ANALYTIC STUDY OF SHELL MODELS OF TURBULENCE

PETER CONSTANTIN, BORIS LEVANT, AND EDRISS S. TITI

Abstract. In this paper we study analytically the viscous sabra shell model of energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant inertial manifolds. Moreover, we establish the existence of exponentially decaying energy dissipation range.

Key words: Turbulence, Dynamic models, Shell models, Navier–Stokes equations.

AMS subject classification: 76F20, 76D05, 35Q30

Contents

1. Introduction 2 2. Preliminaries and Functional Setting 4 3. Existence of Weak Solutions 8 4. Uniqueness of Weak Solutions 11 5. Strong Solutions 11 6. Gevrey class regularity 14 6.1. Preliminaries and Classical Navier-Stokes Equation Results 14 6.2. A Stronger Result 16 7. Global Attractors and Their Dimensions 18 8. Determining Modes 20 9. Existence of Inertial Manifolds 23 9.1. Dimension of Inertial Manifolds 25 10. Conclusions 25 Acknowledgments 25 References 25 Appendix A. Dissipativity in D(A) 28 Appendix B. Differentiability of the semigroup with respect to initial

conditions 30

Date: November 8, 2005.

1. Introduction

Shell models of turbulence have attracted interest as useful phenomenological models that retain certain features of the Navier-Stokes equations (NSE). Their central computational advantage is the parameterization of the fluctuation of a turbulent field in each octave of wave numbers λ^{n} < |k_{n}| ≤ λ^{n+1} by very few representative variables. The octaves of wave numbers are called shells and the variables retained are called shell variables. Like in the Fourier representation of NSE, the time evolution of the shell variables is governed by an infinite system of coupled ordinary differential equations with quadratic nonlinearities, with forcing applied to the large scales and viscous dissipation effecting the smaller ones. Because of the very reduced number of interactions in each octave of wave numbers, the shell models are drastic modification of the original NSE in Fourier space.

The main object of this work is the “sabra” shell model of turbulence, which was introduced in [26]. It is worth noting that the results of this article apply equally well to the well-known Gledzer-Okhitani-Yamada (GOY) shell model, introduced in [31]. For other shell models see [4], [12], [13], [31]. A recent review of the subject emphasizing the applications of the shell models to the study of the energy-cascade mechanism in turbulence can be found in [2].

The sabra shell model of turbulence describes the evolution of complex Fourierlike components of a scalar velocity field denoted by u_{n}. The associated onedimensional wavenumbers are denoted by k_{n}, where the discrete index n is referred to as the “shell index”. The equations of motion of the sabra shell model of turbulence have the following form

\frac{du_{n}}{dt} = i(ak_{n+1}u_{n+2}u_{n+1}^{∗} + bk_{n}u_{n+1}u_{n−1}^{∗} − ck_{n−1}u_{n−1}u_{n−2}) − νk_{n}^{2}u_{n} + f_{n}, (1)

for n = 1, 2, 3, …, and the boundary conditions are u_{−1} = u_{0} = 0. The wave numbers k_{n} are taken to be

k_{n} = k_{0}λ^{n},  (2)

with λ > 1 being the shell spacing parameter, and k_{0} > 0. Although the equation does not capture any geometry, we will consider L = k_{0}^{−1} as a fixed typical length scale of the model. In an analogy to the Navier-Stokes equations ν > 0 represents a kinematic viscosity and f_{n} are the Fourier components of the forcing.

The three parameters of the model a, b and c are real. In order for the sabra shell model to be a system of the hydrodynamic type we require that in the inviscid (ν = 0) and unforced (f_{n} = 0, n = 1, 2, 3, …) case the model will have at least one quadratic invariant. Requiring conservation of the energy

E = ∑_{n=1}^{∞}|u_{n}|^{2}  (3)

leads to the following relation between the parameters of the model, which we will refer as an energy conservation condition

a + b + c = 0.  (4)

Moreover, in the inviscid and unforced case the model possesses another quadratic invariant

W = ∑_{n=1}^{∞} (\frac{a}{c})^{n}|u_{n}|^{2}.  (5)

The physically relevant range of parameters is |a/c| > 1 (see [26] for details). For −1 < \frac{c}{a} < 0 the quantity W is not sign-definite and therefore it is common to associate it with the helicity – in an analogy to the 3D turbulence. In that regime we can rewrite relation (5) in the form

W = ∑_{n=1}^{∞}(−1)^{n}k_{n}^{α}|u_{n}|^{2},  (6)

for

α = log_{λ} |\frac{a}{c}|.  (7)

The 2D parameters regime corresponds to 0 < \frac{c}{a} < 1. In that case the second conserved quadratic quantity W is identified with the enstrophy, and we can rewrite the expression (5) in the form

W = ∑_{n=1}^{∞} k_{n}^{α}|u_{n}|^{2},  (8)

where α is also defined by the equation (7).

The well-known question of global well-posedness of the 3D Navier-Stokes equations is a major open problem. In this work we study analytically the shell model (1). Specifically, we show global regularity of weak and strong solutions of (1) and smooth dependence on the initial data. This is a basic step for studying the long-time behavior of this system and for establishing the existence of a finite dimensional global attractor.

Our next result concerns the rate of the decay of the spectrum (i. e. |u_{n}|) as function of n. Having shown the existence of strong solutions, we prove that |u_{n}| decays exponentially with n for n large enough, depending on the physical parameters of the model. However, if we assume that the forcing f_{n} decays super-exponentially, which is a reasonable assumption, because in most numerical simulations the forcing is concentrated in only finitely many shells, then we can show more. In fact, following the arguments presented in [21] and [20] we show that in this case the solution of (1) belongs to a certain Gevrey class of regularity, namely |u_{n}| also decays super-exponentially in n – an evidence of the existence of a dissipation range associated with this system, which is an observed feature of turbulent flows (see also [11] for similar results concerning 3D turbulence).

Classical theories of turbulence assert that the turbulent flows governed by the Navier-Stokes equations have finite number of degrees of freedom (see, e.g., [13], [25]). Arguing in the same vein one can state that the sabra shell model with nonzero viscosity has finitely many degrees of freedom. Such a physical statement can be interpreted mathematically in more than one way: as finite dimensionality of the global attractor, as existence of finitely many determining modes, and as existence of finite dimensional inertial manifold.

Our first results concern the long time behavior of the solutions of the system (1). We show that the global attractor of the sabra model has a finite fractal and Hausdorff dimensions. Moreover, we show that the number of the determining modes of the sabra shell model equations is also finite. Remarkably, our upper bounds for the number of the determining modes and the fractal and Hausdorff dimensions of the global attractor of the sabra shell model equation coincide (see Remark 1). This is to be contrasted with the present knowledge about Navier-Stokes equations, where the two bounds are qualitatively different.

Finally, and again in contrast with the present knowledge for the Navier-Stokes equations, we show that the sabra shell model equation has an inertial manifold M. Inertial manifolds are finite dimensional Lipschitz globally invariant manifolds that attract all bounded sets in the phase space at an exponential rate and, in particular, contain the global attractor. We show that M = graph(Φ), where Φ is a C^{1} function which slaves the components u_{n}, for all n ≥ N, as a function of {u_{k}}_{k=1}^{N}, for N large enough depending on the physical parameters of the problem, i.e. ν, f, λ, a, b, and c. The reduction of the system (1) to the manifold M yields a finite dimensional system of ordinary differential equations. This is the ultimate and best notion of system reduction that one could hope for. In other words, an inertial manifold is an exact rule for parameterizing the large modes (infinite many of them) in terms of the low ones (finitely many of them). The concept of inertial manifold for nonlinear evolution equations was first introduced in [17] (see also [8], [9], [18], [33], [35]).

2. Preliminaries and Functional Setting

Following the classical treatment of the NSE, and in order to simplify the notation we are going to write the system (1) in the following functional form

\frac{du}{dt} + νAu + B(u, u) = f  (9a)
u(0) = u^{in},  (9b)

in a Hilbert space H. The linear operator A as well as the bilinear operator B will be defined below. In our case, the space H will be the sequences space ℓ^{2} over the field of complex numbers ℂ. For every u, v ∈ H, the scalar product (·, ·) and the corresponding norm |·| are defined as

(u, v) = ∑_{n=1}^{∞} u_{n}v_{n}^{∗}, |u| = ( ∑_{n=1}^{∞}|u_{n}|^{2})^{1/2}.

We denote by {ϕ_{j}}_{j=1}^{∞} the standard canonical orthonormal basis of H, i.e. all the entries of ϕ_{j} are zero except at the place j it is equal to 1.

The linear operator A : D(A) → H is a positive definite, diagonal operator defined through its action on the elements of the canonical basis of H by

Aϕ_{j} = k_{j}^{2}ϕ_{j},

were the eigenvalues k_{j}^{2} satisfy the equation (2). D(A) - the domain of A is a dense subset of H. Moreover, it is a Hilbert space, when equipped with the graph norm

∥u∥_{D(A)} = |Au|, ∀u ∈ D(A).

Using the fact that A is a positive definite operator, we can define the powers A^{s} of A for every s ∈ ℝ

∀u = (u_{1}, u_{2}, u_{3}, …), A^{s}u = (k_{1}^{2s}u_{1}, k_{2}^{2s}u_{2}, k_{3}^{2s}u_{3}, …).

Furthermore, we define the spaces

V_{s} := D(A^{s/2}) = {u = (u_{1}, u_{2}, u_{3}, …) : ∑_{j=1}^{∞} k_{j}^{2s}|u_{j}|^{2} < ∞},

which are Hilbert spaces equipped with the scalar product

(u, v)_{s} = (A^{s/2}u, A^{s/2}v), ∀u, v ∈ D(A^{s/2}),

and the norm |u|_{s}^{2} = (u, u)_{s}, for every u ∈ D(A^{s/2}). Clearly

V_{s} ⊆ V_{0} = H ⊆ V_{−s}, ∀s > 0.

The case of s = 1 is of a special interest for us. We denote V = D(A^{1/2}) a Hilbert space equipped with a scalar product

((u, v)) = (A^{1/2}u, A^{1/2}v), ∀u, v ∈ D(A^{1/2}).

We consider also V′ = D(A^{−1/2}) - the dual space of V. We denote by ⟨·, ·⟩ the action of the functionals from V′ on elements of V. Note that the following inclusion holds

V ⊂ H ≡ H′ ⊂ V′,

hence the H scalar product of f ∈ H and u ∈ V is the same as the action of f on u as a functional in the duality between V′ and V

⟨f, u⟩ = (f, u), ∀f ∈ H, ∀u ∈ V.

Observe also that for every u ∈ D(A) and every v ∈ V one has

((u, v)) = (Au, v) = ⟨Au, v⟩ .

Since D(A) is dense in V one can extend the definition of the operator A : V → V′ in such a way that

⟨Au, v⟩ = ((u, v)), ∀u, v ∈ V.

