Density of global trajectories for filtered Navier Stokes equations*

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 7 (004) NONLINEARITY PII: S (04) Density of global trajectories for filtere Navier Stokes equations* Jesenko Vukainovic Department of Mathematics, Van Vleck Hall, 480 Lincoln Dr., Maison, WI 53706, USA vukaino@math.wisc.eu Receive 8 September 003, in final form 3 January 004 Publishe 7 February 004 Online at stacks.iop.org/non/7/953 (DOI: 0.088/ /7/3/0) Recommene byestiti Abstract For two-imensional perioic Kelvin-filtere Navier Stokes systems, both positively an negatively invariant sets M n, consisting of initial ata for which solutions exist for all negative times an exhibiting a certain asymptotic behaviour backwars in time, are investigate. They are proven to be rich in the sense that they project orthogonally onto the sets of lower moes corresponing to the first n istinct eigenvalues of the Stokes operator. In general, this yiels the ensity in the phase space of trajectories of global solutions, but with respect to a weaker norm. This result applies equally to the two-imensional perioic Navier Stokes equations (NSEs) an the two-imensional perioic Navier Stokes-α moel. We esignate a subclass of filters for which the ensity follows in the strong topology inuce by the (energy) norm of the phase space, as originally conjecture for the NSEs by Baros an Tartar (973 Arch. Ration. Mech. Anal ). Mathematics Subject Classification: 35Q30, 76D05, 35B40, 35B4. Introuction In the Eulerian representation of fluis, Navier Stokes equations (NSEs) express the relationship between velocity v(x, t), ensity ρ(x,t), an pressure p(x, t) at point x R, =, 3, an at time t R. In the case of incompressible homogenous fluis, the ensity is constant in space an time. Assuming for simplicity that ρ, the balance of momentum yiels t v νv + (v )v + p = f, * This work was partially supporte by the NSF grant DMS while the author was a grauate stuent at Iniana University /04/ $ IOP Publishing Lt an LMS Publishing Lt Printe in the UK 953

2 954 J Vukainovic where ν > 0 represents the kinematic viscosity an f represents a given boy forcing. The constraint of incompressibility can be expresse through the ivergence-free conition v = 0. The system is nonlinear an nonlocal. Due to viscosity, the total kinetic energy is issipate forwar in time, proviing the strongest source of information currently known about NSEs. The existence an uniqueness of a strong (regular) solution to the initial value problem v(0) = v 0, subject to aequate bounary conitions, has been establishe efinitively for all positive times only for space imension =. In this case, the equations efine a forwar regularizing flow for which there exists a compact absorbing set. This, in turn, gives rise to the existence of the global attractor the essential object for unerstaning the long time behaviour of the solutions. In the three-imensional case, the energy issipation is use to construct Leray weak solutions for all positive times. In general, however, regularity an uniqueness of such solutions is guarantee for a finite interval of time only. In both theoretical an numerical stuies of NSEs, certain approximations are often useful. Usually, they are solutions of partial ifferential equations that have the NSE as a limiting case. Accoring to the properties of the NSE that they preserve, we istinguish two important classes of approximations. One class, which inclues Galerkin approximations an Leray regularization (mollifie equations), preserves the energy balance equation for the NSEs, but alters the vorticity equation. In the other, which inclues Kelvin-filtere NSEs (KFNSEs), the opposite is the case: the vorticity equation has the same structure as the one for the NSE, whereas the energy balance equation is approximate. The KFNSEs have the following form: t v νv + (u )v + ( u) T v + Q = f, u = Fv, v = 0. The functions v an u enote the Eulerian flui velocity an the filtere Eulerian flui velocity, respectively. The invisci one-imensional version of those equations was originally erive using the Euler Poincaré variational framework for the special choice of filter F = (I α ) (see [0]). The so-calle Navier Stokes-α moel was obtaine in [, ], by generalizing the equation to higher imensions, an aing the viscous term in an a hoc fashion. Due to the special geometric an physical properties, this system is use to moel turbulent flows (see [, 5]). The equations can also be erive in a more general setting by filtering the velocity in the NSE in a manner that preserves a form of the Kelvin circulation theorem, or the structure of the vorticity equation (see [5]). In this paper, the filtering operator F will be given using Fourier multipliers F = ( ) = ( ), where :[0, ) [, ) is a nonecreasing convex function with (0) =. If, the KFNSEs are nothing else but the NSEs. Many issipative ifferential equations of flui ynamics, incluing the NSE an the KFNSE, can be written in the following form: v + νav + B(v, v) = f, (.) where A is a close positive self-ajoint linear operator an B is a bilinear form. Other examples inclue complex Ginzburg Lanau equations, Lorenz equations, Kuramoto Sivashinsky equations an the original Burger s equation. Uner certain conitions, these systems are solve by nonlinear compact forwar regularizing issipative semigroups S(t) in aequate Hilbert spaces H. They possess global attractors compact invariant sets consisting of all omega-limit sets, which can be characterize by A = {u 0 H : S(t)u 0 extens to, an is boune for all negative times}. Attempts to evelop practical algorithms for the stuy of the long time behaviour of those equations, such as various interpolation techniques to etermine whether a solution is on the global attractor, have le to the stuy of solutions for negative times (see [ 4]). There has been a series of results in that respect, an a wie range of phenomena was observe. For the Kuramoto Sivashinsky equations an the complex Ginzburg Lanau equations, it was shown that all global solutions

