On the Backwards Behavior of the Solutions of the 2D Periodic Viscous Camassa Holm Equations

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1 Journal of Dynamics Differential Equations, Vol. 14, No. 1, January 2002 ( 2002) On the Backwards Behavior of the Solutions of the 2D Periodic Viscous Camassa Holm Equations Jesenko Vukadinović 1 Received June 9, 2000 The global behavior of the periodic 2D viscous Camassa Holm equations is studied. The set of initial data for which the solution exists for all negative times grows backwards with an exponential rate no greater then some given value is observed proved to possess some interesting richness density properties. KEY WORDS: Viscous Camassa Holm equations; vorticity quotients; global solutions. 1. INTRODUCTION In this paper we consider the two dimensional viscous Camassa Holm equations d dt v n Dv+(u N) v+v j Nu j +Np=f v=u a 2 Du N u=0 (1.1) where f is a time independent force, n > 0 (representing the kinematic viscosity) a > 0 are given parameters. The functions u: R 2 Q R 2, v: R 2 Q R 2 p: R 2 Q R are unknowns representing velocity, momentum, modified pressure, respectively. We will refer to the equations (1.1) as (VCHE). (VCHE) is a generalization of an one-dimensional equation derived by Camassa Holm (1993), which describes unidirectional surface waves in 1 Department of Mathematics, Indiana University, Bloomington, Indiana /02/ / Plenum Publishing Corporation

2 38 Vukadinović shallow water. Holm et al. (1998) generalized the equations to n dimensions, the so called ideal Camassa Holm equations or Euler alpha model. The parameter a is interpreted as the typical mean amplitude of the fluctuations. Since v in (1.1) represents a momentum, it is plausible to let viscosity act to diffuse this momentum. Chen et al. (1999) proposed the viscous variant of the Camassa Holm equations in which an artificial viscosity term n Dv is introduced into the system. The (VCHE) closely resemble the Navier Stokes equations (NSE): d u n Du+(u N) u+np=f dt N u=0 even coincide with them for a=0. For this reason they are also known as Navier Stokes-a model. In this paper we are interested in the backwards behavior of the solutions of the (VCHE). We will study the set G of initial data u 0 for the (VCHE) for which the solution S(t) u 0 exists for all times t R the rate of the backwards growth for those solutions. The backwards behavior was studied for other related partial ordinary differential equations with a similar structure as the (VCHE) the (NSE). Some examples are the Kuramoto Sivashinsky equation (see Kukavica (1992)), the Ginzburg Lau equation (see Doering et al. (1988)). For a global solution of those equations it was shown that if it grows at most exponentially when t Q., then it is necessarily uniformly bounded. Constantin et al. (1997) studied the backwards behavior for the periodic 2D Navier Stokes equations proved that it demonstrates more similarity with the linear case. For example, it was proved that the set of initial data for which the solution exists for all negative times grows at most exponentially when t Q. is dense in the phase space of the (NSE). In this paper we will show that the backwards behavior for the periodic 2D (VCHE) resembles the (NSE) case. The major object of study in this paper is the set A M n =A 23 1/2 (I+a 2 A) S(t) u u0 G0A : lim 0 2 t Q. (I+a 2 A) 1/2 S(t) u 0 [ l n(1+a 2 l 2 n )4 where denotes the usual L 2 norm, A denotes the Stokes operator, 0 < l 1 < l 2 < its distinct eigenvalues. A sts for the global attractor for the (VCHE). For convenience, we will refer to the quotient in the definition of M n as the vorticity quotient. It emerges naturally from the energy enstrophy balance equations for the (VCHE) is an analogue of the Dirichlet quotient for the (NSE), which was studied in the paper by

3 Backwards Behavior for viscous Camassa Holm Equations 39 Constantin et al. (1997). Most of the results on Dirichlet quotients have an analogue in the present paper, but there are differences in the proofs due to the different structure of the nonlinear terms in the (VCHE). Equivalently, we can define the set M n in the following way: M n ={u 0 G: (I+a 2 A) 1/2 S(t) u 0 =O(e nl n t ) as t Q.} In other words, M n consists of precisely those initial data, for which the solution S(t) u 0 exists for all t R, its kinetic energy (I+a 2 A) 1/2 S(t) u 0 2 increases with an exponential growth rate no greater then nl n when t Q.. In Section 2 we recall some known facts about the (VCHE), we rewrite it in its functional form. We derive the energy enstrophy balance equations some basic inequalities that emerge from them. In this section, we also define the set G consisting of initial data for which there exists a global solution, as well as the sets M n. In Section 3 we prove some useful properties of the sets M n. The central result of this section is Theorem 1, which enables us to control the vorticity quotients of a solution of (VCHE) by controlling the energy of the solution from below. This result is a useful technical tool will be used to obtain all the other results. Another basic result of this section is Theorem 3 that provides us with a tool for producing global solutions with certain properties. In Section 4 we prove that the sets M n are rich in the sense that P n M n =P n H (H being a suitable Hilbert space defined in Section 1, P n H being the spectral space of A corresponding to the eigenvalues l 1, l 2,..., l n ). In Section 5 we use this result to prove some density properties of the sets M n. In Theorem 6 we prove that the set of initial data for which there exists a global solution, which grows at most exponentially, is dense in H. We conclude the paper with a result in which we show that the eigenvectors of the Stokes operator A corresponding to l n for some n N belong to the closure of scaled set M n, even in respect with the stronger energy norm of the (VCHE). The analogue of this result for the (NSE) has been proved by C. Foias I. Kukavica, is yet to be published. 2. PRELIMINARIES We consider the two dimensional viscous Camassa Holm equations d dt v n Dv+(u N) v+ C 2 j=1 v=u a 2 Du N u=0 v j Nu j +Np=f