In particular, it follows that

∥Au∥_{V′} = ∥u∥, ∀u ∈ V.  (10)

Before proceeding and defining the bilinear term B, let us introduce the sequence analogue of Sobolev functional spaces.

Definition 1. For 1 ≤ p ≤ ∞ and m ∈ ℝ we define sequence spaces

w^{m,p} := {u = (u_{1}, u_{2}, …) ∈ : ∥A^{m/2}u∥_{p} = ( ∑_{n=1}^{∞}(k_{n}^{m}|u_{n}|)^{p})^{1/p} < ∞},

for 1 ≤ p < ∞, and

w^{m,∞} := {u = (u_{1}, u_{2}, …) ∈ : ∥A^{m/2}u∥_{∞} = sup_{1≤n≤∞}(k_{n}^{m}|u_{n}|) < ∞}.

For u ∈ w^{m,p} we define it’s norm

∥u∥_{w^{m,p}} = ∥A^{m/2}u∥_{p},

where ∥·∥_{p} is the usual norm in the ℓ^{p} sequence space. The special case of p = 2 and m ≥ 0 corresponds to the sequence analogue of the classical Sobolev space, which we denote by

h^{m} = w^{m,2}.

Those spaces are Hilbert with respect to the norm defined above and its corresponding inner product.

The above definition immediately implies that h^{1} = V. It is also worth noting that V ⊂ w^{1,∞} and the inclusion map is continuous (but not compact) because

∥u∥_{w^{1,∞}} = ∥A^{1/2}u∥_{∞} ≤ ∥A^{1/2}u∥_{2} = ∥u∥.

The bilinear operator B(u, v) will be defined formally in the following way. Let u, v ∈ H be of the form u = ∑_{n=1}^{∞} u_{n}ϕ_{n} and v = ∑_{n=1}^{∞} v_{n}ϕ_{n}. Then

B(u, v) = −i ∑_{n=1}^{∞} (ak_{n+1}v_{n+2}u_{n+1}^{∗} + bk_{n}v_{n+1}u_{n−1}^{∗}+

+ ak_{n−1}u_{n−1}v_{n−2} + bk_{n−1}v_{n−1}u_{n−2})ϕ_{n}, (11)

where here again u_{0} = u_{−1} = v_{0} = v_{−1} = 0. It is easy to see that our definition of B(u, v), together with the energy conservation condition

a + b + c = 0

implies that

B(u, u) = −i ∑_{n=1}^{∞} (ak_{n+1}u_{n+2}u_{n+1}^{∗} + bk_{n}u_{n+1}u_{n−1}^{∗} − ck_{n−1}u_{n−1}u_{n−2})ϕ_{n},

which is consistent with (1). In the sequel we will show that indeed our definition of B(u, v) makes sense as an element of H, whenever u ∈ H and v ∈ V or u ∈ V and v ∈ H. For u, v ∈ H we will also show that B(u, v) makes sense as an element of V′.

Proposition 1.

(i) B : H × V → H and B : V × H → H are bounded, bilinear operators. Specifically,

|B(u, v)| ≤ C_{1}|u|∥v∥, ∀u ∈ H, v ∈ V,  (12)

and

|B(u, v)| ≤ C_{2}∥u∥|v|, ∀u ∈ V, v ∈ H,  (13)

where

C_{1} = (|a|(λ^{−1} + λ) + |b|(λ^{−1} + 1)),

and

C_{2} = (2|a| + 2λ|b|).

(ii) B : H × H → V′ is a bounded bilinear operator and

∥B(u, v)∥_{V′} ≤ C_{1}|u||v|, ∀u, v ∈ H.  (14)

(iii) B : D(A^{d}) × D(A^{d}) → D(A^{d−1/2}) is a bounded operator and

|A^{d−1/2}B(u, v)| ≤ C_{1}|A^{d}u||A^{d}v|, ∀u, v ∈ D(A^{d}).  (15)

as (iv) For every u ∈ H, v ∈ V

Re(B(u, v), v) = 0,  (16)

(v) B : H × D(A) → V is a bounded bilinear operator and for every u ∈ H

and v ∈ D(A)

∥B(u, v)∥ ≤ C_{3}|u||Av|,  (17)

where

C_{3} = (|a|(λ^{−2} + λ^{2}) + |b|(1 + λ^{2})).

Proof. To prove the first statement, let u ∈ H and v ∈ V. Using the form of the eigenvalues k_{n} given in (2) and applying the Cauchy-Schwartz inequality we get

|B(u, v)|^{2} = ∑_{n=1}^{∞}|(B(u, v), ϕ_{n})|^{2} ≤

≤ ∑_{n=1}^{∞}|i(ak_{n+1}v_{n+2}u_{n+1}^{∗} + bk_{n}v_{n+1}u_{n−1}^{∗}+

+ ak_{n−1}u_{n−1}v_{n−2} + bk_{n−1}v_{n−1}u_{n−2}|^{2} ≤

≤ (|a|(λ^{−1} + λ) + |b|(λ^{−1} + 1))^{2}|u|^{2}∥v∥^{2},

where the last inequality follows from the Cauchy-Schwartz inequality. The case of u ∈ V and v ∈ H is proved in the similar way.

As for (ii), let u, v ∈ H and w ∈ V. Then

|⟨B(u, v), w⟩| = |∑_{n=1}^{∞} B_{n}(u, v)w_{n}^{∗}| ≤

≤ ∑_{n=1}^{∞}|ak_{n+1}v_{n+2}u_{n+1}^{∗}w_{n}^{∗} + bk_{n}v_{n+1}w_{n}^{∗}u_{n−1}^{∗}+

+ ak_{n−1}w_{n}^{∗}u_{n−1}v_{n−2} + bk_{n−1}w_{n}^{∗}v_{n−1}u_{n−2}| ≤

≤ (|a|(λ^{−1} + λ) + |b|(λ^{−1} + 1))∥w∥_{w^{1,∞}}|u||v| ≤ C_{1}∥w∥|u||v|. (18)

Similar calculations can be applied to prove the statement (iii).

The statement (iv) follows directly from the energy conservation condition (4). The proof of (v) is very similar to that of the first and second parts of the

proposition. Let u ∈ D(A), then using the form of the eigenvalues k_{n} given in (2) and applying the Cauchy-Schwartz inequality we get

∥B(u, v)∥^{2} = ∑_{n=1}^{∞}|(A^{1/2}B(u, v), ϕ_{n})|^{2} ≤

≤ ∑_{n=1}^{∞}|ak_{n+1}k_{n}v_{n+2}u_{n+1}^{∗} + bk_{n}^{2}v_{n+1}u_{n−1}^{∗}+

+ ak_{n−1}k_{n}u_{n−1}v_{n−2} + bk_{n−1}k_{n}v_{n−1}u_{n−2}|^{2} ≤

≤ (|a|(λ^{−2} + λ^{2}) + |b|(1 + λ^{2}))^{2}|u|^{2}|Av|^{2}.

□

Let us fix m ∈ ℕ and consider H_{m} = span{ϕ_{1}, …, ϕ_{m}} and let P_{m} be an ℓ_{2} orthogonal projection from ℓ^{2} onto H_{m}. If u ∈ H, then

P_{m}u = u^{m} = (u_{1}^{m}, u_{2}^{m}, …, u_{m}^{m}, 0, 0, …),

where u_{n}^{m} = (u, ϕ_{n}). In order to simplify the notation we will write u_{n} instead of u_{n}^{m}, whenever it will not cause any confusion. We would like to obtain the same estimates as above in Proposition 1 for the truncated bilinear term P_{m}B(u^{m}, v^{m}). Setting u_{j} = u_{j}^{m} = 0 for all j = 0, −1 and j > m we get

P_{m}B(u^{m}, v^{m}) = −i ∑_{n=2}^{m−1} (ak_{n+1}v_{n+2}u_{n+1}^{∗}ϕ_{n−1} + bk_{n}v_{n+1}u_{n−1}^{∗}ϕ_{n}+

+ ak_{n−1}u_{n−1}v_{n−2}ϕ_{n+1} + bk_{n−1}v_{n−1}u_{n−2}ϕ_{n+1}).

We observe that similar estimates to (14), (19) and (16) hold for the truncated bilinear operator P_{m}B(u^{m}, v^{m}). Namely, ∀u^{m}, v^{m} ∈ H_{m}

∥P_{m}B(u^{m}, v^{m})∥_{V′} ≤ C_{1}|u^{m}||v^{m}|.  (19)

and ∀u^{m}, v^{m} ∈ H

Re(P_{m}B(u^{m}, v^{m}), v^{m}) = 0.  (20)

3. Existence of Weak Solutions

In this section we prove the existence of weak solutions to the system (9). Our notion of weak solutions is similar to that of the Navier-Stokes equation (see, for example, [5], [34]).

Theorem 2. Let f ∈ L^{2}([0, T), V′) and u^{in} ∈ H. There exists

u ∈ L^{∞}([0, T], H) ∩ L^{2}([0, T], V),  (21)

with

\frac{du}{dt} ∈ L^{2}([0, T], V′),  (22)

satisfying the weak formulation of the equation (9), namely

⟨\frac{du}{dt}, v⟩ + ν ⟨Au, v⟩ + ⟨B(u, u), v⟩ = ⟨f, v⟩  (23a)
u(x, 0) = u^{in}(x),  (23b)

for every v ∈ V.

The weak solution u satisfies

u ∈ C([0, T], H).  (24)

Proof. The proof follows standard techniques for equations of Navier-Stokes type. See, for example, ([5], [34] or [35]). We will use the Galerkin procedure to prove the global existence and to establish the necessary a priori estimates.

The Galerkin approximating system of order m for equation (9) is an m - dimensional system of ordinary differential equations

\frac{du^{m}}{dt} + νAu^{m} + P_{m}B(u^{m}, u^{m}) = P_{m}f  (25a)
u^{m}(x, 0) = P_{m}u^{in}(x).  (25b)

We observe that the nonlinear term is quadratic in u^{m}, thus by the theory of ordinary differential equations, the system (25) possesses a unique solution for a short interval of time. Denote by [0, T_{m}) the maximal interval of existence for positive time. We will show later that in fact, T_{m} = T.

Let us fix m and the time interval [0, T]. On [0, T_{m}) we take the inner product (·, ·)_{H} of the equation (25a) with u^{m}, considering the real part and using (20) we get

\frac{1}{2} \frac{d}{dt}|u^{m}|^{2} + ν∥u^{m}∥^{2} = Re(P_{m}f, u^{m}).

The right hand side could be estimated using the Cauchy-Schwartz and Young inequalities

|(P_{m}f, u^{m})| ≤ ∥f∥_{V′}∥u^{m}∥ ≤ \frac{1}{2ν} ∥f∥_{V′}^{2} + \frac{ν}{2} ∥u^{m}∥^{2}.