3 Density of global trajectories 955 are boune (see [, 9]), an thus containe in the attractor. In contrast, the global solutions to the two-imensional perioic NSEs, which exhibit exponential backwars growth, exist, an are, in fact, quite rich (see [6]). In this paper, we will show that this is the case for two-imensional perioic KFNSEs, in general. Baros an Tartar (973) conjecture that the set of the initial ata for which the solution exists backwars for a given interval of time (t 0, 0] is ense in the phase space (see []). In [6], the authors prove that for the perioic two-imensional NSE, the set of initial ata for which the solutions exten to all negative times an grow backwars at most exponentially are ense in the phase space, but with respect to a weaker norm. In this paper, we will show that the same is generally the case also for the two-imensional perioic KFNSEs. However, if the filter satisfies the conition lim sup n ( n+ )/( n ) = ( i being the ith istinct eigenvalue of ), the ensity follows with respect to the strong topology inuce by the natural energy norm of the phase space (Fv, v) /.. Kelvin-filtere NSEs.. Filtere viscous flui equations We consier the NSEs an relate equations supplemente with space-perioic bounary conitions: the velocity v(x, t) an the pressure p(x, t) will be require to be L-perioic in every space coorinate x i, i =,...,. We enote = [0,L]. Let u Hper ( (0,T)) be a strong solution of the NSE on some interval [0,T). Two important quantities for the solution are the total kinetic energy e(t) := v(x, t) x, an the enstrophy E(t) := v(x, t) x. The strongest source of quantitative information in the stuy of the NSE is the energy balance equation e t + νe = v f x, (.) which we obtain after multiplying the NSE by v, an integrating by parts. The nonlinear term cancels out because of the ivergence-free conition an the ientity u i ( i v j )v j + ( i u i )vj = i (u i vj ). Another important tool is the vorticity equation, obtaine by taking the curl in the NSE. Denoting the vorticity by q = curl v, it reas t q νq + (v )q (q )v = curl f. (.) The existence of solutions of NSE is generally prove by constructing regular approximate solutions, an passing to the limit as the approximation parameter tens to zero (or infinity). Approximations of particular interest are the ones that preserve a certain property of the NSE, for example, the energy balance equation or the vorticity equation. Here, we escribe two ways of constructing regular approximate solutions, the Leray regularization an the KFNSEs. In both cases we moify the nonlinear term of the NSE by introucing a spatially filtere Eulerian flui velocity, while the linear term remains unchange. The velocity can be filtere by mollifying: u = Fv = J δ v, where J is a positive, symmetric, normalize, smooth kernel that ecays sufficiently fast at infinity, an J δ (ξ) = δ J(ξ/δ), or using the Fourier multipliers: u = Fv = ( δ)v, where :[0, ) [, ) is a nonecreasing convex function with (0) =. The Leray regularization of the NSE is given by t v νv + (u )v + p = f, u = Fv, v = 0. (.3) A solution of the initial bounary value problem for (.3) with the initial conition v(0) = v 0 exists for all positive times, an is real analytic an unique (see [4]). One can easily see that solutions satisfy the same energy balance equation (.). The KFNSEs rea t v νv + (u )v + ( u) T v + Q = f, u = Fv, v = 0. (.4)

4 956 J Vukainovic The solution to the initial value problem is smooth an global (see [3, 4, ]). Taking the curl in (.4) yiels the vorticity equation t q νq + (u )q (q )u = curl f, where q = curl v is the vorticity. Observe that it has the same structure as the vorticity equation of NSE. Another important analogy to the NSEs is that KFNSEs have a Kelvin circulation theorem v x = (νv + f) x, t γ(u) γ (u) where γ (u) is a close path moving with the spatially filtere velocity u. In aition, in the two-imensional case, the enstrophy balance equation for the NSE is preserve exactly: E +ν v x = v f x. t In contrast, the Leray regularization oes not have a Kelvin circulation theorem, oes not preserve the structure of the vorticity equation, or the enstrophy balance equation in two imensions. However, KFNSEs o not have the same energy balance as the NSEs. Instea,we have an approximate equation t u v x + ν u i v i x = u f x... Mathematical framework for two-imensional perioic KFNSEs... Function spaces, Helmholtz Leray ecomposition, an the Stokes operator. In orer to stuy the perioic KFNSEs in a functional setting, we introuce suitable function spaces. By L per () an Hper l () we enote the spaces of vector fiels u : R R, which are L-perioic in each space variable, have zero space average u(x) x = 0 an belong, respectively, to L (O) an H l (O) for every boune open set O R. Let { } V = u Cper () : u = 0, u = 0 be the set of test functions. We introuce the classical spaces H an V to be closures of V in the (real) Hilbert spaces L () an H (), respectively. The spaces H an V are also (real) Hilbert spaces with respective scalar proucts (u, v) = j= u j v j an ((u, v)) = j,k= u j x k v j x k, u,v V. The corresponing norms are enote by u =(u, u) / an u =((u, u)) /, respectively. We enote the ual of V by V. By the Rellich embeing theorem, the natural inclusions i : V H an i : H V are compact. The Helmholtz Leray ecomposition resolves any vector fiel w L per () uniquely into the sum of a curl an a graient vector fiel w = u + p with u H, an p H per (). The map w u is well efine, an it is the orthogonal projection (calle the Helmholtz Leray projector) on the space H in L per (). We enote it by P L : L per () H ; observe that H ={ p : p H per ()} is the orthogonal complement of the space H in L per (). By A = P L we efine the Stokes operator with omain D(A) = H per () V. In the space perioic case Au = P L u = u, u D(A). The Stokes operator A is one-to-one from D(A) onto H. The inverse A : H D(A) is compact. Integration by parts verifies

5 Density of global trajectories 957 that the Stokes operator is self-ajoint (Au, v) = (u, Av), an positive (Au, u) = u > 0, u D(A)\{0}. Thus, A is also self-ajoint an positive. From the elementary spectral theory of compact, self-ajoint operators on Hilbert spaces, there exists a sequence of eigenvectors w m,m=,,... of A forming an orthonormal basis in H, such that the corresponing sequence of eigenvalues forms a nonincreasing sequence σ m,m=,,... of real, positive numbers accumulating at 0. Let λ m := /σ m. Note that the Stokes operator A can be extene to an isomorphism between V an V. Then, D(A) ={u V : Au H }. We expan each vector fiel u H in a Fourier or spectral expansion u = m= ûmw m, where û m := (u, w m ). Parseval s ientity reas u = m= û m. For a function : R + R + let us efine { } H := u = û m w m : (λ m ) û m < an m= m= { V := u = û m w m : m= m= } λ m (λ m ) û m <, enowe with inner proucts (u, v) = (λ m )û m ˆv m an ((u, v)) = λ m (λ m )û m ˆv m, m= respectively. The corresponing norms are u = m= (λ m) û m an u = m= λ m(λ m ) û m. Let us also efine Fourier multipliers (A)u = m= (λ m)û m w m, u H. One can verify that D(A / ) = V an D(A / ) = V. For u V, Au = m= λ mû m w m an, thus, u = (Au, u) = A / u = m= λ m û m. We arrange the eigenvalues λ,λ,...of the Stokes operator in strictly increasing orer (π/l) = < <. It is known that n n, asn, an n+ n.an important spectral gap property in the two-imensional case is lim sup n n+ n log n m= > 0 (.5) (see [3]). This implies, in particular, lim n n+ / n =. We efine P n to be the orthogonal projection in H on the spectral space of A corresponing to the eigenvalues,,..., n, respectively; also let Q n := I P n. We have several variations of the Poincaré inequality Q n u n+ Q n u, u V an, if is increasing, Q n u ( n+) Q n u, u H.... Nonlinear terms. Regaring the nonlinear terms that appear in the NSEs an the KFNSEs, we introuce the following trilinear forms: v i b(u, v, w) = u j w i x = (u )v w x xj an b u j (u, v, w) = v j w i x = (( u) T v) w x, xi