4 40 Vukadinović where f L 2 (W) 2 (which is W-periodic > W f=0) is a given function representing body forcing, n > 0 is the constant viscosity, a > 0 is a given parameter. u: R 2 Q R 2, v: R 2 Q R 2 p: R 2 Q R, are unknown functions representing velocity, vorticity modified pressure, respectively. We supplement the system with the periodic boundary condition u(x+le j )=u(x), x R 2, 1 [ j [ 2 F u=0 W where L > 0, W=[0, L] 2, (e j ) 2 j=1 is the canonical basis in R 2. Since we are interested in a functional setting for these equations, we need to introduce suitable functional spaces. First, we define V=3 u : u is a vector valued W-periodic trigonometric polynomial, N u=0, F u=04 W We define H V to be closures of V in the (real) Hilbert spaces L 2 (W) 2 H 1 (W) 2, respectively. H V are also (real) Hilbert spaces with respective scalar products 2 (u, v)= C F u j v j, j=1 W u, v H 2 u ((u, v))= C F j v j, u, v V j=1 W x k x k The corresponding norms are u =(u, u) 1/2 for u H u =((u, u)) 1/2 for u V respectively. We denote the dual of V by VŒ. By the Rellich embedding theorem, the natural inclusions i 1 : V Q H i 2 : H Q VŒ are compact. We denote the orthogonal projection (called the Leray-Projector) on the space H by P L : L 2 (W) 2 Q H; observe that H + ={Np : p H 1 (W)} is the orthogonal complement of H in L 2 (W). By A= P L D we denote the Stokes operator with domain D(A)=H 2 (W) 2 5 V. A is a selfadjoint positive

5 Backwards Behavior for viscous Camassa Holm Equations 41 operator with compact inverse. The eigenvalues of A are of the form (k 2 1+k 2 2)(2p/L) 2 where k 1, k 2 N 0, k 2 1+k 2 2 ] 0. We can arrange them in increasing order 1 2p L 22 =l 1 < l 2 < Note that lim l n =. (2.1) n Q. l n+1 l n \ l 1 For the purposes of this paper, an important property is also lim sup n Q. (l n+1 l n )=. (2.2) We define P n to be the orthogonal projection in H on the spectral space of A corresponding to the eigenvalues l 1, l 2,..., l n ; also let Q n :=I P n. Integration by parts gives u = A 1/2 u, u V We also define [u] := (I+a 2 A) 1/2 u, [u] := A 1/2 (I+a 2 A) 1/2 u, u V u D(A) We will use the Poincaré inequality u 2 \ l 1 u 2, u V, similarly, [u] 2 \ (1+a 2 l 1 ) u 2, u V Regarding the nonlinear terms that appear in the Camassa Holm equations, we introduce the following bilinear forms: B(u, v) :=P L ((u N) v), u H, v V

6 42 Vukadinović B*(u, v) :=P L 1 C 2 j=1 v j Nu j 2, u V, v H By integrating by parts we obtain following important identities: (B(u, v), w)= (B(u, w), v), u H, v, w V (2.3) (B(u, v), Av)=(B(Av, v), u), u H, v D(A) (2.4) The connection between B B* is given by the identity (B*(v, w), u)=(b(u, v), w), u H, v, w V (2.5) Now we can rewrite the viscous Camassa Holm equations in the functional form v +nav+b(u, v)+b*(u, v)=f v=(i+a 2 A) u (2.6) where P L f is replaced by f H. Observe that the term containing modified pressure doesn t occur in this equation since P L (Np)=0. This new functional version of the (VCHE) is understood in VŒ. Classical theorems imply that, for every u 0 V, there exists a unique solution u(t)=s(t) u 0 for t \ 0 of (2.6), which satisfies u(0)=u 0. If the solution u(t) also exists for t [ t 0, 0] for some t 0 > 0, then it is still uniquely determined by u 0, we still denote it by S(t) u 0. It is known that u L. loc((0,.) : H 3 (W)). Also, for any t 0 > 0, the solution operator S(t 0 ): V Q V is continuous. Our aim is now to find balance equations for (VCHE), that do not involve nonlinear terms. To this end, observe that for u, v V (B(u, v), u)+(b*(u, v), u)=(b(u, v), u)+(b(u, u), v) for u V, v D(A) =(B(u, v), u) (B(u, v), u)=0 (B(u, v), Av)+(B*(u, v), Av)=(B(u, v), Av)+(B(Av, u), v) =(B(u, v), Av) (B(Av, v), u)=0

