ of u + v starting at p will connect to a point q L if and only if (4.) holds. For general divergence free vector fields on two-dimensional manifold with nonzero genus, the connection lemma as Lemma 4. is not valid. But, we have the following lemma, which is a corollary of Lemma 4.1. Lemma 4.3. Let M = T, u, v D r (T M) with v sufficiently small, and L M be a (n, m) type closed orbit line. If the equality (4.) is not valid on L, then for any p L, the orbit line of u + v passing through p, if it is closed, is not of the type (n, m) Proof of Theorems We prove only Theorem 4.1, and Theorem 4. can be proved in the same fashion. Let u = u H + k, where u H C 1 ([0, ), H r (T M)) and k = harmonic part of u. By the Hodge decomposition, the problem (.) is equivalent to uh t + (k )u H + (u H )u H = µ u H p + f H, (4.3) u H (x, 0) = ϕ H, dk dt = b(t), k(0) = a. (4.4) It is easy to see that k(t) = a + t b(t) is the solution of (4.4). 0 As in the proof of Theorem 3.1, we can show that from (4.3) that lim u H(, t) C 1 = 0. t Hence the solution u = k(t) + u H (x, t) has no singular points on T provided k(t) C 0 as t. This proves Assertion 1 in the theorem. If k(t) = o(t n ), then we can infer from (4.3) that u H (x, t) = t n u 1 + t n v(x, t) (4.5)

15 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NSE 03 where u 1 satisfies (3.3), and v satisfies v t µ v + p = nt 1 (u 1 + v) t n [(u 1 + v) ](u 1 + v) t n (k(t) )(u 1 + v) + f 1 (x, t). (4.6) As the proof of Theorem 3.1 we can deduce from (4.6) that lim v(, t) C1 = 0. (4.7) t By assumption, u 1 is Hamiltonian structurally stable, which is of the (n, m) type. Hence, from (4.5) and (4.7) we get that u H (x, t) are Hamiltonian structurally stable with the (n, m) type for all t > 0 sufficiently great. Finally, we shall show that the solution u = u H (x, t) + k(t) has the stable block structure, which is isomorphic to the structure that the S blocks of u H are broken and transformed into the D blocks. By Theorem.1, for t > 0 sufficiently great, u H (x, t) have the block decomposition as T = Ω(t) + I i=0d i (t) + K k=1a k (t). Since the D blocks D i (t) of u H are structurally stable, under the small perturbation k(t), the topological structures of D i (t) are invariant. We know that A k consists of two saddle self-connections Γ k i (i = 1, ), and by assumption Γ k 1 = ±(ne 1 + me ), Γ k = ±(ne 1 + me ). Moreover, each closed orbit γ in Ω near Γ k i is homological to Γk i, namely γ = Γk i in H 1 (T, Z). Because the condition below holds k(t) dl = 0 γ if and only if k 1 (t) k (t) = n m. Hence, by Lemma 4.3, there exists a t 0 > 0 sufficiently great such that for any t > t 0, the saddle points q k of u H (x, t) + k(t) near that of u on A k does not have the (n, m) type of saddle self connections. On the other hand we see that the closed orbits l in A k are of (0, 0) type and satisfy k(t) dl = 0. l Hence the saddle points q k of u H + k must have a (0, 0) type of saddle self connection. Thus, the S blocks A k of u H are broken under the perturbation k, and transformed into the D blocks. Therefore, the block structure of u H + k is stable for any t > 0 sufficiently great. The proof is complete. REFERENCES [1] M. Ghil, T. Ma, and S. Wang, Structural bifurcation of -D incompressible flows, Indiana Univ. Math. J., 50 (001), pp Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 000). [], Structural bifurcation of -D incompressible flows with the dirichlet boundary conditions and applications to boundary layer separations, (00).

16 04 T. MA AND S. WANG [3] T. Ma and S. Wang, The geometry of the stream lines of steady states of the Navier-Stokes equations, in Nonlinear partial differential equations (Evanston, IL, 1998), vol. 38 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1999, pp [4], Structural evolution of the Taylor vortices, MAN Math. Model. Numer. Anal., 34 (000), pp Special issue for R. Temam s 60th birthday. [5], Global structure of -D incompressible flows, Discrete Contin. Dynam. Systems, 7 (001), pp [6], Structure of D incompressible flows with the Dirichlet boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 1 (001), pp [7], Block structure and stability of -d incompressible flows, submitted, (00). [8], Structural classification and stability of incompressible vector fields, Physica D, 171 (00), pp [9] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3rd, rev. ed., North Holland, Amsterdam, Received February 003; revised November address: showang@indiana.edu