6 958 J Vukainovic where u, v, w Cper () are ivergence-free. Using the fact that u is ivergence-free, the prouct rule an the partial integration give b(u, v, w) = b(u, w, v), an in particular b(u, v, v) = 0. The two trilinear forms are relate through the ientity: b u j u j (u, v, w) = v j w i x = w i v j x xi xi = b(w, u, v) = b(w, v, u). In the two-imensional case we have the inequality b(u, v, w) c u / u / v / v / w, which allows us to exten b an b on V V V. In particular, this allows us to efine B(u, v) an B (u, v) V for u, v V through (B(u, v), w) = b(u, v, w) an (B (u, v), w) = b (u, v, w) = b(w, v, u), respectively. If u or v is in D(A), one can prove that B(u, v), B (u, v) H. In this case, we have B(u, v) = P L ((u )v), an B (u, v) = P L (v j u j ). We have the following orthogonality property: (B(u, v), v) = 0, u,v V. (.6) In the two-imensional perioic case with zero space average, the inertial term satisfies a further important orthogonality property, known as the enstrophy invariance (B(u, u), Au) = 0, u D(A). Actually, an even stronger form of enstrophy invariance hols: (B(u, v), Av) = (B(Av, v), u), u V, v D(A). (.7) Recall that the connection between B an B is given by the ientity (B (u, v), w) = (B(w, v), u), u, v, w V. (.8) Using ientities (.6) (.8), we obtain the following important orthogonality properties for the nonlinear term B B of the KFNSEs ((B B )(u, v), u) = (B(u, v), u) (B(u, v), u) = 0, u,v V, (.9) an for u V, v D(A) ((B B )(u, v), Av) = (B(u, v), Av) (B(Av, v), u) = 0. (.0).3. Functional form of two-imensional perioic KFNSEs an some elementary facts.3.. Functional form. Let :[0, ) [, ) be a twice ifferentiable function such that 0, 0 an (0) =. Let ψ(ξ) = ξ(ξ), υ(ξ) = (ξ)/ξ an χ(ξ) = ξ(ξ). Applying the Helmholtz Leray projector P L on the KFNSEs (.4), we obtain the following functional form of the equations: v + νav + (B B )(u, v) = f, (.) v = (A)u, where without loss of generality we assume f H. Observe that the term containing moifie pressure oes not occur in this equation since P L ( Q) = 0. This new functional version of the KFNSE is unerstoo in V. Classical theorems imply that, for every u 0 H, there exists a unique solution u(t) = S(t)u 0 for t 0 of (.), which satisfies u(0) = u 0 an u C b ([0, ), H ) C((0, ), V ). If the solution u(t) also exists for t [t 0, 0] for some t 0 < 0, then it is still uniquely etermine by u 0, an therefore we may enote the solution by S(t)u 0 wherever it is efine. Also, for any t 0 > 0, the solution operator S(t 0 ) : H H is continuous.

7 Density of global trajectories Energy an enstrophy inequalities. Our aim is now to fin balance equations for the KFNSE that o not involve terms in which trilinear forms b an b appear. Therefore, keeping (.9) an (.0) in min, if we multiply the KFNSE by u, we obtain the so-calle energy balance equation t u + ν(av, u) = (f, u) (.) an if we multiply it by Av, we obtain the so-calle enstrophy balance equation t v + ν(av, Av) = (f, Av). (.3) Using Young s inequality, these equations yiel, respectively, the following useful inequalities: t u + ν u f ν (.4) an t u + ν u f ν. (.5) Observe that relation (.4) yiels that t u(t) is a ecreasing function as long as u(t) > f /(ν ), an (.5) yiels that t u(t) is ecreasing as long as u(t) > f /(ν / ). If u is a solution of the KFNSE efine on some interval [t 0, ), the Gronwall lemma gives u(t) u(t 0) e ν (t t 0 ) f ( + e ν (t t 0 ) ) ν ) f = ( u(t 0 ) ν e ν (t t 0 ) f + ν, t t 0. (.6) Similarly, u(t) u(t 0 ) e ν (t t 0 ) f ( + e ν (t t 0 ) ), ν t t 0. (.7) Also, if the solution is efine on [t,t 0 ]wehave u(t) u(t 0) eν (t 0 t) f ( e ν (t 0 t) ν ) ) f = ( u(t 0 ) ν e ν (t 0 t) f + ν. (.8) Observe that, if the solution is efine on (,t 0 ], an if u(t 0 ) > f /ν the integral t0 (/ u(t) ) t < Global solutions an attractor. Every solution u(t) = S(t)u 0 of the KFNSE, which can be extene for all t<0, is calle a global solution. The set of initial ata, for which there exists a global solution, can be efine as G := t0 S(t)H. (.9) Let ρ 0 := f /ν. Inequality (.6) implies lim sup u(t) ρ 0. t

