Rafael Granero-Belinch Ón

ON THE DYNAMICS OF 3D ELECTRIFIED FALLING FILMS

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Introduction

The present work is concerned with the full 3D dynamics of a thin fluid film falling along with a flat inclined plate. Besides gravitational effects, we consider the action of an electric field acting normal to the plate. In particular, for the case where the fluid lies on top of the plate (overlying films), the following equation was derived by Tomlin, Papageorgiou & Pavliotis [START_REF] Tomlin | Three-dimensional wave evolution on electrified falling films[END_REF]:

η t + ηη x + (β -1)η xx -η yy -γΛ 3 η + ∆ 2 η = 0 (1)
where β > 0 is the Reynolds number, 0 ≤ γ ≤ 2 measures the electric field strength and Λ is a non-local operator corresponding to the electric field effect given on the Fourier variables as Λu = |ξ|û(ξ) = (ξ 2 1 + ξ 2 2 ) 0.5 û(ξ). We observe that the term corresponding to the electric field, -γΛ 3 (η), always has a destabilizing effect, while the term (β -1)η xx can be stabilizing or destabilizing depending on the value of the Reynolds number. Namely, for subcritical Reynolds numbers 0 < β < 1, (β -1)η xx is a stabilizing term, while for supercritical Reynolds numbers 1 < β, it has a destabilizing effect.

Since falling films have received much attention from many authors, a wide variety of results about their nonlinear stability can be found. In particular, the 2D case (1D interface) was first studied by González & Castellanos [START_REF] Gonzalez | Nonlinear electrohydrodynamic waves on films falling down an inclined plane[END_REF]. These authors identified a critical electric field strength for sub-critical Reynolds number flows above which instability was found. Later on, Tseluiko & Papageorgiou also considered the 2D case (1D interface). In particular, Tseluiko & Papageorgiou performed a numerical study of the 1D analog of (1) and found attractors for the dynamics for certain values of the physical parameters [START_REF] Tseluiko | A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics[END_REF]. The same authors provided analytical bounds on the energy of the solutions and the dimension of the attractor [START_REF] Tseluiko | Wave evolution on electrified falling films[END_REF] (see also [START_REF] Tseluiko | Dynamics of an electrostatically modified Kuramoto-Sivashinsky-Korteweg-de Vries equation arising in falling film flows[END_REF] for the case of vertical film flow). Compared with the case of 1D interface, equation [START_REF] Bronski | Uncertainty estimates and L 2 bounds for the Kuramoto-Sivashinsky equation[END_REF] generalizes previous works by taking transverse dynamics into consideration.

Equation ( 1) is similar to the well-known Kuramoto-Sivashinsky equation in one dimension,

η t + ηη x = -η xx -η xxxx (KS)
which is a model appearing in several applications. For instance, LaQuey, Mahajan, Rutherford & Tang [START_REF] Laquey | Nonlinear saturation of the trapped-ion mode[END_REF] obtained (KS) as a model of collisional trapped-ion mode in tokamak geometry (see also Cohen, Krommes, Tang & Rosenbluth [START_REF] Cohen | Non-linear saturation of the dissipative trapped-ion mode by mode coupling[END_REF]), Kuramoto & Tsuzuki [START_REF] Kuramoto | Persistent propagation of concentration waves in dissipative media far from thermal equilibrium[END_REF] considered the possible instabilities of a two components reaction-diffusion system and also recovered (KS). Furthermore, Sivashinsky [START_REF] Sivashinsky | Nonlinear analysis of hydrodynamic instability in laminar flames-I. derivation of basic equations[END_REF] (see also the companion paper by Michelson & Sivashinsky [START_REF] Michelson | Nonlinear analysis of hydrodynamic instability in laminar flames-II. numerical experiments[END_REF]) derived (KS) as a model of the evolution of a disturbed plane flame front. Later on, Sivashinsky & Michelson [START_REF] Sivashinsky | On irregular wavy flow of a liquid film down a vertical plane[END_REF] linked (KS) to the evolution of a film of viscous liquid flowing down a vertical plane. Several equations sharing some similarities where obtained by Topper & Kawahara [START_REF] Topper | Approximate equations for long nonlinear waves on a viscous fluid[END_REF], Lee & Chen [START_REF] Lee | Nonlinear dynamical models of plasma turbulence[END_REF], Coward & Hall [START_REF] Coward | On the nonlinear interfacial instability of rotating core-annual flow[END_REF], Frenkel & Indireshkumar [START_REF] Frenkel | Wavy film flows down an inclined plane: perturbation theory and general evolution equation for the film thickness[END_REF] and by James & Wilczek [START_REF] James | Vortex dynamics and lagrangian statistics in a model for active turbulence[END_REF] when considering falling fluid films, plasma turbulence and cellular suspensions.

Equation (KS) has rich dynamics. Indeed, applying the Fourier transformation to the linear part of (KS), ∂ t η(ξ) = (ξ 2 -ξ 4 )η(ξ), it results in the stability of high frequencies (|ξ| > 1) and instability of low frequencies (0 < |ξ| < 1). Specifically, the term η xx leads to instability at large scales; the dissipative term η xxxx is responsible for damping at small scales. Then we see that for general initial data, the linear problem is unstable and leads to an exponential growth of certain frequencies. When the nonlinear term ηη x is added, stabilization occurs as this term transfers energy from the long wavelengths to the short wavelengths and balances the exponential growth due to the linear parts. This interaction between the unstable linear parts and a nonlinearity who carries energy between frequencies makes the solution of (KS) to develop chaotic dynamics for certain values of the parameters.

This nonlinear stabilization of the Kuramoto-Sivashinsky equation with L-periodic boundary conditions, η(x + L, t) = η(x, t), for all x and t, was considered mathematically by Nicolaenko, Scheurer & Temam in [START_REF] Nicolaenko | Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors[END_REF] under the hypothesis that the initial data has odd symmetry: η 0 (x) = -η 0 (-x). After that, Ilyashenko [START_REF] Il'yashenko | Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation[END_REF], Collet, Eckmann, Epstein & Stubbe [START_REF] Collet | A global attracting set for the Kuramoto-Sivashinsky equation[END_REF] and Goodman [START_REF] Goodman | Stability of the Kuramoto-Sivashinsky and related systems[END_REF] found new bounds for the L 2 -norm of the solution of the KS equation without oddness condition for the initial data. The fact that the solutions are uniformly bounded leads us to the question of the optimal bound for the radius of the absorbing set in L 2 for arbitrarily large periods L. In that regards, the known bounds are lim sup

t→∞ L 0 u 2 dx 0.5 ≤ O(L p )
where the original p = 5/2 [START_REF] Nicolaenko | Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors[END_REF] was later improved to p = 8/5 [START_REF] Collet | A global attracting set for the Kuramoto-Sivashinsky equation[END_REF] and finally to p = 3/2 [START_REF] Bronski | Uncertainty estimates and L 2 bounds for the Kuramoto-Sivashinsky equation[END_REF]. The global bound has been upgraded recently by Giacomelli & Otto [START_REF] Giacomelli | New bounds for the Kuramoto-Sivashinsky equation[END_REF], where they proved the bound lim sup

t→∞ L 0 u 2 dx 0.5 ≤ o(L 1.5 ).
We observe that the conjectured bound is O(L 0.5 ).

The analyticity of solutions is of great interest not only for KS equation, but also for other nonlinear partial differential equations. For instance, we refer the reader to the seminar paper by Foias & Temam [START_REF] Foias | Gevrey class regularity for the solutions of the Navier-Stokes equations[END_REF] where they show that solutions of the Navier-Stokes equations are analytic in time with values in a Gevrey class of functions (in space). This technique has been extended largely to other nonlinear parabolic equations and, in particular, Collet, Eckmann, Epstein & Stubbe [START_REF] Collet | Analyticity for the Kuramoto-Sivashinsky equation[END_REF] addressed the spatial analyticity of solutions of one-dimensional Kuramoto-Sivashinsky equation. They showed that at large time the solutions are analytic in a strip around the real axis and also gave a rigorous lower bound for its width, i.e. the radius of analyticity is proportional to L -16/25 . Grujić [START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF] used a Gevrey class technique to obtain a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of L. This latter result shed some light on a conjecture in [START_REF] Collet | Analyticity for the Kuramoto-Sivashinsky equation[END_REF] that asks whether there is a α > 0, independent of L, such that the solutions of the KS equation are analytic in space in the complex strip {x + is, s < α} for sufficiently large time. In higher dimensions, the literature on estimating the radius of analyticity for the Kuramoto-Sivashinsky-type equations is more scarce. For example, we refer to the works by Pinto [START_REF] Pinto | Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two[END_REF][START_REF] Pinto | Analyticity and Gevrey class regularity for a Kuramoto-Sivashinsky equation in space dimension two[END_REF] where, among other properties, the author studied the time analyticity of a variant of the twodimensional KS equation. More recently, Ioakim & Smyrlis [START_REF] Ioakim | Analyticity for Kuramoto-Sivashinsky type equations and related systems[END_REF] also studied the analyticity properties of solutions of Kuramoto-Sivashinsky type equations and some related systems.

