Quadratic lifespan and growth of Sobolev norms for derivative Schr\"odinger equations on generic tori

TL;DR

This paper studies the growth of high Sobolev norms for derivative Schrödinger equations on generic tori, establishing control over these norms for long times using normal form and para-differential techniques.

Contribution

It introduces a novel method to control high Sobolev norms of solutions to derivative Schrödinger equations on generic tori over long times without conserved quantities.

Findings

Controlled the growth of Sobolev norms over time O(ε^{-2})

Constructed a modified energy equivalent to low norms

Applied normal form and para-differential calculus to handle small divisors

Abstract

We consider a family of Schr\"odinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension $d\geq1$. We study the behaviour of high Sobolev norms $H^{s}$, $s\gg1$, of solutions with initial conditions in $H^{s}$ whose $H^{\rho}$-Sobolev norm, $1\ll\rho\ll s$, is smaller than $\e\ll1$. We provide a control of the $H^{s}$-norm over a time interval of order $O(\e^{-2})$. %where $\e\ll1$ is the size of the initial condition in $H^{\rho}$. 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" $H^{\rho}$ (when $\rho$ is sufficiently high) over a nontrivial time interval $O(\e^{-2})$. 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 $H^{s}$.