8 960 J Vukainovic Moreover, the balls B H (0,ρ) in H with ρ ρ 0 are positively invariant for the semigroup S(t), an for ρ>ρ 0 they are absorbing balls. Let us efine A := t0 S(t) { u 0 H : u 0 ρ 0 }. This set is the global attractor for the KFNSE. It is the smallest compact, invariant set that attracts all the solutions. Even more, it attracts all the boune sets in H uniformly. It can be characterize in the following way: { } A = u 0 G : lim sup S(t)u 0 < t { = u 0 G : S(t)u 0 f },t R ν { } = u 0 G : S(t)u 0 f ν /,t R The following properties will be neee (see [5]): A is a nonempty compact connecte subset of H. A is positively an negatively invariant, i.e. S(t)A = A for t R. positively invariant set containing A is connecte. F (A) <, where F enotes the fractal imension. 3. Backwar asymptotic behaviour of the global solutions. Moreover, any 3.. Dirichlet quotients. Definition an some properties of the sets M n 3... Filtere Dirichlet quotients. The quotients ( v, v) (u, v) = (Av, v) (u, v) = u u are referre to as filtere Dirichlet quotients. Since both energy an enstrophy satisfy ifferential equations that o not involve the nonlinear term B B (see (.) an (.3)), the Dirichlet quotients o as well. The asymptotic behaviour of the Dirichlet quotients for the NSE when t has been stuie extensively (see [7, 8]). In [6], the asymptotic behaviour of the Dirichlet quotients for the two-imensional perioic NSE when t was examine. Here, we will concentrate on the asymptotic behaviour of the filtere Dirichlet quotients for the two-imensional perioic KFNSE when t. Let us first efine the following locally compact cones. Definition. Let n N, an 0 <κ ( n+ n )/. Let { } u 0 C n,κ := u 0 V : u 0 ψ( n + κ). The following theorem on the behaviour of the Dirichlet quotients will be the main technical tool in the stuy of the backwars behaviour of the two-imensional perioic KFNSE. Theorem. There exists c > 0 such that for any n N, 0 <κ ( n+ n )/, u 0 C n,κ, an T>0: S(t)u 0 > c f νκ, t [0,T) S(t)u 0 C n,κ, t [0,T].

9 Density of global trajectories 96 Remark. Observe that in the case f = 0, the last theorem states that the cones C n,κ are positively invariant uner the solution operator S. Proof. Let us enote ũ := u/ u, v = (A)u an ṽ := v/ u. Also, let f = f m w m an ũ = ˆũ m= m w m. Obviously, ũ =. Let us first assume that v(0) D(A). Applying (.) an (.3) we obtain v t u = ( u 4 t v u v ) t u = u [( ν(av, Av) + (f, Av)) ṽ ( ν(av, u) + (f, u))] = ν(aṽ, Aṽ ṽ ũ) + ( f/ u,aṽ ṽ ũ ). Thus, we obtain t ṽ + ν(aṽ, Aṽ ṽ ũ) = (f/ u,aṽ ṽ ũ). (3.) Foragivent 0 let us efine µ(t) to be the solution of the equation ψ(µ(t)) = µ(t)(µ(t)) = ṽ(t). (3.) Since (ṽ, Aṽ ṽ ũ) = ṽ ṽ ũ = 0, we have ( ) Aṽ, Aṽ ṽ ũ = ( Aṽ µṽ, Aṽ ṽ ũ ) = = (λ m µ)(λ m )(λ m (λ m ) ṽ ) ˆũ m m= (λ m µ)(ψ(λ m ) ψ(µ))(λ m ) ˆũ m. The latter is a positive quantity, since ψ is increasing. With ψ(x) ψ(y), x y, γ(x,y) := x y ψ (x), x = y, (Aṽ, Aṽ ṽ ũ) = (λ m µ) γ(µ,λ m )(λ m ) ˆũ m. m= m= m= ˆ On the other han, applying Young s inequality, an assuming for a moment that µ {λ,λ,...}, we obtain the following inequality for the term involving the force f : ( f/ u,aṽ ṽ ũ ) f = (ψ(λ m ) ψ(µ)) ˆ m ˆũ m u m= ν (λ m µ)(ψ(λ m ) ψ(µ))(λ m ) ˆũ m m= ψ(λ m ) ψ(µ) + ν u (λ m= m )(λ m µ) fˆ m. Thus, t ṽ + ν (λ m µ) γ(µ,λ m )(λ m ) ˆũ m ν u m= m= γ(µ,λ m ) (λ m ) ˆ f m.

10 96 J Vukainovic Since (λ m µ) γ(µ,λ m )(λ m ) ˆũ m [ min (λm µ) γ(µ,λ m ) ], m= we also obtain [ t ṽ + ν min (λm µ) γ(µ,λ m ) ] γ(µ,λ m ) m N ν u (λ m= m ) fˆ m. (3.3) If for some t 0 [0,T]: ũ(t 0 ) = ṽ(t 0 ) = ψ( n + κ), then µ(t 0 ) = n + κ. Since ψ is convex,wehave min [(λ m µ) γ(µ,λ m )] = (µ n ) γ(µ, n ) = κ γ( n + κ, n ). m N Let for m N γ( n + κ, λ m ) c(m, n, κ) := (λ m )γ ( n + κ, n ). In the Navier Stokes case,, so c(m, n, κ) =. Otherwise, because of the convexity of ψ, ifλ m n ψ( n + κ) ψ(λ m ) ψ( n + κ) ψ( n ), n + κ λ m κ so c(m, n, κ). If λ m n+,wehave c(m, n, κ) λ m/(λ m ( n + κ)) (ψ( n + κ) ψ( n ))/κ. Since λ λ/(λ ( n + κ)) is ecreasing, n+ c(m, n, κ) ( n+ ( n + κ))ψ ( n ) n+ ( n+ n )(+ (0) n ). Therefore, c := sup { c(m, n, κ) / : m N, n N, κ (0,( n+ n )/) } <. Now, (3.3) implies ( ) t ṽ(t 0) c(m, n, κ) f γ( n + κ, n ) ν u(t 0 ) νκ ( ) c f γ( n + κ, n ) ν u(t 0 ) νκ. The right-han sie is negative by one of our assumptions. Therefore, t ṽ(t 0) < 0, so ṽ(t) ecreases in a neighbourhoo of t 0. Since ṽ(0) ψ( n + κ), an because of the latter, ṽ(t) cannot excee ψ( n + κ) on the interval [0,T]. This proves the theorem in the case v(0) D(A). Otherwise, we fix ɛ (0,T). There exist 0 <ɛ,ɛ <ɛsuch that v(ɛ ) D(A), an ṽ(ɛ ) ψ( n + κ + ɛ ). Applying the first part of the proof an then letting ɛ 0 completes the proof. m N

