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The stability of Sobolev norms for the linear wave equation with unbounded perturbations

In this paper, we prove that the Sobolev norm of solutions of the linear wave equation with unbounded perturbations of order one stay bounded for the all time. The main proof is based on the KAM reducibility of the linear wave equation. To the best of our knowledge, this is the first reducibility result of the linear wave equation with general quasi-periodic unbounded potentials on the torus.


Introduction
In this paper, we consider a linear wave equation with unbounded quasi-periodic perturbations of the form (1.1) ∂ tt u − ∂ xx u + mu + W(ωt)u = 0, t ∈ R, x ∈ T = R/2πZ, where W(ωt) is a pseudo-differential operator of order one, and quasi-periodic in time with frequencies ω ∈ O := [1,2]  The problem of estimating the high Sobolev norm of linear partial differential equations has been widely studied. The two remarkable results were obtained by Bourgain [16,17] for the free Schrödinger equation with time dependent potential (1.2) i∂ t u = −∆u + V (t, x)u, on the d-dimensional torus. Bougain [16] derived a t ǫ upper bound of the Sobolev norm of solutions for smooth and bounded time dependent potentials. When the potential V is analytic and time quasi-periodic, Bourgain [17] proved that the Sobolev norm of solution grows like a power of log(t). The result obtained in [16] has been extended by Delort [18] and Berti-Maspero citeberti19 to the Zoll manifolds and flat tours. The logarithmic bounds on Sobolev norms in [17] has been extended by Wang [36] to an analytic and bounded time dependent potential on T. However, Bourgain's original method can only deal with the bounded perturbations, especially the multiplicative potential V (t, x). The first result on the Schrödinger-type equation with unbounded, time dependent perturbation of the form (1.3) i∂ t u(t) = Hu(t) + P(t)u, is due to Maspero-Robert [28]. The method in [28] can be applied to the free Schrödinger equation on Zoll manifolds with time smooth perturbations of order m < 1, which provided a t ǫ upper bound of Sobolev norms of solutions. Based on the pseudo-differential operator technique, Bambusi-Grébert-Maspero-Robert [10] extended their results to more Schrödingertype equations, including the free Schrödinger on Zoll manifolds with perturbations of order m < 2. In the meantime Montolto [30,32] has independently studied the maximum order of perturbations for the Schrödinge-type equation on T. It's worth mentioning that, based on a delicate Quantum version Nehoroshev theorem [9], Bambusio-Langella-Montalto [8] proved a t ε upper bound of Sobolev norm of solutions for the free Schrödinger equation on the flat torus with unbounded perturbations of order m < 2.
For the Schrödinger-type equation with small time quasi-periodic perturbations of the form i∂ t u(t) = Hu(t) + εP(ωt)u, the KAM reducibility is a powerful tool to investigate the uniformly boundedness of the solutions in Sobolev space. For the bounded perturbations, we mention the results of Eliasson-Kuksin [19] which proved the reducibility of the Schrödinger equation on T d and Grébert et al. [24,25] which proved the reducibility of the quantum harmonic oscillator on R d .The reducibility results imply that the Sobolev norms of solutions for the equation considered is uniformly bounded. For the unbounded perturbations, there are several papers devoting to the reducibility of some Schrödinger equations, such as the quantum harmonic oscillator [4][5][6][7], duffing oscillator [27], relativistic Schrödinger equation on torus [34] and the free schrödinger equation on Zoll manifolds [22,23]. Compared with the enormous results of Schrödinger-type equations, there are few results concerning the growth of Sobolev nrom of solutions for the wave equation. Estimating the high Sobolev norm of solutions for linear wave equations on compact manifolds is much more subtle than Schrödinger-type equations. Fang-Han-Wang [20] had constructed a small time periodic potential for the wave equation on the torus such that the Sobolev norms of solutions is bounded for all time. While, Bourgain [17] constructed a time periodic potential for the wave equation, which provoke exponential growth of Sobolev norm. In order to avoid such terrible upper bound, people has to pay more attention to the wave equation with time quasi-periodic perturbation. Naturally, the KAM reducibility becomes one of the main research methods. For the bounded perturbations, we mention the results of Li [26] and Liang [35] which proved the reducibility of wave equation on the torus with small time quasi-periodic multiplicative potential. Maspero [21] proved the reducibility of the wave equation with non-small time quasi-periodic multiplicative potential. For the unbounded perturbations, Montolto [31], Sun et al. [33] studied the wave equation with some special unbounded perturbations.
