SCATTERING THEORY FOR THE DEFOCUSING 3D NLS IN THE EXTERIOR OF A STRICTLY CONVEX OBSTACLE

作     者：Liu, Xuan Song, Yilin Zheng, Jiqiang 

作者机构：School of Mathematics Hangzhou Normal University Hangzhou311121 China The Graduate School of China Academy of Engineering Physics Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100088 China National Key Laboratory of Computational Physics Beijing100088 China 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Schrodinger equation 

摘      要：In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schrödinger equation iut + ∆Ωu = |u|αu in the exterior domain Ω of a smooth, compact and strictly convex obstacle in R3. It is conjectured that in Euclidean space, if the solution has a prior bound in the critical Sobolev space, that is, u ∈ L∞t (I;H‧xsc (R3)) with sc := 32 − α2 ∈ (0, 23 ), then u is global and scatters. In this paper, assuming that this conjecture holds, we prove that if u is a solution to the nonlinear Schrödinger equation in exterior domain Ω with Dirichlet boundary condition and satisfies u ∈ L∞t (I;H‧Dsc (Ω)) with sc ∈ ［2123 ）, then u is global and scatters. The proof of the main results relies on the concentration-compactness/rigidity argument of Kenig and Merle [Invent. Math. 166 (2006)]. The main difficulty is to construct minimal counterexamples when the scaling and translation invariance breakdown on Ω. To achieve this, two key ingredients are required. First, we adopt the approach of Killip, Visan, and Zhang [Amer. J. Math. 138 (2016)] to derive the linear profile decomposition for the linear propagator eit∆Ω in H‧ sc(Ω). The second ingredient is the embedding of the nonlinear profiles. More precisely, we need to demonstrate that nonlinear solutions in the limiting geometries, which exhibit global spacetime bounds, can be embedded back into Ω. Finally, to rule out the minimal counterexamples, we will establish long-time Strichartz estimates for the exterior domain NLS, along with spatially localized and frequency-localized Morawetz *** Codes 35Q55 Copyright © 2024, The Authors. All rights reserved.