7 Backwards Behavior for viscous Camassa Holm Equations 43 Therefore, if we multiply (VCHE) by u Av respectively, we obtain the following equations: 1 d 2 dt [u]2 +n(av, u)=(f, u) (2.7) 1 d 2 dt v 2 +n(av, Av)=(f, Av) (2.8) These equations yield, respectively, the following inequalities d dt [u]2 +nl 1 [u] 2 [ f 2 nl 1 (2.9) d dt v 2 +nl 1 v 2 [ f 2 n (2.10) Observe that from the relation (2.9), it follows that t Q [u( )] is a decreasing function as long as [u(t)] > f /nl 1. If u is a solution of (VCHE) defined on some interval [t 0,.), the Gronwall lemma gives [u(t)] 2 [ [u(t 0 )] 2 e nl 1(t t 0) + f 2 (1 e nl 1(t t 0) ), t \ t n 2 l 2 0 (2.11) 1 v(t) 2 [ v(t 0 ) 2 e nl 1(t t 0) + f 2 Also, if the solution is defined on [t, t 0 ] we have n 2 l 1 (1 e nl 1(t t 0) ), t \ t 0 (2.12) [u(t)] 2 \ [u(t 0 )] 2 e nl 1(t 0 t) f 2 (e nl 1(t 0 t) 1) (2.13) n 2 l 2 1 Observe that, if the solution is defined on (., t 0 ], the integral > t 0 1. dt [u(t)] 2 exists. Every solution u(t)=s(t) u 0 of the (VCHE) for an initial datum u 0 V, which is defined for all t R, is called a global solution. u 0 belongs to a

8 44 Vukadinović trajectory of a global solution if only if u 0 4 t \ 0 S(t) V. Therefore, we define G :=3 S(t) V (2.14) t \ 0 to be the set of initial data, for which there exists a global solution. The (VCHE) have a global attractor A={u 0 G : lim sup [S(t) u 0 ] <.} t Q. =3 u0 G : [S(t) u 0 ] [ f nl 1, t R4 A is S( )-invariant, meaning that S(t) A=A for t \ 0. Also, it is a nonempty, compact, connected subset of V. Moreover, every S( )-invariant set containing A is connected. For 0 < o < l 1 /2 let us define the sets (I+a M n, o :=A 23 2 A) S(t) u u0 G0A : lim sup 0 2 t Q. [S(t) u 0 ] 2 [ (l n +o)(1+a 2 (l n +o))4 Some obvious properties of the sets M n, o are S(t) M n, o =M n, o, t \ 0, n N A M 1, o M 2, o In the next section, after proving a crucial theorem, we ll be able to prove some nontrivial properties of these sets. For example, we will prove that the set M n, o doesn t really depend on o. This enables us to define a new set M n as M n, o for an arbitrary choice of 0 < o < l 1 /2. For later purposes let us also define the following sets for n N: C n, o :=3 u0 A 1 V : (I+a2 A) u 0 2 [u 0 ] 2 [ (l n +o)(1+a 2 (l n +o))4

9 Backwards Behavior for viscous Camassa Holm Equations SOME PROPERTIES OF THE SETS M n, o In this section, we will prove some properties of the sets M n, o. To this end, we will need the following crucial result: Theorem 1. Let T > 0 n N. Let u be a solution of (VCHE) for the initial datum u 0 C n, o let v 0 :=(I+a 2 A) u 0. If then [u(t)] > f, t [0, T) (3.1) no v(t) 2 [u(t)] 2 [ (l n+o)(1+a 2 (l n +o)), t [0, T] or, in other words, u(t) C n, o for t [0, T]. Proof. Let us denote ũ := u [u], ṽ := v [u] Applying (2.7) (2.8) we obtain 1 d v 2 2 dt [u] 2= d [u] 4 2 dt v 2 [u] 2 v 2 1 d 2 2 dt [u]2 Thus, we obtain = 1 [u] 2 [( n(av, Av)+(f, Av)) ṽ 2 ( n(av, u)+(f, u))] = n(aṽ, Aṽ ṽ 2 ũ)+(f/[u], Aṽ ṽ 2 ũ) 1 d v 2 2 dt [u] 2+n(Aṽ, Aṽ ṽ 2 ũ)=(f/[u], Aṽ ṽ 2 ũ) (3.2) Let us define m to be the positive solution of the quadratic equation ṽ 2 =m(1+a 2 m) (3.3)