11 Density of global trajectories Definition an properties of sets M n. Definition. For n N an 0 <κ ( n+ n )/, let us efine the set M n,κ := { S(t)u 0 A u 0 G\A : lim sup t S(t)u 0 ψ( n + κ). The set M n,κ is positively an negatively invariant: S(t)M n,κ = M n,κ for t R. Aswe will see, it oes not really epen on κ. This enables us to efine the set M n as M n,κ for an arbitrary choice of 0 <κ ( n+ n )/. Let us begin with two corollaries of theorem. Corollary. Let n N an 0 <κ /. Let u 0 M n,κ.if u 0 c f /νκ, then there exists a unique t 0 0 such that S(t)u 0 > c f νκ, t (,t 0), S(t)u 0 c f νκ, t [t 0, ), an } S(t)u 0 S(t)u 0 ψ( n + κ), t (,t 0 ]. Proof. By our previous remarks, t S(t) is ecreasing as long as S(t)u 0 > ρ 0, an lim sup t S(t)u 0 ρ 0. Since c f /νκ > ρ 0, there exists t 0 > 0 such that S(t 0 )u 0 = c f /νκ. The first two inequalities follow immeiately. Now let t (,t 0 ]. Since lim sup t S(t)u 0 / S(t)u 0 ψ( n + κ), there exists t t such that S(t )u 0 / S(t )u 0 ψ( n + κ). By virtue of theorem, the last inequality follows. Corollary. For each n N, 0 <κ /, { M n,κ u 0 V : u 0 < c } f Cn,κ. νκ Proof. This follows trivially from the last corollary. Theorem. We can characterize the sets M n,κ in the following way: M n,κ = { S(t)u 0 A u 0 G\A : lim t S(t)u 0 {ψ( ), ψ( ),...,ψ( n )} In particular, M n,κ oes not epen on the choice of κ. Moreover, for every u 0 G\A, S(t)u 0 lim {ψ( ), ψ( ),...} { }. t S(t)u 0 Proof. Inequality (3.3) implies t ṽ ν u m= Since γ(µ,λ m ) (λ m ) ˆ f m. t ṽ(t) = t ψ(µ(t)) = ψ (µ(t))µ (t), }.

12 964 J Vukainovic we obtain the inequality µ (t) ν u m= f γ(µ,λ m ) ψ (µ)(λ m ) fˆ m c ν u. (3.4) Let u 0 M n,κ \A an u(t) = S(t)u 0. Let {t n } an {T n } be two sequences of real numbers such that t n <T n an lim n t n = lim n T n =, an so that an u(t n ) lim n u(t n ) = lim sup t u(t) u(t) In particular, u(t n ) lim n u(t n ) = lim inf t lim µ(t n) = lim sup µ(t), n t u(t) u(t). lim µ(t n) = lim inf n t Integrating inequality (3.4) on the interval [t n,t n ], we obtain µ(t n ) µ(t n ) c f ν Letting n, we conclue Tn t n u(t) t 0, µ(t). n. lim sup µ(t) lim inf µ(t), t t so µ = lim t µ(t) < exists. Therefore, by (3.3), since otherwise the limit woul not exist, we have 0 lim sup µ (t) ν lim inf t t = ν min m N [ min m N ] (µ(t) λ m )(ψ(µ(t)) ψ(λ m )) + lim sup ψ (µ(t)) t (µ λ m )(ψ(µ ) ψ(λ m )). ψ (µ ) c f ν u(t) Since ψ is increasing, the latter expression is equal to zero. Therefore, µ {,,..., n } an u(t) lim t u(t) {ψ( ), ψ( ),...,ψ( n )}. If lim sup t u(t) / u(t) =, repeating the above argument as in the first case we conclue that lim t u(t) / u(t) =. This proves the theorem. Definition 3. We efine M n := M n,κ for any κ ]0,( n+ n )/]. Let also M n ={u/ u : u M n }. Corollary 3. For u 0 M n such that u 0 > c f /ν, we have u 0 ( u 0 ψ n + c ) f. (3.5) ν u 0 Proof. This follows trivially from corollary taking κ = c f /ν u 0.

13 Density of global trajectories 965 Remark. For u V, we have because of the convexity of ψ ( ) u ψ u u u. In particular, since ψ is increasing, u u ψ(µ) u u µ. Remark 3. For the solution u of the KFNSE that satisfies the conitions in theorem, we obtain the estimate ( u(t) u(0) + f ) e 4ν(n+κ)t f (3.6) 8ν ( n + κ) 8ν ( n + κ) for t [0,T]. Proof. From the last remark we have u(t) / u(t) n + κ, t [0,T]. Using Young s inequality we have t u = ν u +(f, u) ν( n + κ) u ν( n + κ) u 4ν( n + κ) u f ν( n + κ). By the Gronwall inequality, we then obtain (3.6). Corollary 4. We have f ν( n + κ) M n { u 0 G : S(t)u 0 = O ( e (+ɛ)ν n t ) as t } for any ɛ>0. The next result provies us with a metho for proucing elements of the sets M n. It will be use for all further results. Theorem 3. Let u,u,... H, an let t >t > >t j,j. Suppose that S(t)u j exists on the interval [t j, ). Let us also assume u k M, k N (3.7) for some constant M>0, an S(t k )u k c f, νκ k N. (3.8) Let there exist some n N such that S(t k )u k S(t k )u k ψ( n + κ), k N. (3.9) Then, there exist u M n an a subsequence {u kj } of {u k } such that lim k j S(t)u = 0, j t R. (3.0)

