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

22 Apr 2018  ·  Chaichenets Leonid, Hundertmark Dirk, Kunstmann Peer, Pattakos Nikolaos ·

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 $\frac32<q\leq\frac{18}{11}, s>\frac23-\frac1{q}$ and $2\leq p\leq 3$ or $\frac{18}{11}<q\leq2, s>\frac23-\frac1{q}$ 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 $p\neq2$. read more

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Analysis of PDEs