On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori

We consider a completely resonant nonlinear Schrödinger equation on the d-dimensional torus, for any d≥1, with polynomial nonlinearity of any degree 2p+1, p≥1, which is gauge and translation invariant. We study the behaviour of high Sobolev Hs-norms of solutions, s≥s1+1>d/2+2, whose initial datum u0∈Hs satisfies an appropriate smallness condition on its low Hs1 and L2-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm Hs over finite but long time scale that is exponential in the regularity parameter s1. As a byproduct we get stability of the low Hs1-norm over such time interval. A key ingredient in the proof is the introduction of a suitable “modified energy” that provides an a priori upper bound on the growth. This is obtained by combining para-differential techniques and suitable tame estimates.