On the unconditional uniqueness for NLS in $H^s$

TL;DR

This paper establishes the unconditional uniqueness of solutions in the homogeneous Sobolev space ^s for the nonlinear Schrf6dinger equation with power nonlinearity, extending known results to new cases using advanced analytical techniques.

Contribution

It provides a unified proof of unconditional uniqueness for ^s solutions in both subcritical and critical cases, including previously unresolved scenarios.

Findings

Unified proof covering subcritical and critical cases

Extension of uniqueness results to new parameter ranges

Use of negative order Sobolev and Besov spaces in analysis

Abstract

In this article, we study the unconditional uniqueness of $\dot H^s$, $0<s< 1$, solutions for the nonlinear Schr\"odinger equation $i\partial_t u +\Delta u+ c |u|^\alpha u=0$ in ${\mathbb R}^n$. We give a unified proof of the previously known results in the subcritical cases and critical cases, and we also extend these results to some previously unsettled cases. Our proof uses in particular negative order Sobolev spaces (or Besov spaces), general Strichartz estimates, and the improved regularity property for the difference of two solutions.