TRAJECTORY ATTRACTORS FOR NON-AUTONOMOUS DISSIPATIVE 2D EULER EQUATIONS

We construct the trajectory attractor $\mathfrak{A}_{\Sigma }$ for 
the non-autonomous dissipative 2d Euler systems with periodic 
boundary conditions that contain time dependent dissipation terms 
$-r(t)u$ such that $0<\alpha \le r(t)\le \beta$, for $t\ge 0$. 
External forces $g(x,t),x\in \mathbb{T}^{2},t\ge 0,$ also depend on 
time. The corresponding non-autonomous dissipative 2d 
Navier--Stokes systems with the same terms $-r(t)u$ and $g(x,t)$ 
and with viscosity $\nu >0$ also have the trajectory attractor 
$\mathfrak{A}_{\Sigma }^{\nu }.$ Such systems model large-scale 
geophysical processes in atmosphere and ocean. We prove that 
$\mathfrak{A}_{\Sigma }^{\nu }\rightarrow \mathfrak{A}_{\Sigma }$ 
as viscosity $\nu \rightarrow 0+$ in the corresponding metric 
space. Moreover, we establish the existence of the minimal limit 
$\mathfrak{A}_{\Sigma }^{\min }\subseteq \mathfrak{A}_{\Sigma }$ of 
the trajectory attractors $\mathfrak{A}_{\Sigma }^{\nu }$ as $\nu 
\rightarrow 0+.$ Every set $\mathfrak{A}_{\Sigma }^{\nu }$ is 
connected. We prove that $\mathfrak{A}_{\Sigma }^{\min }$ is a 
connected invariant subset of $\mathfrak{A}_{\Sigma }.$ The problem 
of the connectedness of the trajectory attractor 
$\mathfrak{A}_{\Sigma }$ itself remains open.

In the mentioned above geophysical models, the viscosity term ν∆u is responsible for small-scale dissipation (note that 0 < ν α in the physically relevant cases). We prove that the Hausdorff deviation of the set A ν Σ form the set A Σ (measured in the corresponding metric ρ(·, ·)) approaches zero as viscosity ν vanishes: dist ρ (A ν Σ , A Σ ) → 0 as ν → 0 + . We also study some important properties of the trajectory attractors A Σ and A ν Σ specifying below.
The methods of trajectory attractors for evolution partial differential equations was developed in [6]- [10] (see also the review [25]). This approach is highly fruitful in the study of the long time behaviour of solutions to evolution equations for which the uniqueness theorem of the corresponding initial-value problem is not proved yet (e.g., for the 3D Navier-Stokes system) or does not hold.
The trajectory attractors for autonomous 2d Euler system with dissipation (that is, when r ≡ α and g ≡ g(x) ∈ H 1 are independent of time) and for the corresponding autonomous dissipative 2d Navier-Stokes systems with vanishing viscosity have been studied in [11].
The paper is organized as follows. In Sec. 1, we study the non-autonomous dissipative 2d Euler system with periodic boundary conditions. Using the Galerkin method, we prove that the initial-value problem for this system has at least one weak distribution solution u(x, t) such that u(x, t) ∈ L ∞ (R + ; H 1 ) and ∂ t u(x, t) ∈ L b 2 (R + ; H −1 ). Here H 1 denotes the space of periodic solenoidal vector fields with a finite Sobolev H 1 -norm and the space H −1 = H 1 * is dual for H 1 . Moreover, the constructed solution u(x, t) satisfies the corresponding energy inequality (see the next paragraph) that is important for the subsequent study. Note that the uniqueness theorem for weak solutions to the 2d Euler system in the class L ∞ (R + ; H 1 ) is not proved.
