Improved existence time for a Boussinesq system
Abstract
We consider a strongly dispersive Boussinesq-type system arising as a model for the propagation of weakly nonlinear water waves. The system is characterized by a nonlinearity parameter 0<ε≤ 1 and a shallow water parameter 0<μ≤ 1. In the Boussinesq regime μ∼ ε, Saut, Wang and Xu \cite{SWX2017} used a hyperbolic approach to prove the well-posedness of the system on a time scale of order 1/ε, in both one and two dimensions, under a non-cavitation condition. In this paper, we establish well-posedness on a time scale of order $ ( μ^{ 1/4} ε^{-1} h_0 )^{4/3}$ in one dimension and of order (μ 1/4ε−1 h0)2− in two dimensions, assuming a non-cavitation condition with a parameter h0>0. Our results show quantitatively how the lifespan depends on the balance between nonlinearity and dispersion in the regimes ε≲ μ and μ≪ε. They also make explicit how the existence time deteriorates as h0→0, that is, as the wave approaches cavitation. Our proofs combine dispersive and Strichartz estimates with energy estimates.
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BibTeX
@misc{tesfahun2026improved,
title = {{Improved existence time for a Boussinesq system}},
author = {Achenef Tesfahun and Sigmund Selberg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39891},
eprint = {2609.39891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39891},
}
RIS
TY - UNPB
TI - Improved existence time for a Boussinesq system
AU - Achenef Tesfahun
AU - Sigmund Selberg
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.39891
UR - https://arxiv.org/abs/2609.39891
AB - We consider a strongly dispersive Boussinesq-type system arising as a model for the propagation of weakly nonlinear water waves. The system is characterized by a nonlinearity parameter 0<ε≤ 1 and a shallow water parameter 0<μ≤ 1. In the Boussinesq regime μ∼ ε, Saut, Wang and Xu \cite{SWX2017} used a hyperbolic approach to prove the well-posedness of the system on a time scale of order 1/ε, in both one and two dimensions, under a non-cavitation condition. In this paper, we establish well-posedness on a time scale of order $ ( μ^{ 1/4} ε^{-1} h_0 )^{4/3}$ in one dimension and of order (μ^ 1/4ε^−1 h_0)^2− in two dimensions, assuming a non-cavitation condition with a parameter h_0>0. Our results show quantitatively how the lifespan depends on the balance between nonlinearity and dispersion in the regimes ε≲ μ and μ≪ε. They also make explicit how the existence time deteriorates as h_0→0, that is, as the wave approaches cavitation. Our proofs combine dispersive and Strichartz estimates with energy estimates.
ER -