Plugging it into the equation we get

\frac{d}{dt}|u^{m}|^{2} + ν∥u^{m}∥^{2} ≤ \frac{1}{ν} ∥f∥_{V′}^{2}.  (26)

Integrating the last equation from 0 to s, where 0 < s ≤ T_{m}, we conclude that for every such s,

|u^{m}(s)|^{2} ≤ |P_{m}u^{in}|^{2} + \frac{1}{ν} ∫_{0}^{T} ∥f(t)∥_{V′}dt.

Thus, lim sup_{s→T_{m}^{−}}|u_{m}(s)|^{2} < ∞. Therefore, the maximal interval of existence of solution of the system (25a) for every m is [0, T) - an interval, where the terms of the equation make sense, and in particular, f. If f ∈ L^{2}([0, ∞), V′), then we can take T > 0 arbitrary large and show the existence of weak solutions on [0, T). Now, we observe that

∥u^{m}∥_{L^{∞}([0,T),H)}^{2} ≤ |P_{m}u^{in}|^{2} + \frac{1}{ν} ∫_{0}^{T} ∥f∥_{V′}dt,

and hence

The sequence u^{m} lies in the bounded set of L^{∞}([0, T], H).  (27)

Next, integrating (26) over the interval [0, T] we get

|u^{m}(T)|^{2} + ν ∫_{0}^{T} ∥u^{m}(t)∥^{2}dt ≤ |u^{m}(0)|^{2} + \frac{1}{ν} ∫_{0}^{T} ∥f(t)∥_{V′}dt,

and using the fact that |u^{m}(0)| = |P_{m}u^{in}| ≤ |u^{in}| we conclude

The sequence u^{m} lies in the bounded set of L^{2}([0, T], V).  (28)

Finally, in order to apply Aubin’s Compactness Lemma (see, e.g., [1], [5]) we need to estimate the norm ∥\frac{du_{m}}{dt} ∥_{V′}. Consider the equation

\frac{du^{m}}{dt} = −νAu^{m} − P_{m}B(u^{m}, u^{m}) + P_{m}f

Then, (19) and (27) imply that ∥P_{m}B(u^{m}, u^{m})∥_{V′} is uniformly bounded for every 0 ≤ t ≤ T, hence

P_{m}B(u^{m}, u^{m}) ∈ L^{p}([0, T], V′),

for every p > 1 and T < ∞. Moreover, from the fact that ∥P_{m}f∥_{V′} ≤ ∥f∥_{V′} and the condition on f it follows that P_{m}f ∈ L^{2}([0, T], V′).

The relations (10) and (28) imply that the sequence {Au^{m}} is uniformly bounded in the space L^{2}([0, T], V′). Therefore, using the fact that f ∈ L^{2}([0, T], V′) we conclude,

The sequence \frac{du^{m}}{dt} lies in the bounded set of L^{2}([0, T], V′).  (29)

At last, we are able to apply Aubin’s Compactness Lemma. According to (27),

(28) and (29), there exists a subsequence u^{m′}, and u ∈ H satisfying

u^{m′} ⟶ u weakly in L^{2}([0, T], V)  (30a)
u^{m′} ⟶ u strongly in L^{2}([0, T], H)  (30b)

We are left to show that the solution u that we found, satisfies the weak formulation of the equation (9) together with the initial condition. Let v ∈ V be an arbitrary function. Take the scalar product of (25a) with v and integrate for some 0 ≤ t_{0} < t ≤ T to get

(u^{m′}(t), v) − (u^{m′}(t_{0}), v) + ν ∫_{t_{0}}^{t}((u^{m′}(τ), v))dτ+
+ ∫_{t_{0}}^{t}(P_{m′}B(u^{m′}, u^{m′}), v)dτ = ∫_{t_{0}}^{t} ⟨P_{m′}f, v⟩ dτ. (31)

The conclusion (30) implies that there exists a set E ⊂ [0, T] of Lebesgue measure 0 and a subsequence u^{m′′} of u^{m′} such that for every t ∈ [0, T] \ E, u^{m′′}(t) converges to u(t) weakly in V and strongly in H. As for the third term

lim_{m′→∞} ∫_{t_{0}}^{t}((u^{m′}(τ), v))dτ ⟶ ∫_{t_{0}}^{t}((u(τ), v))dτ,

because of (30a). The same holds for the forcing term. We are left to evaluate the nonlinear part. It is enough to check that

∫_{t_{0}}^{t}|(P_{m′}B(u^{m′}, u^{m′}), v) − B(u, u), v)|dτ ⟶ 0, as m′ → ∞.  (32)

Let us define Q = Q_{m′} = I − P_{m′}. Then, by definition of B(u, u) and arguments similar to (14) we have

|(P_{m′}B(u^{m′}, u^{m′}), v) − B(u, u), v)| =

= | ∑_{n=m}^{∞} (ak_{n+1}v_{n+2}u_{n+1}^{∗}v_{n}^{∗} + bk_{n}v_{n+1}v_{n}^{∗}u_{n−1}^{∗} + ak_{n−1}v_{n}^{∗}u_{n−1}v_{n−2}+

+ bk_{n−1}v_{n}^{∗}v_{n−1}u_{n−2})| ≤ C∥Qv∥|Qu|^{2} ≤ C∥v∥|u − P_{m′}u|^{2}.

We see that (32) holds, because of (30b). Passing to the limit in (31) we finally conclude that u is weakly continuous in H and that it satisfies the weak form of the equation, namely

(u(t), v) − (u(t_{0}), v) + ν ∫_{t_{0}}^{t}((u(τ), v))dτ+
+ ∫_{t_{0}}^{t}(B(u, u), v)dτ = ∫_{t_{0}}^{t} ⟨f, v⟩ dτ.

In order to finish the proof we need to show that u satisfies (24). First, recall that u ∈ L^{2}([0, T], V) and \frac{du}{dt} ∈ L^{2}([0, T], V′). Therefore, it follows that |u(t)|^{2} is continuous. Moreover, u ∈ C_{w}([0, T], H) – the space of weakly continuous functions with values in H. Now, let us fix t ∈ [0, T] and consider the sequence {t_{n}} ⊂ [0, T] converging to t. We know that |u(t_{n})| converges to |u(t)| and u(t_{n}) converges weakly to u(t). Then, it follows that u(t_{n}) converges strongly to u(t), and we finally conclude that

u ∈ C([0, T], H).
□

4. Uniqueness of Weak Solutions

Next we show that the weak solutions depend continuously on the initial data and in particular the solutions are unique.

Theorem 3. Let u(t), v(t) be two different solutions to the equation (9) on the time interval [0, T] with the corresponding initial conditions u^{in} and v^{in} in H. Then, for every t ∈ [0, T] we have

|u(t) − v(t)| ≤ e^{K}|u^{in} − v^{in}|,  (33)

where

K = C_{1} ∫_{0}^{t}∥u(s)∥ds.  (34)

Proof. Consider u, v – two solutions of the equation (9a) satisfying the initial conditions u(0) = u^{in} and v(0) = v^{in}. The difference w = u − v satisfies the equation

\frac{dw}{dt} + νAw + B(u, w) + B(w, u) − B(w, w) = 0,

with the initial condition w(0) = w^{in} = u^{in} − v^{in}.

According to Theorem 2, \frac{dw}{dt} ∈ L^{2}([0, T), V′) and w ∈ L^{2}([0, T), V). Hence, the particular case of the general theorem of interpolation due to Lions and Magenes [28] (see also [34], Chap. III, Lemma 1.2), implies that ⟨\frac{dw}{dt} , w⟩_{V′} = \frac{1}{2} \frac{d}{dt}|w|^{2}. Taking the inner product < ·, w > and using (14) and (16) we conclude that

\frac{1}{2} \frac{d}{dt}|w|^{2} ≤ \frac{1}{2} \frac{d}{dt}|w|^{2} + ν∥w∥^{2} = −Re < B(w, u), w >≤ C_{1}∥u∥|w|^{2}.

Gronwall’s inequality now implies

|w(t)|^{2} ≤ |w^{in}|^{2} exp (C_{1} ∫_{0}^{t}∥u(s)∥ds).  (35)

Since w^{in} = u^{in} −v^{in} and u ∈ L^{2}([0, T), V) the equation (33) follows. In particular, in the case that u^{in} = v^{in}, it follows that u(t) = v(t) for all t ∈ [0, T].

□

5. Strong Solutions

In the analogy to the Navier-Stokes equations we would like to show that the shell model equation (9) possesses even more regular solutions under an appropriate assumptions.

Let u^{m} be the solution of the Galerkin system (25). Taking the scalar product of the equation (25a) with Au^{m}, and using Young and Cauchy-Schwartz inequalities we obtain

\frac{1}{2} \frac{d}{dt}∥u^{m}∥^{2} + ν|Au^{m}|^{2} ≤ |(P_{m}B(u^{m}, u^{m}), Au^{m})| + |(f^{m}, Au^{m})| ≤

≤ C_{1}∥u^{m}∥|u^{m}||Au^{m}| + \frac{1}{ν} |f|^{2} + \frac{ν}{4} |Au^{m}|^{2} ≤

≤ \frac{C_{1}^{2}}{ν} ∥u^{m}∥^{2}|u^{m}|^{2} + \frac{ν}{4} |Au^{m}|^{2} + \frac{1}{ν} |f|^{2} + \frac{ν}{4} |Au^{m}|^{2}.

Finally, we get

\frac{d}{dt}∥u^{m}∥^{2} + ν|Au^{m}|^{2} ≤ \frac{2C_{1}^{2}}{ν} ∥u^{m}∥^{2}|u^{m}|^{2} + \frac{2}{ν} |f|^{2}.

By omitting for the moment the term ν|Au^{m}|^{2} from the left hand side of the equation, we can multiply it by the integrating factor exp(−\frac{2C_{1}^{2}}{ν} ∫_{t_{0}}^{t}|u^{m}(s)|^{2}ds) for some 0 ≤ t_{0} ≤ t to get

\frac{d}{dt}(∥u^{m}∥^{2} exp(−\frac{2C_{1}^{2}}{ν} ∫_{t_{0}}^{t}|u^{m}(s)|^{2}ds)) ≤ \frac{2}{ν} |f|^{2} exp(−\frac{2C_{1}^{2}}{ν} ∫_{t_{0}}^{t}|u^{m}(s)|^{2}ds) ≤ \frac{2}{ν} |f|^{2}.

