Low-regularity global well-posedness for the defocusing cubic wave equation outside a ball with radial initial data
Abstract
We study the three-dimensional defocusing cubic nonlinear wave equation with homogeneous Dirichlet boundary condition in the exterior of the unit ball, Ω=R3backslashB(0,1)̅, for radial initial data. For every 1/2≤ s≤3/4, we establish global well-posedness in ḢsD,rad(Ω)×Ḣs−1D,rad(Ω), including the critical endpoint s=1/2, without an additional L4 assumption on the initial displacement. The proof uses the distorted Fourier transform associated with the Dirichlet Laplacian to establish radial Sobolev embeddings, including the quantitative estimate |f|Lp(Ω)≤ C√p |f|ḢD1/2(Ω) for 4≤ p<∞, with C independent of p. We combine a nonlinear Fourier truncation argument with the critical Strichartz estimate to construct a global nonlinear component with high-frequency-truncated initial data. For the remainder, we construct a local energy solution in an affine energy space and derive a logarithmic energy inequality that prevents finite-time energy blow-up. Persistence of the original Sobolev regularity, uniqueness, and local Lipschitz dependence on each finite time interval complete the proof.
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BibTeX
@misc{lin2026lowregularity,
title = {{Low-regularity global well-posedness for the defocusing cubic wave equation outside a ball with radial initial data}},
author = {Ze Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38495},
eprint = {2610.38495},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38495},
}
RIS
TY - UNPB
TI - Low-regularity global well-posedness for the defocusing cubic wave equation outside a ball with radial initial data
AU - Ze Lin
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38495
UR - https://arxiv.org/abs/2610.38495
AB - We study the three-dimensional defocusing cubic nonlinear wave equation with homogeneous Dirichlet boundary condition in the exterior of the unit ball, Ω=R^3backslashB(0,1)̅, for radial initial data. For every 1/2≤ s≤3/4, we establish global well-posedness in Ḣ^s_D,rad(Ω)×Ḣ^s−1_D,rad(Ω), including the critical endpoint s=1/2, without an additional L^4 assumption on the initial displacement. The proof uses the distorted Fourier transform associated with the Dirichlet Laplacian to establish radial Sobolev embeddings, including the quantitative estimate |f|_L^p(Ω)≤ C√p |f|_Ḣ_D^1/2(Ω) for 4≤ p<∞, with C independent of p. We combine a nonlinear Fourier truncation argument with the critical Strichartz estimate to construct a global nonlinear component with high-frequency-truncated initial data. For the remainder, we construct a local energy solution in an affine energy space and derive a logarithmic energy inequality that prevents finite-time energy blow-up. Persistence of the original Sobolev regularity, uniqueness, and local Lipschitz dependence on each finite time interval complete the proof.
ER -