Reducibility and quasi-periodic solutions for a two dimensional beam equation with quasi-periodic in time potential

: This article is devoted to the study of a two-dimensional (2 D ) quasi-periodically forced beam equation under periodic boundary conditions, where ε is a small positive parameter, φ ( t ) is a real analytic quasi-periodic function in t with frequency vector ω = ( ω 1 , ω 2 . . . , ω m ). We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to ﬁnite dimensional invariant tori of an associated inﬁnite dimensional Hamiltonian system. The proof is based on an inﬁnite dimensional KAM theorem and Birkho ﬀ normal form. By solving the measure estimation of inﬁnitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coe ﬃ cients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkho ﬀ normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.


Introduction and main result
In this paper, we will are concerned with existence of quasi-periodic solutions for a two-dimensional (2D) quasi-periodically forced beam equation with periodic boundary conditions u(t, x 1 , x 2 ) = u(t, x 1 + 2π, x 2 ) = u(t, x 1 , x 2 + 2π) (1.2) where ε is a small positive parameter, φ(t) is a real analytic quasi-periodic function in t with frequency vector ω = (ω 1 , ω 2 . . . , ω m ) ⊂ [ , 2 ] m for some constant > 0. Such quasi-periodic functions can be written in the form φ(t) = ϕ(ω 1 t, . . . , ω m t), where ω 1 , . . . , ω m are rationally independent real numbers, the "basic frequencies" of φ, and ϕ is a continuous function of period 2π in all arguments, called the hull of φ. Thus φ admits a Fourier series expansion where k · ω = m j=1 kˆj · ωˆj. We think of this equation as an infinite dimensional Hamiltonian system and we study it through an infinite-dimensional KAM theory. The KAM method is a composite of Birkhoff normal form and KAM iterative techniques, and the pioneering works were given by Wayne [25], Kuksin [15] and Pöschel [19]. Over the last years the method has been well developed in one dimensional Hamiltonian PDEs. However, it is difficult to apply to higher dimensional Hamiltonian PDEs. Actually, it is difficult to draw a nice result because of complicated small divisor conditions and measure estimates between the corresponding eigenvalues when the space dimension is greater than one. In [11,12] the authors obtained quasi-periodic solutions for higher dimensional Hamiltonian PDEs by means of an infinite dimensional KAM theory, where Geng and You proved that the higher dimensional nonlinear beam equations and nonlocal Schrödinger equations possess small-amplitude linearly-stable quasi-periodic solutions. In this aspect, Eliasson-Kuksin [9], C.Procesi and M.Procesi [20], Eliasson-Grebert-Kuksin [5] made the breakthrough of obtaining quasi-periodic solutions for more interesting higher dimensional Schrödinger equations and beam equations. However, all of the work mentioned above require artificial parameters, and therefore it cannot be used for classical equations with physical background such as the higher dimensional cubic Schrödinger equation and the higher dimensional cubic beam equation. These equations with physical background have many special properties, readers can refer to [4,16,[22][23][24] and references therein.
Fortunately, Geng-Xu-You [10], in 2011, used an infinite dimensional KAM theory to study the two dimensional nonlinear cubic Schrödinger equation on T 2 . The main approach they use is to pick the appropriate tangential frequencies, to make the non-integrable terms in normal form as sparse as possible such that the homological equations in KAM iteration is easy to solve. More recently, by the same approach, Geng and Zhou [13] looked at the two dimensional completely resonant beam equation with cubic nonlinearity u tt + ∆ 2 u + u 3 = 0, x ∈ T 2 , t ∈ R. (1.3) All works mentioned above do not conclude the case with forced terms. The present paper study the problem of existence of quasi-periodic solutions of the equation (1.1)+(1.2). Let's look at this problem through the infinite-dimensional KAM theory as developed by Geng-Zhou [13]. So the main step is to convert the equation into a form that the KAM theory for PDE can be applied. This requires reducing the linear part of Hamiltonian system to constant coefficients. A large part of the present paper will be devoted to proving the reducibility of infinite-dimensional linear quasi-periodic systems. In fact, the question of reducibility of infinite-dimensional linear quasi-periodic systems is also interesting itself.