(36)

Assuming that the forcing f is essentially bounded in the norm of H for all 0 ≤ t ≤ T, namely that f ∈ L^{∞}([0, T], H) and denoting |f|_{∞} = ∥f∥_{L^{∞}([0,T],H)}, we obtain by integrating equation (36)

∥u^{m}(t)∥^{2} ≤ e^{( \frac{2C^{2}_{1}}{ν} ∫_{t_{0}}^{t} |u^{m}(s)|^{2}ds)}(∥u^{m}(t_{0})∥^{2} + \frac{2}{ν} |f|_{∞}^{2}(t − t_{0})).  (37)

In order to obtain a uniform bound on ∥u^{m}(t)∥^{2} for all 0 ≤ t ≤ T and for all m we need to estimate both the exponent and ∥u^{m}(t_{0})∥^{2}. The last quantity is of course bounded for t_{0} = 0 if we assume that P_{m}u^{in} ∈ V for all m, however we want to put a less strict assumptions. Let us first refine the estimates we made in the proof of the existence of weak solutions. Take the scalar product of equation (25a) with u^{m} to get

\frac{1}{2} \frac{d}{dt}|u^{m}|^{2} + ν∥u^{m}∥^{2} = (f^{m}, u^{m}) ≤ \frac{1}{2k_{1}^{2}ν} |f|^{2} + \frac{k_{1}^{2}ν}{2} |u^{m}|^{2},

and using the inequality k_{1}^{2}|u^{m}|^{2} ≤ ∥u^{m}∥^{2} we get

\frac{d}{dt}|u^{m}|^{2} + ν∥u^{m}∥^{2} ≤ \frac{1}{k_{1}^{2}ν} |f|^{2}.  (38)

Integrating from 0 to t, and once again using the fact that f ∈ L^{∞}([0, T], H), we get

ν ∫_{0}^{t}∥u^{m}(s)∥^{2}ds ≤ |u^{m}(0)|^{2} + \frac{1}{k_{1}^{2}ν} |f|_{∞}^{2}t.  (39)

On the other hand, Gronwall’s inequality, applied to (38) gives us

|u^{m}(t)|^{2} ≤ |u^{m}(0)|^{2}e^{−νk_{1}^{2}t} + ∫_{0}^{t} e^{−νk_{1}^{2}(t−s)} \frac{|f|^{2}}{k_{1}^{2}ν} ds,

and implies

|u^{m}(t)|^{2} ≤ |u^{m}(0)|^{2}e^{−k_{1}^{2}νt} + \frac{|f|_{∞}^{2}}{k_{1}^{4}ν^{2}} (1 − e^{−k_{1}^{2}νt}),  (40)

were the last expression yields for every t > 0

|u^{m}(t)|^{2} ≤ |u^{m}(0)| + \frac{|f|_{∞}^{2}}{k_{1}^{4}ν^{2}} .  (41)

Using the equation (41) we immediately can evaluate the exponent in (37) by

e^{( \frac{2C^{2}_{1}}{ν} ∫_{t_{0}}^{t} |u^{m}(s)|^{2}ds)} ≤ e^{\frac{2C^{2}_{1}}{ν} (t−t_{0})(|u^{m}(0)|^{2}+_{\frac{1}{k^{4}_{1}ν^{2}}} |f|_{∞}^{2})}.  (42)

The last step is to get a uniform bound for the ∥u^{m}(t_{0})∥^{2}. Let us integrate equation (38) from t to t + τ. The result, after applying (41) is

ν ∫_{t}^{t+τ} ∥u^{m}∥^{2}ds ≤ |u^{m}(t + τ)|^{2} − |u^{m}(t)|^{2} + \frac{τ}{k_{1}^{2}ν} |f|^{2} ≤

≤ |u^{m}(0)|^{2} + \frac{1}{k_{1}^{2}ν} |f|_{∞}^{2}(\frac{1}{k_{1}^{2}ν} + τ)

The standard application of Markov’s inequality implies that on every interval of length τ there exists a time t_{0} satisfying

∥u^{m}(t_{0})∥^{2} ≤ \frac{2}{τ} (\frac{1}{ν} |u^{m}(0)|^{2} + \frac{1}{k_{1}^{2}ν^{2}} |f|_{∞}^{2}(\frac{1}{k_{1}^{2}ν} + τ)).  (43)

Let us set τ ≤ \frac{1}{k_{1}^{2}ν}. Then, for every t ≥ τ and an appropriate t_{0} chosen in the interval [t−τ, t] we get a uniform bound for all m ∈ ℕ and all t ≥ τ, by substituting (40) and (43) into (37)

τ∥u^{m}(t)∥^{2} ≤ 2e^{\frac{2C^{2}_{1}}{k^{2}_{1}ν^{2}} (|u^{m}(0)|^{2}+_{\frac{1}{k^{4}_{1}ν^{2}}} |f|_{∞}^{2})} · (\frac{1}{ν} |u^{m}(0)|^{2} + \frac{3}{k_{1}^{4}ν^{3}} |f|_{∞}^{2}) (44)

To expand the estimate for all times 0 ≤ t ≤ \frac{1}{k_{1}^{2}ν}, we can apply the last inequality to the dyadic intervals of the form [\frac{1}{k_{1}^{2}ν} \frac{1}{2^{k+1}} , \frac{1}{k_{1}^{2}ν} \frac{1}{2^{k}}] with τ = \frac{1}{k_{1}^{2}ν} \frac{1}{2^{k+1}} . Then, if t ∈ [\frac{1}{k_{1}^{2}ν} \frac{1}{2^{k+1}} , \frac{1}{k_{1}^{2}ν} \frac{1}{2^{k}}], it satisfies τ ≤ t ≤ 2τ, and for such t inequality (44) implies

t∥u^{m}(t)∥^{2} ≤ 4e^{\frac{2C^{2}_{1}}{k^{2}_{1}ν^{2}} (|u^{m}(0)|^{2}+_{\frac{1}{k^{4}_{1}ν^{2}}} |f|_{∞}^{2})} · (\frac{1}{ν} |u^{m}(0)|^{2} + \frac{3}{k_{1}^{4}ν^{3}} |f|_{∞}^{2})  (45)

Since the last estimate does not depend on k we can conclude that it holds for all 0 ≤ t ≤ \frac{1}{k_{1}^{2}ν}.

Finally, passing to the limit in the Galerkin approximation we obtain

Theorem 4. Let T > 0 and f ∈ L^{∞}([0, T], H). Then if u^{in} ∈ H, the equation (9) possesses a solution u(t) satisfying

u ∈ L_{loc}^{∞}((0, T], V) ⋂ L_{loc}^{2}((0, T], D(A)) ⋂ L^{∞}([0, T], H) ⋂ L^{2}([0, T], V).

Moreover, the following bound holds

sup_{0<t≤min\{_{\frac{1}{k^{2}_{1}ν}} ,T \}} νk_{1}^{2}t∥u(t)∥^{2} + sup_{_{\frac{1}{k^{2}_{1}ν}} ≤t≤T}∥u(t)∥^{2} ≤

≤ 6e^{\frac{2C^{2}_{1}}{k^{2}_{1}ν^{2}} (|u^{in}|^{2}+_{\frac{1}{k^{4}_{1}ν^{2}}} |f|_{∞}^{2})} · (k_{1}^{2}|u^{in}|^{2} + \frac{3}{k_{1}^{2}ν^{2}} |f|_{∞}^{2}). (46)

If the initial condition u^{in} ∈ V, then the solution u(t) satisfies

u ∈ C([0, T], V) ⋂ L^{2}([0, T], D(A)),  (47)

6. Gevrey class regularity

6.1. Preliminaries and Classical Navier-Stokes Equation Results. We start this section with a definition

Definition 2. Let us fix positive constants τ, p > 0, q ≥ 0 and define the following class of sequences

G_{τ}^{p,q} = {u ∈ ℓ_{2} : |e^{τA^{p/2}}A^{q/2}u| = ∑_{n=1}^{∞} e^{2τk_{n}^{p}}k_{n}^{q}|u_{n}|^{2} < ∞}.

We will say that the sequence u ∈ ℓ_{2} is Gevrey regular if it belongs to the class G_{τ}^{p,q} for some choice of τ, p > 0 and q ≥ 0.

In order to justify this definition let u = (u_{1}, u_{2}, u_{3}, …) and suppose that u_{n}, n = 1, 2, 3, …, are the Fourier coefficients of some scalar function g(x), with corresponding wavenumbers k_{n}, defined by relation (2). Then, if u ∈ G_{τ}^{p,q}, for some p, q and τ, then the function g(x) is analytic. The concept of the Gevrey class regularity for showing the analyticity of the solutions of the two-dimensional Navier-Stokes equations, was first introduced in [20], simplifying earlier proofs. Later this technique was extended to the large class of analytic nonlinear parabolic equations in [21].

For q = 0 we will write G_{τ}^{p} = G_{τ}^{p,0}. Denote by (·, ·)_{p,τ} the scalar product in G_{τ}^{p} and

|u|_{p,τ} = ∥u∥_{G_{τ}^{p}}, ∀u ∈ G_{τ}^{p}.

In the case of q = 1 we will write

∥u∥_{p,τ} = ∥u∥_{G_{τ}^{p,1}}, ∀u ∈ G_{τ}^{p,q},

and the scalar product in G_{τ}^{p,1} will be denoted by ((·, ·))_{p,τ}.

Following the tools introduced in [20] and generalized in [21] we prove the following:

Theorem 5. Let us assume that u(0) = u^{in} ∈ V and the force f ∈ G_{σ_{1}}^{p_{1}} for some σ_{1}, p_{1} > 0. Then, there exists T, depending only on the initial data, such that for every 0 < p ≤ min\{p_{1}, log_{λ} \frac{1+\sqrt{5}}{2}\} the following holds

(i) For every t ∈ [0, T] the solution u(t) of the equation (9) remains bounded in G_{ψ(t)}^{p,1}, for ψ(t) = min{t, σ_{1}}.

(ii) Furthermore, there exists σ > 0 such that for every t > T the solution u(t) of the equation (9) remains bounded in G_{σ}^{p,1}.

Before proving the theorem, we prove the following estimates for the nonlinear term:

Lemma 6. Let u, v, w ∈ G_{τ}^{p,1} for some τ > 0 and p ≤ log_{λ} \frac{1+\sqrt{5}}{2}. Then B(u, v) belongs to G_{τ}^{p} and satisfies

|(A^{1/2}e^{τA^{p/2}}B(u, v), A^{1/2}e^{τA^{p/2}}w)| ≤ C_{4}∥u∥_{G_{τ}^{p,1}}∥v∥_{G_{τ}^{p,1}}∥w∥_{G_{τ}^{p,1}},  (48)

for an appropriate constant C_{4} > 0.