It should be mentioned that the unbounded perturbations consider in [31,33] is in the special form of V (ωt)∆, which can be obtained by linearizing some nonlinear equations [32]. People are more concerned with general unbounded perturbations. From the viewpoint of KAM theory, if the order of perturbations is strictly smaller than one, the KAM reducibility for such wave equation is straightforward. If the order of perturbations is equal to one, some serious problems are occurs in the measure estimate in KAM iteration. The similar problems are resolved by Berti-Biasco-Procesi [11], in which they obtained some quasi-periodic solutions of the Hamiltonian derivative wave equation. In order to estimate the number of non-resonance conditions, they introduced the "quasi-Töplitz" property of the perturbations to get the higher order asymptotic decay estimate of normal frequencies. However, the property of momentum conservation of the equation is indispensable for preserveing the "quasi-Töplitz" property in KAM iteration. The property is missing from the wave equation (1.1) in the present paper. Therefore, this paper adopts a completely different method, that is the method of pseudodifferential operator.
For the Schrödinger-type equation with unbounded perturbations, the method of pseudodifferential operator can effectively smoothing the perturbations, so as to avoid a series of difficulties caused by the order of perturbations. However, such skills are almost useless for the wave equation. One of the main reasons is that the wave equation with Hamiltonian form can be seen as a 2 × 2 matrix valued, Schrödinger-type equation (2.16). For the matrixvalued pseudo-differential operator, the commutator of two matrix-valued pseudo-differential operators can not gains one derivative. Therefore, we can not transform the whole perturbation in equation (2.16) into a smoothing one. The main novelty of the present paper is that we find a delicate bounded transformation such that the original perturbations K(ωt) in equation (2.16) can be transformed into a new one P(ωt) in equation (4.17), where the diagonal part are smoothing operators and the off-diagonal part are still bounded operators. Furthermore, such structure can be maintained in the KAM iteration. Under these conditions we can also get the higher order asymptotic decay estimate of eigenvalues.
Remark 1.1. If we have a good control of the matrix decay norm of the operator e iG , where G is a self-adjoint, pseudo-differential operator of order 0 < m < 1 2 , the method presented in this paper may be extended to the case that the order of perturbations is less than 3/2. Form the Lemma 3.4 in [8], we known that the operator e iG is bounded in Sobolev space H r x . But the information about its matrix decay norm is missing, which is essential to the KAM iteration.
Remark 1.2. In addition, the reducibility problem of wave equation (1.1) in this paper is the cornerstone of some further works. Inspired by [1,2], we can use the quantitative reducibility result in this paper to explore the existence of Sobolev, linearly stable, quasi-periodic solutions of the following derivatives wave equations.
• The Hamiltonian derivatives wave equations with quasi-periodic force • The autonomous derivatives wave equations The paper is organized as follows: In section 2, we introduce some important definitions of pseudo-differential operator, so that we can precisely state our main results. In section 3, we introduce some norms of infinite dimensional matrix, such that the KAM iterations in section 5 can be well understood. In section 4, we introduce the the symbolic calculus of pseudo-differential operators in [32], such that the diagonal part of the perturbation K(ωt) in equation (2.16) can be reduced to a operator of order −1. In section 5, we give a block-diagonal reducibility result for the equation (2.16). In Section 6, we conclude the proof of Theorem 2.7 and the Corollary 2.9.
