Global well-posedness for the cubic fractional NLS on the unit disk

Abstract

In this paper, we prove that the cubic nonlinear Schrödinger equation with the fractional Laplacian on the unit disk is globally well-posed for certain radial initial data below the energy space. The result is proved by extending the I-method in the fractional nonlinear Schrödinger equation setting.

Global well-posedness and long-time behavior of the fractional NLS

“…In this section, we present a deterministic local well-posedness result for σ ∈ [ 1 2 , 1] in (1.1), which heavily replies on a bilinear Strichartz estimate obtained in Subsection 5.1. See also Section 3 in [51]. We also show a convergence from Galerkin approximations of FNLS to FNLS.…”

“…On the circle, Gibbs measures were constructed in [46] and the dynamics on full measure sets with respect to these measures was studied. The authors of the present paper proved global well-posedness below the energy space for FNLS posed on the unit disc by extending the I-method (introduced by Colliander, Keel, Staffilani, Takaoka and Tao [14]) to the fractional context [51]. 1 Here weak turbulence refers to an energy cascade from low to high frequencies as time evolves.…”