Proof. Let us fix an arbitrary p ≤ log_{λ} \frac{1+\sqrt{5}}{2}. Observe, that the operator e^{τA^{p/2}} is diagonal in the standard canonical basis {ϕ_{n}}_{n=1}^{∞} of H with corresponding eigenvalues e^{τk̃_{n}}, n = 1, 2, …, where k̃_{n} = λ̃^{n} for some λ̃ ≤ \frac{1+\sqrt{5}}{2}. Therefore,

|(A^{1/2}e^{τA^{p/2}}B(u, v), A^{1/2}e^{τA^{p/2}}w)| ≤

≤ ∑_{n=1}^{∞} e^{2τk̃_{n}} |ak_{n+1}k_{n}^{2}v_{n+2}u_{n+1}^{∗}w_{n}^{∗} + bk_{n}^{3}v_{n+1}w_{n}^{∗}u_{n−1}^{∗}

+ ak_{n}^{2}k_{n−1}w_{n}^{∗}u_{n−1}v_{n−2} + bk_{n}^{∗}k_{n−1}w_{n}^{∗}v_{n−1}u_{n−2}| ≤

≤ sup_{n≥1}|e^{τk̃_{n}}k_{n}w_{n}| ∑_{n=1}^{∞} e^{τk̃_{n}}(|ak_{n+1}k_{n}v_{n+2}u_{n+1}^{∗}| + |bk_{n}^{2}v_{n+1}u_{n−1}^{∗}|+

+ |ak_{n}k_{n−1}u_{n−1}v_{n−2}| + |bk_{n}k_{n−1}u_{n−1}v_{n−2}|) ≤ C_{4}∥w∥_{G_{τ}^{p,1}}∥u∥_{G_{τ}^{p,1}}∥v∥_{G_{τ}^{p,1}}, (49)

were we can choose C_{4} = (|a|(λ^{−2} + λ^{2}) + |b|(1 + λ^{2})). The last inequality is the result of the Cauchy-Schwartz inequality and the fact, that e^{τ(λ̃^{n}−λ̃^{n+2}−λ̃^{n+1})}, e^{τ(λ̃^{n}−λ̃^{n+1}−λ̃^{n−1})} and e^{τ(λ̃^{n}−λ̃^{n−1}−λ̃^{n−2})} are less than or equal to 1 for all τ > 0 and our specific choice of λ̃.

□

Now we are ready to prove the Theorem:

Proof. (Of Theorem 5) Let us take an arbitrary 0 < p ≤ min\{p_{1}, log_{λ} \frac{1+\sqrt{5}}{2}\} and denote φ(t) = min(t, σ_{1}). For a fixed time t take the scalar product of the equation (9a) with Au(t) in G_{φ(t)}^{p} to get

(u′(t), Au(t))_{p,φ(t)} + ν|Au(t)|_{p,φ(t)}^{2} =
= (f, Au(t))_{p,φ(t)} − (B(u(t), u(t)), Au(t))_{p,φ(t)}.  (50)

The left hand side of the equation could be transformed in the following way:

(u′(t),Au(t))_{p,φ(t)} = (e^{φ(t)A^{p/2}}u′(t), e^{φ(t)A^{p/2}}Au(t)) =

= (A^{1/2}(e^{φ(t)A^{p/2}}u(t))′ − φ′(t)A^{(p+1)/2}e^{φ(t)A^{p/2}}u(t), A^{1/2}e^{φ(t)A^{p/2}}u(t)) =

= \frac{1}{2} \frac{d}{dt}∥u(t)∥_{p,φ(t)}^{2} − φ′(t)(A^{(p+1)/2}u(t), A^{1/2}u(t))_{p,φ(t)} ≥

≥ \frac{1}{2} \frac{d}{dt}∥u(t)∥_{p,φ(t)}^{2} − |Au(t)|_{p,φ(t)}∥u(t)∥_{p,φ(t)} ≥

≥ \frac{1}{2} \frac{d}{dt}∥u(t)∥_{p,φ(t)}^{2} − \frac{ν}{4} |Au(t)|_{p,φ(t)}^{2} − \frac{1}{ν} ∥u(t)∥_{p,φ(t)}^{2}.  (51)

The right hand side of the equation (50) can be bounded from above in the usual way using Lemma 6, Cauchy-Schwartz and Young inequalities

|(f, Au(t))_{p,φ(t)} − (B(u(t), u(t)), Au(t))_{p,φ(t)}| ≤
≤ |f|_{p,φ(t)}|Au(t)|_{p,φ(t)} + C_{4}∥u∥_{p,φ(t)}^{3} ≤
≤ \frac{1}{ν} |f|_{p,φ(t)}^{2} + \frac{ν}{4} |Au(t)|_{p,φ(t)}^{2} + C_{4}∥u∥_{p,φ(t)}^{3}.  (52)

Combining (51) and (52), the equation (50) takes the form

\frac{d}{dt}∥u(t)∥_{p,φ(t)}^{2} + ν|Au(t)|_{p,φ(t)}^{2} ≤ \frac{2}{ν} |f|_{p,φ(t)}^{2} + \frac{2}{ν} ∥u(t)∥_{p,φ(t)}^{2} + C_{4}∥u∥_{p,φ(t)}^{3} ≤

≤ \frac{2}{ν} |f|_{p,φ(t)}^{2} + C_{5} + C_{4}∥u∥_{p,φ(t)}^{3}.

Denoting y(t) = 1 + ∥u(t)∥_{p,φ(t)}^{2} and K = \frac{2}{ν} |f|_{p,φ(t)}^{2} + C_{5} + C_{4}^{2/3} we get an inequality

ẏ ≤ Ky^{3/2},

It implies

y(t) = 1 + ∥u(t)∥_{p,φ(t)}^{2} ≤ 2y(0) = 2(1 + ∥u^{in}∥^{2}),

for

t ≤ T_{1}(σ_{1}, ∥u^{in}∥, |f|_{p,φ(t)}) = \frac{2 − \sqrt{2}}{K}y^{−1/2}(0) = \frac{2 − \sqrt{2}}{K}(1 + ∥u^{in}∥^{2})^{−1/2}.

According to Theorem 4, the solution u(t) of the equation (9) remains bounded in V for all time if we start from u^{in} ∈ V. Moreover, from the relation (47) we conclude that there exists a constant M, such that for all t > 0

∥u(t)∥ ≤ M.

Hence we can repeat the above arguments starting from any time t > 0 and find that

|e^{φ(T_{2})A^{p/2}}A^{1/2}u(t)| ≤ 2 + 2M^{2},  (53)

for all t ≥ T_{2} = \frac{2−\sqrt{2}}{K}(1 + M^{2})^{−1/2}, and the Theorem holds.

□

6.2. A Stronger Result. We prove the following stronger version of Theorem 5; it is not known whether this holds for the Navier-Stokes equations.

Theorem 7. Let us assume that u(0) = u^{in} ∈ H and the force f ∈ G_{σ_{1}}^{p_{1}} for some σ_{1}, p_{1} > 0. Then, there exists T, depending only on the initial data, such that for every 0 < p ≤ min\{p_{1}, log_{λ} \frac{1+\sqrt{5}}{2}\} the following holds

(i) For every t ∈ [0, T] the solution u(t) of the equation (9) remains bounded in G_{ψ(t)}^{p}, for ψ(t) = min{t, σ_{1}}.

(ii) For every t > T there exists σ > 0 such that the solution u(t) of the equation (9) remains bounded in G_{σ}^{p}.

Before proving the theorem, we obtain the following estimates for the nonlinear term:

Lemma 8. Let u, v ∈ G_{τ}^{p}, w ∈ G_{τ}^{p,1} for some τ > 0 and p ≤ log_{λ} \frac{1+\sqrt{5}}{2}. Then B(u, v) belongs to G_{τ}^{p} and satisfies

|(e^{τA^{p/2}}B(u, v), e^{τA^{p/2}}w)| ≤ C_{6}∥u∥_{G_{τ}^{p}}∥v∥_{G_{τ}^{p}}∥w∥_{G_{τ}^{p,1}},  (54)

for an appropriate constant C_{6} > 0.

Proof. Let us fix an arbitrary p ≤ log_{λ} \frac{1+\sqrt{5}}{2}. Then we note that the operator e^{τA^{p/2}} is diagonal with eigenvalues e^{τk̃_{n}}, n = 1, 2, …, where k̃_{n} = λ̃^{n} for λ̃ ≤ \frac{1+\sqrt{5}}{2}. Hence,

|(e^{τA^{p/2}}B(u, v), e^{τA^{p/2}}w)| ≤

≤ ∑_{n=1}^{∞} e^{2τk̃_{n}} |ak_{n+1}u_{n+2}v_{n+1}^{∗}w_{n}^{∗}+

+ bk_{n}u_{n+1}w_{n}^{∗}v_{n−1}^{∗} − ck_{n−1}w_{n}^{∗}v_{n−1}u_{n−2}| ≤

≤ sup_{n≥1}|e^{τk̃_{n}}k_{n}w_{n}| ∑_{n=1}^{∞} e^{τk̃_{n}}(|aλu_{n+2}v_{n+1}^{∗}|+

+ |bu_{n+1}v_{n−1}^{∗}| + |cλ^{−1}v_{n−1}u_{n−2}|) ≤

≤ C_{6}∥w∥_{G_{τ}^{p,1}}∥u∥_{G_{τ}^{p}}∥v∥_{G_{τ}^{p}},  (55)

were we can choose C_{6} = (λa + b + λ^{−1}c). The last inequality is the result of the Cauchy-Schwartz inequality and the fact, that e^{τ(λ̃^{n}−λ̃^{n+2}−λ̃^{n+1})}, e^{τ(λ̃^{n}−λ̃^{n+1}−λ̃^{n−1})} and e^{τ(λ̃^{n}−λ̃^{n−1}−λ̃^{n−2})} are less than or equal to 1 for all τ and the specific choice of λ̃.

□

Now we are ready to prove the Theorem:

Proof. (Of Theorem 7) Let us take an arbitrary 0 < p ≤ min\{p_{1}, log_{λ} \frac{1+\sqrt{5}}{2}\} and denote φ(t) = min(t, σ_{1}). For a fixed time t take the scalar product of the equation (9a) with u(t) in G_{φ}^{p}(t) to get

(u′(t), u(t))_{p,φ(t)} + ν∥u(t)∥_{p,φ(t)}^{2} =
= (f, u(t))_{p,φ(t)} − (B(u(t), u(t)), u(t))_{p,φ(t)}.  (56)

The left hand side of the equation could be transformed in the following way:

(u′(t),Au(t))_{p,φ(t)} = (e^{φ(t)A^{p/2}}u′(t), e^{φ(t)A^{p/2}}u(t)) =

= ((e^{φ(t)A^{p/2}}u(t))′ − φ′(t)A^{p/2}e^{φ(t)A^{p/2}}u(t), e^{φ(t)A^{p/2}}u(t)) =

= \frac{1}{2} \frac{d}{dt}|u(t)|_{p,φ(t)}^{2} − φ′(t)(A^{p/2}u(t), u(t))_{p,φ(t)} ≥

≥ \frac{1}{2} \frac{d}{dt}|u(t)|_{p,φ(t)}^{2} − ∥u(t)∥_{p,φ(t)}|u(t)|_{p,φ(t)} ≥

≥ \frac{1}{2} \frac{d}{dt}|u(t)|_{p,φ(t)}^{2} − \frac{ν}{4} ∥u(t)∥_{p,φ(t)}^{2} − \frac{1}{ν} |u(t)|_{p,φ(t)}^{2}.  (57)

The right hand side of the equation (56) could be bounded from above in the usual way using Lemma 8, Cauchy-Schwartz and Young inequalities

|(f, u(t))_{p,φ(t)} − (B(u(t), u(t)), u(t))_{p,φ(t)}| ≤
≤ |f|_{p,φ(t)}|u(t)|_{p,φ(t)} + C_{6}∥u∥_{p,φ(t)}|u|_{p,φ(t)}^{2} ≤
≤ \frac{3}{4}|f|_{p,φ(t)}^{4/3} + \frac{1}{4}|u(t)|_{p,φ(t)}^{4} + \frac{ν}{4} ∥u∥_{p,φ(t)}^{2} + \frac{C_{6}^{2}}{ν} |u|_{p,φ(t)}^{4}.  (58)

Combining (57) and (58), the equation (56) takes the form

\frac{d}{dt}|u(t)|_{p,φ(t)}^{2} + ν∥Au(t)∥_{p,φ(t)}^{2} ≤

≤ \frac{3}{2}|f|_{p,φ(t)}^{4/3} + \frac{2}{ν} |u(t)|_{p,φ(t)}^{2} + (\frac{1}{2} + \frac{2C_{6}^{2}}{ν})|u|_{p,φ(t)}^{4} ≤

≤ \frac{3}{2}|f|_{p,φ(t)}^{4/3} + C_{7}|u|_{p,φ(t)}^{4},

for some positive constant C_{7}.