In Sec. 2, we construct the trajectory attractor for the non-autonomous dissipative 2d Euler systems with symbols (r, g) = σ ∈ Σ = H + (σ 0 ). We define spaces F b + and F loc + (F b + ⊂ F loc + ) that contain the weak solutions u(x, t) constructed in Sec. 1. Then, we define the space of trajectories (solutions) K + Σ (N ) ⊂ F b + depending on N > 0. The set K + Σ (N ) consists of the weak solutions u(x, t) of the system that satisfy the following energy inequality: u(t) 2 N e −αt + γ −1 g 0 2 L b 2 (R+;H 1 ) , ∀t ∈ R + , where γ = γ(α) > 0 and · := · H 1 denotes the norm in H 1 and g 0 is the original external force in the dissipative Euler system. The space F loc + is equipped with the local weak topology Θ loc + generated by the weak convergence of sequences {v n (x, t)} ⊂ F loc + . We prove that the trajectory space K + Σ (N ) is bounded in F b + and closed in the topology Θ loc + . These assertions are very important for the subsequent study. Consider the translation semigroup {T (h), h 0} acting on a trajectory (solution) u(x, t) by the formula: T (h)u(x, t) = u(x, h + t). It follows from the definition of the trajectory space that K + Σ (N ) is invariant with respect to {T (h)} : T (h)K + Σ (N ) ⊆ K + Σ (N ) for all h 0. Using these facts and applying the theory of trajectory attractors, we prove that the translation semigroup {T (h)} acting on K + Σ (N ) has the global attractor A Σ (N ) which we call the trajectory attractor of the system. Recall that T (h)A Σ (N ) = A Σ (N ) for all h 0. We then prove that the set In Sec. 3 and 4, we study the non-autonomous dissipative 2d Navier-Stokes systems with periodic boundary conditions and with viscosity ν > 0 that contains the dissipative terms −r(t)u and the external forces g(x, t) such that (r, g) = σ ∈ Σ. The corresponding initial-value problem is well-posed and we construct the trajectory attractor A ν Σ for these equations. We prove that dist ρ (A ν Σ , A Σ ) → 0 as viscosity ν → 0 + . The trajectory attractor A ν Σ is a connected set in the topology Θ loc + for all ν > 0. In Sec. 5, we prove the existence of the minimal limit A min Σ of the trajectory at- is the minimal set that satisfies these properties. We prove that the set A min Σ is connected in the topology Θ loc + and strictly invariant with respect to the translation semigroup. The question of whether or not the trajectory attractor A Σ by itself is a connected subset of Θ loc + remains an open problem.
1. Non-autonomus 2d Euler systems with dissipation. We consider the following equations: and P is the orthogonal Leray operator, which projects the space [L 2 (T 2 )] 2 onto the subspace ([X] E denotes the closure of the set X in the topological space E). We define similarly the space In particular by the Gagliardo-Nirenberg inequality, B(u, v) ∈ H for u, v ∈ H 2 (see, for example, [23]). Moreover, the trilinear form b(u, v, w) : The unknown pressure function p(x, t) is eliminated form the first equation of the system (1.1) by applying the operator P to both sides. In (1.1), the time-dependent dissipation coefficient r(t) satisfies the inequalities 0 < α r(t) β, t 0, (1.3) and, for simplicity, we assume that r(t) ∈ C b (R + ) where R + := {0 t < +∞}. We also assume that the external force The autonomous dissipative 2D Euler system (1.1), with r(t) ≡ α and g(x, t) ≡ g 0 (x) for t 0, was considered in a number of papers (see, for example, [4], [22], [15]). Equations (1.1) describe large-scale geophysical processes in atmosphere and ocean when the main dissipation occurs in the planetary boundary layer and is parameterized by the term −ru (see, for example, [21,Ch.4]).
We now consider the time dependent terms r(t) and g(x, t) in greater details. We denote the pair of function σ(s) := (r(s), g(·, s)), s 0, and we call this function the time symbol (or symbol ) of the system (1.1). Here, it is convenient to use the letter s as the time variable instead t.
We consider the translation operators T (h), h 0, acting in the space Ξ + by the formula T (h)ξ(s) := ξ(h+s), s 0. It is clear that the set of operators {T (h), h 0} forms a semigroup that maps Ξ + to itself.
We now come back to the system (1.1). Let we be given an original time symbol σ 0 (s) := (r 0 (s), g 0 (·, s)) ∈ Ξ + that satisfies inequalities (1.3) and (1.5). Moreover, we assume that the function σ 0 (s) is translation compact in Ξ + . (For translation compactness criteria see [10].) As examples of translation compact functions in Ξ + , one can consider periodic, quasiperiodic or almost periodic functions σ 0 (s), s 0, with values in the space R + × H 1 (for more examples, see [10]). We shall use for H + (σ 0 ) a short notation Σ = H + (σ 0 ). We note that every time symbol σ(s) = (r(s), g(·, s)) ∈ Σ has the following property. The function σ(s) is also translation compact in Ξ + and H + (σ) ⊆ Σ. Moreover, the function r(s) satisfies inequalities (1.3) and the function g(s) := g(·, s) satisfies the inequality In what follows, we formulate several propositions concerning weak solutions of system (1.1) with a fixed time symbol that satisfies (1.3) and (1.5). It is clear that all these proposition hold for every symbol σ ∈ Σ.
