2024/12/14 by Liu, Xuan, Song, Yilin, Zheng, Jiqiang
#35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.13215
In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schrödinger equation iut + ΔΩu = |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 ∈ Lt^∞(I; Hxsc(ℝ3)) with sc := (3)/(2) - \frac2α ∈ (0, (3)/(2)), 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 ∈ Lt^∞(I; HscD(Ω)) with sc ∈ [(1)/(2), (3)/(2)), 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. \bf 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. \bf 138 (2016)] to derive the linear profile decomposition for the linear propagator eitΔΩ in Hsc(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 estimates.