On the Normal Form of the Kirchhoff Equation

Consider the Kirchhoff equation ∂ttu-Δu(1+∫Td|∇u|2)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{tt} u - \Delta u \Big ( 1 + \int _{\mathbb {T}^d} |\nabla u|^2 \Big ) = 0 \end{aligned}$$\end{document}on the d-dimensional torus Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}^d$$\end{document}. In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.


Introduction
We consider the Kirchhoff  where the Hamiltonian is and ∇ u H, ∇ v H are the gradients with respect to the real scalar product namely H ′ (u, v)[f, g] = ∇ u H(u, v), f + ∇ v H(u, v), g for all u, v, f, g. More compactly, (1.2) is ∂ t w = J∇H(w), (1.5) where w = (u, v), ∇H = (∇ u H, ∇ v H) and The Cauchy problem for the Kirchhoff equation is given by (1.1) with initial data at time t = 0 u(0, x) = α(x), u t (0, x) = β(x). (1.7) Such a Cauchy problem is known to be locally well posed in time for initial data (α, β) in the Sobolev space H 3 2 (T d ) × H 1 2 (T d ) (see the work of Dickey [18]). However, the conserved Hamiltonian (1.3) only controls the H 1 × L 2 norm of the couple (u, v). Since the local well-posedness has only been established in regularity higher than the energy space H 1 × L 2 , it is not trivial to determine whether the solutions are global in time. In fact, the question of global well-posedness for the Cauchy problem (1.1)-(1.7) with periodic boundary conditions (or with Dirichlet boundary conditions on bounded domains of R d ) has given rise to a long-standing open problem: while it has been known for eighty years, since the pioneering work of Bernstein [7], that analytic initial data produce global-in-time solutions, it is still unknown whether the same is true for C ∞ initial data, even of small amplitude.
For initial data of amplitude ε, the linear theory immediately gives existence of the solution over a time interval of the order of ε −2 . In [4], we performed one step of quasilinear normal form and established a longer existence time, of the order of ε −4 ; indeed, all the cubic terms giving a nontrivial contribution to the energy estimates are erased by the normal form. One may wonder whether the same type of mechanism works also for (one or more) subsequent steps of normal form.
In this paper, we give a negative answer to such a question, as we explicitly compute the second step of normal form for the Kirchhoff equation on T d , erasing all the nonresonant terms of degree five. It turns out that, differently from what happens for cubic terms, the contribution to the energy estimates of the resonant terms of degree five is different from zero. This, of course, leaves open the question whether for small amplitude initial data the time of existence can be extended beyond the lifespan ∼ ε −4 (partial results in this direction are in preparation [5]). The presence of resonant terms of degree five that give a nontrivial contribution to the energy estimates can, however, be interpreted as a sign of non-integrability of the equation. Another interesting open question is whether these "non-integrable" terms in the normal form can somehow be used to construct "weakly turbulent" solutions pushing energy from low to high Fourier modes, in the spirit of the works [11], [24], [25], [23], [22] for the semilinear Schrödinger equation on T 2 . Proving existence of such solutions may be a very hard task, but one may at least hope to use the normal form that we compute in this paper to detect some genuinely nonlinear behavior of the flow, over long time-scales (as in [20], [27]) or even for all times (as in [26]).