Denoting y(t) = 1 + |u(t)|_{p,φ(t)}^{2} and K = \frac{3}{2}|f|_{p,φ(t)}^{4/3} + C_{7} we get an inequality

ẏ ≤ Ky^{2},

It implies

y(t) = 1 + ∥u(t)∥_{p,φ(t)}^{2} ≤ 2y(0) = 2(1 + |u^{in}|^{2}),

for

t ≤ T_{1}(σ_{1}, |u^{in}|, |f|_{p,φ(t)}) = \frac{1}{2K} y^{−1}(0) = \frac{1}{2K} (1 + |u^{in}|^{2})^{−1}.

According to Theorem 2 the solution of the equation (9) remains bounded in H for all time, if we start from u^{in} ∈ H. Moreover, from the estimate (24), it follows that there exists a constant M, such that for all t > 0

|u(t)| ≤ M.

Hence we can repeat the above arguments starting from any time t > 0 and find that

|e^{φ(T_{2})A^{p/2}}A^{1/2}u(t)| ≤ 2 + 2M^{2},  (59)

for all t ≥ T_{2} = \frac{1}{2K} (1 + M^{2})^{−1}, and the Theorem holds.

□

7. Global Attractors and Their Dimensions

The first mathematical concept which we use to establish the finite dimensional long-term behavior of the viscous sabra shell model is the global attractor. The global attractor, A, is the maximal bounded invariant subset of the space H. It encompasses all of the possible permanent regimes of the dynamics of the shell model. It is also a compact subset of the space H which attracts all the trajectories of the system. Establishment of finite Hausdorff and fractal dimensionality of the global attractor implies the possible parameterization of the permanent regimes of the dynamics in terms of a finite number of parameters. For the definition and further discussion of the concept of the global attractor see, e.g., [14] and [35].

In analogy with Kolmogorov’s mean rate of dissipation of energy in turbulent flow we define

ϵ = ν ⟨∥u∥^{2}⟩ ,

the mean rate of dissipation of energy in the shell model system. ⟨·⟩ represents the ensemble average or a long-time average. Since we do not know whether such long-time averages converge for trajectories we will replace the above definition of ϵ by

ϵ = ν sup_{u^{in}∈A} lim sup_{t→∞} \frac{1}{t} ∫_{0}^{t}∥u(s)∥^{2}ds.

We will also define the viscous dissipation length scale l_{d}. According to Kolmogorov’s theory, it should only depend on the viscosity ν and the mean rate of energy dissipation ϵ. Hence, pure dimensional arguments lead to the definition

l_{d} = (\frac{ν^{3}}{ϵ} )^{1/4},

which represents the largest spatial scale at which the viscosity term begins to dominate over the nonlinear inertial term of the shell model equation. In analogy with the conventional theory of turbulence this is also the smallest scale that one needs to resolve in order to get the full resolution for turbulent flow associated with the shell model system.

We would like to obtain an estimate of the fractal dimension of the global attractor for the system (9) in terms of another non-dimensional quantity – the generalized Grashoff number. Suppose the forcing term satisfies f ∈ L^{∞}([0, T], H) and denote ∥f∥_{L^{∞}([0,∞),H)} = |f|_{∞}. Then we define the generalized Grashoff number for our system to be

G = \frac{|f|_{∞}}{ν^{2}k_{1}^{3}} .  (60)

The generalized Grashoff number was first introduced in the context of the study of the finite dimensionality of long-term behavior of turbulent flow in [15]. To check that it is indeed non-dimensional we note that ∥f∥_{L^{∞}([0,T],H)} has the dimension of \frac{L}{T^{2}} , were L is a length scale and T is a time scale. k_{1} has the dimension of \frac{1}{L} and the kinematic viscosity ν has the dimension of \frac{L^{2}}{T}. In order to obtain an estimate of the generalized Grashoff number, which will be used in the proof of the main result of this section, we can apply inequality get

lim sup_{t→∞} \frac{1}{t} ∫_{0}^{t}∥u(s)∥^{2}ds ≤ \frac{|f|_{∞}^{2}}{ν^{2}k_{1}^{2}} = ν^{2}k_{1}^{4}G^{2}.  (61)

Theorem 9. The Hausdorff and fractal dimensions of the global attractor of the system of equations (9), d_{H}(A) and d_{F} (A) respectively, satisfy

d_{H}(A) ≤ d_{F} (A) ≤ 4 log_{λ} (\frac{L}{l_{d}} ) + 2 log_{λ} (C_{1} \frac{λ^{2} − 1}{λ^{2}}) − 2.  (62)

In terms of the Grashoff number G the upper bound takes the form

d_{H}(A) ≤ d_{F} (A) ≤ 2 log_{λ}(C_{1}G) − 2.  (63)

Proof. We follow [6] (see also [5], [35]) and linearize the shell model system of equations (9) about the trajectory u(t) in the global attractor. In the Appendix B we show that the solution u(t) is differentiable with respect to the initial data, hence the resulting linear equation takes the form

\frac{dU}{dt} + νAU + B_{0}(t)U = 0  (64a)
U(0) = U^{in},  (64b)

where B_{0}(t)U = B(u(t), U(t)) + B(U(t), u(t)) is a linear operator. In order to simplify the notation, we will denote

Λ(t) = −νA − B_{0}(t).

Let U_{j}(t) be solutions of the above system satisfying U_{j}(0) = U_{j}^{in}, for j = 1, 2, …, N. Assume now that U_{1}^{in}, U_{2}^{in}, …, U_{N}^{in} are linearly independent in H, and consider Q_{N}(t) – an H-orthogonal projection onto the span {U_{j}(t)}_{j=1}^{N}. Let {φ_{j}(t)}_{j=1}^{N} be the orthonormal basis of the span of {U_{j}(t)}_{j=1}^{N}. Notice that φ_{j} ∈ D(A) since span{U_{1}(t), …, U_{N}(t)} ⊂ D(A).

Using the definition of Λ(t) and the inequalities of the Proposition 1 we get

Re(T race[Λ(t) ∘ Q_{N}(t)]) = Re ∑_{j=1}^{N}(Λ(t)φ_{j}, φ_{j}) =

= ∑_{j=1}^{N} −ν∥φ_{j}∥^{2} − Re(B(φ_{j}, u(t)), φ_{j}) ≤

≤ −ν ∑_{j=1}^{N}(k_{j}^{2} + |(B(φ_{j}, u(t)), φ_{j})|) ≤

≤ −νk_{0}^{2}λ^{2} \frac{λ^{2N} − 1}{λ^{2} − 1} + C_{1}N∥u(t)∥.  (65)

According to the trace formula of [6] (see also [5], [35]), if N is large enough, such that

lim sup_{t→∞} \frac{1}{t} ∫_{0}^{t} Re(T race[Λ(t) ∘ Q_{N}(s)])ds < 0,

then the fractal dimension of the global attractor is bounded by 2N. We need therefore to estimate N in terms of the energy dissipation rate. Using (65), the definition of ϵ and applying Jensen’s inequality we find that it is sufficient to require N to be large enough such that it satisfies

νk_{0}^{2}λ^{2} \frac{λ^{2N} − 1}{λ^{2} − 1} > C_{1}N( lim sup_{t→∞} \frac{1}{t} ∫_{0}^{t}∥u(t)∥^{2})^{1/2} = C_{1}N(\frac{ϵ}{ν} )^{1/2}.

Finally, ϵν^{−3} = l_{d}^{−4} implies

λ^{N+1} > \frac{λ^{2N} − 1}{N} > C_{1} \frac{λ^{2} − 1}{λ^{2}}(\frac{L}{l_{d}} )^{2},

which proves (62). Applying the estimate (61) to the inequality (65) we get the estimation of N in terms of the generalized Grashoff number

λ^{N+1} > C_{1}G.

□

8. Determining Modes

Let us consider two solutions of the the shell model equations u, v corresponding to the forces f, g ∈ L^{2}([0, ∞), H)

\frac{du}{dt} + νAu + B(u, u) = f,  (66)
\frac{du}{dt} + νAu + B(u, u) = g.  (67)

We give a slightly more general definition of a notion of the determining modes, than the one that is introduced previously in literature [16] (see also [14], [15], [23] and references therein).

Definition 3. We define a set of determining modes as a finite set of indices M ⊂ ℕ, such that whenever the forces f, g satisfy

|f(t) − g(t)| → 0, as t → ∞,  (68)

and

∑_{n∈M}|u_{n}(t) − v_{n}(t)|^{2} → 0, as t → ∞  (69)

it follows that

|u(t) − v(t)| → 0, as t → ∞.  (70)

The number of determining modes N of the equation is the size of the smallest such a set M.

We would like to recall the following generalization of the classical Gronwall’s lemma which was proved in [23], [24] (see also, [14]).

Lemma 10. Let α = α(t) and β = β(t) be locally integrable real-valued functions on [0, ∞) that satisfy the following condition for some T > 0:

lim inf_{t→∞} \frac{1}{T} ∫_{t}^{t+T} α(τ)dτ > 0,  (71)
lim sup_{t→∞} \frac{1}{T} ∫_{t}^{t+T} α^{−}(τ)dτ < ∞,  (72)
lim_{t→∞} \frac{1}{T} ∫_{t}^{t+T} β^{+}(τ)dτ = 0,  (73)

where α^{−}(t) = max{−α(t), 0} and β^{+}(t) = max{β(t), 0}. Suppose that ξ = ξ(t) is an absolutely continuous nonnegative function on [0, ∞) that satisfies the following inequality almost everywhere on [0, ∞):

\frac{dξ}{dt} + αξ ≤ β.  (74)

Then ξ(t) → 0 as t → ∞.

