Breakdown of Regularity of Scattering for Mass-Subcritical NLS
Keyword(s):
Abstract We study the scattering problem for the nonlinear Schrödinger equation $i\partial _t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $\Sigma$ holds and the wave operator is well defined on $\Sigma$. We show that there exists $0<\beta <p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+\beta }$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology.
2021 ◽
Vol 382
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pp. 87-121
2012 ◽
Vol 219
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pp. 3187-3197
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2020 ◽
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pp. 124188
2018 ◽
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pp. 1405-1419
2012 ◽
Vol 50
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pp. 492-521
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