Main result
To give a precise statement of our main result, we introduce here the functional setting. Function space. On the torus T d , it is not restrictive to set the problem in the space of functions with zero average in space, for the following reason. Given initial data α(x), β(x), we split both them and the unknown u(t, x) into the sum of a zero-mean function and the average term, α(x) = α 0 +α(x), β(x) = β 0 +β(x), u(t, x) = u 0 (t) +ũ(t, x), Then the Cauchy problem (1.1)-(1.7) splits into two distinct, uncoupled Cauchy problems: one is the problem for the average u 0 (t), which is and has the unique solution u 0 (t) = α 0 +β 0 t; the other one is the problem for the zero-mean componentũ(t, x), which is Thus one has to study the Cauchy problem for the zero-mean unknownũ(t, x) with zeromean initial dataα(x),β(x); this means to study (1.1)-(1.7) in the class of functions with zero average in x.
For any real s ≥ 0, we consider the Sobolev space of zero-mean functions and its subspace of real-valued functions u, for which the complex conjugates of the Fourier coefficients satisfy u j = u −j . For s = 0, we write L 2 0 instead of H 0 0 the space of square-integrable functions with zero average.
In this paper we prove the following normal form result.
Theorem 1.1. There exists δ > 0 and a map Φ : B m 1 sym (δ) → B m 1 (2δ), "close to identity" (see Remark 1.2), injective and conjugating system (1.2) to (see (5.5)-(5.6) and the whole Section 5 for the precise definition of W ). The transformation Φ maps B s sym (δ) to B s (2δ) for all s ≥ m 1 . The transformed vector field W is the sum of linear terms, cubic terms, quintic terms, and a remainder of homogeneity ≥ 7. The linear and cubic terms of W give zero contribution to the energy estimates (i.e. the estimates for the time evolution of Sobolev norms), while the quintic terms give a nonzero contribution to the energy estimates (see (5.36)-(5.39)). Remark 1.2. In Section 2 we will introduce the transformations Φ (1) and Φ (2) , which symmetrize the system and introduce complex coordinates. These transformations are not close to identity. By saying that the map Φ is "close to identity" we mean that There is a certain similarity between our computation and the one performed by Craig and Worfolk [14] for the normal form of gravity water waves. In both cases one deals with an equation whose vector field is strongly unbounded (quasilinear here, fully nonlinear in [14]) and in both cases the first steps of normal form show an "integrable" behavior, while after few steps some genuinely non-integrable terms show up. However, there is an important difference: while the normal form computed in [14] is only the result of a formal computation, the transformation Φ that we construct here to put the Kirchhoff equation in normal form is a bounded transformation that is well defined between Sobolev spaces. This is obtained thanks to the "quasilinear symmetrization" performed in [4], following the strategy for quasilinear normal forms introduced by Delort in the papers [16]- [17] on quasilinear Klein-Gordon equation on T.

Related literature
Equation (1.1) was introduced by Kirchhoff [29] to model the transversal oscillations of a clamped string or plate, taking into account nonlinear elastic effects. The first results on the Cauchy problem (1.1)-(1.7) are due to Bernstein. In his 1940 pioneering paper [7], he studied the Cauchy problem on an interval, with Dirichlet boundary conditions, and proved global wellposedness for analytic initial data (α, β).
After that, the research on the Kirchhoff equation has been developed in various directions, with a different kind of results on compact domains (bounded subsets of R d with Dirichlet boundary conditions, or periodic boundary conditions T d ) or non compact domains (R d or "exterior domains" Ω = R d \ K, with K ⊂ R d compact domain).
On R d , Greenberg and Hu [21] in dimension d = 1 and D'Ancona and Spagnolo [15] in higher dimension proved global wellposedness with scattering for small initial data in weighted Sobolev spaces.
On compact domains, dispersion, scattering and time-decay mechanisms are not available, and there are no results of global existence, nor of finite time blowup, for initial data (α, β) of Sobolev, or C ∞ , or Gevrey regularity. The local wellposedness in the Sobolev class H 3 2 × H 1 2 has been proved by Dickey [18] (see also Arosio and Panizzi [2]), Beyond the question about the global wellposedness for small data in Sobolev class, another open question concerns the local wellposedness in the energy space H 1 × L 2 or in H s × H s−1 for 1 < s < 3 2 . We also mention the recent results [3], [32], [12], which prove the existence of time periodic or quasi-periodic solutions of time periodically or quasi-periodically forced Kirchhoff equations on T d , using Nash-Moser and KAM techniques.
For more details, generalizations and other open questions, we refer to Lions [30], to the surveys of Arosio [1], Spagnolo [33], Matsuyama and Ruzhansky [31], and to other references in our previous paper [4].