10 46 Vukadinović Now observe that (Aṽ, Aṽ ṽ 2 ũ)=(aṽ, Aṽ m(1+a 2 m) ũ) =(Aṽ, Aṽ mṽ)+m(aṽ, ṽ (1+a 2 m) ũ) =(Aṽ, Aṽ mṽ)+ma 2 (Aṽ, Aũ mũ) = Aṽ mṽ 2 +ma 2 (Aṽ mṽ, Aũ mũ) +m(ṽ, Aṽ mṽ)+m 2 a 2 (ṽ, Aũ mũ) (ṽ, Aṽ mṽ)+ma 2 (ṽ, Aũ mũ)=(ṽ, Aṽ) m(ṽ, ṽ)+ma 2 (ṽ, Aũ) a 2 m 2 (ṽ, ũ) Therefore, we have Similarly, =m+a 2 m 2 m(ṽ, ṽ)+ma 2 (ṽ, Aũ) a 2 m 2 =m(1 (ṽ, ṽ)+a 2 (Aũ, ṽ)) =m(1 (ũ, ṽ))=0 (Aṽ, Aṽ ṽ 2 ũ)= Aṽ mṽ 2 +ma 2 [Aũ mũ] 2 (f/ u, Aṽ ṽ 2 ũ)=(f/ u, Aṽ mṽ)+(f/ u, mṽ ṽ 2 ũ) Thus, =(f/ u, Aṽ mṽ)+ma 2 (f/ u, Aũ mũ) =(f/ u, Aṽ mṽ) +ma 2 ((I+a 2 A) 1/2 f/ u, (I+a 2 A) 1/2 (Aũ mũ)) [ f 2 2n[u] 2+n 2 Aṽ mṽ 2 +a m1 2 (1+a2 A) 1/2 f 2 + n 2 2n[u] 2 2 [Aũ mũ]2 f 2 [ (1+a 2 m) 2n[u] 2+n 2 ( Aṽ mṽ 2 +ma 2 [Aũ mũ] 2 ) d dt ṽ 2 +n( Aṽ mṽ 2 +ma 2 [Aũ mũ] 2 ) [ (1+a 2 m) f 2 n[u] 2 (3.4)

11 Backwards Behavior for viscous Camassa Holm Equations 47 Since we obtain Let us suppose that Aṽ mṽ 2 +ma 2 [Aũ mũ] 2 \ (1+a 2 m)[aũ mũ] 2 \ (1+a 2 m) min n N m l n 2 d dt ṽ 2 +n(1+a 2 m) min m l n 2 [ (1+a 2 m) f 2 (3.5) n N n[u] 2 ṽ(t 0 ) 2 =(l n +o)(1+a 2 (l n +o)) for some t 0 [0, T). In that case m(t 0 )=l n +o, so (3.5) implies d dt ṽ(t f 0) 2 [ (1+a m)1 2 2 n[u(t 0 2 )] 2 no2 The right-h side is negative by one of our assumptions. Therefore, d ṽ(t dt 0) 2 < 0, so ṽ(t) decreases in a neighborhood of t 0. This proves the theorem. i Corollary 1. Let 0 < o < l 1 /2, u 0 M n, o, v 0 :=(I+a 2 A) u 0. If [u 0 ] > f /no, then there exists a unique t 0 \ 0 such that (I+a 2 A) S(t) u 0 2 [S(t) u 0 ] > f no, t (., t 0) [S(t) u 0 ] [ f no, t [t 0,.) [S(t) u 0 ] 2 [ (l n +o)(1+a 2 (l n +o)), t [0, t 0 ] Proof. This follows from the Theorem 1, the inequality (2.13). i Corollary 2. For each n N, 0 < o < l 1 /2, M n, o 3 u0 V : [u 0 ] [ f 4 2 C no n, o

12 48 Vukadinović Proof. This follows trivially from the last corollary. i Theorem 2. Let u 0 G0A, let u(t) :=S(t) u 0, v(t) :=(I+a 2 A) S(t) u 0. Then, Therefore, v(t) 2 lim t Q. [u(t)] {l 1(1+a 2 l 2 1 ), l 2 (1+a 2 l 2 ),...} 2 {.} (I+a M n, o =A 23 2 A) S(t) u u0 G0A : lim 0 2 t Q. [S(t) u 0 ] 2 {l 1 (1+a 2 l 1 ), l 2 (1+a 2 l 2 ),..., l n (1+a 2 l n )}4 ={u 0 G : [S(t) u 0 ]=O(e (1+E) nl n t ) as t Q., -E > 0} Proof. the limes If we rewrite (3.5) in its integral form, we can easily see that lim t Q. v(t) 2 2 [0,.] [u(t)] exists. Since u 0 A, lim t Q. [u(t)]=.. Suppose that the above limes is finite. Then, lim t Q. m(t) is also finite. Therefore, by (3.5), 0 [ lim sup t Q. d dt ṽ 2 [ n lim inf (1+a 2 m) min m l n 2 t Q. n N +lim sup t Q. (1+a 2 m) f 2 n[u] 2 = n lim (1+a 2 m(t)) min lim m(t) l n 2 t Q. n N t Q. since otherwise the limes would not exist. This proves the theorem. i The previous theorem tells us that the set M n, o doesn t depend on the choice of 0 < o < l 1 /2. Therefore, we define M n :=M n, o for any choice of 0 < o < l 1 /2.