A weaker version of the above statement was introduced in [15]. In fact this weaker version would be sufficient for our purposes.

Theorem 11. Let G be a Grashoff number defined in (60). Then,

(i) The first N modes are determining modes for the shell model equation provided

N > \frac{1}{2} log_{λ}(C_{1}G).  (75)

(ii) Let u, v be two solutions of the equations (66) and (67) correspondingly. Let the forces f, g satisfy (68) and integer N be defined as in (75). If

lim_{t→∞}|u_{N}(t) − v_{N}(t)|, and lim_{t→∞}|u_{N−1}(t) − v_{N−1}(t)| = 0,

then

lim_{t→∞}|Q_{N}u(t) − Q_{N}v(t)| = 0.

Proof. Denote w = u − v, which satisfies the equation

\frac{dw}{dt} + νAw + B(u, w) + B(w, v) = f − g.  (76)

Let us fix an integer m > 1 and define P_{m} : H → H to be an orthogonal projection onto the first m coordinates, namely, onto the subspace of H spanned by m vectors of the standard basis {e_{i}}_{i=1}^{m}. Also denote Q_{m} = I − P_{m}.

It is not hard to see that the second part of the Theorem implies the first statement. Therefore it is enough for us to prove that if

lim_{t→∞}|w_{N}(t)|, and lim_{t→∞}|w_{N−1}(t)| = 0,

then for every m > \frac{1}{2} log_{λ}(C_{1}G)

lim_{t→∞}|Q_{m}w| = 0.

Multiplying the equation (76) by Q_{m}w in the scalar product of H we get

\frac{1}{2} \frac{d}{dt}|Q_{m}w|^{2} + ν∥Q_{m}w∥^{2} =
= Re(f − g, Q_{m}w) − Re(B(u, w), Q_{m}w) − Re(B(w, v), Q_{m}w). (77)

For the left hand side of the equation we can use the inequality k_{m+1}^{2}|Q_{m}w| ≤ ∥Q_{m}w∥. For the right hand side of the equation we observe that

Re(B(u, w), Q_{m}w) = Im ∑_{n=m+1}^{∞} (ak_{n+1}w_{n+2}u_{n+1}^{∗}w_{n}^{∗} + bk_{n}w_{n+1}w_{n}^{∗}u_{n−1}^{∗}+

+ ak_{n−1}w_{n}^{∗}u_{n−1}w_{n−2} + bk_{n−1}w_{n}^{∗}w_{n−1}u_{n−2}) ≤
≤ |ak_{m}w_{m+1}^{∗}u_{m}w_{m−1} + ak_{m+1}w_{m+2}^{∗}u_{m+1}w_{m} + bk_{m}w_{m+1}^{∗}w_{m}u_{m−1}|.

Moreover,

Re(B(w, v), Q_{m}w) = Im ∑_{n=m+1}^{∞} (ak_{n+1}v_{n+2}w_{n+1}^{∗}w_{n}^{∗} + bk_{n}v_{n+1}w_{n}^{∗}w_{n−1}^{∗}+

+ ak_{n−1}w_{n}^{∗}w_{n−1}v_{n−2} + bk_{n−1}w_{n}^{∗}v_{n−1}w_{n−2}) ≤
≤ |ak_{m}w_{m+1}^{∗}w_{m}v_{m−1} + ak_{m+1}w_{m+2}^{∗}w_{m+1}v_{m}+
+ bk_{m}w_{m+1}^{∗}v_{m}w_{m−1}| + C_{1}∥v∥|Q_{m}w|^{2}.

Finally denoting ξ = |Q_{m}w|^{2} we can rewrite the equation in the form

\frac{1}{2} \frac{dξ}{dt} + αξ ≤ β,

were

α(t) = νk_{m+1}^{2} − C_{1}∥v(t)∥,

and

β(t) = (f − g, Q_{m}w)+
+ |ak_{m}w_{m+1}^{∗}u_{m}w_{m−1} + ak_{m+1}w_{m+2}^{∗}u_{m+1}w_{m} + bk_{m}w_{m+1}^{∗}w_{m}u_{m−1}|+
+ |ak_{m}w_{m+1}^{∗}w_{m}v_{m−1} + ak_{m+1}w_{m+2}^{∗}w_{m+1}v_{m} + bk_{m}w_{m+1}^{∗}v_{m}w_{m−1}|.

To see that β(t) satisfies condition (73) of Lemma 10 it is enough to show that β(t) → 0 as t → ∞. It is clearly true, if we assume that the forces f, g satisfy (68) and |w_{m}(t)|, |w_{m−1}(t)| → 0 as t → ∞. Moreover, it is easy to see that α(t) satisfies the condition (72). Hence, in order to apply Lemma 10 we need to show that α(t) also satisfies the conditions (71), namely, that

lim inf_{t→∞} \frac{1}{T} ∫_{t}^{t+T} α(τ)dτ ≥ νk_{m+1}^{2} − C_{1}( lim sup_{t→∞} \frac{1}{T} ∫_{t}^{t+T} ∥v(τ)∥^{2}dτ)^{1/2} > 0.

To estimate the last quantity, we can use the bound (61) to conclude that if m satisfies

λ^{2m} > C_{1}G,

then

|Q_{m}w| → 0 as t → ∞.
□

Remark 1. The existence of the determining modes for the Navier-Stokes equations both in two and three dimensions is known (see, e.g., [7] [14], [24] and reference therein). However, there exists a gap between the upper bounds for the lowest number of determining modes and the dimension of the global attractor for the two-dimensional Navier-Stokes equations both for the no-slip and periodic boundary conditions. Our upper bounds for the dimension of the global attractor and for the number of determining modes for the sabra shell model equation coincides. Recently it was shown in [22] that a similar result is also true for the damped-driven NSE and the Stommel-Charney barotropic model of ocean circulation.

9. Existence of Inertial Manifolds

In this section we prove the existence of a finite-dimensional inertial manifold (IM). The concept of inertial manifold for nonlinear evolution equations was first introduced in [17] (see also, e.g., [5], [8], [9], [18], [33], [35]). An inertial manifold is a finite dimensional Lipschitz, globally invariant manifold which attracts all bounded sets in the phase space at an exponential rate and, consequently, contains the global attractor. In fact, one can show that the IM is smoother, in particular C^{1} (see, e.g., [10], [32], [33]). The smoothness and invariance under the reduced dynamics of the IM implies that a finite system of ordinary differential equations is equivalent to the original infinite system. This is the ultimate and best notion of system reduction that one could hope for. In other words, an IM is an exact rule for parameterizing the large modes (infinite many of them) in terms of the low ones (finitely many of them).

In this section we use Theorem 3.1 of [18] to show the existence of inertial manifolds for the system (9) and to estimate its dimension. Let us state Theorem 3.1 of [18] in the following way (see also [35], Chapter VIII, Theorem 3.1)

Theorem 12. Let the nonlinear term of the equation (80) R(·) be a differential map from D(A) into D(A^{1−β}) satisfying

|R′(u)v| ≤ C_{8}|u||A^{β}v|  (78a)
|A^{1−β}R′(u)v| ≤ C_{9}|Au||Av|,  (78b)

for some 0 ≤ β < 1, for all u, v ∈ D(A) and appropriate constants C_{8}, C_{7} > 0, which depend on physical parameters a, b, c, ν and ∥f∥.

Assume in addition that there exists an N large enough, such that the eigenvalues of A satisfy

k_{N+1} − k_{N} ≥ max\{\frac{4}{l}(1 + l)K_{2}, \frac{b}{K_{1}} \}  (79a)
k_{N+1} ≥ \frac{K_{1}}{b} ,  (79b)

where

K_{1} = ∥f∥ + 4C_{7}ρ^{2},

K_{2} = \frac{8}{ρ}∥f∥ + 6C_{7}ρ,

ρ is the radius of the absorbing ball in D(A), which existence is provided by Proposition 14, and b > 0, l < 1 are arbitrary constants.

Then the equation (80) possesses an inertial manifold of dimension N.

In order to apply the Theorem, we need to show that the sabra model equation satisfies the properties of the abstract framework of [18]. Let us rewrite our system (9) in the form

\frac{du}{dt} + νAu + R(u) = 0,  (80)

where the nonlinear term R(u) = B(u, u) − f.

First of all, we need to show that the nonlinear term R(u) of the equation (80) satisfies (78) for β = \frac{1}{2}. However, it is worth noting that in our case we could have chosen any 0 ≤ β ≤ \frac{1}{2}. Indeed, and based on (15) R(u) is a differentiable map from D(A) to D(A^{1/2}) = V, where

R′(u)w = B(u, w) + B(w, u), ∀u, w ∈ D(A).

Moreover, the estimates (78) are satisfied according to Proposition 1, namely

|R′(u)v| = |B(u, v) + B(v, u)| ≤ C_{8}|u|∥v∥,

where C_{8} = (|a|(λ^{−1} + λ) + |b|(λ^{−1} + 1)) + (2|a| + 2λ|b|). In addition,

∥R′(u)v∥ = ∥B(u, v) + B(v, u)∥ ≤ C′_{7}|u||Av| ≤ C_{7}|Au||Av|,

where the last inequality results from the fact that k_{1}^{2}|u| ≤ |Au| and the constants C_{7}, C′_{7} satisfy C_{7} = C′_{7}/k_{1}^{2} = (|a|(λ^{−1} + λ) + |b|(λ^{−1} + 1))/k_{1}^{2}.

Finally, because of the form of the wavenumbers k_{n} (2), the spectral gap condition (79) is satisfied. Therefore, we can apply the Theorem 12 to equation (9), and conclude that the sabra shell model possesses an inertial manifold.

9.1. Dimension of Inertial Manifolds. Let us calculate the dimension of the inertial manifold of the the sabra shell model equation for the specific choice of parameters

l = \frac{1}{2}, b = ρ.

In that case, the conditions (79) take the form

k_{N+1} ≥ (λ − 1)k_{N} = k_{N+1} − k_{N} ≥ 12K_{2}.  (81a)
k_{N+1} ≥ K_{2} ≥ \frac{K_{1}}{ρ} ,  (81b)

which become a single condition and

Corollary 13. The equation (9) possesses an inertial manifold of dimension

N ≥ log_{λ}(12K_{2}) − 1 = log_{λ} (24(\frac{4}{ρ}∥f∥ + 3C_{7}ρ)) − 1.  (82)

10. Conclusions

We have established the global regularity of the sabra shell model of turbulence. We shown analytically that the shell model enjoys some of the commonly observed features of real world turbulent flows. Specifically, we have established using the Gevrey regularity technique the existence of an exponentially decaying dissipation range. Moreover, we have provided explicit upper bounds, in terms of the given parameters of the shell model, for the number asymptotic degrees of freedom. Namely, we presented explicit estimates for the dimension of the global attractor and for the number of determining modes. We have also shown that the shell model possess a finite dimensional inertial manifold. That is, there exists an exact rule which parameterizes the small scales as a function of the large scales. The existence of inertial manifolds is not known for the Navier-Stokes equations. It is worth mentioning that the tools presented here can be equally applied to other shell models.