The goal of the present work is to mathematically study the initial value problem for nonlocal two-dimensional Kuramoto-Sivashinsky-type equation with periodic boundary conditions and initial data with zero mean

L 0 L 0 η 0 (x, y)dxdy = 0.
Of course, the zero average condition is propagated by the PDE. We organize this paper as follows. In section 2, we give some notations, definitions and classical results. In section 3, we show the global existence of solutions to initial value problem (1) and in section 4, we prove the existence of an absorbing set in L 2 and in higher Sobolev norms. In section 5, we prove that these solutions are analytic in a strip based on a priori estimates in a Gevrey class. Finally, in section 6, we establish a bound for the number of spatial oscillations which are a manifestation of the spatial chaos that this PDE evidences.

1.1. Notation. We will use C to denote a universal constant that can change from one line to another. We will make frequent use of the usual homogeneous Sobolev spaces Ḣs :

Ḣs (T 2 ) = η ∈ L 2 (T 2 ) : ξ∈Z 2 |ξ| 2s | η(ξ)| 2 < ∞
where η(ξ) is the Fourier series of η :

η(ξ) def = (2π) -2 T 2 e -iξ 1 x-iξ 2 y η(x, y)dxdy, ξ = (ξ 1 , ξ 2 ) ∈ Z 2 .

Rescaling of the equation

We assume that η is L-periodic, η(x + L, y) = η(x, y), η(x, y + L) = η(x, y), and define T 2 = [0, 2π] 2 and λ = 2π L . We rescale our variables according to x = λx, ỹ = λy, η = λ -1 η, t = λ 2 t, which gives

λ 3 ηt + λ 3 η ηx + (β -1)λ 3 ηxx -λ 3 ηyy -γλ 4 Λ 3 (η) + λ 5 ∆ 2 η = 0.
Then we obtain

ηt + η ηx + (β -1)η xx -ηyy -γλΛ 3 (η) + λ 2 ∆ 2 η = 0. ( 2
)
Denoting δ = γλ and = λ 2 , we can equivalently consider the following initial-value problem (3)

η t + ηη x + (β -1)η xx -η yy -δΛ 3 (η) + ∆ 2 η = 0, (x, y) ∈ T 2 , t > 0 with initial data η(x, y, 0) = η 0 (x, y), (x, y) ∈ T 2 .
In what follows, we will drop the tilde notation.

Global existence of strong solutions

In this section, we will state the global well-posedness result of the initial-value problem (3):

Theorem 1. If η 0 ∈ H 2 (T 2 ), then for every 0 < T < ∞ the initial value problem (3) has a unique solution η ∈ C([0, T ]; H 2 (T 2 )) ∩ L 2 (0, T ; H 4 (T 2 )).
Proof.

Step 1 : L 2 estimate. We multiply (3) by η and integrate by parts to obtain 1 2

d dt η 2 L 2 = (β -1) η x 2 L 2 -η y 2 L 2 + δ Λ 3 2 η 2 L 2 -∆η 2 L 2 ≤ C(β, δ) η 2 H 3 2 -η 2 H 2 ≤ C(β, δ) η 3/2 H 2 η 1/2 L 2 -η 2 H 2
By Young's inequality, we find that

d dt η 2 L 2 ≤ -∆η 2 L 2 + C( , β, δ) η 2 L 2
where C( , β, δ) is a constant depending on , β, δ and may change line by line. An application of Gronwall's inequality leads us to

η 2 L 2 + t 0 exp (C( , β, δ) (t -s)) ∆η 2 L 2 ds ≤ C( , β, δ) η 0 2 L 2 e t .
Hence,

η 2 L 2 + t 0 η 2 H 2 ds ≤ C( , β, δ) η 0 2 L 2 e t .
Step 2 : H 1 estimate. Now we multiply (3) by -∆η and integrate by parts to obtain that 1 2

d dt η 2 H 1 = (β -1) η x 2 H 1 -η y 2 H 1 + δ Λ 3 2 η 2 H 1 -∆η 2 H 1 + η L ∞ η H 1 ∆η L 2 .
Using the same method as in step 1, we get 1 2

d dt η 2 H 1 ≤ - 3 4 ∆η 2 H 1 + C( , β, δ) η 2 H 1 + 1 2 η 2 L ∞ η 2 H 1 + 2 ∆η 2 L 2 ≤ - 4 ∆η 2 H 1 + C( , β, δ) + 1 2 η 2 L ∞ η 2 H 1 , which implies d dt η 2 H 1 ≤ - 2 ∆η 2 H 1 + C( , β, δ) + 1 η 2 L ∞ η 2 H 1 .
Using Gronwall's inequality, we find that

η 2 H 1 + 2 t 0 exp C( , β, δ)(t -s) + t s 1 η 2 L ∞ ∆η 2 H 1 ds ≤ η 0 2 H 1 exp C( , β, δ)t + t 0 1 η 2 L ∞ .
From step 1, we already have that

t 0 η 2 H 2 ds ≤ C( , β, δ) η 0 2 L 2 e t .
Using the Sobolev embedding, we get that

t 0 η 2 L ∞ ds ≤ C( , β, δ) η 0 2 L 2 e t .
Inserting this into the inequality above, we obtain

η 2 H 1 + 2 t 0 exp C( , β, δ)(t -s) + t s 1 η 2 L ∞ ∆η 2 H 1 ds ≤ η 0 2 H 1 exp C( , β, δ)t + 1 2 η 0 2 L 2 exp (C( , β, δ)t) ≤ η 0 2 H 1 exp (exp (C( , β, δ η 0 H 1 )t))
. Hence, we conclude that

η 2 H 1 + 2 t 0 exp (exp (C( , β, δ η 0 H 1 )(t -s))) ∆η 2 H 1 ds ≤ η 0 2 H 1 exp (exp (C( , β, δ η 0 H 1 )t)) . In particular, η 2 H 1 + 2 t 0 ∆η 2 H 1 ds ≤ η 0 2 H 1 exp (exp (C( , β, δ η 0 H 1 )t)) .
Step 3 : H 2 estimate. We multiply (3) by ∆ 2 η and integrate by parts to obtain that 1 2

d dt η 2 H 2 = (β -1) η x 2 H 2 -η y 2 H 2 + δ Λ 3 2 η 2 H 2 -∆η 2 H 2 + η L ∞ η H 1 ∆ 2 η L 2 .
Using the same method as in step 1 and step 2, we can obtain

d dt η 2 H 2 ≤ - 2 ∆η 2 H 2 + C( , β, δ) + 1 η 2 L ∞ η 2 H 1 ≤ - 2 ∆η 2 H 2 + C( , β, δ) + 1 η 2 L ∞ η 2 H 2 ,
Using Gronwall's inequality again, we obtain that

η 2 H 2 + 2 t 0 exp C( , β, δ)(t -s) + t s 1 η 2 L ∞ ∆η 2 H 2 ds ≤ η 0 2 H 2 exp C( , β, δ)t + t 0 1 η 2 L ∞ .
Then, the Sobolev embedding implies that

η 2 H 2 + 2 t 0 exp (exp (C( , β, δ, η 0 H 2 )(t -s))) ∆η 2 H 2 ds ≤ η 0 2 H 2 exp (exp (C( , β, δ, η 0 H 2 )t))
. Finally, we have

η 2 H 2 + 2 t 0 ∆η 2 H 2 ds ≤ η 0 2 H 2 exp (exp (C( , β, δ, η 0 H 2 )t)) .
Step 4 : Existence of solution We consider a positive, symmetric mollifier J (such as the periodic heat kernel), to approximate the initial value problem (3) by the regularized problem

∂ t η + J * ∂ x (J * η ) 2 2 = J * (1 -β)∂ xx (J * η ) + ∂ yy (J * η ) + δΛ 3 (J * η ) -∆ 2 (J * η ) , with initial data η (0) = J * η 0 .
By the Picard's theorem, these sequence of regularized problems have a unique solution η = C 1 ([0, T ], H 2 (T 2 )). Moreover, these problems verify the same energy estimates as in steps 1-3 and, as a consequence, we can take T = T (η 0 ) independent of . Passing to the limit we conclude the existence of at least one solution in η ∈ L ∞ (0, T ; H 2 ) ∩ L 2 (0, T ; H 4 ) and since we have a priori estimates, these solutions exist for arbitrary long time T .