14 966 J Vukainovic Proof. We first want to prove that, for every t R, there exists a constant C(t) > 0 such that S(t)u k C(t) (3.) for k large enough. Without loss of generality, we may assume M> f /ν. First, we fix k N. By lemma, there exists a unique β k t k such that S(t)u k c f νκ, t [t k,β k ], S(t)u k c f νκ,tβ k, (3.) S(t)u k S(t)u k ψ( n + κ), t [t k,β k ]. (3.3) Since we are intereste in extracting a subsequence, without loss of generality we may assume that either β k 0 for all k N, or that β k > 0 for all k N. Assuming that the latter is true, by (.6) an (3.7), we obtain c f = S(β ν κ k )u k f u k e ν β k + ν ( e ν β k ) ( ) e ν β k M f f ν + ν. We obtain an upper boun on β k, which covers both of the cases β k log κ (ν M f ) ν (c =: t κ ) f M. (3.4) With C,C,...being various constants, (3.6) implies now ( ) S(t)u k S(β k )u k f + 8ν e 4ν n+(β k t) ( ) c f = f + ν κ 8ν e 4ν n+(β k t) = C e 4ν n+(β k t), t [t k,β k ]. From here, using (3.3) an (3.4), we conclue S(t)u k ψ( n + κ)c e 4ν n+(β k t) C e 4ν n+(t M t), t [t k,β k ]. (3.5) On the other han, for t β k from (.8) an the previous calculations, we obtain S(t)u k S(β k )u k e ν (t β k ) f + ( e ν (t β k ) ) ν ψ( n + κ) S(β k )u k e ν (t β k ) f + ( e ν (t β k ) ) ν ψ( n + κ) c f f + =: C ν κ ν 3. This an (3.5) together imply S(t)u k max { C e 4ν n+(t M t) },C 3 =: C(t) t tk. (3.6)

15 Density of global trajectories 967 Since H is compactly imbee in V, for every i N, we can extract a subsequence {S(t i )u k i j } j N of {S(t i )u k } ki which converges in H. Without loss of generality, we may assume that {kj i } {kl j } for i l. Now we may use the Cantor iagonal process to extract a subsequence {u kj } of {u k } such that lim j S(t ki )u kj =: w i H, i N exists. Actually, w i V. Since S(t) : H H is a continuous mapping for t 0, we obtain w i = S(t ki t kj )w j, j i, i, j N, which means that w i belongs to the trajectory of a solution. Letting u := S( t k )w, we obtain u = S( t kj )w j, j N. Therefore, u G. Again, by the continuity of S(t) : H H, we obtain lim S(t)u k j S(t)u = 0, t R. j It remains to prove that u M n. To this en, we consier two cases. If lim inf k β k =, then (3.0) an (3.) imply S(t)u c f νκ, t R, an, thus, u A. If, on the other han, lim inf k β k = β >, (3.0) an (3.3), give S(t)u S(t)u In both cases u M n. ψ( n + κ), t β. Theorem 4. For each n N, M n is a connecte, locally compact, both positively an negatively invariant subset of H. Proof. As an invariant set containing A, M n is connecte. In orer to prove the local compactness of M n, because of the compactness of A, it suffices to check that every sequence u,u,... M n \A, which is boune in H, has a subsequence converging to an element of M n. Since lim t S(t)u k = for k N, there exist t > t > such that lim t t j = an S(t k )u k c f / κ. By virtue of corollary we have S(t k )u k / S(t k )u k ψ( n + κ), k N. Applying theorem 3, there exists a subsequence of {u k } which converges to an element of M n in H. 3.. 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 wors, P n H = P n M n. First, we nee to prove a series of lemmas. Lemma. Let u C n,κ for some n N, an 0 <κ ( n+ n )/. Then, Q n u ψ( n + κ) ψ( n+ ) ψ( n + κ) P nu. (3.7) Also, Q n u n+ n κ (( n + κ) P n u P nu ). (3.8)

16 968 J Vukainovic In particular, an Q n u n + n+ n+ n P n u (3.9) u n+ n+ n P n u. (3.0) Proof. Inequality (3.7) follows from Q n u ψ( n+ ) Q nu ψ( n + κ) ( Pn u ψ( n+ ) + Q nu ). Observe that by lemma u / u n + κ. Therefore, Q n u ( u P nu ) n+ n + κ Q n u + (( n + κ) P n u n+ P nu ), n+ an (3.8) follows. The other two inequalities follow from (3.8) an the fact that κ ( n+ n )/. Lemma. Let u M n, for some n N. Let 0 < κ /. Then, the following estimates hol: { ( ) } c f Q n u max νκ, ψ( n + κ) / P n u (3.) ψ( n+ ) ψ( n + κ)) an { ( ) } c f n + / n+ Q n u max, P n u. (3.) ν n+ n Proof. If u c f /νκ, the statement is obvious. If, on the other han, u >c f /νκ, we get u ψ(( n + n+ )/) u by corollary. Applying lemma, we obtain (3.) an (3.). In orer to prove the next important step towars the proof of theorem 5 we will nee the well-known theorem by Brouwer, which we will state here. Theorem 6. Let R n be enowe with a norm, an let B(r) R n be a close ball with raius r>0. Ifg : B(r) B(r) is a continuous mapping an if g(x) = x for all x B(r), then g is onto. Lemma 3. 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. In other wors, the operator P n S(t 0 ) : P n H P n H is onto. Proof. For r>0we efine B H (r) := {u 0 H : u 0 r}, an let B n (r) = B H (r) P n H. Let us fix a r 0 >c f /νκ large enough that p 0 B n (r 0 ). In orer to prove the lemma, let us choose a continuous function θ : R R such that θ(x) = for x r 0 an θ(x) = 0 for x r 0. Define g(u 0 ) = P n S(θ( u 0 )t 0 )u 0, u 0 P n H.