We consider the initial condition for (1.1) at t = 0 : Recall that a function u = u(x, t), x ∈ T 2 , t 0, is said to be a weak distribution solution of (1.1) if u ∈ L ∞ (0, M ; H 1 ) for every M > 0 and u = u(x, t) satisfies the system in the distribution sense of the space D(0, M ; H −1 ). We prove the existence of a weak distribution solution of the problem (1.1), (1.6) by the Galerkin method. Consider as a basis in H an orthogonal (in H) system of eigenfunctions {e j (x) = (e 1 j (x), e 2 j (x)) ∈ H 2 , j = 0, 1, 2, . . .} of the Stokes operator −P ∆e j (x) = λ j e j (x), (∇, e j (x)) = 0, x ∈ T 2 , j = 0, 1, 2, . . .
We point out that P ∆ ≡ ∆ in the space H 2 with periodic boundary conditions (see, for example, [24]). Recall that e 0 (x) ≡ e 0 is a constant vector and 0 = λ 0 < λ 1 < λ 2 · · · λ j → +∞ as j → ∞. We seek a Galerkin approximation in the form  where the c jn (t) are unknown scalar functions, and the function u n (x, t) satisfies the equation ∂ t u n + Π n B(u n , u n ) + r(t)u n = Π n g(·, t).
(1.7) Here, Π n denotes the orthogonal projector from H onto the finite-dimensional subspace [e 0 (x), . . . , e n (x)]. This equation is equivalent to the corresponding system of ODE with respect to the functions c jn (t), j = 0, 1, 2, . . . , n. The initial condition is given at t = 0, where u 0 is the same as in (1.6). Clearly, the problem (1.7), (1.8) has a unique solution u n (x, t) ∈ C 1 ([0, τ n ); H 1 ) for some τ n > 0. We take the inner product in H of each side of equation (1.7) with u n (t) := u n (·, t) and use the classical identity (see [24], [13]). After elementary transformations, we get that (1.10) Here, recall, |u n (t)| 2 = u n (t) 2 H . Taking the inner product in H of equation (1.7) with −P ∆u n (t) = −∆u n (t) and using the standard identities −(u n , ∆u n ) = |∇u n | 2 and −(g(t), ∆u n ) = (∇g(t), ∇u n ), we obtain the equality As is well known, in the case of periodic boundary conditions (x ∈ T 2 ), we have the identity (B(u, u), ∆u) = 0, ∀u ∈ H 2 (1.12) (see [24, Ch. VI, Lemma 3.1] and [15]), which plays a key role in our future reasonings. Equations (1.11) and (1.12) imply the equality We now sum equalities (1.10) and (1.13) where u n 2 = |u n | 2 + |∇u n | 2 and g(t), u n H 1 = (g(t), u n ) + (∇g(t), ∇u n ). Since r(t) α > 0, the differential equality (1.14) implies that Multiplying both sides of (1.15) by e αt and integrating in t we obtain Estimating the last integral, we have (1.17) Using (1.16) and (1.17), we arrive at the main estimate It follows from the inequality (1.18) that the solution u n (t) of the problem (1.7), (1.8) can be extended to the entire half-line R + (that is, τ n = +∞ for every n), (1.20) Inequality (1.19) implies the existence of a subsequence {n } ⊂ {n} such that for some function u(·, t) ∈ L ∞ (R + ; H 1 ). We assert that u(x, t) is a weak solution of the problem (1.1), (1.6). Indeed, from equations (1.7), the assumption (1.3), and the estimate (1.19), we find that Here, we have used the inequalities that follow from the identity (1.2) and the embedding Relations (1.21) and (1.24) imply that Using now (1.21), (1.25), and the Aubin compactness theorem (see [2,14,20]), we obtain that It follows from (1.26) (by using the routine reasoning similar to [23,20,19]) that Now, with regard to relations (1.21), (1.25), and (1.27), we can pass to the limit as n → ∞ in equation (1.7) in the space of distributions D (R + ; H −1 ) (see [20]). We obtain that the function u(x, t) is a weak distribution solution of equation (1.1) in the space D (R + ; H −1 ) and ∂ t u(·, t) belongs to L b 2 (R + ; H −1 ) while it follows from (1.20) that u(x, t) satisfies the initial condition (1.8). Finally, from the inequality (1.18) we conclude that the constructed weak solution u(x, t) satisfies the estimate (1.28) We have proved the following assertion.
