Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

Abstract

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n − 1 with n ≥ 3 and d ≥ 2. If ε ≪ 1 is the size of the initial datum, we prove that the lifespan Tε of solutions is O(ε −A(n−2) −) where A ≡ A(d, n) = 1 + 3 d−1 when n is even and A = 1 + 3 d−1 + max(4−d d−1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain Tε = O(ε −6 −), much better than O(ε −1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of √ ∆ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step. Contents 1. Introduction 1 2. Small divisors 9 3. The Birkhoff normal form step 19 4. The modified energy step 30 5. Proof of Theorem 1 33 References 36 2010 Mathematics Subject Classification. 35Q35, 35Q53, 37K55. Key words and phrases. Lifespan for semi-linear PDEs, Birkhoff normal forms, modified energy, irrational torus. Felice Iandoli has been supported by ERC grant ANADEL 757996. Roberto Feola, Joackim Bernier and Benoit Grébert have been supported by the Centre Henri Lebesgue ANR-11-LABX-0020-01 and by ANR-15-CE40-0001-02 "BEKAM" of the ANR.

Long-time stability of the quantum hydrodynamic system on irrational tori

“…If σ 1 j 1 + σ 2 j 2 + σ 3 j 3 = 0, σ i ∈ {±1} we have the estimate The proof of this proposition is divided in several steps and it is postponed to the end of the section. The main ingredient is the following standard proposition which follows the lines of [3,6]. Here we give weak lower bounds of the small divisors, these estimates will be improved later.…”

“…Unfortunately we have also a power of the highest frequency µ 1 which represents, in principle, a loss of derivatives. However, this loss of derivatives may be transformed in a loss of length of the lifespan through partition of frequencies, as done for instance in [10,17,12,6].…”

Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity

“…We point out that the gain between the standard small divisor estimates (6) and the new ones (8) is huge: we have replaced polynomial losses of derivatives by logarithmic ones. Moreover, using technics inspired by [Del09,BFGI21], these logarithms losses can almost be considered as constants. So it means that we have no losses in our small divisor estimates.…”

Sobolev norms explosion for the cubic NLS on irrational tori

“…It is reasonable then to argue that the more complicated the resonant structure of a system is, the richer should be its resonant dynamics. This is why in general we expect to be more difficult to appreciate instability phenomena for equations on 1-dimensional spatial domains 1 . In higher dimension a similar situation occurs when we consider irrational tori, namely…”

“…Another evidence of obstructions to instability of NLS equations on irrational tori is provided by the works of Deng [10] and Deng-Germain [11], where the authors prove that on 3-dimensional irrational tori the polynomial in time upper bounds for the growth of Sobolev norms have a smaller degree compared to the rational case. We also mention [2] for polynomial time estimates on the growth of Sobolev norms of solutions of linear Schrödinger equation with time-dependent potential on irrational tori and recent works of long time stability in Sobolev spaces, based on normal form methods [13], [12], [1], [9].…”