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arXiv · 1802.10464

Nonlinear Schr\"odinger equation, differentiation by parts and modulation spaces

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

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the modulation space $M_{p,q}^{s}(\mathbb R)$ where $1\leq q\leq2$, $2\leq p<\frac{10q'}{q'+6}$ and $s\geq0$. Moreover, for either $1\leq q\leq\frac32, s\geq0$ and $2\leq p\leq 3$ or $\frac32 \frac23-\frac1{q}$ and $2\leq p\leq 3$ or $\frac{18}{11} \frac23-\frac1{q}$ and $2\leq p<\frac{10q'}{q'+6}$ we show that the Cauchy problem is unconditionally wellposed in $M_{p,q}^{s}(\mathbb R).$ This improves \cite{NP}, where the case $p=2$ was considered and the differentiation by parts technique was introduced to a problem with continuous Fourier variable. Here the same technique is used, but more delicate estimates are necessary for $p\neq2$.

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Leonid Chaichenets, Dirk Hundertmark, Peer Kunstmann, Nikolaos Pattakos. 2018-02-27. Nonlinear Schr\"odinger equation, differentiation by parts and modulation spaces. https://doi.org/10.1007/s00028-019-00501-z

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