Linear transformations
We start by recalling the first standard transformations in [4], which transforms system (1.2) into another one (see (2.6)) where the linear part is diagonal, preserving both the real and the Hamiltonian structure of the problem. These standard transformations are the symmetrization of the highest order and then the diagonalization of the linear terms.
Symmetrization of the highest order. In the Sobolev spaces (1.8) of zero-mean functions, the Fourier multiplier To symmetrize the system at the highest order, we consider the linear, symplectic trans- which is the Hamiltonian system ∂ t (q, p) = J∇H (1) (q, p) with Hamiltonian H (1) = H •Φ (1) , namely The original problem requires the "physical" variables (u, v) to be real-valued; this corresponds to (q, p) being real-valued too. Also note that Λ System (2.3) and the Hamiltonian H (1) (q, p) in (2.4) are also meaningful, without any change, for complex functions q, p. Thus we define the change of complex variables (q, p) = Φ (2) where the pairing ·, · denotes the integral of the product of any two complex functions The map Φ (2) : of pairs of complex functions. When (q, p) are real, (f, g) are complex conjugate. The restriction of Φ (2) to the space of pairs of complex conjugate functions is an R-linear isomorphism onto the space L 2 0 (T d , R)× L 2 0 (T d , R) of pairs of real functions. For g = f , the second equation in (2.6) is redundant, being the complex conjugate of the first equation. In other words, system (2.6) has the following "real structure": it is of the form where the vector field F(f, g) satisfies Under the transformation Φ (2) , the Hamiltonian system (2.3) for complex variables (q, p) becomes (2.6), which is the Hamiltonian where J is defined in (2.4), ·, · is defined in (2.7), and ∇H (2) is the gradient with respect to ·, · . System (2.3) for real (q, p) (which corresponds to the original Kirchhoff equation) becomes system (2.6) restricted to the subspace To complete the definition of the function spaces, for any real s ≥ 0 we define

Diagonalization of the order one
In [4] (Section 3) the following global transformation Φ (3) is constructed. Its effect is to remove the unbounded operator Λ from the "off-diagonal" terms of the equation, namely those terms coupling f andf .
ρ is the function

4)
and ϕ is the inverse of the function Then, for all real s ≥ 1 2 , the nonlinear map In [4] it is proved that system (2.6), under the change of variable (f, (3.6) Note that system (3.6) is diagonal at the order one, i.e. the coupling of η and ψ (except for the coefficients) is confined to terms of order zero. Also note that the coefficients of (3.6) are finite for η, ψ ∈ H 1 0 , while the coefficients in (2.6) are finite for f, g ∈ H 1 2 0 : the regularity threshold of the transformed system is 1 2 higher than before. The real structure is preserved, namely the second equation in (3.6) is the complex conjugate of the first one, or, in other words, the vector field in (3.6) satisfies property (2.8).
Quintic terms. By Taylor's expansion, The transformed Hamiltonian. Even if Φ (3) is not symplectic, nonetheless it could be useful to calculate the transformed Hamiltonian, because it is still a prime integral of the equation. By definition (3.3), one has

Normal form: first step
The next step is the cancellation of the cubic terms contributing to the energy estimate. Following [4], we write (3.6) as and R ≥5 (η, ψ) is the bounded remainder of higher homogeneity degree In [4] the term B 3 (and not D ≥3 , as it gives no contribution to the energy estimate) is removed by the following normal form transformation. Let where A 12 , C 12 are the bilinear maps .7), (4.8), (4.9). For all complex functions u, v, h, all real s ≥ 0, where M (w, z) is defined in (4.6), and To estimate matrix operators and vectors in H s 0 (T d , c.c.), we define (w, z) s := w s = z s for every pair (w, z) = (w, w) of complex conjugate functions.
where m 0 is defined in (4.9). For w m 0 < 1 2 , the operator (I + K(w, z)) : The nonlinear, continuous map Φ (4) is invertible in a ball around the origin.
The terms (1 + P)D 1 and X + 3 in (4. 19) give no contributions to the energy estimate, because, as one can check directly, Similarly, also PX + 3 gives no contribution to the energy estimate, because  26) and, for w m 0 ≤ 1 2 , for all complex functions h, where R ≥5 is defined in (4.4) and C is a universal constant.
where C is a universal constant.
We analyze the terms in (4.31). By (4.11), (4.12), the first component of K(w, z)X + 3 (w, z) is and its second component is the conjugate of the first one. Recalling (4.3), the first component of the last term in (4.31) is In Fourier series, with all indices in Z d \ {0}, one has Thus the first component of the quintic term X + 5 (w, z) is and, in Fourier series, Notation. In the coefficients of the vector field X + 5 there appear several denominators, which imply the corresponding restrictions on the indices j, k, ℓ to prevent the denominators from vanishing. From now on, we will stop indicating explicitly the restrictions on the indices in summations and adopt instead the convention 0/0 = 0 in the coefficients. For instance, instead of j,k,ℓ |k| =|j| In this example, when |j| = |k| the denominator of the coefficient vanishes; the numerator also vanishes because of the factor (1 − δ |k| |j| ); this has to be interpreted as We collect similar monomials, and we get that (X + 5 (w, z)) 1 is the sum of the following eight terms:

