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

We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension d≥1. We study the behavior of high Sobolev norms Hs, s≫1, of solutions with initial conditions in Hs whose Hρ-Sobolev norm, 1≪ρ≪s, is smaller than ε≪1. We provide a control of the Hs-norm over a time interval of order O(ε−2). Due to the lack of conserved quantities controlling high Sobolev norms, the key ingredient of the proof is the construction of a modified energy equivalent to the “low norm” Hρ (when ρ is sufficiently high) over a nontrivial time interval O(ε−2). This is achieved by means of normal form techniques for quasi-linear equations involving para-differential calculus. The main difficulty is to control the possible loss of derivatives due to the small divisors arising form three waves interactions. By performing “tame” energy estimates we obtain upper bounds for higher Sobolev norms Hs.