13 Backwards Behavior for viscous Camassa Holm Equations 49 Corollary 3. (I+a 2 A) u 0 2 [u 0 ] 2 [1 For u 0 M n such that [u 0 ] > 4 f /nl 1, we have ln + 2 f 21 1+a n[u ] ln + 2 f 22 (3.6) n[u 0 ] Proof. This follows trivially from Corollary 2 taking o=2 f /n[u 0 ]. i Lemma 1. For u A 1 V, v :=(I+a 2 A) u, we have v 2 v \ [u] 2 2 [u] 2 Also v 2 [u] [ 2 m(1+a2 m) S [u] 2 [u] [ m 2 for m > 0. Since Proof. Observe that v 2 v 2 [u] 2 [u] 2 = 1 v 2 1 = 1 v 2 1 = a2 v 2 1 = a2 v 2 1 = a2 v 2 5 ṽ 2 = v 2 v 2 v 2 (v, v) (Av, v) 2 (v, u) (Av, u) (v, v) Av, v 2 (v, u) u (Av, u) Av, Au 2 (v, u) u (Av, u) (Av, u) Av v, Au (v, u) (Av, u) Au 6 (v, u) u 2 \ 0 1 [u] 2= v 2 v 2 1+a 2 [u] 2 [u] 2 2 (v, u) u 2 the last statement follows. i

14 50 Vukadinović Remark 1. For the solution u of (VCHE) that satisfies the conditions in Theorem 1, we obtain the estimate [u(t)] \1 2 [u(0)]2 + f 2 8n 2 (l n 2 e 4n(l n +o) t f 2 (3.7) +o) 2 8n 2 (l n +o) 2 for t [0, T]. Proof. inequality For t [0, T], we obtain using the last remark Young s d dt [u]2 = 2n(Av, u)+2(f, u) f 2 \ 2n(l n +o)[u] 2 2n(l n +o) u 2 2n(l n +o) f 2 \ 4n(l n +o)[u] 2 2n(l n +o) By the Gronwall inequality, we obtain then (3.7). The next result provides us with a method for producing elements of the sets M n. It will be used for all further results. But first, we need to state the following elementary fact: Remark 2. If {u k } is a bounded sequence in V lim k Q. u k u 0 =0 for some u 0 H, then u 0 V u 0 [ lim inf k Q. u k. Theorem 3. Let u 1, u 2,... V, let t 1 > t 2 > be such that i lim t Q. t j =.. Suppose that for the initial data u j exists on the interval [t j,.). Let us also assume the solution S(t) u j for some constant M > 0, Let there exist some n N such that (I+a 2 A) S(t k ) u k 2 [u k ] [ M, k N (3.8) [S(t k ) u k ] \ f no, k N (3.9) [S(t k ) u k ] 2 [ (l n +o)(1+a 2 (l n +o)), k N (3.10)

15 Backwards Behavior for viscous Camassa Holm Equations 51 Then, there exist u. M n a subsequence {u kj } of {u k } such that lim [S(t) u kj S(t) u. ]=0, t R (3.11) j Q. Proof. We first want to prove that, for every t R, there exists a constant C(t) > 0 such that (I+a 2 A) S(t) u k 2 [ C(t) (3.12) for k big enough. Without loss of generality, we may assume M > f /nl 1. First, we fix k N. By Lemma 1, there exists a unique b k \ t k such that [S(t) u k ] \ f no, t [t k, b k ] [S(t) u k ] [ f no, t \ b k (3.13) (I+a 2 A) S(t) u k 2 [S(t) u k ] 2 [ (l n +o)(1+a 2 (l n +o)), t [t k, b k ] (3.14) Observe that by (2.11) (3.22), we obtain f 2 n 2 o 2=[S(b k) u k ] 2 [ [u k ] 2 e nl 1b k + f 2 (1 e nl 1b k ) n 2 l 2 1 [ e nl 1b k1 M2 f 2 n 2 l 2 1 Therefore, we obtain an upper bound on b k s 2 + f 2 n 2 l 2 1 b k [ 1 nl 1 log o2 (n 2 l 2 1M 2 f 2 ) (l 2 1 o 2 ) f 2 =: t M (3.15)

16 52 Vukadinović With C 1, C 2,... being various constants, (3.7) implies now [S(t) u k ] [1 2 [S(bk ) u k ] f 2 e 4nl n+1 (t b k ) 8n 2 l 2 1 =1 f 2 f 2 n 2 o2+ 8n 2 l e 4nl n+1 (t b k ) =C 1 e 4nl n+1(t b k), t [t k, b k ] From here, using (3.14) (3.15), we conclude (I+a 2 A) S(t) u k 2 [ (l n +o)(1+a 2 (l n +o)) C 1 e 4nl n+1(t b k ) [ C 2 e 4nl n+1(t t M), t [t k, b k ] (3.16) On the other h, for t \ b k from (2.13) the previous calculations, we obtain (I+a 2 A) S(t) u k 2 [ (I+a 2 A) S(b k ) u k 2 e nl 1(t b k ) This (3.16) together imply + f 2 n 2 l 1 (1 e nl 1(t b k ) ) [ (l n +o)(1+a 2 (l n +o))[s(b k )] 2 e nl 1(t b k ) + f 2 n 2 l 1 (1 e nl 1(t b k ) ) [ (l n +o)(1+a 2 (l n +o)) f 2 n 2 o 2=: C 3 (I+a 2 A) S(t) u k 2 [ max{c 2 e 4nl n+1(t t M ), C 3 }=: C(t), t \ t k (3.17) Now we may use the Cantor diagonal process to extract a subsequence {u kj } of {u k } such that the limits lim S(t ki ) u kj =: w i V, j Q. i N exist in the space V. Since S(t): V Q V is a continuous mapping for t \ 0, we obtain w i =S(t ki t kj ) w j, j [ i, i, j N

