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arXiv · 2404.15775

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

Ryosuke Hyakuna
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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 $s$.

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Ryosuke Hyakuna. 2024-04-24. Unconditional well-posedness for the nonlinear Schr\"odinger equation in Bessel potential spaces. https://arxiv.org/abs/2404.15775

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