Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

Abstract

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $M_{p,q}^s(\mathbb{R})$ where $1\le q\le 2$, $2\le p<\frac{10q^{\prime }}{q^{\prime }+6}$ and $s\ge 0$. Moreover, for either $1\le q\le \frac{3}{2},s\ge 0$ and $2\le p\le 3$ or $\frac{3}{2}<q\le \frac{18}{11},s>\frac{2}{3}-\frac{1}{q}$ and $2\le p\le 3$ or $\frac{18}{11}<q\le 2,s>\frac{2}{3}-\frac{1}{q}$ and $2\le p<\frac{10q^{\prime }}{q^{\prime }+6}$ 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 $p\ne 2$.

arXiv commentWrong statements and claims of the previous version have been fixed, unconditional wellposedness result has been added and the reference list has been expanded. arXiv admin note: substantial text overlap with arXiv:1802.08274

References in corpus (2)

• Local well-posedness of nonlinear dispersive equations on modulation spaces
• NLS in the modulation space $M_{2,q}(\mathbb{R})$

Cited by in corpus (7)

• On smoothing estimates in modulation spaces and the nonlinear Schrödinger equation with slowly decaying initial data
• Unconditional uniqueness of solutions for nonlinear dispersive equations
• Knocking out teeth in one-dimensional periodic NLS
• Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces $M_{p,q}^s({\mathbb{R}}^d)\cap M_{\infty ,1}({\mathbb{R}}^d)$
• Nonlinear wave equations with slowly decaying initial data
• Global well-posedness and equicontinuity for mKdV in modulation spaces
• Unconditional uniqueness of higher order nonlinear Schrödinger equations