Step 5 : Uniqueness. We can prove the uniqueness of the solutions by contradiction, i.e. assuming that there exists two solutions of the problem (3), η 1 and η 2 , corresponding to the same initial data η 0 . We denote their difference by η. Then we have (4)

η t + 1 2 η 2 1 -η 2 2 x + (β -1)η xx -η yy -δΛ 3 (η) + ∆ 2 η = 0.
As the proof of L 2 estimate in step 1, we multiply (4) by η and integrate by parts:

1 2 d dt η 2 L 2 = (β -1) η x 2 L 2 -η y 2 L 2 + δ Λ 3 2 (η) L 2 -∆η 2 L 2 - T 2 1 2 (η 2 1 -η 2 2 ) x η ≤ C( , β, δ) η 2 L 2 - 4 ∆η 2 L 2 + 1 2 η 1 + η 2 L ∞ η H 1 η L 2 ≤ C( , β, δ) η 2 L 2 - 4 ∆η 2 L 2 + 4 η 2 H 1 + 1 4 η 1 + η 2 2 L ∞ η 2 L 2 ≤ C( , β, δ) + 1 4 η 1 + η 2 2 L ∞ η 2 L 2 ≤ C( , β, δ) + 1 2 η 1 2 L ∞ + η 2 2 L ∞ η 2 L 2
Using Gronwall's inequality, we have that

η 2 L 2 ≤ η 0 2 L 2 exp C( , β, δ)t + 1 t 0 η 1 2 L ∞ + η 2 2 L ∞ .
From step 1, we already have that

t 0 η 1 2 H 2 ds ≤ η 0 2 L 2 exp (C( , β, δ)t) ,
and t 0 η 2 2 H 2 ds ≤ η 0 2 L 2 exp (C( , β, δ)t) .
Thus, the uniqueness of solution follows from the inequality

η 2 L 2 ≤ η 0 2 L 2 exp (exp (C(δ, , β, η 0 L 2 )t)) .
Step 6 : Endpoint continuity in time. To conclude the endpoint continuity, we can perform a standard argument using the parabolic gain of regularity L 2 (0, T ; H 4 ). Indeed, we can take 0 < σ 1 as small as desired and there exists a 0 < σ < σ such that u(σ ) ∈ H 4 . Repeating the same argument as before, we find a solution

η σ ∈ L ∞ (σ , T ; H 4 ) ∩ C([σ , T ], H 2 ).
Because of the uniqueness of solution we obtain the continuity of the original solution

η ∈ C((0, T ], H 2 ).
Finally, the continuity at the origin is a consequence of the energy estimates.

Large time dynamics

The goal of this section is to prove uniform boundedness of solutions

η ∈ L ∞ ([0, ∞); L 2 (T 2 )).
In other words, we establish the existence of an absorbing ball in L 2 by collecting global bounds showing the dissipative character of the equation. We start by proving the following two Gagliardo-Nirenberg inequalities Lemma 1. For smooth enough periodic functions with zero mean, we have that the following two inequalities hold true

∇η 2 L 4 (T 2 ) ≤ C η L ∞ (T 2 ) ∆η L 2 (T 2 ) , ∆η 2 L 2 (T 2 ) ≤ C η L ∞ (T 2 ) ∆η L 2 (T 2 )
. proof. We start proving the first inequality:

∇η 4 L 4 (T 2 ) = T 2 (∇η • ∇η) 2 = - T 2 η∇ • ∇η|∇η| 2 = - T 2 η∆η|∇η| 2 + η∇η • ∇|∇η| 2 ≤ C η L ∞ (T 2 ) ∆η L 2 (T 2 ) |∇η| 2 L 2 (T 2 ) ≤ C η L ∞ (T 2 ) ∆η L 2 (T 2 ) ∇η 2 L 4 (T 2 )
. Therefore, we conclude our result by noticing that

∆η 2 L 2 (T 2 ) ≤ C η∆η + |∇η| 2 L 2 (T 2 ) ≤ C η∆η L 2 (T 2 ) + |∇η| 2 L 2 (T 2 ) ≤ C η L ∞ (T 2 ) ∆η L 2 (T 2 ) + ∇η 2 L 4 (T 2 ) ≤ C η L ∞ (T 2 ) ∆η L 2 (T 2 ) .
Remark 1. We observe that the previous constants C can be computed explicitly.

The rest of this section is devoted to prove that the solutions of problem (3) remain uniformly bounded in L 2 . The following background flow method was first used by Nicolaenko, Scheurer, & Temam [START_REF] Nicolaenko | Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors[END_REF] and then improved by Collet, Eckmann, Epstein & Stubbe [START_REF] Collet | A global attracting set for the Kuramoto-Sivashinsky equation[END_REF], Goodman [START_REF] Goodman | Stability of the Kuramoto-Sivashinsky and related systems[END_REF] and Bronski & Gambill [START_REF] Bronski | Uncertainty estimates and L 2 bounds for the Kuramoto-Sivashinsky equation[END_REF]. Before stating the main result of this section, let us first define the following subspace of H 2 (T 2 ) :

H 2 od (T 2 ) = {η ∈ H 2 (T 2 ) : -η(-x, y) = η(x, y), ∀(x, y) ∈ T 2 }
In terms of the rigorous results, a global bound on the solution is given by the following theorem :

Theorem 2. Let η 0 ∈ H 2 od (T 2 ).
Then the solution η of the initial-value problem (3) satisfies

lim sup t→∞ η(t) L 2 (T 2 ) ≤ R ,δ,β . ( 5
)
where R ,δ,β depends on , δ, β.

proof. The proof is based on the construction of a Lyapunov functional, F(t), such that

d dt F(t) ≤ 0, if F(t) ≥ R ,δ,β , i.e.
implying the existence of an absorbing set in L 2 . We first let φ be a smooth, 2πperiodic function, which we will choose later. Then, we multiply equation (3) by η -φ, and integrate by parts:

(η -φ) t + ηη x + (β -1)η xx -η yy -δΛ 3 (η) + ∆ 2 η (η -φ) = 0, thus, 1 2 d dt η -φ 2 L 2 + 1 2 T 2 φ x η 2 -(β -1) η x 2 L 2 + η y 2 L 2 + ∆η 2 L 2 - T 2 (β -1)η xx φ + T 2 η yy φ -δ T 2 Λ 3 η(η -φ) - T 2 ∆η∆φ = 0.
For the term corresponding to the nonlocal self-adjoint operator Λ, we have that

T 2 Ληφ = T 2 ηΛφ.
Hence, by the Young's inequality and the Hölder inequality, we have that, 1 2

d dt η -φ 2 L 2 = - T 2 φ x 2 η 2 + (β -1) η x 2 L 2 -η y 2 L 2 -∆η 2 L 2 + δ T 2 Λ 3 ηη +(1 -β) T 2 η x φ x + T 2 η y φ y -δ T 2 Λ 3 ηφ + T 2 ∆η∆φ ≤ - T 2 φ x 2 η 2 + 2|β -1| η x 2 L 2 - 1 2 η y 2 L 2 - 2 ∆η 2 L 2 + δ Λ 3 2 η 2 L 2 + 1 2 η 2 L 2 + |β -1| φ x 2 L 2 + 1 2 φ y 2 L 2 + 2 ∆φ 2 L 2 + δ 2 2 φ 2 H 3 (6)
Now, we define the following function

f (|ξ|) = f (z) := 1 2 + 2(β + 2)z 2 + δz 3 - 4 z 4 , z ≥ 0.
Observe that f (z) has at most three zeros. Since > 0, we find that f (z) is bounded and has global maximum 1 2 + (8(β + 2) + 4δ 2 )δ 2 / 3 at point z 1 = 2δ/ . Then we have the following inequality

1 2 + 2|β -1||ξ 1 | 2 - 1 2 |ξ 2 | 2 + δ|ξ| 3 - 4 |ξ| 4 ≤ f (z 1 ) = 1 2 + (8(β + 2) + 4δ 2 )δ 2 / 3 := C(β, δ, ).
Inserting this relation into [START_REF] Coward | On the nonlinear interfacial instability of rotating core-annual flow[END_REF], we obtain that 1 2

d dt η -φ 2 L 2 (T 2 ) = - 4 ∆η 2 L 2 (T 2 ) -η 2 L 2 + T 2 λ - φ x 2 η 2 + F (φ), (7)
where λ = C(β, δ, ) + 1, and

F (φ) = |β -1| φ x 2 L 2 + 1 2 φ y 2 L 2 + 2 ∆φ 2 L 2 + δ 2 2 φ 2 H 3 . Now, we choose φ(x, y) such that φ x 2 = -λ 0<|ξ 1 |≤A/ e -ixξ 1 ,
which is possible since the right-hand side has zero horizontal mean value. Here, A is a constant independent of , which will be determined later. Then, we claim that

| T 2 λ - φ x 2 η 2 | ≤ 8 ∆η 2 L 2 (T 2 ) . (8)
Indeed, with the choice of φ, we have that