Normal form: second step
We consider a transformation of the form where M(u, v) is a matrix operator of homogeneity degree 4. In particular, and similarly for the other terms and for M(u, v). We assume that for all u, v, u (n) , v (n) , n = 1, 2, 3, 4. We also assume that (1) , u (2) , v (1) , v (2) −ℓ h k c 11 (j, ℓ, k) e ik·x for some coefficient c 11 (j, ℓ, k) to be determined, and similarly for all the other terms. One has and The transformed equation is where W (u, v) := (I + K(u, v)) −1 X + (Φ (5) (u, v)).

(5.5)
Recalling (4.32), we decompose where (1 + P(Φ (5) ))(D 1 + X + 3 ) give no contribution to the energy estimate, and W ≥7 (u, v) is defined by difference and contains only terms of homogeneity at least seven in (u, v). We calculate each term of the first component (W 5 ) 1 of W 5 . First, one has Thus the terms in ( Hence we choose a 11 (j, ℓ, k) := |j| 2 |ℓ| 2 128(|j| + |ℓ|) does not contain monomials of the type u j u −j u ℓ u −ℓ u k e ik·x . Next, since (X + 5 ) 1 (u, v) does not contain monomials u j u −j u ℓ v −ℓ u k e ik·x , we fix so that (W 5 ) 1 (u, v) also does not contain such monomials.

Hence we fix
|k| − |j| − |ℓ| , (5.13) and the terms in ( Next, the terms in ( Hence we fix 14) and the terms in (W 5 ) 1 (u, v) containing the monomials u j u −j u ℓ v −ℓ v k e ik·x become j,ℓ,k |j|=|k| Next, the terms in ( Hence we fix 15) and the terms in ( Next, the terms in ( Hence we fix Next, the terms in ( Hence we fix Summarizing, it remains With similar calculations, or deducing the formula from the real structure, the second (5.19) where C is a universal constant.
Remark 5.3. The bound |p| ≥ 1 in the proof of Lemma 5.2 is sharp. Indeed, it is enough to show that there are infinitely many choices of k, j, ℓ ∈ Z d \ {0} such that the triple (|k| 2 , |j| 2 , |ℓ| 2 ) is of the form (n, n + 1, 4n + 2) for some n ∈ N. In dimension d ≥ 3, this is trivial.
In dimension d = 2, recall that the set of integers that can be written as the sum of two squares is closed under multiplication, by Brahmagupta's identity (x 2 + y 2 )(z 2 + w 2 ) = (xz + yw) 2 + (xw − yz) 2 .
All the operators G ∈ {A 11 , C 11 , F 11 , A 12 , B 12 , C 12 , D 12 , F 12 } satisfy for all complex functions u, v, w, z, h, all real s ≥ 0, where C is a universal constant.
Proof. It is an immediate consequence of Lemma 5.4.
where m 1 is defined in (5.29) and C is a universal constant. There exists a universal δ > 0 such that, for u m 1 < δ, the operator (I + K(u, v)) : satisfying for all s ≥ 0.
The nonlinear, continuous map Φ (5) is invertible in a ball around the origin.
Lemma 5.7. There exists a universal constant δ > 0 such that, for all (w, If, in addition, w ∈ H s 0 for some s > m 1 , then u also belongs to H s 0 , and u s ≤ 2 w s . This defines the continuous inverse map (Φ (5) ) −1 : Proof. Using the estimates of Lemma 5.6, the proof of Lemma 5.7 is a straightforward adaptation of the proof of Lemma 4.3 in [4].
Spheres in Fourier space. We observe that the system (or some relevant aspects of it concerning the evolution of Sobolev norms) can be described by taking sums over all frequencies k ∈ Z d with a fixed (Euclidean) length |k| = λ. For each λ ∈ Γ, S λ ≥ 0 and B λ ∈ C. By (5.21)-(5.22), neglecting the terms from W ≥7 , one has ∂ t S λ = 3i 32 α,β∈Γ α+β=λ Equations (5.41)-(5.42) form a closed system in the variables (S λ , B λ ) λ∈Γ . They play the role of an "effective equation" for the dynamics of the Kirchhoff equation. This will be the starting point for further analysis in the forthcoming paper [5].