Scattering theory for the defocusing 3d NLS in the exterior of a strictly convex obstacle
Xuan Liu, Yilin Song, Jiqiang Zheng
TL;DR
This work extends the scattering theory for defocusing NLS to the exterior of a strictly convex obstacle in ℝ^3 under a Euclidean critical-norm conjecture. Building on Kenig–Merle’s concentration-compactness/rigidity framework, the authors develop a linear profile decomposition for the exterior-domain propagator e^{itΔ_Ω}, and they prove embedding results for nonlinear profiles from limiting geometries back into the exterior domain. A stability theory and a nonlinear profile decoupling mechanism are then employed to reduce any potential counterexample to an almost periodic solution, which is subsequently ruled out via long-time Strichartz estimates and Morawetz inequalities, with frequency-localized variants addressing the half-space-like limits. Assuming the Euclidean conjecture holds, they prove global well-posedness and scattering for s_c ∈ [½, 3/2) in the exterior Domain, thereby connecting exterior-domain dynamics to the Euclidean theory and providing a rigorous pathway to transfer critical-norm scattering results to obstacle problems. The work advances the understanding of dispersive behavior in nontrivial geometries and highlights robust techniques—profile decompositions in variable geometries, stability, and spatially localized Morawetz estimates—for exterior-domain NLS.
Abstract
In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schrödinger equation $iu_t + Δ_Ωu = |u|^αu$ in the exterior domain $Ω$ 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 $s_c := \frac{3}{2} - \frac{2}α \in (0, \frac{3}{2})$, 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 Schrödinger equation in exterior domain $Ω$ with Dirichlet boundary condition and satisfies $u \in L_t^\infty(I; \dot{H}^{s_c}_D(Ω))$ 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 $Ω$. 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 $e^{itΔ_Ω}$ in $\dot{H}^{s_c}(Ω)$. 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.