Notations: In the present paper, we denote the notation A s B as A ≤ C(s)B, where C(s) depends on the data of the problem, namely the Sobolev index s, the number d of time frequencies, the diophantine exponent τ > 0 in the non-resonance conditions, which will be required along the proof.

Main result
Given a Function f : We define the sup-norm and lipschitz semi-norm as For any γ > 0, we define the Lipschitz-norm For notation convenience, we omit to write the set O.
2.1. Function space and pseudo-differential operators.
Notation: In the rest of the paper, we fix where for any real number x ∈ R, we denote by [x] its integer part. Pseudo-differential operators: Definition 2.1. (Pseudo-differential operators and symbols) Let m ∈ R, s ≥ s 0 , α ∈ N, we say that an operator A = A(θ) is in the class OPS m s,α , if there exists a function a = a(θ, x, ξ) : In that case, we say that a(θ, x, ξ) is in the class S m s,α . The operator A is said to be a pseudodifferential operator of order m, and the function a is symbol.
If A := A(λ) is depending in a Lipschitz way on some parameter ω ∈ O ⊆ R d , we set where the Reminder R 2,AB satisfies (2.6) Adjoint of pseudo-differential operator Considering a θ-dependent families of pseudo-differential operator A(θ), the symbol of the adjoint operator A * = Op(a * (θ, x, ξ)) is Lemma 2.4. (Lemma 2.16 in [14]) Let A = Op(a) ∈ OPS m s+s0+|m|,0 , and dependent on the parameters ω ∈ O. Then, the adjoint operator A * satifies s+|m ′ |+2,α be a self-adjoint operator and G = Op(g(ξ)) be a real Fourier multiplies of order m ′ (independent of parameters ω). We define a new operator B = Op(g(ξ) · a(θ, x, ξ)), that Also, the operator B − B * ∈ OPS m+m ′ −1 s,α and satisfies The estimate (2.9) is a direct corollary of Definition 2.1.
In order to state our main results, we rewritten the wave equation (1.1) as new form, by introducing the new variables, In the new variables , the equation (1.1) is transformed to Taking its complex conjugate, we can obtain the following matrix valued, Schrödinger-type system such that the following hold true. For any ω ⊆ O ǫ , there exists a liner bounded and invertible operator T (ωt, ω) ∈ B(H r x × H r x ) for any 0 ≤ r ≤ S s , such that the change of coordinates q = T (ωt, ω)v conjugates the equation (2.16) to the block diagonal time-independent system Remark 2.8. From Lemma 2.2, we known that K(ωt) ∈ OPS −1 s−3,2 is a real and self-adjoint operator, and satisfies |K| γ for some absolute constants c r , C r > 0.

Linear operators
3.1. Matrix representation of linear operator. Consider a linear operator A : L 2 (T) → L 2 (T), it action on a function u ∈ L 2 (T) as Also, we can identify the linear operator A as a infinite-dimensional block matrix We also consider a smooth θ− dependent families of linear operator θ → A(θ), T d → B(L 2 (T)), which can be expanded in Fourier series as It action on a function u ∈ L 2 (T d × T) as Remark 3.1. From the matrix representation (3.3), we can regard the linear operator A as a pseudo-differential operator Op(a(θ, x, ξ)), for any ξ = j ∈ Z, one has · 0 stands for L 2 operator norm. If the operator A is Lipschitz depending on the parameters If the linear operator A is independent of θ and has the block matrix representation (3.1), the matrix decay block norm becomes Let s ≥ s 0 , and A, B ∈ M s . Then, 3 : Let s ≥ s 0 , and A ∈ M s . There exists a constant C(s) > 0, such that for any integer n ≥ 1, 5 : Items 1 − 4 hold, replacing | · | s by | · | γ s and · s by · γ s . Proof. The proof is similar to Lemmata 2.7, 2.8 in [29].
