An inviscid dyadic model of turbulence: the global attractor

Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. In a companion paper [6] it is proved that every solution for the system with forcing blows up in finite time in the Sobolev $H^{5/6}$ norm. In this present paper, it is proved that after the blow-up time all solutions stay in $H^s$, $s<5/6$ for almost all time and the energy dissipates. Moreover, it is proved that the unique equilibrium is an exponential global attractor.


Introduction
One of the outstanding open questions in fluid dynamics is existence and uniqueness of solutions to the Cauchy problem for the three-dimensional Euler equations (1.1) ∂u ∂t = −(u · ∇)u − ∇p, ∇ · u = 0.
In last few decades simplified models that capture some properties of fluid equations have been proposed and studied. In this article we analyze one of these models, a so called "dyadic" model for the equations of fluid motion. We study the following dyadic model: where the force f is chosen so that f 0 > 0 and f j = 0 for j > 0 for simplicity. The model (1.2) without forcing is a special case of the infinite dimensional dynamical system Such an inviscid model has been studied recently in a number of articles including [12,15,17,25]. Variants of the model that include viscosity are Date: October 2, 2018.
1 discussed in [3,16]. Analysis of more general "shell" models and motivation in terms of turbulence modeling can be found in [2,8,9,14,20,21]. In a companion paper [6] we presented a motivation for the model (1.2) from the Fourier space Euler equations (1.1) in 3-dimensions. The coefficient a 2 j (t) is the total energy in the frequency space shell 2 j ≤ |k| < 2 j+1 . In this context l 2 and H s , respectively the energy and Sobolev norms, are defined as The nonlinear terms on the right hand side of (1.2) retain important features of the advective term in the Euler equation, namely bilinearity and skewsymmetry. The presence of the specific quadratic term 2 5(j−1) 2 a 2 j−1 ensures a certain monotonicity (see also, [1] and [23]) in the cascade of energy through the scales j. We defined a regular solution for the model (1.2) to be a solution with bounded H 5/6 norm and such solutions satisfy the energy equality. In [6] we proved that: As Mattingly et al [19] observe in their recent analysis of an infinite linear dynamical system, the most interesting features of such models belong to solutions after the time of blow-up when some norm becomes infinite. This is particularly true in the context of models that illustrate behavior that has been proposed to characterize hydrodynamic turbulence in the works of, for example, Kolmogorov [18], Onsager [22], Frisch [13], Robert [24], Constantin et al [7] and Eyink-Sreenivasan [10].
In our present paper we study the solutions of (1.2) after the time of blow-up in H 5/6 . We prove: (1) The H s norms for s < 5/6 are locally square integrable in time.
The existence of a global attractor for an inviscid system is, perhaps, surprising. However it is exactly consistent with the concept of anomalous or turbulent dissipation conjectured by Onsager [22]. As we discuss in Section 6, after the time of blow up in H 5/6 the energy spectrum exactly reproduces Kolmogorov's law: whereǭ is the average of the energy dissipation rate.
Notation. For notational convenience we adopt λ j as the scaling parameter in the analysis performed in sections 2 -5. We do this to illustrate that the results are qualitatively independent of the exact choice of λ (which depends on the spatial dimension and the construction of the model). The proofs of results in section 4 require λ < 2 3 . As we discussed in [6] the relevant λ for the 3-dimensional model is 2 5/2 . This exponent determines the values of the exponent of the fixed point and the critical Sobolev space exponent H 5/6 . The exponent also reproduces the exponents in the Kolmogorov's law (1.5).
Acknowledgements. The authors would like to thank Marie Farge, Jonathan Mattingly, Kai Schneider and Eric Vanden-Eijnden for very helpful discussions. S.F. was partially supported by NSF grant number DMS 0503768. N.P. was partially supported by NSF grant number DMS 0304594.