17 Backwards Behavior for viscous Camassa Holm Equations 53 Letting we obtain u. :=S( t k1 ) w 1 u. =S( t kj ) w j, j N Therefore, u. G. Again, by the continuity of S(t): V Q V, we obtain lim [S(t) u kj S(t) u. ]=0, j Q. t R It remains to prove that u. M n. To this end, we consider two cases. If lim inf k Q. b k =., then (3.11) (3.13) imply [S(t) u. ] [ f no, t R, thus, u. A. If, on the other h, lim inf k Q. b k =b. >., Remark 2 together with (3.11) (3.14), gives (I+a 2 A) S(t) u. 2 [S(t) u. ] 2 [ (l n +o)(1+a 2 (l n +o)), t [ b. In both cases u. M n. i Theorem 4. For each n N, M n is a connected, locally compact subset of V. Proof. Since M n is a S( )-invariant set containing A, it is connected as well. In order to prove the local compactness, it suffices to check that every sequence u 1, u 2,... M n 0A, which is bounded in V, has a subsequence converging to an element of M n. Since lim t Q. [S(t) u k ]=. for k N, there exist t 1 > t 2 > such that lim t Q. t j =. [S(t k ) u k ] \ f /l 1 o. Theorem 3 Corollary 1 imply the wanted. i 4. RICHNESS OF THE SETS M n One of the main results of this paper is the following theorem on the richness of the sets M n. Theorem 5. Let n N. For every p 0 P n H, there exists u. M n such that P n u. =p 0. In other words, P n H=P n M n.

18 54 Vukadinović First, we need to prove a series of lemmas. Lemma 2. Let u C n, o for some n N, let v :=(I+a 2 A) u. Then 1 [Q n u] 2 [ l n+1 l n o ((l n+o)[p n u] 2 [P n u] 2 ) (4.1) In particular [Q n u] 2 [ l n+l n+1 l n+1 l n [P n u] 2 (4.2) [u] 2 [ 2l n+1 l n+1 l n [P n u] 2 (4.3) Proof. Observe that by Lemma 3, we have [u] 2 [u] [ l n+o 2 Therefore, [Q n u] 2 [ 1 l n+1 [Q n u] 2 = 1 l n+1 ( [u] 2 [P n u] 2 ) [ l n+o l n+1 [Q n u] l n+1 ((l n +o)[p n u] 2 [P n u] 2 ) (4.1) follows. The other two inequalities follow from (4.1) the fact that o < l 1 [ l n+1 l n both inequalities follow. i Lemma 3. hold: Let u M n, for some n N. Then, the following estimates [Q n u] [ max3 f 1 no, l n+l n+1 l n+1 l n 2 1/2 [Pn u]4 (4.4)

19 Backwards Behavior for viscous Camassa Holm Equations 55 Q n u [ a n +b n [P n u] (4.5) where 2 f a n := nl 1 (1+a 2 l n+1 ) 1/2 2l b n :=1 n+1 1+a 2 l n+1 2 1/2 1 (l n+1 l n ) 1/2 Also, lim n Q. a n =0 lim inf n Q. b n =0. Proof. If [u] [ f no, then [Q n u] [ [u] [ f no If, on the other h, [u] \ f no, we get (I+a 2 A) u 2 [ (l n +o)(1+a 2 (l n +o))[u] 2 by Corollary 1. Applying Lemma 2, we obtain (4.4). Equation (4.5) follows from (4.4), Q n u [ (1+a 2 l n+1 ) 1/2 [Q n u] By the properties (2.1) (2.2) of the eigenvalues of the Stokes operator A, we obtain lim n Q. a n =0 lim inf n Q. b n =0. i Lemma 4. Let p 0 P n H for some n N. Then, for every t 0 > 0, there exists w 0 P n H such that P n S(t 0 ) w 0 =p 0. Proof. there exists First observe that, for every u 0 P n H such that [u 0 ] > f no, y(u 0 )=min3 y > 0 : [Pn S(y) u 0 ]= f no 4 > 0