17 Density of global trajectories 969 By (.6), we have S(t)B(r) B(r), r f /ν, t 0. Therefore, g(b n (r 0 )) B n (r 0 ). Also, g is continuous, an it satisfies g(u 0 ) = u 0 for u 0 = r 0. By the previous Brouwer s theorem, there exists w 0 B n (r 0 ) such that g(w 0 ) = p 0. Let us assume for a moment that r 0 was chosen in such a way that w 0 r 0. Then, by the efinition of g, we woul have P n S(t 0 )w 0 = g(w 0 ) = p 0, an this is exactly the claim of this lemma. In orer to complete the proof, let 0 < κ /. For u 0 P n H such that u 0 >c f /νκ we have u 0 / u 0 ψ( n) ψ( n + κ). By corollary, there exists a unique τ u0 > 0, such that S(t)u 0 >c f /νκ, t [0,τ u0 ), S(t)u 0 c f /νκ, t (τ u0, ), an for t [0,τ u0 ]wehave S(t)u 0 / S(t)u 0 ψ( n + κ) an S(t)u 0 u 0 e 4νn+t f /8ν n. Lemma now implies S(t)u 0 n+ n+ n P n S(t)u 0 for t [0,τ u0 ]. Therefore, P n S(t)u 0 n+ n S(t)u 0 n+ ( u 0 e 4ν n+t n+ n n+ 8ν n We now want to prove that there exists r 0 > c f /νκ such that P n S(t)u 0 > p 0, f ), t [0,τ u0 ]. (3.3) t [0,t 0 ], u 0 P n H \B n (r 0 ). Let { l = max p 0, c } f. (3.4) νκ We fix any r 0 >c f /νκ such that ) n+ n (r 0 e 4ν n+t 0 f >l. (3.5) n+ 8ν n For u 0 >r 0, we obtain ( ) l c f = S(τ u0 νκ )u 0 P n S(τ u0 )u 0 n+ n (r 0 e 4ν n+τ u0 f ). (3.6) n+ 8ν n Combining (3.5) an (3.6), we get t 0 <τ u0. In particular, (3.3) hols for t [0,t 0 ], an so oes (3.5). From (3.3), (3.5) an (3.4), we obtain P n S(t)u 0 > p 0, t [0,t 0 ] (3.7) for all u 0 P n H \B n (r 0 ). In orer to complete the proof it remains to show that w 0 r 0. Assuming the opposite, we woul have g(w 0 ) = P n S(θ( w 0 )t 0 )w 0 > p 0. This is a contraiction, since g(w 0 ) = p 0. The lemma is proven. Now we have the tools to prove theorem 5. Proof of theorem 5. Let p 0 P n H. Choose a sequence 0 >t >t >t 3 > such that lim k t k =. By lemma 3, there exist p,p,... P n H such that P n S( t k )p k = p 0, k N. Let u k := S( t k )p k, k N. Therefore, P n u k = p 0, k N. Let us efine U k (t) := S(t)u k, t t k, an V k (t) := (A)U k (t), t t k. Let us assume first that p k c f ν (3.8)

18 970 J Vukainovic 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 ) = (A)p k ψ( n ) / p k ψ( n ) / c f ν ( = (c )( n ) / n Since (c )( n ) / ( n / ) /, inequality (.7) gives Also, V k (t) (c )( n ) / ( n ) / f ) / f ν /. ν /, t t k. U k (t) ψ( n) / c f, t t ψ( ) / k. (3.9) ν Repeating the arguments from theorem 3, passing to a subsequence, an using Cantor s iagonal process, we can conclue that lim U k(t) S(t)u = 0, t R k for some u G. From here because of the continuity of the projection P n, it follows that P n u = P n u k = p 0 an, using (3.9), u A M n. This proves the lemma uner assumption (3.8). Let us now assume that p k c f /ν for infinitely many k N. Again, by passing to a subsequence, we may assume that this is true for all k N. Again, for each k N, either u k c f /ν or u k > c f /ν. If the latter is true, since u k = S( t k )p k, we have U k (t) = S(t)u k > c f, t [t k, 0] ν an since p k ( ) n + n+ p k ψ( n ) ψ, by theorem, we have u 0 u 0 = U k(0) ( ) n + n+ U k (0) ψ. In any case, by lemma, since P n u k = p 0, { ( ) } c f / n+ u k max, p 0, k N. ν n+ n Therefore, the sequence {u k } satisfies the assumptions of theorem 3. We conclue that there exists a u M n, so that lim U k(t) S(t)u = 0, t R. k In particular, lim k u k u = 0 an, as in the previous case, P n u = p 0. This completes the proof of the theorem.

19 Density of global trajectories Density properties of the sets M n This section is evote to the ensity properties of the sets M n an G. In [], the authors postulate that for the solution operator S of the NSEs, the set S(t)H for t>0 is ense in the phase space H. This conjecture was partly proven in [6]. It was shown that the set of initial ata for which there exists a global solution, the set G, is ense in the phase space H, but with respect to the weaker norm of the space V. Here, we will prove that a similar result hols for two-imensional perioic KFNSEs for any choice of the nonecreasing convex. However, if emonstrates exponential asymptotic behaviour, the ensity is shown with respect to the norm of the phase space H. Theorem 7. The set n N M n( G) is ense in H with respect to the norm υ. Proof. Let u 0 H be arbitrary. By theorem 5, for each n N there exists u n M n such that P n u n = P n u 0, an from (3.) we have { ( ) } c f n + / n+ Q n u n max, P n u 0. ν n+ n This implies u n u 0 υ = Q n (u n u 0 ) υ / n+ Q n(u n u 0 ) / ( ) n+ Qn u n + Q n u 0 / n+ max ( max { c f ν / n+ { ( c f n + n+, ν n+ n By virtue of (.5), we obtain lim inf n u n u 0 υ = 0., ( n+ n ) / P nu 0 ) / P n u 0 } + Q n u 0 ) } + / n+ Q nu 0. Theorem 8. Let (ξ) C (ξ), ξ>0 for some positive constant C. The set n N M n( G) is ense in H with respect to the norm. Proof. Let u 0 H be arbitrary. By theorem 5, for each n N there exists u n M n such that P n u n = P n u 0. By (.5) there exists an increasing sequence of integers {n k } k N such that lim k ( nk + nk ) =. Let 0 <κ /. First, let us assume that for infinitely many k N, u nk A. Since A is compact, there exist a subsequence of {u nk } that converges in H to some ũ 0 A. Without loss of generality, we can assume that u nk ũ 0 0, as k. Also, u nk ũ 0 υ 0, as k. From the last theorem we know that u nk u 0 in the norm υ, so the two limits have to coincie. Thus, u 0 A an u nk u 0 in the norm. This shows the ensity in this case. Let us now assume that for infinitely many k N, u nk M n \A. Without loss of generality we may assume that this is the case for all k N. We consier two cases. First, let us assume that for infinitely many k N, u nk / u nk ψ( n k + κ). Without loss of generality, let us assume that this is the case for all k N. Then, by (3.7) we have Q nk u nk ψ( nk + κ) ψ( nk +) ψ( nk + κ) P n k u nk.