Given N ∈ N, we define a smoothing operator Π N A for any operator A with block -matrix representation (3.1), From the Definition 2.1,3.2 and Remark 3.1, we can state a link between the pseudodifferential norms and matrix block decay norms.
We can endow the M s,ρ with the norm If the operator A is Lipschitz depending on the parameters ω ∈ O, we say that that A(ω) ∈ Lip(O, M s,ρ ). For any γ > 0, we endow it with the norm Definition 3.10. Let A be a θ−dependent linear operator. Given s ≥ s 0 and ρ > 0, we say that A ∈ L s,ρ if We can endow the L s,ρ with the norm If the operator A is Lipschitz depending on the parameters ω ∈ O, we say that that A(ω) ∈ Lip(O, L s,ρ ). For any γ > 0, we endow it with the norm The operator e iB Ae −iB belongs to M s,ρ and with the quantitative bounds The analogous assertions hold true, if A ∈ Lip(O, M s,ρ ) and B ∈ Lip(O, M s ).
Proof. The item 1 is a direct corollary of Lemmata 3.4, 3.7 and Definition 3.8.

3.2.
The real and self-adjoint operators. Definition 3.13. (1): Given a θ-dependent operator A(θ) : T d → B(L 2 (T)), we define its conjugate operatorĀ byĀu = Aū. The conjugate operatorĀ has the matrix representation The adjoint operator A * has the matrix representation is real, if it maps real functions to real andĀ(θ) = A(θ). Then, for any i, j ∈ Z, α, β ∈ N, θ ∈ T d , the following assertions hold true, Then, for any i, j ∈ Z, α, β ∈ N, θ ∈ T d , the following assertions hold true, Lemma 3.15. Let A = Op(a) ∈ OPS m s,α . Then, A is real if and only if a(θ, x, ξ) = a(θ, x, −ξ). 3.3. 2 × 2 operator matrix. In this section, we will describe a special structure of operator matrix.
It needs emphasize that the diagonal operator A d and anti-diagonal operator A a have different symmetry properties and matrix decay properties. For the details, we show it in the following definition.
Definition 3.16. Given a 2 × 2 operator matrix A of the form (5.12), and ρ, o ∈ R, s ≥ s 0 , we say that A belongs to N s (ρ, o) if If the operator A d and A o are Lipschitz depending on the parameters ω ∈ O, we say that that N s (ρ, o)). For any γ > 0, we endow it with the norm Remark 3.17.
• The symmetry properties of the space N s (ρ, o) is equivalent to ask that the operator matrix A is the Hamiltonian vector field of a real valued quadratic Hamilton. For the details, we refer to [29,31].
• The decay properties (3.29) of the diagonal operator A d is essential to the measure estimate in section 4, we will show that the decay properties of the diagonal operator can be maintained in KAM iteration.    0) and B ∈ N s (ρ, ρ). Then, the following assertions hold true. 1 : For any r ∈ [0, s] and θ ∈ T d , the operator e iB(θ) ∈ B(H r x × H r x ) with the standard operator norm uniformly bounded in θ.

:
The operator e iB Ae −iB belongs to N s (ρ, 0) and e iB Ae −iB − A belongs to N s (ρ, ρ) with the quantitative bounds The analogous assertions hold true, if A ∈ Lip(O, N s (ρ, 0)) and B ∈ Lip (O, N s (ρ, ρ)).
Proof. The proof is similar to Lemma 2.23 in [21] and Lemma 3.11.