In 1960s, Bogoliubov-Mitropolsky-Samoilenko [3] found that KAM technique can be applied to study reducibility of non-autonomous finite-dimensional linear systems to constant coefficient equations. Subsequently, the technique is well developed for the reducibility of finite-dimensional systems, and we don't want to repeat describing these developments here. Comparing with the finite-dimensional systems, the reducibility results in infinite dimensional Hamiltonian systems are relatively few. Such kind of reducibility result for PDE using KAM technique was first obtained by Bambusi and Graffi [1] for Schrödinger equation on R. About the reducibility results in one dimensional PDEs and its applications, readers refer to [2,7,17,18,21] and references therein.
Recently there have been some interesting results in the case of systems in higher space dimensions. Eliasson and Kuksin [6] obtained the reducibility for the linear d-dimensional Schrödinger equatioṅ Grébert and Paturel [14] proved that a linear d-dimensional Schrödinger equation on R d with harmonic potential |x| 2 and small t-quasiperiodic potential reduced to an autonomous system for most values of the frequency vector ω ∈ R n . For recent development for high dimensional wave equations, Eliasson-Grébert-Kuksin [8] , in 2014, studied reducibility of linear quasi-periodic wave equation.
However, the reducibility results in higher dimension are still very few. The author Min Zhang of the present paper has studied the two dimensional Schrödinger equations with Quasi-periodic forcing in [27]. However, it would seem that the result cannot be directly applied to our problems because of the difference in the linear part of Hamiltonian systems and the Birkhoff normal forms. As far as we know, the reducibility for the linear part of the beam equation (1.1) is still open. In this paper, by utilizing the measure estimation of infinitely many small divisors, we construct a symplectic change of coordinates which can reduce the linear part of Hamiltonian system to constant coefficients. Subsequently, we construct a symplectic change of coordinates which can transform the Hamiltonian into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we show that there are many quasi-periodic solutions for the equation (1.1) via KAM theory. Remark 1.1. Similar to [13], we introduced a special subset of Z 2 for the small divisor problem could be simplified. Then we define subspace U in L 2 (T 2 ) as follows We only prove the existence of quasi-periodic solutions of the equation (1.1) in U.
The following definition quantifies the conditions the tangential sites satisfy. It acquired from Geng-Xu-You [10].
Any three different points of them are not vertices of a rectangle (if n > 2) or n = 2. (ii). For any d ∈ Z 2 odd \ S , there exists at most one triplet {i, j, l} with i, j ∈ S , l ∈ Z 2 odd \ S such that d − l + i − j = 0 and |i| 2 − | j| 2 + |d| 2 − |l| 2 = 0. If such triplet exists, we say that d, l are resonant in the first type and denote all such d by L 1 . (iii). For any d ∈ Z 2 odd \ S , there exists at most one triplet {i, j, l} with i, j ∈ S , l ∈ Z 2 odd \ S such that d + l − i − j = 0 and |d| 2 + |l| 2 − |i| 2 − | j| 2 = 0. If such triplet exists, we say that d, l are resonant in the second type and denote all such d by L 2 . (iv). Any d ∈ Z 2 odd \ S should not be in L 1 and L 2 at the same time. It means that L 1 ∩ L 2 = ∅.
Remark 1.2. We can give an example to show the admissible set S above is non-empty. For example, for any given positive integer n ≥ 2, the first point (x 1 ,ỹ 1 ) ∈ Z 2 odd is chosen asx 1 > n 2 ,ỹ 1 = 2x 5 n 1 , and the second one is chosen asx 2 =x 5 1 ,ỹ 2 = 2x 5 n 2 , the others are defined inductively bỹ The choice of the admissible set is same to that in [13], where the proof of such admissible set is given.
In this paper, we assume that (H) φ(t) is a real analytic quasi-periodic function in t with frequency vector ω, and [φ] 0 where [φ] denotes the time average of φ, coinciding with the space average.

