On the Normal Form of the Kirchhoff Equation
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On the Normal Form of the Kirchhoff Equation Pietro Baldi1 · Emanuele Haus2 In memory of Walter Craig Received: 1 June 2020 / Revised: 8 September 2020 / Accepted: 10 October 2020 © The Author(s) 2020
Abstract Consider the Kirchhoff equation ∂tt u − u 1 +
Td
|∇u|2 = 0
Td .
on the d-dimensional torus 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. Keywords Kirchhoff equation · Quasilinear wave equations · Quasilinear normal forms Mathematics Subject Classification 35L72 · 35Q74 · 37J40 · 70K45
1 Introduction We consider the Kirchhoff equation on the d-dimensional torus Td , T := R/2πZ (periodic boundary conditions) |∇u|2 d x = 0. (1.1) ∂tt u − u 1 + Td
B
Emanuele Haus [email protected] Pietro Baldi [email protected]
1
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Naples, Italy
2
Dipartimento di Matematica e Fisica, Università Roma Tre, Largo San Leonardo Murialdo 1, 00146 Rome, Italy
123
Journal of Dynamics and Differential Equations
Equation (1.1) is a quasilinear wave equation, and it has the structure of a Hamiltonian system ∂t u = ∇v H (u, v) = v, (1.2) ∂t v = −∇u H (u, v) = u 1 + Td |∇u|2 d x , where the Hamiltonian is H (u, v) =
1 2
Td
v2 d x +
1 2
Td
|∇u|2 d x +
1 2
Td
|∇u|2 d x
and ∇u H , ∇v H are the gradients with respect to the real scalar product f , g := f (x)g(x) d x ∀ f , g ∈ L 2 (Td , R), Td
2
,
(1.3)
(1.4)
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 0 J= −1
1 . 0
(1.6)
The Cauchy problem for the Kirchhoff equation is given by (1.1) with initial data at time t =0 u(0, x) = α(x), ∂t u(0, x) = β(x). (1.7) Such a Cauchy problem is known to be locally wellposed in time for initial data (α, β) in 3 1 the Sobolev space H 2 (Td )× H 2 (Td ) (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 wellposedness 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 wellposedness for the Cauchy problem (1.1)–(1.7) with periodic boundary conditions (or with Dirichlet boundary conditions on bounded domains of Rd ) has given rise to a longstanding open problem: while it h
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