Regularization procedure
The goal of this section is smoothing the diagonal part of the 2 × 2 operator matrix K(ωt) in equation (2.16). In the following lemmas, we will conjugate the diagonal parts of the perturbation into smoothing operators, while the anti-diagonal parts remain bounded operators. The results of this section is essential to the KAM iteration in the following section. For any γ ∈ (0, 1) and τ 0 > d, we introduce the set Lemma 4.1. For any ω ∈ O γ , and symbol k ∈ S 0 s−3,2 with k = k * , there exists a symbol g ∈ S 0 s−τ0−5,2 , with g = g * , such that Furthermore, one has Proof. Taking a cut-off function χ 1 (ξ) ∈ C ∞ (R, R), it is satisfying For the symbol (4.2), one has (4.5) From the Lemma 2.5, we known that Our goal is determine a symbol g, such that Since we require that g = g * , we look for a symbol of the form g = q+q * 2 . From (4.5), one has (4.7) Next, we look for a symbol q that satisfies (4.8) For any ω ∈ O γ , the symbol q defined as is the characteristic function of the set {ξ ∈ R : ξ > 0}(resp.{ξ ∈ R : ξ ≤ 0}).

KAM iteration
After the preliminary transformation in the previous section, we can get the following equation and h 0 j is linear operator from E j to E j . To be more precise, for j ∈ N + and a ∈ {1, −1}, one has and h 0 0 = √ m + k θ,x (0).

General step of KAM iteration.
In this section, we are going to perform an KAM iteration reducibility scheme for the linear equation (5.1). The main goal is to block-diagonalize the linear equation (5.1), and the key is to constantly square the size of the perturbation.
In the following, we show the outline of k th KAM iteration. For notional convenience, in the subsection, we drop the index n and write + instead of n + 1. Through where H + (t) = e iU + (ωt,ω) H(t)e −iU + (ωt,ω) − 1 0 e isU + (ωt,ω)U+ e −isU + (ωt,ω) ds, (5.6) We determine the operator matrix U + by solving the homological equation The new Hamiltonian becomes H + (t) = H + 0 + P + , where P + = R + Π ⊥ N P and In order to get a nice solution of the homological equation (5.10) and ensure the convergence of the KAM iteration, the second-order Melnikov conditions are required to be imposed. Denoting λ j,a , a ∈ {1, −1} as the eigenvalues of the block h j , we choose the frequency vector ω from the following set Proof. The homological equation (5.10) is split in the two equation Considering the block matrix representation and expanding the Fourier series in time, for any ℓ ∈ Z d , |ℓ| ≤ N and i, j ∈ N, we can get Takeing I as unite matrix, one see that, for any i, j ∈ N, Since the block h j is self-adjoint, one sees spec(h j ) = spec(h j ) = ω · ℓ + λ j,a | a ∈ {1, −1} . From Lemma 7.2, one get immediately that (5.20) [ Let . For the equation (5.18), we can get that From Lemma 7.2 again, we can get Thus, (5.20),(5.22) imply that can be bounded by the same norms of P d .
The bounds control of U a is more delicate. From Lemma 7.2, we can get that We also need control the bounds of [ D U a (ℓ)] j i , [ U a D (ℓ)] j i and [ ¤ D σ U a D −σ (ℓ)] j i , (σ = ±1). Considering the term [ D U a (ℓ)] j i , and applying Lemma 7.2 again, one gets The similar bounds hold for [ U a D (ℓ)] j i and [ ¤ D σ U a D −σ (ℓ)] j i . Applying the difference operator ∆ ω to equation (5.19), one gets Applying Lemma 7.2 again, we can get  N s+b (1, 0)). Assume (5.14) and, for some fixed constant C s ,

The similar bounds hold for ∆
Then, P + = Π N P + R is defined on O + γ and satisfies the quantitative bounds Proof. From homological equation (5.10), the operator R defined in (5.8) can be rewritten as (ℓ, i, j) = (0, j, j), |ℓ| ≤ N n , and |ω · ℓ + λ n i,a + λ n j,a ′ | ≥ II : There exists a operator matrix U n (ωt, ω) ∈ Lip(O n γ , N s (1, 1)) and satisfies (5.42) U n γ s−b,1,1 ≤ P 0 γ s,1,0 γ −1 N 2τ +1 n−1 N −a n−2 . The change of coordinate e −iU n conjugate H n−1 to H n := H n 0 (ω) + P n (ωt, ω) such that III : The operator H n 0 is block diagonal, self-adjoint and time independent, where (5.46) P n γ s−b,1,0 ≤ P 0 γ s,1,0 N −a n−1 . Proof. We prove these assertions by inductive. It's easy to verified that the properties in items I − IV hold true for n = 0. Let us suppose that the statements hold true for a fixed n ∈ N and define the set O n+1 γ in item I. We prove these assertions also hold true for n + 1. In order to apply Lemmata 5.1, 5.2, we need check the assumptions in the two Lemmata. For the assumption in Lemma 5.1, one has For the assumption in Lemma 5.2, if N 0 is sufficient large, one has since 3 2 (2τ + 1) − a < 0. Now, we can apply Lemma 5.1 with P := P n and U + := U n+1 , The inequality (5.52) can be verified by (5.38), by takeing N 0 larger enough and δ small enough. So, the item IV is valid for n + 1.