The Hamiltonian setting
Let's rewrite the beam equation (1.1) as follows 3) The equation can be written as the Hamiltonian equationq = i ∂H ∂q and the corresponding Hamiltonian functions is The eigenvalues and eigenfunctions of the linear operator −∆ with the periodic boundary conditions are respectively λ j = | j| 2 and φ j (x) = 1 2π e i< j,x> . Now let's expand q into a Fourier series q = the coordinates belong to some Hilbert space l a,s of sequences q = (· · · , q j , · · · ) j∈Z 2 odd that has the finite norm q a,s = j∈Z 2 odd |q j || j| s e | j|a (a > 0, s > 0).
The corresponding symplectic structure is i j∈Z 2 odd dq j ∧ dq j . In the coordinates, the Hamiltonian equation (2.3) can be written asq (q i q j q dql +q iq jqd q l ).
Denote ϕ(ϑ) be the shell of φ(t), we introduce the action-angle variable (J, ϑ) ∈ R m × T m , then (2.6) can be written as followsθ and the corresponding Hamiltonian function is and

Reducibility via KAM theory
Now We are going to study the reducibility of the Hamiltonian (2.8). To make this reducibility, we introduce the notations and spaces as follows.
Assume B(η; ω) be an operator from D a,s to D a,s for (η; ω) ∈ D a,s × R * , then we denote |∂ˆj ω B| a,s,D a,s ×R * .
Reducibility of the autonomous Hamiltonian equation corresponding to the Hamiltonian (2.8) will be proved by an KAM iteration which involves an infinite sequence of change of variables. By utilizing the measure estimation of infinitely many small divisors, we will prove that the composition of these infinite many change of variables converges to a symplectic change of coordinates, which can reduce the Hamiltonian equation corresponding to the Hamiltonian (2.8) to constant coefficients.

Convergence and reducibility theorem
The reducibility of the linear Hamiltonian systems can be summarized as follows.
We obtain the following sequences: D a,s 0 ⊃ D a,s 1 ⊃ · · · ⊃ D a,s ν ⊃ · · · ⊃ D a,s ∞ . From (3.30), (3.32) and (3.33), denote Similar to [27], it can be seen that the limiting transformation T 0 • T 1 • · · · converges to a symplectic coordinate transformation Σ 0 ∞ . And there exists an absolute constant C > 0 independent of j such that with id is identity mapping.
In view of the Hamiltonian (2.8) satisfies the conditions (3.1) − (3.4), (3.6), (3.7) with ν = 0, the above iterative procedure can run repeatedly. Thus the transformation Σ 0 ∞ changes the Hamiltonian (2.8) toH We present the following lemma which has been used in the above iterative procedure. The proof is similar to Lemma 3.1 in [15]. Lemma 3.1. For any given k ∈ Z m , j ∈ Z 2 odd ,l ∈ N, denote where δ(x) = 1 as x = 0 and δ(x) = 0 as x 0. Then the sets R 1 , R 2 l is measurable and

The Hamiltonian after the iterative procedure
In view of the symplectic coordinate transformation Σ 0 ∞ is linear, and (3.36), then
For a ≥ 0 and s > 0, the gradientsG 4 q ,G 4 q are real analytic as maps from some neighborhood of origin in l a,s × l a,s into l a,s with G 4 q a,s = O( q 3 a,s ), G 4 q a,s = O( q 3 a,s ).

