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∈ℝ:x≥0}×ℝ2y,z, subject to homogeneous Dirichlet boundary conditions, for arbitrarily large initial data in the coercive energy space H01(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 Xs,b built from the spectral decomposition of the Dirichlet realization of the model Friedlander operator Δg=∂x2+(1+x)∂y2+∂z2. 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 H0s(Ω) for 1/2<s<1. The initial high-frequency datum satisfies a quantitative sub-energy decay estimate of the form |P>λu0|H0s(Ω) labeleq:absdecay ≲ λs−1|u0|H01(Ω), providing a small parameter to counteract the derivative loss.
Related
- Global well-posedness and scattering for the defocusing nonlinear Schrödinger equation below energy space
- Global well-posedness and scattering for the three-dimensional defocusing cubic Schrödinger equation in Hs, s>1/2
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
- Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system
- Low-regularity global well-posedness for the defocusing cubic wave equation outside a ball with radial initial data
BibTeX
@misc{meas2026largedata,
title = {{Large-Data Global Well-Posedness for the Defocusing Cubic Schrödinger Equation in 3D Convex Domains}},
author = {Len Meas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30920},
eprint = {2609.30920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30920},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Large-Data Global Well-Posedness for the Defocusing Cubic Schrödinger Equation in 3D Convex Domains
AU - Len Meas
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30920
UR - https://arxiv.org/abs/2609.30920
AB - 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∈ℝ:x≥0}×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=∂_x^2+(1+x)∂_y^2+∂_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 1/2<s<1. The initial high-frequency datum satisfies a quantitative sub-energy decay estimate of the form |P_>λu_0|_H_0^s(Ω) labeleq:abs_decay ≲ λ^s−1|u_0|_H_0^1(Ω), providing a small parameter to counteract the derivative loss.
ER -