Nonlinear Schroedinger equations with radially symmetric data of critical regularity

TL;DR

This paper proves the global existence of small solutions for radially symmetric nonlinear Schrödinger equations with critical regularity initial data, using weighted smoothing estimates to handle negative-order Sobolev spaces.

Contribution

It establishes global solutions for critical regularity data with radial symmetry, extending previous results to negative-order Sobolev spaces.

Findings

Global existence of small solutions under radial symmetry.

Effective use of weighted smoothing estimates.

Solutions exist despite initial data having negative-order differentiability.

Abstract

This paper is concerned with the global existence of small solutions to pure-power nonlinear Schroedinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where the power of nonlinearity is somewhat smaller than the pseudoconformal power and the initial data belong to the scale-invariant homogeneous Sobolev space. In spite of the negative-order differentiability of initial data the nonlinear Schroedinger equation has global in time solutions provided that the initial data have the small norm. The key ingredient in the proof of this result is an effective use of global weighted smoothing estimates specific to radially symmetric solutions.