Partial Birkhoff normal form
As in [13], Let S is an admissible set. We define Z 2 * = Z 2 odd \ S . For simplicity, we define the following three sets: and Obviously, the set (i, j, d, l) ∈ (Z 2 odd ) 4 : is empty. Similar to [13], the set (i, j, d, l) ∈ (Z 2 odd ) 4 : is empty. For Proposition 4.1, we give the following lemma that will be proved in the "Appendix".
Now we introduce the parameter vectorξ = (ξ j ) j∈S and the action-angle variable by setting q j = I j +ξ j e iθ j ,q j = I j +ξ j e −iθ j , j ∈ S .
Let x = ϑ ⊕ θ with θ = (θ j ) j∈S , y = J ⊕ I, z = (z j ) j∈Z 2 * and ζ = ω ⊕ (ξ j ) j∈S , and let's introduce the phase space P a,s = T m+n × C m+n × l a,s × l a,s (x, y, z,z) where T m+n is the complexiation of the usual (m + n)-torus T m+n . Let D a,s (s , r) := {(x, y, z,z) ∈ P a,s : |Imx| < s , |y| < r 2 , z a,s + z a,s < r}, for W = (x, y, z,z) ∈ P a,s . Set α ≡ (. . . , α j , . . .) j∈Z 2 * , β ≡ (. . . , β j , . . .) j∈Z 2 * , α j and β j ∈ N with finitely many nonzero components of positive integers. The product z αzβ denotes j z α j jz β j j . Let where P αβ = k,b P kbαβ y b e i<k,x> are C 4 W functions in parameter ζ in the sense of Whitney. Let P D a,s (s ,r),Σ ≡ sup z a,s <r, z a,s <r α,β where, if P α,β = k∈Z m+n ,b∈N m+n P kbαβ (ζ)y b e i<k,x> , P αβ is short for the derivatives with respect to ζ are in the sense of Whitney. Denote by X P the vector field corresponding the Hamiltonian P with respect to the symplectic structure dx ∧ dy + idz ∧ dz, namely, X P = (∂ y P, −∂ x P, i∇zP, −i∇ z P).
Its weighted norm is defined by X P D a,s (s ,r),Σ ≡ P y D a,s (s ,r),Σ + 1 r 2 P x D a,s (s ,r),Σ s (s ,r),Σ e | j|a + j∈Z 2 * Pz j D a,s (s ,r),Σ e | j|a ).
The following Lemma can be obtained and the proof is similar to Lemma 3.2 in [27].

An infinite-dimensional KAM theorem for partial differential equations
In order to prove our main result (Theorem 1.1), we need to state a KAM theorem which was proved by Geng-Zhou [13]. Here we recite the theorem from [13].
Let us consider the perturbations of a family of Hamiltonian in n-dimensional angle-action coordinates (x, y) and infinite-dimensional coordinates (z,z) with symplectic structure j∈S dx j ∧ dy j + i The tangent frequencies ω = ( ω j ) j∈S and normal ones Ω = ( Ω j ) j∈Z 2 * depend on n parameters ξ ∈ Π ⊂ R n , with Π a closed bounded set of positive Lebesgue measure.
For each ξ there is an invariant n-torus T n 0 = T n × {0, 0, 0} with frequencies ω(ξ). The aim is to prove the persistence of a large portion of this family of rotational torus under small perturbations H = H 00 + P of H 00 . To this end the following assumptions are made.
Assumption A5. (Zero-momentum condition): The normal form part A + B +B + P satisfies the following condition A + B +B + P = k∈Z n ,b∈N n ,α,β (A + B +B + P) kbαβ (ξ)y b e i<k,x> z αzβ and we have Now we state the basic KAM theorem which is attributed to Geng-Zhou [13], and as a corollary, we get Theorem 1.1.
Verifying (A3) : For (5.2), B d is defined as follows, where (i, j, l) is uniquely determined by d. In the following, we only prove (A3) for det[< k, ω(ξ) > We need to prove that |Z(ξ)| ≥ γ |k| τ , (k 0). For this purpose, we need to divide into the following two cases.
Case 1. When k 1 0, notice that then all the eigenvalues of Z(ξ) are not identically zero. Case 2. When k 1 = 0, then We assert that all the eigenvalues of Z(ξ) are not identically zero. Here we're just proving it for d, d ∈ L 1 , and everything else is similar. Let In view of |i| 2 + |d| 2 = | j| 2 + |l| 2 and (2.10),(3.45), Thus, √˜ξ i ξ j 8π 2 λ i λ j
Hence all eigenvalues of Z(ξ) are not identically zero for k 0. According to Lemma 3.1 in [10], det(Z(ξ)) is polynomial function in ξ of order at most four. Thus