where the function |u(t)| 2 is absolutely continuous (cf. (1.10)). However, an analogous identity (see (1.13)) for the function |∇u(t)| 2 , t 0, does not hold since a weak solution is no longer smooth enough.
) under the condition that ∇ × u 0 and ∇ × g belong to L ∞ (T 2 ) (see [26], [27], [5]). These results can be extended to equations (1.1). However, it will be shown in the next section that no uniqueness theorem is required in the study of trajectory attractors for non-autonomous 2d Euler system with dissipation (1.1).
2. Trajectory attractor of non-autonomous 2d Euler equations with dissipation . We introduce the spaces F b + and F loc + : The topology Θ loc + is metrizable on any ball The corresponding metric for B R is denoted by ρ(·, ·). Moreover, any ball B R is a compact set in the topology Θ loc + and thus B R is a compact metric space (see, for example, [10], [1]).
We consider the non-autonomous dissipative 2d Euler system (1.1) with symbols σ(s) = (r(s), g(s)), s 0, belonging to the hull Σ : , and σ 0 (s) is a translation compact function in Ξ + . We now define the trajectory set K + σ (N ) of the system (1.1) with symbol σ that depend on a number N 0.
We note that the set K + σ (N ) is non-empty for all σ ∈ Σ and for any N 0. Indeed, if u 0 ∈ H 1 and u 0 2 N, then the solution u(·, t) of (1.1), (1.6) with specified initial data u 0 obtained by the Galerkin method (see Proposition 1.1) is a weak solution of (1.1) in D (R + ; H −1 ) and satisfies inequality (2.1) (see (1.28 This semigroup also acts in K + σ (N ) and the following translation identity holds: Indeed, if u(·) ∈ K + σ (N ) is a weak solution of (1.1) with symbol σ(s), then the function T (h)u(s) = u(h + s) is a weak solution of (1.1) with the shifted symbol σ(h + s) = T (h)σ(s). Moreover, since u(·) satisfies inequality (2.1), we see that, for all h 0, is called the trajectory space of the system (1.1). It is clear that K + Σ (N ) ⊆ F loc + .
Proposition 2.1. For any fixed N 0, the space K + Σ (N ) is bounded in F b + and closed in the topology Θ loc + .
Proof. The boundedness of K + Σ (N ) in F b + follows from the inequality (2.1) and the estimate similar to (1.22): be a sequence that converges to a function w(·) ∈ F b + in Θ loc + , that is, It is clear that u n ∈ K + σn (N ) for some σ n ∈ Σ. Since Σ is compact in Ξ + , there exists a subsequence of {σ n } (for which we preserve the notation {σ n }) such that σ n → σ as n → ∞ in the space Ξ + for some σ ∈ Σ. Let σ n (t) = (r n (t), g n (t)) and σ(t) = (r(t), g(t)). Therefore, The sequence {u n (t)} is bounded in F b + since K + Σ (N ) is a bounded set. We state that w(·) ∈ K + Σ (N ). The functions u n (x, t) satisfy the equations ∂ t u n + B(u n , u n ) + r n (t)u n = g n (x, t), (∇, u n ) = 0. (2.10) First, we show that w(t) is a weak solution of the system (1.1) with symbol σ(t) = (r(t), g(t)). We fix an arbitrary > 0. Using (2.6), (2.7), and applying the Aubin compactness theorem (see [2,14,20]), we obtain that, passing to a subsequence of {u n (t)} (for which we preserve the notation {u n (t)}), we may assume that Then, using (2.8), we clearly have Recall that L 2 (0, ; H) ⊂ L 2 (T 2 × [0, ]) 2 and therefore we may assume that We now study the behavior of the nonlinear term B(u n , u n ) in (2.10). Identity (1.2) implies that B(u n , u n ) = P ∂ x1 u 1 n u n + ∂ x2 u 2 n u n . and in L 2 (T 2 × [0, ]) 2 as well. Applying the known lemma on the weak convergence from [20], we conclude from (2.14) and (2.15) that weakly in L 2 (T 2 × [0, ]) 2 and * -weakly in L ∞ (0, ; H) since (2.15) holds. Therefore, due to (2.13), . We now observe that, using (2.7), (2.16), (2.11), and (2.9), we can pass to the limit as n → ∞ in every term of equation (2.10) in the distribution space D (0, ; H −1 ) and find that the function w(x, t) satisfies the equation Since the number was arbitrary, the function w(x, t) is a weak solution of (1.1) in the space D (R + ; H −1 ).
