WELL-POSEDNESS OF QUADRATIC HARTREE TYPE EQUATIONS BELOW L2

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Title
WELL-POSEDNESS OF QUADRATIC HARTREE TYPE EQUATIONS BELOW L2
Creator
KAMEI RYOSUKE
Source Title
島根大学総合理工学部紀要.シリーズB
Memoirs of the Graduate School of Science and Engineering, Shimane University. Series B, Mathematics
Volume 57
Start Page 27
End Page 38
Journal Identifire
ISSN 1342-7121
Descriptions
This paper studies the Cauchy problem for the nonlinear Schr¨odinger equation i∂tu − ∂2 xu = f(u) in one space dimension. The nonlinear interaction f(u) is a linear combination of (V ∗x u)u, (V ∗x ¯u)u, (V ∗x u)¯u and (V ∗x ¯u)¯u, where V (x) is a locally integrable function whose Fourier transform satisfies | ˆ V (ξ)| ≲ ⟨ξ⟩−m for some m ≥ 0. The Cauchy problem is well-posed in Hs for s > −(m/2+1/4); furthermore, if f(u) contains only the first and the last types of nonlinear terms, then the Cauchy problem is well-posed for s > −(m/2+3/4). The proof is based on bilinear estimates in Xs,b spaces.
Language
eng
Resource Type departmental bulletin paper
Publisher
総合理工学部
The Interdisciplinary Graduate School of Science and Engineering
Date of Issued 2024
Publish Type Version of Record
Access Rights open access
Relation
ソウゴウ リコウ ガクブ