Local and global well-posedness in $L^{2}(\mathbb R^{n})$ for the inhomogeneous nonlinear Schr\"{o}dinger equation

Local and global well-posedness in $L^{2}(\mathbb R^{n})$ for the inhomogeneous nonlinear Schr\"{o}dinger equation

This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $iu_{t} +\Delta u=\lambda \left|x\right|^{-b} \left|u\right|^{\sigma } u, u(0)=u_{0} \in L^{2}(\mathbb R^{n})$, where $\lambda \in \mathbb C$, $0

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Hauptverfasser: An, JinMyong, Kim, JinMyong
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description This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $iu_{t} +\Delta u=\lambda \left|x\right|^{-b} \left|u\right|^{\sigma } u, u(0)=u_{0} \in L^{2}(\mathbb R^{n})$, where $\lambda \in \mathbb C$, $0
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We prove the local well-posedness and small data global well-posedness of the INLS equation in the mass-critical case $\sigma =\frac{4-2b}{n} $, which have remained open until now. We also obtain some local well-posedness results in the mass-subcritical case $\sigma &lt;\frac{4-2b}{n} $. In order to obtain the results above, we establish the Strichartz estimates in Lorentz spaces and use the contraction mapping principle based on Strichartz estimates.</description><identifier>DOI: 10.48550/arxiv.2107.00790</identifier><language>eng</language><subject>Mathematics - Analysis of PDEs</subject><creationdate>2021-07</creationdate><rights>http://creativecommons.org/licenses/by/4.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2107.00790$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2107.00790$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>An, JinMyong</creatorcontrib><creatorcontrib>Kim, JinMyong</creatorcontrib><title>Local and global well-posedness in $L^{2}(\mathbb R^{n})$ for the inhomogeneous nonlinear Schr\"{o}dinger equation</title><description>This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $iu_{t} +\Delta u=\lambda \left|x\right|^{-b} \left|u\right|^{\sigma } u, u(0)=u_{0} \in L^{2}(\mathbb R^{n})$, where $\lambda \in \mathbb C$, $0&lt;b&lt;\min \left\{2,{\rm \; }n\right\}$ and $0&lt;\sigma \le \frac{4-2b}{n} $. We prove the local well-posedness and small data global well-posedness of the INLS equation in the mass-critical case $\sigma =\frac{4-2b}{n} $, which have remained open until now. We also obtain some local well-posedness results in the mass-subcritical case $\sigma &lt;\frac{4-2b}{n} $. 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We prove the local well-posedness and small data global well-posedness of the INLS equation in the mass-critical case $\sigma =\frac{4-2b}{n} $, which have remained open until now. We also obtain some local well-posedness results in the mass-subcritical case $\sigma &lt;\frac{4-2b}{n} $. In order to obtain the results above, we establish the Strichartz estimates in Lorentz spaces and use the contraction mapping principle based on Strichartz estimates.</abstract><doi>10.48550/arxiv.2107.00790</doi><oa>free_for_read</oa></addata></record>
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title Local and global well-posedness in $L^{2}(\mathbb R^{n})$ for the inhomogeneous nonlinear Schr\"{o}dinger equation
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