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arXiv · 1904.04030

On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R}) + H^1(\mathbb{T})$

Leonid Chaichenets·Dirk Hundertmark·Peer Christian Kunstmann·Nikolaos Pattakos
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Abstract

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $|u|^{α- 1} u$ for $α\in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($α= 2$) we obtain global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb R) + H^1(\mathbb T))$ via Gronwall's inequality.

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Leonid Chaichenets, Dirk Hundertmark, Peer Christian Kunstmann, Nikolaos Pattakos. 2019-10-10. On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R}) + H^1(\mathbb{T})$. https://doi.org/10.1007/s00030-020-00670-8

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