T 2 λ - φ x 2 η 2 = T 2 λ 1 + 0<|ξ 1 |≤A/ e -ixξ 1 η 2 = λ |ξ 1 |≤A/ T 2 e -ixξ 1 η 2 = λ |ξ 1 |≤A/ η 2 (ξ 1 , 0).
Since the odd symmetry in the x-direction is preserved by the equation and η 0 ∈ H 2 od (T 2 ), we have the fact that η 2 (0, y, t) = 0 for any y ∈ T and t ≥ 0, so that

ξ 1 ∈Z η 2 (ξ 1 , 0) = 0, which implies |ξ 1 |≤A/ η 2 (ξ 1 , 0) = - |ξ 1 |>A/ η 2 (ξ 1 , 0).
Then, by Cauchy-Schwartz, we bound

|ξ 1 |≤A/ η 2 (ξ 1 , 0) ≤ |ξ 1 |>A/ |ξ 1 | 2 η 2 (ξ 1 , 0) 1 |ξ 1 | 2 ≤ A |ξ 1 |>A/ |ξ 1 | 2 η 2 (ξ 1 , 0) 1 |ξ 1 | ≤ A |ξ 1 |>A/ |ξ 1 | 4 η 2 (ξ 1 , 0) 2 1 2 ξ 1 >A/ 1 |ξ 1 | 2 1 2 ≤ C A ξ 2 ∈Z ξ 1 ∈Z |ξ 1 | 2 + |ξ 2 | 2 2 η 2 (ξ 1 , ξ 2 ) 2 1 2 ≤ C A ∆(η 2 ) L 2 (T 2 ) ≤ C A ∆η 2 L 2 (T 2 ) ,
where we have used Lemma 1 in the last step as well as the Sobolev embedding

H 2 (T 2 ) into L ∞ (T 2 ). Hence, | T 2 λ - φ x 2 η 2 | ≤ λ C A ∆η 2 L 2 (T 2 )
The choice A > 8Cλ justifes the claim [START_REF] Foias | Gevrey class regularity for the solutions of the Navier-Stokes equations[END_REF].

We consider the functional

F(t) = η -φ 2 L 2 (T 2 )
. Inserting this into (7), we obtain 1 2

d dt η -φ 2 L 2 (T 2 ) ≤ - 8 ∆η 2 L 2 (T 2 ) -η 2 L 2 + F (φ) Hence 1 2 d dt η -φ 2 L 2 (T 2 ) ≤ -η -φ 2 L 2 (T 2 ) + φ 2 L 2 + F (φ), (9)
or, equivalently, 1 2

d dt F ≤ -F + φ 2 L 2 + F (φ)
, which allows us to conclude the uniform boundedness of F. Indeed, using Gronwall inequality, we immediately obtain

η -φ 2 L 2 (T 2 ) ≤ η 0 -φ 2 L 2 (T 2 ) -φ 2 L 2 -F (φ) e -2t + φ 2 L 2 + F (φ)
. where

F (φ) = |β -1| φ x 2 L 2 + 1 2 φ y 2 L 2 + 2 ∆φ 2 L 2 + δ 2 2 φ 2 H 3 . Thus, if 1 η(t) L 2 , we conclude that η L 2 (T 2 ) ≤ η -φ L 2 (T 2 ) + φ L 2 (T 2 ) ≤ η 0 -φ 2 L 2 (T 2 ) + φ 2 L 2 + F (φ) 1 2 e -t + 2 φ L 2 + F (φ) 1 2 ≤ η 0 L 2 + 2 φ L 2 + (β + 2) φ H 1 + √ φ H 2 + δ φ H 3 e -t + 2 φ L 2 + (β + 2) φ H 1 + √ φ H 2 + δ φ H 3 := R ,δ,β .
This completes the proof of Theorem 2.

Similarly as we have obtained that there exists an absorbing set in L 2 , we can conclude the existence of an absorbing set in higher Sobolev norms.

Theorem 3. Let η 0 ∈ H 2 od (T 2 ).
Then the solution η of the initial-value problem (3)

satisfies lim sup t→∞ η(t) H 2 (T 2 ) ≤ R ,δ,β .
where R ,δ,β is a constant depending on , δ, β.

proof. Recalling the existence of an absorbing set in the L 2 -norm and the regularity results in Theorem 1, so the proof is straightforword by using a bootstrap argument. We first show that there exists an absorbing set in the H 1 -norm. Inequality (5) implies that for a T > 0 large enough, we have

η(t) L 2 (T 2 ) ≤ R ,β,δ + 1, ∀t > T.
Combining this inequality with the L 2 energy estimate in the proof of Theorem 1, we obtain that

η 2 L ∞ ([0,T ];L 2 ) ≤ η 0 2 L 2 exp (C( , β, δ)T
) , which results in that there exists a constant depending on initial data, δ and such that

max 0≤t<∞ η(t) 2 L 2 (T 2 ) ≤ C( η 0 L 2 , , β, δ).
We multiply (3) by -∆η and integrate by parts to obtain that 1 2

d dt η 2 H 1 ≤ (β -1) η x 2 H 1 -η y 2 H 1 + δ Λ 3 2 η 2 H 1 -∆η 2 H 1 + η 2 L 4 ∆η x L 2 ≤ c η 4 L 4 - 2 η 2 H 3 + |β -1| η x 2 H 1 -η y 2 H 1 + δ Λ 3 2 η 2 H 1 ≤ C ,δ,η 0 η 2 H 1 - 2 η 2 H 3 + |β -1| η x 2 H 1 -η y 2 H 1 + δ Λ 3 2 η 2 H 1 , ≤ C ,β,δ,η 0 η 2 H 1 - 4 η 2 H 3 + C ,δ η 2 L 2 ≤ - 8 η 2 H 3 + C ,β,δ,η 0 ≤ - 16 η 2 H 1 - 16 η 2 H 3 + C ,β,δ,η 0 .
where we used the Plancherel Theorem, the Poincaré inequality and the Sobolev inequality ,β,δ). Using the Gronwall inequality, we immediately obtain the uniform bound

η L 4 ≤ C η H 1/2 ≤ C η 1/2 L 2 η 1/2 H 1 . It follows that d dt η 2 H 1 + 8 η 2 H 1 ≤ C( η 0 L 2 ,
η(t) 2 H 1 ≤ C( η 0 H 1 , , β, δ). Recall that the H 2 energy estimate is d dt η 2 H 2 + 2 η 2 H 4 ≤ C( , β, δ) + 1 η 2 L ∞ η 2 H 1 ,
so we can mimic the previous proof to obtain that there exists an absorbing set in H 2 . The proof is completed.

Remark 2. It is worth to point out that in the two theorems above, we show the large time dynamic of the equation for initial data belonging to H 2 od (T 2 ), which requires an odd symmetry condition in x-direction. Since the odd symmetry in x-direction is preserved by equation (3), working in H 2 od (T 2 ) is reasonable. To remove this condition, we require more regularity on the test function φ and this causes the case to become more delicate to handle. This case will be shown in our upcoming work.

Analyticity

The aim of this section is to show instant analyticity for the solutions of (3). We shall prove that the solutions of ( 3) are analytic in a strip. In order to do this, we use the method developed by Collet, Eckmann, Epstein & Stubbe in [START_REF] Collet | Analyticity for the Kuramoto-Sivashinsky equation[END_REF] (see also [START_REF] Foias | Gevrey class regularity for the solutions of the Navier-Stokes equations[END_REF]). Roughly speaking, our proof is based on a priori estimates for functions in certain Gevrey class.

Given a function σ(t) positive (see its formula explicit below), we consider the weighted exponential operators

e σ(t)Λ η = ξ∈T 2 η(ξ)e σ(t)|ξ| e iξ•x
for functions in the space

G := {η ∈ L 2 (T 2 ) : ξ∈Z 2 e 2σ(t)|ξ| |η(ξ)| 2 < ∞}.
We observe that the functions in G are analytic. We also define the inner product and norm on this Hilbert space by

µ, η σ(t) = T 2 e σ(t)Λ µe σ(t)Λ η = 4π 2 ∈Z 2 e 2σ(t)| | µ( ) η( ), η 2 σ(t) = e σ(t)Λ η 2 L 2 .
With these previous definitions, we can state the main result of this section.

Theorem 4. Let η 0 be given in H 2 od (T 2 ). Then, there exists T 0 depending on η 0 , , β, δ such that the solution of (3) satisfies

e σ(t)Λ η(t) 2 L 2 ≤ 1 + 2C 2 ,β,δ,η 0 , ∀ t > 0
where σ(t) = min{tanh(t), tanh T 0 2 }. In particular, it becomes analytic for t > 0.

Before proving theorem 4, we first state some auxiliary lemmas:

Lemma 2. For every b > a ≥ 0, Λ a 2 η 2 σ(t) ≤ Λ b 2 η 2a b σ(t) η 2-2a b σ(t) .
proof.