20 56 Vukadinović Note also that with v 0 :=(I+a 2 A) u 0 we have v 0 2 [u 0 ] 2 [ l n(1+a 2 l n ) [ (l n +o)(1+a 2 (l n +o)) Therefore, by Theorem 1, we have for t [0, y 0 ] Lemma 2 now implies (I+a 2 A) S(t) u 0 2 [ (l [S(t) u 0 ] 2 n +o)(1+a 2 (l n +o)) [S(t) u 0 ] 2 \ [u 0 ] 2 e 4nl n+1t f 2 8n 2 l 2 n for t [0, y 0 ]. Therefore, [S(t) u 0 ] 2 [ 2l n+1 l n+1 l n [P n S(t) u 0 ] 2 [P n S(t) u 0 ] 2 \ l n+1 l n 2l n+1 [S(t) u 0 ] 2 \ l n+1 l n 2l n+1 1 [u0 ] 2 e 4nl n+1t f 2 8n 2 ln2, t [0, 2 y0 ] (4.6) For r > 0 we define B(r) :={u 0 V : [u 0 ] < r} B n (r)=b(r) 5 P n H. We now want to prove that there exists r 0 > f no (4.7) such that [P n S(t) u 0 ] > [p 0 ], u 0 P n H0B n (r 0 ), t [0, t 0 ] (4.8) It is sufficient to consider only the case [p 0 ] \ f. We fix any r no 0 > f no that l n+1 l n 2l n+1 1 r 2 such 0e 4nl n+1t 0 f 2 8n 2 ln2 > 2 [p0 ] 2 (4.9)

21 Backwards Behavior for viscous Camassa Holm Equations 57 For [u 0 ] \ r 0, we obtain [p 0 ] \1 f 2 no 22 =[P n S(y 0 ) u 0 ] 2 \ l n+1 l n 2l n+1 1 r 2 0e 4nl n+1y 0 f 2 (4.10) 8n 2 ln2 2 Combining (4.9) (4.10), we get t 0 < y 0. In particular, (4.6) holds for t [0, t 0 ]. From (4.6) (4.9) we obtain [P n S(t) u 0 ] > [p 0 ] (4.11) for all u 0 P n H0B n (r 0 ), t [0, t 0 ]. In order to prove the lemma, let us choose a continuous function h: R Q R such that h(x)=1 for x [ r 0 h(x)=0 for x \ 2r 0. Define g(u 0 )=P n S(h([u 0 ] t 0 ) u 0, u 0 P n H By (2.11) we have S(t) B(r) B(r), r \ f nl 1, t \ 0 Therefore, g(b n (2r 0 )) B n (2r 0 ) Also, g is continuous, it satisfies g(u 0 )=u 0 for [u 0 ]=2r 0. We can choose r 0 large enough that p 0 B n (2r 0 ). By Brouwer s Fixed Point Theorem, there exists w 0 B n (2r 0 ) such that g(w 0 )=p 0. If we show that [w 0 ] [ r 0, by the definition of g, we would have P n S(t 0 ) w 0 =g(w 0 )=p 0 this is exactly the claim of this lemma. To this end, let s assume [w 0 ] > r 0. Then, [g(w 0 )]=[P n S(h([w 0 ] t 0 ) w 0 ] > [p 0 ] by (4.8). This is a contradiction, the lemma is proven. i Now we have the tools to prove Theorem 5.

22 58 Vukadinović Proof of Theorem 5. Let p 0 P n H. Choose a sequence 0 > t 1 > t 2 > t 3 > such that lim k Q. t k =.. By Lemma 4, there exist u 1, u 2,... V p 1, p 2,... P n H such that S( t k ) p k =u k, k N P n u k =p 0, k N Let us define U k (t) :=S(t) u k, t \ t k V k (t) :=(I+a 2 A) U k (t), t \ t k Let us assume first that [p k ] [ f no (4.12) for infinitely many k N. Without loss of generality, we may assume that this is true for all k N. By the Poincaré inequality, we have V k (t k ) = (1+a 2 A) p k [ l 1/2 n (1+a 2 l n ) 1/2 [p k ] [ l 1/2 n (1+a 2 1/2 f l n ) no The inequality (3.21) gives V k (t) [ l 1/2 n (1+a 2 1/2 f l n ) no, t \ t k Therefore, [U k (t)] [ l1/2 n (1+a 2 l n ) 1/2 f l 1/2 1 (1+a 2 l 1 ) 1/2 no, t \ t k (4.13) Similarly, as in the Theorem 3, we may, by passing to a subsequence, assume that lim [U k (t) S(t) u. ]=0, t R k Q.

23 Backwards Behavior for viscous Camassa Holm Equations 59 for some u. G. From here it follows that P n u. =P n u 1 =p 0, using (4.13), u. A M n. This proves the Lemma under the assumption (4.12). Let us now assume that [p k ] \ f no for infinitely many k N. Again, by passing to a subsequence, we may assume that this is true for all k N. Since either [u k ] [ f or [u no k] > f for no each k N, by Lemma 2 [u k ] [ max3 f no, 1 2l n+1 l n+1 l n 2 [p0 ]4, k N Since V k (t k ) 2 [U k (t k )] 2= (I+a2 A) p k 2 [p k )] 2 [ (l n +o)(1+a 2 (l n +o)), k N the assumptions of the Theorem 3 are satisfied. We obtain u. M n, so lim [U k (t) S(t) u. ]=0, t R k Q. Exactly as in the previous case we obtain P n u. =p 0. This completes the proof of the theorem. i 5. SOME DENSITY PROPERTIES OF THE SETS M n This section is concerned with questions on the density of the sets M n. It consists of two results. The first one is an analogue of a result by Constantin et al. (1997), which positively answers the conjecture by Bardos Tartar (1973). It concerns the (NSE), postulates that S(t) H is dense in H for t > 0. However, the density is shown in a weaker norm then the natural energy norm of (NSE). Here, we find ourselves in a similar situation. We prove that 1 n N M n is dense in H, but again in a weaker norm then the natural energy norm for the (VCHE). In the second result we show that the eigenvectors of the Stokes operator A corresponding to l n for some n N belong to the closure of scaled set M n. This time, the closure is taken with respect to the natural