Acknowledgments

The authors would like to thank Professors I. Procaccia and V. Lvov for the very stimulating and inspiring discussions. The work of P.C. was partially supported by the NSF grant No. DMS-0504213. The work of E.S.T. was supported in part by the NSF grants No. DMS–0204794 and DMS–0504619, the MAOF Fellowship of the Israeli Council of Higher Education, and by the USA Department of Energy, under contract number W–7405–ENG–36 and ASCR Program in Applied Mathematical Sciences.

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[34] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.

[35] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New-York, 1988.

Appendix A. Dissipativity in D(A)

In this section we prove that the equation (9) is dissipative in different norms. First of all, we start with the space H. Consider the relation (40), and by letting t → ∞ we obtain

lim sup_{t→∞}|u(t)| ≤ \frac{|f|_{∞}}{νk_{1}^{2}} = Gνk_{1},  (83)

where |f|_{∞} = ∥f∥_{L^{∞}([0,∞),H)} and G is the generalized generalized Grashoff number, defined in (60).

The existence of the absorbing ball in the space V readily follows from Theorem 4. However, the exact formula for the radius of the absorbing ball is much more involved.

We already mentioned that if we assume that the forcing f belongs to some Gevrey class for all t ≥ 0, then the existence of absorbing balls for solutions of the equation (9) in the norms of V and D(A) follows from the fact that in that case all the functions belonging to the global attractor are in some Gevrey class and hence are bounded in all weaker norms. In the current section we will show that the strong assumption on the force f to be in the Gevrey class could be dropped. Our proof follows [19] (see also [5], ch. 12).

Fix m ∈ ℕ, let P_{m} be the projection of H onto the first m coordinates and denote

P_{m}u = u^{m}.

Recall that the Galerkin approximating system (25) of order m for equation (9) is an m - dimensional system of ordinary differential equations.

Let us consider time as a complex variable t = se^{iθ}, θ ∈ (−\frac{π}{2} , \frac{π}{2} ) and s ∈ ℝ_{+}. Then

\frac{d}{ds}∥u^{m}(se^{iθ})∥^{2} = \frac{d}{ds}(u^{m}(se^{iθ}), Au^{m}(se^{iθ})) = 2Re(e^{iθ} \frac{d}{dt}u^{m}, Au^{m}).

Then, by taking the scalar product of (25a) with Au^{m}, multiplying by e^{iθ} and taking the real part we obtain

\frac{1}{2} \frac{d}{ds}∥u^{m}(se^{iθ})∥^{2} + ν cos θ|Au^{m}(se^{iθ})|^{2} =

= Re(e^{iθ}(f, Au^{m}) − e^{iθ}(P_{m}B(u^{m}, u^{m}), Au^{m})) ≤

≤ \frac{ν cos θ}{2} |Au^{m}(se^{iθ})|^{2} + \frac{|f|^{2}}{ν cos θ} + \frac{C_{1}^{2}}{ν cos θ} ∥u^{m}(se^{iθ})∥^{4},

where the last line follows from (19) and Young’s inequality. Finally we deduce

\frac{d}{ds}∥u^{m}(se^{iθ})∥^{2} + ν cos θ|Au^{m}(se^{iθ})|^{2} ≤ \frac{2|f|^{2}}{ν cos θ} + \frac{2C_{1}^{2}}{ν cos θ} ∥u^{m}(se^{iθ})∥^{4}.

Now we are able to derive the bound

∥u^{m}(t)∥^{2} ≤ 2(∥u^{m}(0)∥^{2} + 1) ≤ 2(∥u^{in}∥^{2} + 1),  (84)

provided t = se^{iθ} satisfies

s ≤ \frac{1}{2K} (∥u^{in}∥^{2} + 1)^{−1},  (85)

where K = \frac{2|f|^{2}}{ν cos θ} + \frac{2C_{1}^{2}}{ν cos θ}.

Given the sequence u = (u_{1}, u_{2}, …) ∈ H let us define the function g : ℝ×ℂ → ℂ by the Fourier series

g(x, t) = ∑_{n=1}^{∞} u_{n}(t)e^{ik_{n}x},

and

g_{m}(x, t) = ∑_{n=1}^{m} u_{n}(t)e^{ik_{n}x}.

If u is a weak solution of the equation (9) for t ∈ [0, T], then g(x, t) ∈ L_{2}(ℝ) for the same t. Moreover, the bounds (84) and (85) show that the functions g_{m}(x, t) are analytic for every x ∈ ℝ and for every m ∈ ℕ in the domain

D = D(ν, ∥u^{in}∥, |f|) = \{t = se^{iθ} : |θ| < \frac{π}{2} , 0 < s ≤ \frac{1}{2K} (∥u^{in}∥^{2} + 1)^{−1}\}.

We would like to obtain a bound for ∥\frac{d}{dt}u∥ = ∥\frac{d}{dt}g′(x, t)∥_{L_{2}(ℝ)}, where g′(x, t) is the derivative of g(x, t) with respect to x. Let γ be a small circle contained in D. Then, according to Cauchy formula

\frac{d}{dt}g′_{m}(x, t) = \frac{1}{2πi} ∫_{γ} \frac{g′_{m}(x, z)}{z − t}dz.

Hence, using (84) we get

∥\frac{d}{dt}u^{m}∥ = ∥\frac{d}{dt}g′_{m}(x, t)∥_{L_{2}(ℝ)} ≤ \frac{1}{\sqrt{2}πr_{γ}} (∥u^{m}(0)∥^{2} + 1)^{1/2},

where r_{γ} is the radius of γ. Let M ⊂ D be a compact subset of D and denote r_{M} = dist(M, ∂D), then

∥\frac{d}{dt}u^{m}∥ ≤ \frac{1}{\sqrt{2}πr_{M}} (∥u^{m}(0)∥^{2} + 1)^{1/2},

for all t ∈ M and all m ∈ ℕ.

Now let us take the scalar product of (25a) with Au^{m} we get the inequality

2ν|Au^{m}(t)|^{2} ≤ ∥\frac{d}{dt}u^{m}∥^{2} + 2|(f, Au^{m})| + 2|(B(u, u), Au^{m})|.

Since all the summands at the right hand side are bounded and those bounds do not depend on m we conclude that there exists a constant E = E(ν, |f|, ∥u^{in}∥) such that

|Au(t)| ≤ E,

for all t ∈ M. Repeating this procedure starting instead of 0 from some t_{1} ∈ M we may conclude that |Au| is bounded for all t > 0. Moreover, using the fact that the solution possesses an absorbing ball in V we finally prove the following statement

Proposition 14. Let f ∈ L^{∞}((0, ∞), H) and u^{in} ∈ H. Then, the solution of the equation (9) possesses absorbing balls in the spaces H, V and D(A).

Appendix B. Differentiability of the semigroup with respect to initial conditions

Theorems 2 and 3 shows in fact, that the initial-value problem (9) is well posed. This allows us to define the semigroup S(t), i.e., the one parameter semigroup family of operators

S(t) : u^{in} ∈ H → u(t) ∈ H,

which are bounded for almost all t ≥ 0. According to the Theorem 3, the mapping S(t) : H → H is Lipschitz continuous with respect to the intial data. The purpose of this section is to show that this mapping is also Fréchet differentiable with respect to the initial data (see, e.g., [35] (ch. VI, 8)).

First of all we wish to linearize the non-linear term. Let us fix T > 0 and let u, v ∈ L^{∞}([0, T], H) ∩ L^{2}([0, T], V) be solutions of the equation (9) with the initial conditions u(0) = u^{in} and v(0) = v^{in}. Then for almost every t ∈ [0, T] we can write

B(u(t), u(t)) − B(v(t), v(t)) = B_{0}(t)(u(t) − v(t)) + B_{1}(t, u − v),

where B_{0}(t) : H → V′ is a linear operator defined as

B_{0}(t)w = B(u(t), w(t)) + B(w(t), u(t)),

and

B_{1}(t, w) = −B(w(t), w(t)).

Let us consider a solution U(t) of the linearized equation

\frac{dU}{dt} + νAU + B_{0}(t)U = 0  (86a)
U(0) = U^{in} = u^{in} − v^{in},  (86b)

satisfying U ∈ L^{∞}([0, T], H) ∩ L^{2}([0, T], V). Denote

φ = u − v − U.

Using the fact that u, v satisfy equation (9) and U is the solution of (86) we can directly check that φ satisfies

\frac{dφ}{dt} + νAφ + B_{0}(t)φ = −B_{1}(t, u(t) − v(t)),  (87)

and φ(0) = 0. Multiplying by φ we get

\frac{1}{2} \frac{d}{dt}|φ|^{2} + ν∥φ∥^{2} + ⟨B_{0}(t)φ, φ⟩ = − ⟨B_{1}(t, u(t) − v(t)), φ⟩ .  (88)

According to the Proposition 1 we can rewrite the last equation as the inequality

\frac{1}{2} \frac{d}{dt}|φ|^{2} + ν∥φ∥^{2} ≤ C′_{8}|φ| · |u| · ∥φ∥ + C_{9}|u − v|^{2} · ∥φ∥.

The fact that u ∈ L^{∞}([0, T], H) together with Young’s inequality imply

\frac{d}{dt}|φ|^{2} ≤ \frac{d}{dt}|φ|^{2} + ν∥φ∥^{2} ≤ \frac{C_{8}}{ν} |φ|^{2} + \frac{C_{9}}{ν} |u − v|^{4}.

Our next step is to use Gronwall’s inequality to get

|φ(t)|^{2} ≤ \frac{C_{9}}{ν} ∫_{0}^{T} e^{\frac{C_{8}}{ν} (T −s)}|u(s) − v(s)|^{4}ds,

for every t ∈ (0, T]. Once again applying Theorem 3 we finally get

|φ(t)|^{2} ≤ C_{10}|u^{in} − v^{in}|^{4},

where 0 < C_{10} = \frac{C_{9}^{2}e^{4K}}{ν} ∫_{0}^{T} e^{\frac{C_{8}}{ν} (T −s)}ds and K is the constant from the statement of the Theorem 3. It follows that

\frac{|u(t) − v(t) − U(t)|}{|u^{in} − v^{in}|} ≤ |u^{in} − v^{in}| ⟶ 0, 0 < t ≤ T,

as |u^{in} − v^{in}| tends to 0. That exactly means that U(t) is a differential of S(t) with respect to u^{in} ∈ H.

(P. Constantin) Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA

E-mail address: const@math.uchicago.edu

(B. Levant) Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

E-mail address: boris.levant@weizmann.ac.il

(E. S. Titi) Department of Mathematics and Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697, USA, Also, Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

E-mail address: etiti@math.uci.edu