Λ a 2 η 2 σ(t) = e σ(t)Λ Λ a 2 η 2 L 2 = ξ∈Z 2 e 2σ(t)|ξ| |ξ| a |η(ξ)| 2 = ξ∈Z 2 e σ(t)|ξ| 2a b |ξ| a |η(ξ)| 2a b e σ(t)|ξ|(2-2a b ) |η(ξ)| 2-2a b ≤   ξ∈Z 2 e σ(t)|ξ| 2a b |ξ| a |η(ξ)| 2a b b a   a b   ξ∈Z 2 e σ(t)|ξ|(2-2a b ) |η(ξ)| 2-2a b b b-a   b-a b ≤   ξ∈Z 2 e 2σ(t)|ξ| |ξ| b |η(ξ)| 2   a b   ξ∈Z 2 e 2σ(t)|ξ| |η(ξ)| 2   b-a b ≤ Λ b 2 η 2a b σ(t) η 2-2a b σ(t) .
And an auxiliary lemma estimating the nonlinear term:

Lemma 3. | ηη x , η σ(t) | ≤ c Λη σ(t) Λ 1 2 η 2 σ(t) . proof. We first denote η * = e σ(t)Λ η, then η * (j) = e σ(t)|j| η(j).
By the definition of Fourier series, we have

η = j∈Z 2 η(j)e ij•x , e σ(t)Λ η = j∈Z 2 η(j)e ij•x e σ(t)|j| .
In fact,

ηη x , η σ(t) = (2π) 2 i ∈Z 2 j∈Z 2 ( η(j) η( -j)j 1 η( )e 2σ(t)| | = (2π) 2 i ∈Z 2 j∈Z 2 η * (j) η * ( -j) j 1 η * ( )e σ(t)(| |-|j|-| -j|) . Since | | ≤ |j| + | -j| leads to | | -|j| -| -j| ≤ 0, we have that e σ(t)(| |-|j|-| -j|) ≤ 1. Moreover, ηη x , η σ(t) ≤ (2π) 2 i ∈Z 2 j∈Z 2 η * (j) η * ( -j) j 1 η * ( )e σ(t)(| |-|j|-| -j|) ≤ (2π) 2 ∈Z 2 j∈Z 2 | η * (j)|| η * ( -j)||j|| η * ( )| = (2π) 2 k+j= | η * (j)||j|| η * (k)|| η * ( )| = T 2 φ(x)θ(x)ψ(x)dx where φ(x) = j∈Z 2 | η * (j)||j|e ijx , θ(x) = k∈Z 2 | η * (k)|e ikx , ψ(x) = ∈Z 2 | η * ( )|e i x . Notice that | φ(j)| = |j|| η * (j)| and | θ(k)| = | η * (k)|.
We now bound the last integral by the Hölder inequality,

T 2 φ(x)θ(x)ψ(x)dx ≤ φ(x) L 2 θ(x) L 4 ψ(x) L 4 ≤ c φ(x) L 2 θ(x) 2 H 1 2 = c   j∈Z 2 | φ(j)| 2   1 2 j∈Z 2 |k| 1 2 | θ(k)| 2 = c   j∈Z 2 |j| 2 | η * (j)| 2   1 2 j∈Z 2 |k| 1 2 | η * (k)| 2 = c Λη * L 2 Λ 1 2 η * 2 L 2 ≤ c Λ 1 2 η 2 σ(t) Λη σ(t)
where we used the Sobolev embedding H

1 2 (T 2 ) → L 4 (T 2
) and the Plancherel theorem in the computation above.

We begin now the proof of theorem 4:

Proof of Theorem 4. We first take inner product of (3) with η(t) in Gevrey class G, [START_REF] Giacomelli | New bounds for the Kuramoto-Sivashinsky equation[END_REF] η t , η σ(t) + ηη x , η σ(t) + (β -1)η xx -η yy -δΛ 3 (η) + ∆ 2 η, η σ(t) = 0

Note that 1 2

d dt η, η σ(t) = σ (t)Λ η, η σ(t) + η t , η σ(t)
then we have

η t , η σ(t) = 1 2 d dt η, η σ(t) -σ (t)Λ η, η σ(t) .
Substituting this into (10), 1 2

d dt η, η σ(t) = σ (t) Λη, η σ(t) -(β -1)η xx -η yy -δΛ 3 (η) + ∆ 2 η, η σ(t) -ηη x , η σ(t) ≤ σ (t) Λ 1 2 η 2 σ(t) + β Λη 2 σ(t) + δ Λ 3 2 η 2 σ(t) -Λ 2 η 2 σ(t) + c Λη σ(t) Λ 1 2 η 2 σ(t) ≤ σ (t) Λ 2 η 1 2 σ(t) η 3 2 σ(t) + β Λ 2 η σ(t) η σ(t) + δ Λ 2 η 3 2 σ(t) η 1 2 σ(t) -Λ 2 η 2 σ(t) + c Λ 2 η σ(t) η 2 σ(t)
. By the Young inequality, we have

(11) d dt η 2 σ(t) ≤ D 1 σ (t) 4 3 + D 2 + D 3 η 2 σ(t) + D 4 η 4 σ(t)
where

D 1 = ( 3 2 ) 1/3 , D 2 = 9 4 β 2 , D 3 = ( 27 4 ) 3 δ 4 , D 4 = 9 4 c 2 . By the definition of σ(t) = min{tanh(t), tanh T 0 2 }, we have σ (t) ≤ 1.
Inserting this into [START_REF] Gonzalez | Nonlinear electrohydrodynamic waves on films falling down an inclined plane[END_REF], we obtain

d dt 1 + η 2 σ(t) ≤ K 1 + η 2 σ(t) 2 with K = D 1 + D 2 + D 3 + D 4 . Define y 1 (t) = 1 + η 2 σ(t)
, it turns out to deal with the following ordinary differential inequality

y 1 (t) ≤ Ky 1 (t) 2 y 1 (0) = 1 + η 0 2 L 2 .
After solving this ODI, we obtain

η(t) 2 σ(t) ≤ 1 + 2 η 0 2 L 2 for t ∈ (0, T 0 ], where T 0 = 1 2K 1 + η 0 2 L 2 .
Moreover, according to theorem 1 and theorem 2, we know that the solution is unique, global in time and stays in a ball of Radius R ,β,δ once it has entered it, that is to say, η(t) L 2 remains bounded for all time, lim sup

t→∞ η(t) L 2 (T 2 ) ≤ R ,β,δ .
So up to now, we already prove local analyticity of η(t)

(12) η(t) 2 σ(t) ≤ 1 + 2C 2
,β,δ,η 0 for t ∈ (0, T 0 ]. In order to obtain global analyticity, we follow the previous idea and repeat the argument above starting at T 0 2 . We consider time t ∈ [ T 0 2 , 3T 0 2 ], and let y 2 (t) = 1 + η(t) 2 σ(t-

T 0 2 )
, so

y 2 T 0 2 = 1 + η T 0 2 2 L 2 .
Thus, solving the following ordinary differential inequality

     y 2 (t) ≤ Ky 2 (t) 2 y 2 T 0 2 = 1 + η T 0 2 2 L 2 ,
we have that,

η(t) 2 σ t- T 0 2 ≤ 1 + 2 η T 0 2 2 L 2 ≤ 1 + 2C 2 ,β,δ,η 0 , for time t ∈ [ T 0 2 , 3T 0 2 ]
. By the definition of σ(t) and observe that tanh(t) is strictly increasing, we know that σ(t) remains being a constant after time T 0 2 , and this constant is σ T 0 2 = tanh T 0 2 , so we choose t = T 0 in the inequality above, then

η(t) 2 σ T 0 2 ≤ 1 + 2C 2 ,β,δ,η 0 for t ∈ [T 0 , 3T 0 2 ].
We mimic this argument by adding T 0 2 each time, so we obtain that η(t) is analytic in the time invervals [ 3T 0 2 , 2T 0 ], [2T 0 , 5T 0 2 ]... Recalling local analyticity [START_REF] Goodman | Stability of the Kuramoto-Sivashinsky and related systems[END_REF], we finally obtain η(t) 2 σ(t) ≤ 1 + 2C 2 ,β,δ,η 0 for any t > 0. This completes our proof. Remark 3. It is of interest to point out that the global analyticity of the solution which we show here is better than the result in [START_REF] Granero-Belinchón | On a nonlocal analog of the Kuramoto-Sivashinsky equation[END_REF], where the argument given by the authors can be extended to prove the global analyticity of solutions of the Kuramoto-Sivashinsky equation, outside a set of time instants with zero measure.

