Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic tori

Sobolev norms explosion for the cubic NLS on irrational tori

“…Another evidence of obstructions to instability of NLS equations on irrational tori is provided by the works of Deng [15] and Deng-Germain [16], 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 [4] 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 [19], [18], [3], [13].…”

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

“…The estimate (1) means that, to control the growth of the H s norm for very long times, the initial datum only needs to be small in H s 0 . In particular, as recently highlighted in [FM22], contrary to what is usually assumed, it does not have to be small in H s (moreover note that we do not have to assume that s ≥ s 0 ).…”

Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori

“…On the other hand for the case of higher dimensional manifolds only particular examples are known [4,11,23,[26][27][28]33] and for PDEs in higher space dimension with unbounded perturbations only partial results have been obtained [31,35,43]. A slightly different point of view is the one developed in [46] in which the authors give some upper bounds on the possible energy transfer to high modes, for initial data Fourier supported in a box for the cubic NLS on the irrational square torus in dimension two.…”

“…, | j r | goes to infinity), and this could in principle create a loss of derivatives in the construction of the map used to put the system in Birkhoff normal form. We refer for instance to [11,35,43] (and reference therein) and where this problem is dealt with to prove partial long time stability results, by imposing (1.4) for small r (say r = 3, 4). By partial results we mean that, in the latter papers, the time scales of stability are of order at most −q with a strong limitation on q ≤ 4, and they left open the case q large.…”