Unconditional well-posedness for the nonlinear Schr\"odinger equation in Bessel potential spaces

TL;DR

This paper redefines unconditional well-posedness for the nonlinear Schrödinger equation to include $L^p$-based Sobolev spaces and proves it for 1D cubic NLS in Bessel potential spaces $H^s_p$ for specific $p$ and $s$.

Contribution

It introduces a new definition of unconditional well-posedness applicable to $L^p$-based Sobolev spaces and establishes this property for the 1D cubic NLS in Bessel potential spaces.

Findings

Unconditional well-posedness is extended to $L^p$-based Sobolev spaces.

Proves unconditional well-posedness for 1D cubic NLS in $H^s_p$ spaces.

Valid for $4/3<p extless=2$ with certain regularity conditions.

Abstract

The Cauchy problem for the nonlinear Schr\"odinger equation is called unconditionally well posed in a data space $E$ if it is well posed in the usual sense and the solution is unique in the space $C([0,T]; E)$. In this paper, this notion of unconditional well-posedness is redefined so that it covers $L^p$-based Sobolev spaces as data space $E$ and it is equivalent to the usual one when $E$ is an $L^2$-based Sobolev space $H^s$. Next, based on this definition, it is shown that the Cauchy problem for the 1D cubic NLS is unconditionally well posed in Bessel potential spaces $H^s_p$ for $4/3<p\le 2$ under certain regularity assumptions on $s$.