Existence of attractor and the number of peaks

In this section, we are interested in the existence of the attractor and its properties. By applying the Theorem 1.1 in [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF], we can prove that the initial value problem (3) possesses a compact global attractor in H 2 (T 2 ). First, we denote by S(t) the solution operator, where S(t)η 0 = η(x, y, t).

Definition 1. The solution operator S(t)η 0 = η(x, y, t) defines a compact semiflow in H 2 , if for every initial data η 0 ∈ H 2 , the following four statements holds: (i) S(0)η 0 = η 0 ; (ii) S(t + s)η 0 = S(t)S(s)η 0 , for all t, s; (iii) For every t > 0, S(t)(•) : H 2 → H 2 is continuous; (iv) There exists T * > 0 such that S(T * ) is a compact operator, i.e. for every bounded set B ⊂ H 2 , S(T * )B ⊂ H 2 is a compact set. Definition 2. An attractor A ⊂ H 2 is a set that satisfies the following properties: (i) A is an invariant set, i.e., S(t)A = A, ∀t ≥ 0; (ii) there is an open neighborhood B ⊂ H 2 of A such that, for every η 0 in B, S(t)η 0 converges to A as t → ∞ : dist (S(t)η 0 , A) → 0 as t → ∞.

We shall need the following lemma: Lemma 4. Let η 0 ∈ H 2 od (T 2 ) be the initial data. Then S(•)η 0 ∈ C([0, T ]; H 2 od (T 2 )) defines a compact semiflow in H 2 od (T 2 ). proof. In order to show that S(t)η 0 = η(•, •, t) defines a compact semiflow, we must verify (i)-(iv) in definition 1. If we fix a t 0 , the continuity of S(t 0 )(•) from H 2 od to H 2 od is strightforward by energy estimates. Then, as in Theorem 3, we have the existence of an absorbing set in H 2 od -norm, so there exists

T * such that η(t) H 2 od ≤ R ,δ,β,η 0 , ∀t ≥ T * .
Since (i) and (ii) are obvious, we conclude our proof by invoking the analyticity of solutions.

Theorem 5. The system (3) has a maximal, connected, compact attractor in the space H 2 od (T 2 ). proof. By applying Theorem 1.1 in [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF] and Lemma 4.

The rest of this section is devoted to studying a particular feature of the chaotic behavior of (3), namely, the number of spatial oscillations. We shall need the lemma proved by Grujić in [START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF], which gives us an effective method to study the number of peaks (see also [START_REF] Granero-Belinchón | On a nonlocal analog of the Kuramoto-Sivashinsky equation[END_REF][START_REF] Burczak | On a generalized doubly parabolic Keller-Segel system in one spatial dimension[END_REF]). We cannot use directly the method in [START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF], mainly because Lemma 8.1 in [START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF] is quite suitable to bound the number of peaks in one space dimension, but not appropriate for our two-dimensional model.

We first let η(x, y) = f y (x), where f y (x) depends on both x and y. Then our original problem (3) deduces to [START_REF] Granero-Belinchón | On a nonlocal analog of the Kuramoto-Sivashinsky equation[END_REF] ∂

t f y + f y ∂ x f y + (β -1)∂ 2 x f y + F y (x) = 0,
where F y (x) = -η yy (x, y) -δΛ 3 (η)(x, y) + ∆ 2 η(x, y) can be seen as a forcing term.

In the previous section, we have shown that η(x, y, t) is analytic in a growing complex strip

S σ(t) = {(x, y) + i(x, ỹ) : (x, y) ∈ T 2 , |(x, ỹ)| < σ(t)}. Recalling that η(x, y, t) = f y (x, t), then f y (x, t) is analytic in S σ(t) = {x + ix : x ∈ T, |x| < σ(t)}.
Now we use the following Lemma (from [START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF]): Lemma 5. Fix y in problem [START_REF] Granero-Belinchón | On a nonlocal analog of the Kuramoto-Sivashinsky equation[END_REF]. Let σ > 0, and let f y (x) be analytic in the neighbourhood of {z = x + ix : | z| ≤ σ} and be 2π-periodic in x-direction. Then, for any µ > 0,

T = I µ ∪ R µ , where I µ is a union of at most [ 4π σ ] intervals open in T, and • |∂ x f y (x)| ≤ µ, for all x ∈ I µ ; • card {x ∈ R µ : ∂ x f y (x) = 0} ≤ 2 log 2 2π σ log max | z|≤σ |∂xfy(z)| µ .
With the help of the lemma above, we have our main result. ] open intervals in T and the following estimates hold for t ≥ T 0 2 ,

|∂ x η(x, y, t)| ≤ 1, for all x ∈ I, y ∈ T and card{x ∈ R : |∇η(x, y, t)| = 0} ≤ 4π log 2 log C ,β,δ,η 0 tanh T 0 2 .
where C ,β,δ,η 0 is a constant depending on , β, δ, η 0 . proof. From the results of Theorem 4 and Theorem 5, we know that the system has an attractor and the solution η is analytic at least in the strip of width σ(t). log C e σ(t)Λ η L 2 .

In the last inequality above, we used the following estimate ≤ C e σ(t)Λ η L 2 where we used the fact that where C R ,β,δ,η 0 depends on R ,δ,β and η 0 .

∂ x η(z,
Theorem 6 gives us a bound of the number of wild spatial oscillations of the solution, then the following corollary is a direct result of it.

Corollary 1. Let η be a solution corresponding to the initial data η 0 ∈ H 2 od (T 2 ), then for t ≥ T 0 2 , the number of peaks for η can be bounded as card {peaks for η} ≤ 4π log 2 log C ,β,δ,η 0 tanh T 0 2 . where C ,β,δ,η 0 depends on , β, δ, η 0 and T 0 is defined as before.

Numerical simulations

In this section, we show some numerical solutions of the initial-value problem (3) with the following initial condition [START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF] η 0 (x, y) = -sin x sin y + e -y 2 cos y . We then compute the numerical solution by using a fourth-order exponential time differencing (ETD-RK4) method that was first derived by Cox and Mathews in [START_REF] Cox | Exponential time differencing for stiff systems[END_REF] and then was improved by Kassam and Trefethen in [START_REF] Kassam | Fourth-order time-stepping for stiff PDEs[END_REF].

In the following figures, we show a numerical solution of [START_REF] Il'yashenko | Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation[END_REF] with parameters β = 2, δ = 0.5, = 1 and initial condition [START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF]. We can see that the equation ( 3) is very interesting from a dynamical systems point of view, as it is a PDE that can exhibit chaotic solutions.

Theorem 6 .

 6 Let η be a solution of system (3) for initial data η 0 ∈ H 2 od (T 2 ) and define T 0 as in Theorem 4. Then, T = I ∪ R, where I is a union of at most [

  Now, we can apply Lemma 5 with µ = 1 and boundcard{x ∈ R : |∇η(x, y, t)| = 0} ≤ card{x ∈ R : ∂ x η = 0}

|ξ| 3 e σ(t) 2

 2 |ξ| ≤ Ce σ(t)|ξ| . Since η has global analyticitye σ(t)Λ η(t) L 2 ≤ 1 + 2C ,β,δ,η 0 , ∀ t ≥ 0, we can conclude that for t ≥ T 0 2 card{x ∈ R : |∇η(x, y, t)| = 0} ≤ 4π log 2 log C ,β,δ,η 0 tanh T 0 2 .

Figure 1 . 2 η 2 +where |ξ| = ξ 2 1 + ξ 2 2 .-ξ 2 2 + 2 FF

 1222222 Figure 1. Initial data[START_REF] Grujić | Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation[END_REF]

Figure 2 .

 2 Figure 2. Numerical solution of (15) for β = 2, δ = 0.5, = 1 at t = 40.

Figure 3 .

 3 Figure 3. Numerical solution profile along x-direction for y = 0 of (15) with β = 2, δ = 0.5, = 1 and the same initial data as in Figure 2 at t = 0, 20, 40.

Figure 4 .

 4 Figure 4. Numerical solution profile along y-direction for x = π/2 of (15) with β = 2, δ = 0.5, = 1 and the same initial data as in Figure 2 at t = 0, 20, 40.

  y, t) L ∞ (| z|≤ σ 2 ) = ∂ x η(x + ix, y, t) L ∞ (| z|≤ σ ,ξ 2 )∈Z 2 |ξ 1 | η(ξ 1 , ξ 2 , t)e i(xξ 1 +yξ 2 ) e -xξ 1 ,ξ 2 )∈Z 2 |ξ| 3 | η(ξ 1 , ξ 2 , t)|e (ξ 1 ,ξ 2 )∈Z 2 η(ξ 1 , ξ 2 , t)|e σ(t)|ξ| 1 |ξ| 2≤ e σ(t)Λ η L 2

					2 )
	≤	(ξ 1 L ∞ (| z|≤ σ 2 )
	≤	(ξ 1 ,ξ 2 )∈Z 2	|ξ|| η(ξ 1 , ξ 2 , t)|e |x||ξ|	L ∞ (| z|≤ σ 2 )
	≤	|ξ|| η(ξ 1 , ξ 2 , t)|e	σ(t) 2 |ξ|
		(ξ 1 ,ξ 2 )∈Z 2
	≤	(ξ 1 σ(t) 2 |ξ| 1 |ξ| 2
	≤
						1 2
					1
			 (ξ 1 ,ξ 2 )∈Z 2	|ξ| 4	

Acknowledgment

The authors would like to express sincere gratitude to Ruben Tomlin for calling our attention to an error in our first manuscript. J.H would like to thank Vladimir Georgiev for helpful discussions, and also thank Lorenzo Brandolese and Dragoş Iftimie for providing suggestions and comments. R.GB was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). J.H was partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01.


