Polynomial growth of Sobolev norms of solutions of the fractional NLS equation on \T^d

Abstract

In this paper, we prove polynomial growth bounds for the Sobolev norms of solutions to the fractional nonlinear Schrödinger equation on the torus \T^d (d \ge 2), following and extending a result of Joseph Thirouin on \T [Thi17]. The key ingredient is the establishment of Strichartz estimates for the fractional Schrödinger equation on \T^d. To this end, we employ uniform estimates for oscillatory integrals to overcome the lack of uniformity that arises in higher dimensions.

32 pages