20 97 J Vukainovic By our assumptions, we have now Q nk u nk ψ( nk + κ) ( nk + nk κ)ψ ( nk + κ) P n k u nk ( nk + κ) ( nk + nk κ) ( nk + κ) P n k u nk C ( nk + nk ) u 0 0, k. Let us now assume that u nk / u nk > ψ( n k + κ) for infinitely many k N. By corollary, we know that u nk <c f /νκ. Since lim t S(t)u nk =, there exists t k < 0 such that S(t k )u nk / S(t k )u nk ψ( n k + κ) an S(t k )u nk = c f /νκ, so S(t k )u nk ψ( nk + κ) c f /ν κ.now, Q nk u nk ψ( nk +) Q n k u nk ( S(t k )u nk e ν t k + ψ( nk +) ψ( n k + κ) c f + ψ( nk +) ν κ ψ( nk +) ) f ( e ν t k ) ν f ν ψ( nk + κ) c f + ψ( nk + κ) + ψ ( nk + κ)( nk + nk κ) ν κ f ψ( nk +) f ν C c f +. nk + nk ν κ ψ( nk +) ν Therefore, in both cases Q nk u nk 0 when k. Thus, u nk u 0 = Q nk (u nk u 0 ) Q nk u nk + Q nk u 0 0, (4.) when k. This completes the proof. Remark 4. The conition on in theorem 8 can be replace by the following property: lim sup n ( n+ ) ( n ) =. (4.) Because of the asymptotic behaviour of the spectral gaps lim sup n ( n+ n )/ log n>0, the proof of theorem 8 oes not work for any polynomial choice of the function, an in particular for the two-imensional perioic NSEs, =, an the two-imensional perioic Navier Stokes-α moel, (ξ) = +α ξ. In those cases we only have the weaker result from theorem 7. In orer for (4.) to be satisfie, we nee exponential asymptotic behaviour of the function, for example, (ξ) = e αξ. The next theorem, which resulte from a private communication with Ciprian Foias an Igor Kukavica, is a ensity result in the energy Norm for any choice of nonecreasing, convex. Theorem 9. Let W n :={w : Aw = n w, w = }. Then, W n M n an ( n ) P n H RM k, k= where the closure is taken in H.

21 Density of global trajectories 973 Proof. Let r>c f /ν. By theorem 5, there exists u r M n such that P n u r = rw. Also, by lemma an corollary 3, we have Q n u r (c f /ν u r ) P n u r c f r n+ n c f /ν u r νr( n+ n ) c f. Thus, Q n u r c f r νr( n+ n ) c f 0 an u r = P nu r + Q nu r, r r when we let r. Finally, ( u r w u r = u r rw w r ) Q nu r + u r u r r r u r 0, when r. This proves the theorem. References [] Baros C an Tartar L 973 Sur l unicité rétrograe es équations paraboliques et quelques questions voisines Arch. Ration. Mech. Anal [] Chen S, Foias C, Holm D D, Olson E, Titi E S an Wynne S 998 A connection between the Camassa Holm equations an turbulence flows in pipes an channels Phys. Fluis [3] Constantin P 003 Filtere viscous flui equations Comput. Math. Appl [4] Constantin P 003 Near ientity transformations for the Navier Stokes equations Hanbook of Mathematical Flui Dynamics vol (Amsteram: Elsevier) [5] Constantin P an Foias C 988 Navier Stokes Equations (Chicago: Lectures in Mathematics) (Chicago, IL: Chicago University Press) [6] Constantin P, Foias C, Kukavica I an Maja A J 997 Dirichlet quotients an D perioic Navier Stokes equations J. Math. Pure. Appl [7] Constantin P, Foias C, Nicolaenko B an Temam R 988 Spectral barriers an inertial manifols for issipative partial ifferential equations J. Dyn. Diff. Eqns [8] Constantin P, Foias C an Temam R 988 On the imension of attractors in two-imensional turbulence Physica D [9] Doering C, Gibbon J D, Holm D D an Nicolaenko B 988 Low-imensional behavior in the complex Ginzburg Lanau equation Nonlinearity [0] Dascaliuc R 003 On backwar-time behavior of Burger s original moel for turbulence Nonlinearity [] Foias C, Holm D D an Titi E S 00 The three-imensional viscous Camassa Holm equations, an their relation to the Navier Stokes equations an turbulence theory J. Dyn. Diff. Eqns 4 [] Foias C, Jolly M S an Kukavica I 996 Localisation of attractors Nonlinearity [3] Foais C, Jolly M S, Kukavica I an Titi E S 00 The Lorenz equation as a methafor for some analytic, geometric, an statistical properties of the Navier Stokes equations Descrete Continuous Dyn. Syst [4] Foais C, Jolly M S an Lee W S 00 Nevanlinna Pick interpolation of global attractors Nonlinearity [5] Foias C, Holm D D an Titi E S 00 The Navier Stokes-alpha moel of flui turbulence Physica D [6] Foias C an Kukavica I 00 private communication [7] Foias C an Saut J C 98 Limite u rapport e l enstrophie sur l énergie pour une solution faible es équations e Navier Stokes C. R. Aca. Sci. Paris, Sér I Math [8] Foias C an Saut J C 984 Asymptotic behavior, as t, of solutions of Navier Stokes equations an nonlinear spectral manifols Iniana Univ. Math. J [9] Foias C, Sell G R an Temam R 988 Inertial manifols for nonlinear evolutionary equations J. Diff. Eqns

22 974 J Vukainovic [0] Holm D D, Marsen J E an Ratiu T S 998 Euler Poincaré moels of ieal fluis with nonlinear ispersion Phys. Rev. Lett [] Holm D D, Marsen J E an Ratiu T S 998 Euler Poincaré equations an semiirect proucts with applications to continuum theories Av. Math [] Kukavica I 99 On the behavior of the solutions of the Kuramoto Sivashinsky equations for negative time J. Math. Anal. Appl [3] Richars I 98 On the gaps between numbers that are sums of two squares Av. Math. 46 [4] Vukainovic J 00 On the backwars behavior of the solutions of the D perioic viscous Camassa Holm equations J. Dyn. Diff. Eqns 4