TY - JOUR
T1 - On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D
AU - Georgiev, Vladimir
AU - Li, Chunhua
N1 - Funding Information:
The authors would like to thank the unknown referees for some helpful comments. V. Georgiev was supported in part by Project 2017 “Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari” of INDAM, GNAMPA—Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni , by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University, by the University of Pisa, Project PRA 2018 49 and project “Dinamica di equazioni nonlineari dispersive”, “Fondazione di Sardegna”, 2016. C. Li was partially supported by the Education Department of Jilin Province [ JJKH20180892KJ ] and NNSFC under Grant Number 11461074 .
PY - 2019/11
Y1 - 2019/11
N2 - We consider the scattering problem for the nonlinear Schrödinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator A(s) appearing in commutator relations. The equivalence between the operators −ΔV [Formula presented] and −Δ [Formula presented] in L2 norm sense for 0≤s<1 is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schrödinger operator −ΔV. Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.
AB - We consider the scattering problem for the nonlinear Schrödinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator A(s) appearing in commutator relations. The equivalence between the operators −ΔV [Formula presented] and −Δ [Formula presented] in L2 norm sense for 0≤s<1 is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schrödinger operator −ΔV. Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.
KW - Nonlinear Schrödinger equation
KW - Resolvent estimates
KW - Scattering problem
KW - Strichartz estimates
KW - Time decay estimates
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U2 - 10.1016/j.physd.2019.03.010
DO - 10.1016/j.physd.2019.03.010
M3 - Article
AN - SCOPUS:85065538482
VL - 398
SP - 208
EP - 218
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
ER -