On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schrödinger equation on the torus:i∂tu=|D|αu+|u|2u,u(0,⋅)=u0, where α is a real parameter. We show that, apart from the case α=1, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.