24 60 Vukadinović energy norm for the (VCHE). The analogue of this result for the (NSE) has been proved by C. Foias I. Kukavica, is yet to be published. Theorem 6. The set 1 n N M n ( G) is dense in H. Proof. Let u 0 H be arbitrary. By Theorem 3, for each n N there exists u n M n such that P n u n =P n u 0. From Lemma 3, we have the estimate This implies Q n u n [ a n +b n [P n u n ] u n u 0 = Q n (u n u 0 ) [ Q n u n + Q n u 0 By Lemma 3, we obtain [ a n +b n [P n u n ]+ Q n u 0 =a n +b n [P n u 0 ]+ Q n u 0 lim inf n Q. u n u 0 =0 the theorem is proven. i Theorem 7. Let w be an eigenvector of the Stokes operator A corresponding to the eigenvalue l n for some n N, i.e., let Aw=l n w, let [w]=1. Then w 3 u [u] : u M n4 where the closure is taken in the norm [ ]. In particular, P n H M n +M n + +M z n kn where k n :=dim P n H=m 1 +m 2 + +m n, m j being the multiplicity of l j. Proof. Let r > 4 f /nl 1. By Theorem 6, the exists u r M n, such that P n u r =rw. Also, by Lemma 2, Corollary 3, we have [Q n u r ] 2 [ 2 f n[u r ] [P nu r ] 2 l n+1 l n 2 f n[u r ] 2 f r 2 [ nr(l n+1 l n ) 2 f

25 Backwards Behavior for viscous Camassa Holm Equations 61 Thus, [Q n u r ] 2 2 f [ r 2 nr(l n+1 l n ) 2 f Q 0 [u r ] 2 r 2 = [P nu r ] 2 +[Q n u r ] 2 r 2 Q 1 when we let r Q.. Finally 5 u r [u r 6 ] w = 5 u r rw [u r 1 ] w r 26 1 [u r ] [ [Q nu r ] r r +: : 1 Q 0 [u r ] when r Q.. This proves the theorem. i ACKNOWLEDGMENTS I would like to thank C. Foias for many useful ideas comments, the careful reading through the manuscript. REFERENCES 1. Bardos, C., Tartar, L. (1973). Sur l unicité rétrograde des équations paraboliques et queliques questions voisines. Arch. Rational Mech. Anal. 50, Camassa, R., Holm, D. D. (1993). A completely integrable dispersive shallow water equation with peaked solutions. Phys. Rev. Lett. 71, Camassa, R., Holm, D. D., Hyman, J. M. (1994). A new integrable shallow water equation with peaked solutions. Adv. Appl. Mech. 31, Chen, S., Foias, C., Holm, D. D., Olson, E., Titi, E. S., Wynn, S. (1998). A connection between the Camassa Holm equations turbulent flows in channels pipes. Phys. Fluids 11, Constantin, A., Foias, C. (1988). Navier Stokes Equations, Chicago Lectures in Mathematics, Chicago/London. 6. Constantin, A., Foias, C., Kukavica, I., Majda, A. J. (1997). Dirichlet quotients 2D periodic Navier Stokes equations. J. Math. Pures Appl. 76, Constantin, A., Foias, C., Temam, R. (1988). On the dimension of attractors in twodimensional turbulence. Physica D 30, Doering, C., Gibbon, J. D., Holm, D., Nicolaenko, B. (1988). Low-dimensional behavior in the complex Ginzburg Lau equation. Nonlinearity 1, Foias, C., Kukavica, I., private communications. 10. Foias, C., Saut, J. C. (1981). Limite du rapport de l enstrophie sur l énergie pour une solution faible des équations de Navier Stokes. C. R. Acad. Sci. Paris, Sér. I Math. 298,

26 62 Vukadinović 11. Foias, C., Saut, J. C. (1984). Asymptotic behavior, as t Q., of solutions of Navier Stokes equations nonlinear spectral manifolds. Indiana Univ. Math. J. 33, Foias, C., Sell, G. R., Temam, R. (1988). Inertial manifolds for nonlinear evolutionary equations. J. Diff. Eq. 73, Holm, D. D., Marsden, J. E., Ratiu, T. S. (1998). Euler Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80, Holm, D. D., Marsden, J. E., Ratiu, T. S. (1998). Euler Poincaré equations semidirect products with applications to continuum theories. Adv. Math. 137, Kukavica, I. (1992). On the behavior of the Solutions of the Kuramoto Sivashinsky Equations for negative time. J. Math. Anal. Appl. 166, Temam, R. (1984). Navier Stokes Equations: Theory Numerical Analysis, North- Holl, Amsterdam.