Scattering theory for the defocusing 3d NLS in the exterior of a strictly convex obstacle

TL;DR

This paper proves global well-posedness and scattering for the defocusing 3D nonlinear Schrödinger equation in the exterior of a convex obstacle, assuming a conjecture about Euclidean space solutions, using concentration-compactness and profile decomposition methods.

Contribution

It extends scattering results to exterior domains by constructing minimal counterexamples and embedding nonlinear profiles, relying on advanced linear and nonlinear analysis techniques.

Findings

Proves global existence and scattering under certain Sobolev bounds.

Develops linear profile decomposition for exterior domain Schrödinger propagator.

Establishes long-time Strichartz and Morawetz estimates in exterior domains.

Abstract

In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schr\"odinger equation $iu_t + \Delta_\Omega u = |u|^\alpha u$ in the exterior domain $\Omega$ of a smooth, compact and strictly convex obstacle in $\mathbb{R}^3$. It is conjectured that in Euclidean space, if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^3))$ with $s_c := \frac{3}{2} - \frac{2}{\alpha} \in (0, \frac{3}{2})$, 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 Schr\"odinger equation in exterior domain $\Omega$ with Dirichlet boundary condition and satisfies $u \in L_t^\infty(I; \dot{H}^{s_c}_D(\Omega))$ with $s_c \in \left[\frac{1}{2}, \frac{3}{2}\right)$, 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 $\Omega$. 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 $e^{it\Delta_\Omega}$ in $\dot{H}^{s_c}(\Omega)$. 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 $\Omega$. 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.