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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2012.14356 |
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| _version_ | 1866915118042841088 |
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| author | Soffer, Avy Wu, Xiaoxu |
| author_facet | Soffer, Avy Wu, Xiaoxu |
| contents | This paper establishes the $L^p$ boundedness of wave operators for linear Schrödinger equations in $\mathbb{R}^3$ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application based on this method, combined with Strichartz estimates is the existence and scattering for nonlinear dispersive equations. For example, we prove global existence and uniform boundedness in $L^{\infty}$, for a class of Hartree nonlinear Schrödinger equations in $L^2(\mathbb{R}^3),$ allowing the presence of solitons. We also prove the existence of free channel wave operators in $L^p(\mathbb{R}^n)$ for $p>p_c(n)$, with $p_c(3)=6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_14356 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | L^p Boundedness of the Scattering Wave Operators of Schroedinger Dynamics with Time-dependent Potentials and applications Soffer, Avy Wu, Xiaoxu Analysis of PDEs 35P25, 35Q55, 47A40 This paper establishes the $L^p$ boundedness of wave operators for linear Schrödinger equations in $\mathbb{R}^3$ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application based on this method, combined with Strichartz estimates is the existence and scattering for nonlinear dispersive equations. For example, we prove global existence and uniform boundedness in $L^{\infty}$, for a class of Hartree nonlinear Schrödinger equations in $L^2(\mathbb{R}^3),$ allowing the presence of solitons. We also prove the existence of free channel wave operators in $L^p(\mathbb{R}^n)$ for $p>p_c(n)$, with $p_c(3)=6$. |
| title | L^p Boundedness of the Scattering Wave Operators of Schroedinger Dynamics with Time-dependent Potentials and applications |
| topic | Analysis of PDEs 35P25, 35Q55, 47A40 |
| url | https://arxiv.org/abs/2012.14356 |