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QUBIC: using NbSi TESs with a bolometric interferometer to characterize the polarisation of the CMB

Keywords: TESs, TDM, CMB

QUBIC: using NbSi TESs with a bolometric interferometer to characterize the polarisation of the CMB

Introduction

The measurement of the curl component of the polarization of the Cosmic Microwave Background (CMB), the so called B-modes, is one of the most difficult challenge of the Observationnal Cosmology. Its detection would provide valuable information about the primordial universe, inflation theory as well as quantum gravity. Currently observing or planned CMB experiments share the same detection strategy: they are imagers able to perform high sensitivity observations of the sky thanks to background-limited broadband detectors but are all subject to similar instrumental systematic effects. In interferometers what is formed on the focal plane are the Fourier modes of the sky radiation produced by an array of spatially correlated antennas. Generally, such instrumental configurations suffer low sensitivity and 1 Astroparticule et Cosmologie (CNRS-IN2P3), Paris, France; 2 Centre de nanosciences et de nanotechnologies, Orsay, France; 3 Centre de Spectrométrie Nucléaire et de Spectrométrie de Masse (CNRS-IN2P3), Orsay, France; [START_REF] Mennella | [END_REF] Ctr. Atómico Bariloche e Instituto Balseiro, CNEA, Argentina; 5 Laboratoire de l'Accélérateur Linéaire (CNRS-IN2P3), Orsay, France; 6 Institut de Recherche en Astrophysique et Planétologie (CNRS-INSU), Toulouse, France; 7 Kavli Institute for Particle Astrophysics and Cosmology, Stanford, California, USA; 8 Università di Roma La Sapienza, Roma, Italy; 9 Istituto Nazionale di Fisica Nucleare Roma 1 Section, Roma, Italy; 10 Cardiff University, Cardiff, UK; 11 Univ. Nacional de la Plata, Argentina; 12 Instituto de Tecnologías en Detección y Astropartículas, Argentina; 13 29 Brown University, Providence, RI, USA; 30 Regional Noroeste (CNEA), Argentina. E-mail: piat@apc.univ-paris7.fr the impossibility of exploiting background-limited broadband direct detectors. The great advantage of interferometric instruments resides in their capability of controlling systematic effects. Measurements of the scalar component of the CMB polarization, the E-modes, have been performed in the past with such instrumental configuration (CBI [1], DASI [2]).

In this context, the Q and U Bolometric Interferometer for Cosmology (QUBIC) currently presents a unique working strategy, combining together the sensitivity of bolometric detectors together with the control of systematic effects typical of interferometers [3].

This paper focus on the description and characterization of the QUBIC detection chain. In section 2, we present the architecture of the instrument and its basic principle of operation while section 3 describe the detection chain in more details. The recent results, which demonstrated the operation of bolometric interferometry, are presented in section 4.

The QUBIC instrument

QUBIC is an international ground based experiment dedicated to the observation of CMB polarisation. It will be deployed in Argentina, at the Alto Chorrillos mountain site (altitude of 4869 m a.s.l.) near San Antonio de los Cobres, in the Salta province. A full description of the QUBIC instrument is given in these proceedings [3] and in [4, 5, 6, 7, 8, 9]. A schematic of QUBIC is shown in Fig. 1. The signal from the sky enters the cryostat through a UHMW-PE (Ultra-High Molecular Weight Polyethylene) window [10]. Then, a rotating Half-Wave Plate (HWP) modulates the polarization, and a polarizing grid selects one of the two linear polarization components. An array of 400 back-to-back corrugated horns collects the radiation and re-images it onto a dual-mirror optical combiner that focuses the signal onto two orthogonal TES detectors focal planes. A dichroic filter placed between the optical combiner and the focal planes selects the two frequency bands, centered at 150 GHz and 220 GHz. The detection chain is described in details in section 3. In QUBIC, the optical combiner focuses the radiation emitted by the secondary horns onto the two focal planes so that the image that forms on the detector arrays is the result of the interference arising from the sum of the fields radiated from each of the 400 apertures. In this sense a bolometric interferometer can be considered a synthesized imager: the image on the focal plane is the sky signal in intensity and polarization (modulated by the HWP orientation) convolved with the so-called synthetic beam.

QUBIC has two configurations: the "Technological Demonstrator" (TD) and the "Full Instrument" (FI). The TD and FI share the same cryostat and cryogenics but the TD has only one-quarter of the 150-GHz TES focal plane (256 TESs), an array of 64 horns and switches and a smaller optical combiner. The QUBIC TD is used to demonstrate the feasibility of the bolometric interferometry. It has been extensively tested in the last months at the APC laboratory in Paris. The instrument in now being upgraded with an improved sub-K cryogenic system and toward the FI hardware. QUBIC will be shipped to Argentina and installed at the site for a first-light foreseen at the beginning of 2020.

3 The QUBIC detection chain

The QUBIC detection chain architecture is shown on Fig.

TES

The detectors are Transition Edge Sensors (TES) made with a Nb x Si 1-x amorphous thin film (x≈0.15 in our case). Its transition temperature T c of about 500 mK (Fig. 3) can be adapted by changing the composition x of the compound. The normal state resistance R n is adjusted to about 1Ω with interleaved electrods for optimum performances. To adapt to the optics, the pixels have 3 mm spacing while the membranes structure is 2.7 mm wide without any sensitivity to polarization. The low thermal coupling between the TES and the cryostat is obtained using 500 nm thin SiN suspended membranes, which exhibit thermal conductivities in the range 50-500 pW/K depending on the precise pixel geometry and temperature. The Noise Equivalent Power (NEP) is of the order of 5.10 -17 W / √ Hz at 150GHz with a natural time constant of about 100ms [11]. Light absorption is achieved using a Palladium metallic grid placed in a quarter wave cavity in order to optimize the absorption efficiency. The backshort distance of 400 µm has been chosen after electro-magnetic simulations in order to have absorption higher than 94% at both 150 and 220GHz. The 256-pixel array is finally integrated within the focal plane holder and electrically connected to a Aluminium Printed Circuit Board (PCB, provided by Omni Circuit Boards1 ) using ultrasonic bonding of Aluminium wires (Fig. 3). The detailed fabrication process of the QUBIC detectors is given in these proceedings [12]. The latest upgrade of the production process allows excellent fabrication quality with a dead pixels yield as low as 5%.

SQUIDs

The SQUID multiplexer is composed of 4 columns of 32 SQUIDs biased with capacitors in order to reduce power dissipation and noise (Fig. 4). The SQUIDs are based on a slightly modified SQ600S commercial design provided by StarCryoelectronics2 in order to reduce the area of each dies. Visual inspections and room temperature tests with a probe-station are used to select the SQUIDs before integration on a specific PCB. One SQUID PCB is composed of 32 SQUIDs and is integrated in an aluminium box. The architecture therefore uses 4 of this PCB boxes to readout 128 pixels. As shown in Fig. 2, a stack of 8 SQUID boxes is installed at 1K below the TESs in the cryo-mechanical structure, surrounded with a Cryophy3 magnetic shield.

ASIC

The ASIC is designed in full-custom using CADENCE CAD tools. The used technology is a standard 0.35µ BiCMOS SiGe from Austria MicroSystem (AMS). This technology consists of p-substrate, 4-metal and 3.3 V process. It includes standards complementary MOS transistors and high speed vertical SiGe NPN Heterojunction Bipolar Transistors (HBT). Bipolar transistors are preferentially used for the design of analog parts because of their good performances at cryogenic temperature [13]. The design of the ASIC is based on pre-experimental characterizations results, and its performance at cryogenic temperature is extrapolated from simulation results obtained at room temperature, using CAD tools. Each ASIC board for QUBIC (shown on Fig. 5) has a power dissipation of typically 16 mW and is placed on the 40K stage. The ASIC integrates all parts needed to achieve the readout, the multiplexing and the control of an array of up to 128 TESs/SQUIDs. It includes a differential switching current source to address sequentially 32 lines of SQUIDs, achieving a first level of multiplexing of 32:1. In this configuration, the SQUID are AC biased through capacitors which allows a good isolation (low crosstalk between SQUID columns) and no power dis-sipation. A cryogenic SiGe low noise amplifier (e n =0.3 nV/ √ Hz, gain=70, bandwidth of about 6MHz in simuations) with 4 multiplexed inputs, performs a second multiplexing stage between each column.