Proof. The convergence of the transformations is a standard argument, we skip the details.
Corollary 5.5. For all j ∈ N and ω ∈ O γ , the self-adjoint block {h n j } n≥0 is convergence in · γ 0 to a block h ∞ j , which is fulfils Proof. The convergence of the block is standard. For the bound (5.1), from (5.44), one gets Proof. It's suffice to show that for any For any n > 0, from Theorem 5.3 and Lemma 7.1, one sees that If ω ∈ O ∞ 2γ , for any (ℓ, i, j) ∈ Z d × N × N, (ℓ, i, j) = (0, j, j), a, a ′ ∈ {1, −1} and |ℓ| ≤ N n−1 , one gets The last inequality holds true, because 4εγ −1 N τ n−1 ≤ i − j jN a n−1 .
We can obtain From the Fubini theorem, we can obtain (5.61).
For any i ∈ Z, we known that k θ,x (i) = Op( k ) i i . Thus, from the condition II and Definition 2.1, one has where c * ∈ Γ * := {c * 1 , · · · , c * q }. Also, there exist an absolute positive constant C, such that |b * (i, ω)| ≤ C i e . Take the set Γ as {1, · · · , q}, we can define the set Lemma 5.8. Let 0 < γ 0 < 1 4 , τ 0 > d, one has where C is a positive constant depending on q.
For any j ∈ Z, and a, a ′ ∈ {1, −1}, we take d j,a := j 2 + m + c * (aj) = j + c(m, j) j + c * (aj), and define the set Lemma 5.9. Let 0 < γ 1 ≤ γ0 2 and τ 1 > τ 0 + d, one has Proof. We define the set We consider the measure estimate of set P, estimateing the measure of set Q is relatively simple. Case 1: If ℓ = 0 and i = j, one has Case 2: If ℓ = 0 and i = j, one has Case 3: If ℓ = 0, i = j and |i − j| > 8|ℓ|, one can obtain Case 4: Let |i − j| ≤ 8|ℓ| and i < j, we assume that Therefore, we restrict ourself to the case i < j and i i − j ≤ 4m ℓ τ 0 γ0 . The same arguments can be extended to the symmetric case j > i and j i − j ≤ 4m ℓ τ 0 γ0 . From Lemma 5.7, we known that for any ℓ = 0 Now, we define the index set of (ℓ, i, j), that is Since U = (ℓ,i,j)∈E U ℓ,i,j , from (5.80), one gets provided τ > d + τ 0 . The same computation hold for the set V. Hence, we conclude the estimate (5.72).
The same computation hold for the set Q. Hence, we conclude the measure estimate (5.84).
Since the operator H ∞ 0 is block-diagonal and self-adjoint, we can verified that In this section, we recall some well known facts about self-adjoint operator in the finite dimension Hilbert space H. Let H be a finite dimensional Hilbert space of dimension n equipped by the inner product (, ) H . For any self-adjoint operator A, we order its eigenvalues as spec(A) := λ 1 (A) ≤ λ 2 (A) ≤ · · · ≤ λ n (A).