Second, let us prove that w(x, t) satisfies inequality (2.1). Recall that u n (x, t) belongs to K + σn (N ) and, thus, u n (x, t) satisfies the inequality: (2.17) It follows from (2.6) that, for all t 0, and hence (with regard to (2.17)) We have established that w ∈ K + σ (N ) and, thereby, K + Σ (N ) is closed in Θ loc + . Proposition 2.1 is proved.
Consider the action of the translation semigroup {T (t)} in the trajectory space K + Σ (N ). We conclude from (2.2) that where B R is a ball in F b + with sufficiently large radius R. We denote by ρ(·, ·) the corresponding metric in B R (and in its metric subspace K + Σ (N )). Recall that the semigroup {T (t)} is continuous in this metric space. These facts imply that the semigroup {T (t)}| K + Σ (N ) has a global attractor A Σ (N ) ⊆ K + Σ (N ), called the trajectory attractor of equations (1.1) (see [8,10,24,3]). Recall that is the Hausdorff deviation of a set X from a set Y in a metric space M with metric µ.
Remark 2.1. The following embedding is continuous: [20,10]). The trajectory attractor A Σ of equations (1.1) satisfies (2.20) and hence, for any set B ∈ K + Σ (N ), (2.23) Remark 2.2. We note that we cannot put δ = 1 in formula (2.23). We have constructed the trajectory attractor A Σ in the "weak" topology Θ w,loc + of the space F loc + . It seems natural to consider also the "strong" topology Θ s,loc + in F loc + . However, the attraction to the trajectory attractor A Σ in the topology Θ s,loc + could not be proved without additional assumptions. Dealing with autonomous equations (1.1), this fact has been proved in [12] under the assumption that the curl of the external force ∇×g belongs to L ∞ (T 2 ). The proof is based on the technique of Yudovich (see [26,27]). In the more general case considered in this paper, where ∇ × g ∈ L 2 (T 2 ), the question remains open.
3. Non-autonomous 2d Navier-Stokes systems with dissipation . Consider the following non-autonomous 2d Navier-Stokes system with dissipation: (3.1) We use the same notations as in the Euler system (1.1) with dissipation. In equations (3.1), ν > 0 denotes the kinematic viscosity. The pressure p(x, t) is eliminated from the system by applying the Leray operator P to both sides.
The system (3.1) also has a geophysical interpretation (see [21]). The main dissipation acts in the planetary boundary layer which is described by the time dependent term −r(t)u, while a viscosity term ν∆u is responsible for small-scale dissipation (note that in physically relevant cases 0 < ν α).
Remark 3.1. Studying the classical 2d Navier-Stokes system (r = 0) with periodic boundary conditions, one usually assumes that the functions u(x, t) and g(x, t) have zero means in x over the torus T 2 , in order to avoid the linear growth of the solutions. When r > 0, this assumption can be dropped since the term −r(t)u introduces additional dissipation.