Functional setting
Let us denote H = l 2 with the usual scalar product and norm: The norm |a| will be called the energy norm. Let We fix λ = 2 5/2 and let Here, d s is a strong distance, and d w is a weak distance that induces a weak topology on any bounded subset of H. Hence, a bounded sequence {a k } ⊂ H converges to a ∈ H weakly, i.e., if and only if

Weak solutions
and a j (t) satisfies (1.2) for all j.
Note that since the nonlinear term has a finite number of terms, the notions of a weak solution and a classical solution (of a system of ODEs) coincide. Hence, the weak solutions will be called solutions in the remainder of the paper. Note that if a(t) is a solution on [T, ∞), then automatically a j ∈ C ∞ ([T, ∞)).
Proof. Let u 0 ∈ H and T > 0 be arbitrary. We will show the existence of a solution on [0, T ] by taking a limit of the Galerkin approximation a where a j −1 = 0 and λ = 2 5/2 . From the theory of ordinary differential equations we know that there exists a unique solution a k (t) to (3.1) on [0, T ]. We will show that a sequence of the Galerkin approximations {a k } is weakly equicontinuous. Indeed, it is clear that there exists M , such that for all t ∈ [0, T ] and all j, k. Therefore, for all j, k and all 0 ≤ t ≤ s ≤ T . Thus, for some constant c independent of k. Hence, {a k } is an equicontinuous sequence of functions in C([0, T ]; H w ) with bounded initial data. Therefore, the Ascoli-Arzela theorem implies that {a k } is relatively compact in C([0, T ]; H w ). Hence, passing to a subsequence, we obtain that there exists a weakly continuous H-valued function a(t), such that In particular, a kn j (t) → a j (t) as k n → ∞, for all j, t ∈ [0, T ]. Thus, a(0) = a 0 . In addition, note that for j ≤ k n − 1. Taking the limit as k n → ∞, we obtain Since a j (t) is continuous, it follows that a j ∈ C 1 ([0, T ]) and satisfies (1.2).
Then a j (t) > 0 for all t > 0, and a(t) satisfies the energy inequality Proof. A general solution of (1.2) can be written as where λ = 2 5/2 . Recall that f j ≥ 0 for all j. Since a j (0) ≥ 0 for all j, then a j (t) ≥ 0 for all j, t > 0. Moreover, since f 0 > 0, we have a 0 (t) > 0 for all t > 0 and, consequently, a j (t) > 0 for all j, t > 0. Hence, multiplying (1.2) by a j , taking a sum from 0 to N , and integrating between t 0 and t, we obtain (3.10) Taking the limit as N → ∞, we obtain (3.8).