This cryogenic ASIC includes also the digital synchronization circuit of the overall multiplexing switching (AC current sources and low noise amplifier). A serial protocol allows to focus on sub-array as well as to adjust the amplifiers and current sources with a reduced number of control wires. As the digital side takes a large part, we have developed a full custom CMOS digital library dedicated to cryogenic application and ionizing environments (rad-hard full custom digital library) [13].

Warm electronics and acquisition software

The warm electronics is based on FPGA boards called NetQuiC, one for each ASIC. These boards are connected to the acquisition computer via a network switch. Each NetQuiC board is based on a differential amplifier (gain=100, bandwidth limited to 1MHz with a second order low-pass filter), a 2MHz 16 bits ADC, 7 16 bits DACs and a Xilinx Spartan 6 FPGA (XEM6010 board from Opal Kelly). The FPGA firmware programmed in VHDL takes in charge the following tasks: ASICs control, management of the TCP/IP connection with the acquisition computer, acquisition of scientific signal with the ADC, bias generation and digital Flux Locked Loop (FLL) control. The acquisition software called QUBIC Studio is the single interface to deal with the readout, the control command software and the data storage. Its core is the generic tool called "dispatcher" developed at IRAP that is a realtime-oriented acquisition system widely used on various experiments such as Solar Orbiter, SVOM/ECLAIRS and PILOT.

Characterisation of the QUBIC detection chain

The QUBIC TD test campaign has been done first in blind configuration (the 40K filters being closed) in December 2018 and afterward with all filters open from mid-January to beginning of July 2019. Unfortunately, the 1K fridge was out of order during the second run which imply a 1K stage temperature of only 4K. The detector stage therefore reached only 370mK, not low enough to be in the superconducting state for the detectors. The TESs were nevertheless responsive which allow us to make a first complete optical characterization of the instrument (see [3] for further details). We report here the characterizations made on the detection chain.

ASICs

Both ASICs 1 and 2 have been functionally tested and characterized at low temperature during the TD test campaigns. Low noise multiplexed amplifier characterizations have been investigated using a vector analyzer. A white noise level of 0.3 nV/ √ Hz and a knee frequency of about 400Hz were measured at 70K with a differential voltage gain of 70 (measured in a specific cryogenic test bench).

SQUIDs

The SQUID bias current source in the ASIC is coded on 4 bits and so we have 16 possible values spanning the range (0÷40.8)µA. The feedback coil being biased with a sine wave signal produced by the FPGA, this allows to measure both the the V-I and the V-Φ curves (see Fig. The optimal bias current is selected as the value which maximizes the SQUIDs response for the greatest number of SQUIDs read out with the same ASIC. As expected, the optimal bias current is close to the critical current value (26 µA for our SQ600S SQUIDs). The final choice in bias current allows to readout more than 86% of the SQUIDs with a voltage swing higher than 10µV.

TESs

The TESs have been characterized both in the blind and open configuration (Fig. 7). In the later case, the power background and the limited temperature of the detector stage did not allow to reach the superconducting state. In the closed configuration, it demonstrate the operation of the TES in ETF mode in the lowest bias voltage as expected. The dynamic thermal conductance is measured at about 250 pW/K leading to a NEP of 5 -6.10 -17 W/ √ Hz at 350mK. Moreover, a yield of about 74% is obtained on this array. In the open configuration, the fact that the R(T) curve still has a slope in the higher part as seen in Fig. 3 allow us to get a high enough response to do a full optical characterization with a calibration source3 without the nominal sensitivity.

Conclusions

The QUBIC detection chain is based on two 1024 NbSi TES field arrays and an original 128:1 TDM readout scheme using a cold SiGe ASIC. This detection chain was functionally validated during the QUBIC TD calibration campaigns at APC. Although a cryogenic problem did not allow to cool the 1K stage at its nominal temperature, a first complete optical characterizations of this new instrument has been performed. These measurements open the path toward the self-calibration technique and the spectro-imaging capabilities of QUBIC. The instrument is now being upgraded toward its final configurations, with an expected first light on site in 2020.

Fig. 1

 1 Fig. 1 Left: Schematic of the QUBIC instrument. The window aperture is about 40 cm; the cryostat is about 1.41 m in diameter and 1.51 m in height; Right: 3D rendering of the inner part of the cryostat. TES, Transition Edge Sensor. (Color figure online.)

  2. Each focal plane is composed of four 256-pixel arrays assembled together to obtain 1024-pixel detector. For each quarter focal plane, two blocks of 128 SQUIDs are used at 1K in a 128:1 Time Domain Multiplexing (TDM) scheme. Each block is controlled and amplified by an ASIC cooled to 40K while a warm FPGA board ensure the control and acquisition of the signal to the acquisition computer.

Fig. 2

 2 Fig. 2 Left: QUBIC cryo-mechanical structure which supports one TES focal plane at 350mK on top and the SQUID boxes at 1K below. Right: Architecture of the QUBIC detection chain for one focal plane of 1024 channels. (Color figure online.)

Fig. 3

 3 Fig. 3 Left: Superconducting transition characteristics of four Nb 0.15 Si 0.85 TESs distributed far away from each other on a 256 pixel array. Right: Pictures of the 256 TES array being integrated. (Color figure online.)

Fig. 4

 4 Fig. 4 Left: Topology of the 128 to 1 multiplexer sub-system (4×32 SQUIDs + 1 ASIC). Right: Integration of 32 SQUIDs (1 column) with bias capacitors and filter devices. (Color figure online.)

Fig. 5

 5 Fig. 5 Left: Microphotography of cryogenic ASIC designed to readout 4×32 TES/SQUID pixels. Right: ASIC module assembly used for QUBIC experiment. (Color figure online.)

Fig. 6

 6 Fig. 6 Left: typical flux response of a QUBIC SQUID for different currents. Right: histogram of the SQUID response as a function of current (expressed in digital unit). (Color figure online.)

Fig. 7

 7 Fig. 7 Left: I-V curves of the QUBIC P87 TES array measured in the blind configuration at 348mK. The color code indicate the turn-over bias voltage. Right: same array measured in open configuration at 382mK. (Color figure online.)

  University of Manchester, Manchester, UK; 14 Università degli Studi di Milano Bicocca, Milano, Italy; 15 Istituto Nazionale di Fisica Nucleare Milano Bicocca section, Milano, Italy; 16 National University of Ireland, Maynooth, Ireland; 17 University of Milan, Dept. of Physics, Milano, Italy; 18 Comisión Nacional De Energia Atómica, Argentina; 19 Richmond University, Richmond, VA, USA; 20 Università di Roma Tor Vergata, Roma, Italy; 21 Universidad Nacional de San Martin, San Martin, Argentina; 22 Institut d'Astrophysique Spatiale (CNRS-INSU), Orsay, France; 23 Agenzia Spaziale Italiana, Rome, Italy; 24 Pontificia Universidad Catolica de Chile, Santiago, Chile; 25 Instituto Argentino de Radioastronomia, Argentina; 26 California Institute of Technology, Pasadena, California, USA; 27 Istituto Nazionale di Fisica Nucleare Pisa Section, Pisa, Italy; 28 University of Wisconsin, Madison, WI, USA;

www.omnicircuitboards.com

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Acknowledgements QUBIC is funded by the following agencies. France: ANR (Agence Nationale de la Recherche) and 2014, DIM-ACAV (Domaine dInteret MajeurAstronomie et Conditions dApparition de la Vie), CNRS/IN2P3 (Centre national de la recherche scientifique/Institut national de physique nuclaire et de physique des particules), CNRS/INSU (Centre national de la recherche scientifique/Institut national de sciences de lunivers), UnivEarthS Labex program at Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). Italy: CNR/PNRA (Consiglio Nazionale delle Ricerche/Programma Nazionale Ricerche in Antartide) until 2016, INFN (Istituto Nazionale di Fisica Nucleare) since 2017. Argentina: Secretara de Gobierno de Ciencia, Tecnologa e Innovacin Productiva, Comisin Nacional de Energa Atmica, Consejo Nacional de Investigaciones Cientficas y Tcnicas. UK: the University of Manchester team acknowledges the support of STFC (Science and Technology Facilities Council) grant ST/L000768/1. Ireland: James Murphy and David Burke acknowledge postgraduate scholarships from the Irish Research Council. Duc Hoang Thuong acknowledges the Vietnamese government for funding his scholarship at APC. Andrew May acknowledges the support of an STFC PhD Studentship.