4. Trajectory attractors of dissipative 2d Navier-Stokes systems with vanishing viscosity . In this section, we study the convergence of the trajectory attractors A ν Σ of the non-autonomous dissipative 2d Navier-Stokes systems (3.1) to the trajectory attractor A Σ of the non-autonomous dissipative 2d Euler equations (1.1) as viscosity ν vanishes. The spaces F b + , F loc + , and Θ loc + were defined in Sec. 2. Consider an arbitrary symbol σ ∈ Σ. Similarly to Sec. 2, we define the trajectory set of the system (3.1) having the symbol σ = (r, g) and with a fixed viscosity coefficient ν > 0. By definition, K + σ (ν) consists of all strong solutions u(s) := u(·, s), s 0, of this system. Here, we denote the time variable s instead t. We have proved in Sec. 3 that, for any data u 0 = u(0) ∈ H 1 , there exists a unique function u(·) ∈ K + σ (ν) such that u(0) = u 0 . We clearly have the translation identity that is similar to the identity (2.2). Analogously to (2.3), we define the trajectory space of the system (3.1) with fixed viscosity ν > 0 by the formula It is clear that K + Σ (ν) ⊆ F b + and it is easy to prove that the trajectory space K + Σ (ν) is closed in Θ loc + . The translation semigroup {T (t)} acts on the trajectory space K + Σ (ν) by the formula T (t)u(s) = u(t + s). It is clear that T (t)K + Σ (ν) ⊆ K + Σ (ν) for all t 0. We claim that the translation semigroup {T (t)} acting on K + Σ (ν) has the trajectory attractor A ν Σ ⊂ K + Σ (ν) which attracts bounded (in F b + ) families of solutions to system (3.1) in the topology Θ loc + . (See the similar proof in [7,10] for the case r = 0 and Sec. 2 for ν = 0).
Recall that the set A ν Σ is strictly invariant with respect to {T (t)} : and, for any bounded Comparing Corollary 2.1 and Proposition 4.1, we observe that the trajectory attractor A Σ of the dissipative 2d Euler system (1.1) and the trajectory attractors A ν Σ of the 2d Navier-Stokes system (3.1) belong to the same ball B R0 in F b + having the radius R 0 , R 2 We now study the Hausdorff deviation of A ν Σ from A Σ as ν approaches zero in the topology Θ loc + generated by the metric ρ in B R0 described in Sec. 2. The main result of this section is the following theorem.
This contradicts (4.9). Therefore, (4.7) is true. Finally to prove (4.5), we apply (4.7) for B ν = A ν Σ . Remark 4.1. Recall that Θ loc + ⊂ C loc (R + ; H δ ), 0 δ < 1, and the convergencies In conclusion, we formulate two assertions that follow from the well-posedness of the Cauchy problem for the dissipative 2d Navier-Stokes system (see, e.g., [24,3]). We shall use these facts in the next section. 5. On minimal zero viscosity limit of trajectory attractors A ν Σ . Let A Σ be the trajectory attractor of the non-autonomous Euler system (1.1) with dissipation and A ν Σ be the trajectory attractor of the dissipative 2d Navier-Stokes system (3.1) for ν > 0. We have proved in Sec. 4 that A Σ ⊂ B R0 and A ν Σ ⊂ B R0 for all ν ∈ (0, 1], where B R0 is the ball in F b + (see (4.4)) whose radius R 0 is independent of ν: Recall that the ball B 0 with topology Θ loc + is a compact metric space with the metric ρ(·, ·). It was proved in Theorem 4.1 that dist ρ (A ν Σ , A Σ ) → 0 as ν → 0 + . (5.1) Note that, in fact, the limit relation (5.1) is stronger than that of (4. 16).
Recall that the set A Σ ⊂ B R0 is closed in B R0 . Let A min Σ be the minimal closed subset of A Σ which satisfies the attracting property (5.1), that is, by definition, We refer to the set A min Σ as the minimal limit of the trajectory attractors A ν Σ as ν → 0 + .
We state that the set A min Σ exists and is unique. We have just to prove that The set on the right-hand side of (5.2) is clearly nonempty. It is easy to prove that a point w belongs to the right-hand side of (5.2) if and only if there are points w νn ∈ A νn Σ , n = 1, 2, . . . , ν n → 0+ as n → ∞ such that ρ(w νn , w) → 0 as n → ∞. Due to (5.1), such a limit point w always belongs to A Σ and, moreover, it belongs to every closed attracting set A . The set (5.2) is attracting for A ν Σ as ν → 0 + . Indeed, assuming the converse, we see that there is a sequence w νn ∈ A νn Σ , such that ν n → 0+ and dist ρ w νn , A min