Large-Data Global Well-Posedness for the Defocusing Cubic Schrödinger Equation in 3D Convex Domains

Abstract

We establish the global well-posedness for the energy-subcritical defocusing cubic nonlinear Schrödinger equation (NLS) on the three-dimensional Friedlander model domain \( Ω=\{x\in\mathbb{R}:x\geq0\}\times\mathbb{R}^2_{y,z}, \) subject to homogeneous Dirichlet boundary conditions, for arbitrarily large initial data in the coercive energy space $H_0^1(Ω)$. The phase-space geometry of this configuration features a strictly convex boundary that admits a non-empty glancing set, trapping high-frequency wave packets within a boundary layer via generalized whispering-gallery caustics. These concentration phenomena induce an intrinsic derivative loss in the sharp linear Strichartz estimates, establishing a major microlocal obstruction to closing the nonlinear Duhamel iteration directly at the energy level via classical perturbative frameworks. To bridge the regularity deficit between the conservation laws and the linear theory, we construct a boundary-adapted family of continuous--discrete Bourgain--Strichartz restriction spaces $X^{s,b}$ built from the spectral decomposition of the Dirichlet realization of the model Friedlander operator \( Δ_g=\partial_x^2+(1+x)\partial_y^2+\partial_z^2. \) The normal variable is resolved through discrete Airy spectral modes, while the tangential variables are continuously Fourier analyzed. Within this functional framework, we implement a refined high--low frequency decomposition to formulate a perturbed nonlinear equation for the high-frequency remainder in the sub-energy space $H_0^s(Ω)$ for \( \frac{1}{2} λ}u_0\|_{H_0^s(Ω)} \label{eq:abs_decay} \lesssim λ^{s-1}\|u_0\|_{H_0^1(Ω)}, \) providing a small parameter to counteract the derivative loss.