On the normal form of the Kirchhoff equation

Abstract

Consider the Kirchhoff equationIn 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.

Longer lifespan for many solutions of the Kirchhoff equation

“…In other words, the solutions of such an effective system have essentially the same behavior in dimension 1 or higher. (The only thing that changes substantially with the dimension regards the regularity required by the normal forms, because denominators like |k| − |j|, |k| + |j| − |ℓ|, k, j, ℓ ∈ Z d , accumulates to zero if d ≥ 2, while they are nonzero integers in dimension d = 1; see [2], [3] for more details).…”

“…We start from the normal form of degree five, computed in the previous paper [3]. The first remark is that the time evolution of the Sobolev norms of solutions of (1.1) is fully described by the evolution of the "superactions" S λ in (2.2).…”

“…In [2]- [3] we proved that there exists a change of variable Φ = Φ (1) • • • • • Φ (5) (a bounded quasilinear normal form transformation) that transforms the Cauchy problem (1.1), (1.2) for the Kirchhoff equation into the problem…”

“…In the recent paper [3], we computed the second step of quasilinear normal form for the Kirchhoff equation and showed that there are resonant terms of degree five that cannot be erased and give a nontrivial contribution to the time evolution of Sobolev norms. Here we show that for a suitable set of "nonresonant" initial data the effect of these terms can be neglected on a longer timescale and the lifespan of the corresponding solution is at least ε −6 (Theorem 1.1 below).…”

“…The quasilinear normal form performed in [2]- [3] (summarized in the appendix below), which is particularly simple because of the "already paralinearized" structure of the Kirchhoff equation, overcomes the problem that the standard Birkhoff normal form construction, even at its first step, gives unbounded transformations. However, employing the quasilinear normal form to deduce a longer existence time for the Cauchy problem presents all the other mentioned difficulties.…”