Fixed point
Given a solution a(t) of (1.2) with arbitrary initial data a j (0) ≥ 0, let We sum the expressions for d l , with 0 ≤ l ≤ j and thanks to (4.2) obtain: .
for all k > 1.
Proof. Note that b(t) satisfies the following system of equations: (4.6) Multiplying it by b j and taking a sum from j = 0 to j = k we obtain It now follows that Also we can rewrite (4.8) as Hence (4.9) gives (4.10) Now note that since the initial condition of the solution satisfies a j (0) ≥ 0 for all j, we have that a j (t) > 0 for all j and t > 0. Therefore, Thus, for all k, t ≥ 0. Hence, (4.13) Note that from (4.2) it follows that Now we rewrite (4.13) using (4.14) and (4.2) as follows: Note that we also have Adding equations (4.15) and (4.16) we get For every solution a(t) with the initial data a(0) ∈ l 2 , a j (0) ≥ 0, and every time interval [t 1 , t 2 ], 0 ≤ t 1 ≤ t 2 , we have that ¿From (4.21) it follows that (4.23) Hence, (4.24) Therefore,  for all j ≥ 2. Note that the right hand side goes to −∞ as j → ∞, contradicting (4.19). Therefore, we have shown that for any N > 0 there exists k > N , such that By the definition of a weak solution, b(t) ∈ l 2 for all time t. Therefore, taking a limit as N → ∞ and using Levi's convergence theorem, we obtain that |d(t)| 2 is locally integrable and which concludes the proof.
Let a(t) be a solution of (1.2) with the initial data a(0) ∈ l 2 , a j (0) ≥ 0. Then a(t) 2 s is locally integrable on [0, ∞) for all s < 5/6. Proof. Thanks to (4.3), we have that where to obtain the last line we used λ = 2 5/2 . Due to Theorem 4.2, |d(t)| is locally integrable. Therefore, b(t) 2 s is locally integrable, provided s < 5/6. Hence, a(t) 2 s is locally integrable for s < 5/6. Theorem 4.4. Let a(t) be a solution of (1.2) with a j (0) ≥ 0. Then a(t) exponentially converges in l 2 to the fixed point as t → ∞. More precisely, for some universal constant β > 0.
Using Granwall's inequality, we conclude that Theorem 4.5. Let a(t) be a solution of (1.2) for which a(t) 3 5/6 is integrable on some interval [T 1 , T 2 ]. Then a(t) satisfies the energy equality Proof. Let a(t) be a solution satisfying the hypothesis of the theorem. First, we recall the property of l p spaces which states that if p ≥ q then for all h ∈ l q . We shall apply this property with h j = λ 2j 3 a 2 j and p = 3/2 to obtain where T 1 ≤ t 0 ≤ t ≤ T . However the expression (4.36) with λ = 2 5/2 combined with the assumption of the theorem implies that we can take the limit of (3.10) as N → ∞ to obtain As a consequence, we can now show that every solution (with any initial data in l 2 ) blows up in finite time in H 5/6 norm. Proof. Assume that a(t) 3 5/6 is locally integrable on [0, ∞). Note that Theorem 4.4 implies that a(t) converges to the fixed point in l 2 . Therefore, In particular, we have that Definition 5.1. A set A ⊂ X is d • -attracting set (• = s, w) if it uniformly attracts X in d • -metric, i.e., for any ǫ > 0 there exists t 0 , such that The following result was proved in [5]: Theorem 5.2. The evolutionary system E always possesses a weak global For the dyadic model we define E in the following way.
Clearly, E satisfies properties (1)-(4). Then Theorem 5.2 immediately yields that the weak global attractor A w exists. In order to infer that A w is the maximal invariant set, we need the following result. Proof. Take any sequence a k ∈ E([0, ∞)). Thanks to (5.1), there exists R > 0, such that a k j (t) ≤ R, ∀n, t ≥ 0. Therefore, for some constant c independent of k. Hence, {a k } is an equicontinuous sequence of functions in C([0, ∞); X w ) with bounded initial data. Therefore, Ascoli-Arzela theorem implies that {a k } is relatively compact in C([0, T ]; X w ), for all time T > 0. Using a diagonalization process, we obtain that {a k } is relatively compact in C([0, ∞); X w ). Hence, there exists a weakly continuous X-valued function a(t) on [0, ∞), such that for some subsequence k n . In particular, since X is weakly compact, a(0) ∈ X.
In addition, since a kn (t) is a solution to (1.2), we have for all j. Taking the limit as k n → ∞, we obtain for all j. Since a j (t) is continuous, a j ∈ C 1 ([0, ∞)) and satisfies (1.2).
Finally, we show that the weak global attractor is the fixed point.
Theorem 5.4. A strong global attractor A s of (1.2) exists, A s = A w , and it is the fixed point: Proof. The statement of the theorem immediately follows from Theorem 4.4.

Onsager's Conjecture and Kolmogorov's 5/3 law
As we discussed in [6] much attention has been given to the statistical theories of turbulence developed by Kolmogorov [18] and Onsager [22]. It is suggested that an appropriate mathematical description of 3-dimensional turbulent flow is given by weak solutions of the Euler equations which are not regular enough to conserve energy. Onsager conjectured that for the velocity Hölder exponent h > 1/3 the energy is conserved and that this ceases to be true for h ≤ 1/3. This phenomenon is now called turbulent or anomalous dissipation. Kolmogorov's theory predicts that in a fully developed turbulent flow the energy spectrum E(|k|) in the inertial range is given by (6.1) E(|k|) = c 0ǭ 2/3 |k| −5/3 , whereǭ is the average of the energy dissipation rate. The model system that we study in this present paper exactly reproduces the phenomena described above. The appropriate choice of λ for the 3-dimensional model is 2 5/2 . Interpreting the results of sections 4 and 5 with λ = 2 5/2 , we proved that regular solutions, i.e. solutions with bounded H 5/6 norm, satisfy the energy equality whereas solutions after the time of blow-up loose regularity and dissipate energy. The model (1.2) is derived under the assumption that a 2 j (t) is the total energy in the shell 2 j ≤ |k| < 2 j+1 . Hence the energy spectrum for the fixed point is: Furthermore, since the support of any time average measure belongs to the global attractor, the average dissipation rate is equal to the dissipation rate of the fixed point. This result is proved in [11] for the 3D Navier-Stokes equations, but it also holds for the inviscid dyadic model due to the anomalous dissipation. Thus Kolmogorov's law is valid for the system.