Enhanced lifespan of solutions to Whitham-Boussinesq system with surface tension
Abstract
We consider a Whitham-Boussinesq type system with surface tension arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water, which is characterized by a nonlinearity parameter 0<ε≤ 1, a shallow water parameter 0<μ≤ 1 and a nonnegative surface tension parameter β. In this paper, we establish well-posedness of the associated Cauchy problem for β>1/3 on a time scale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} h_0 ε^{-1} )^{4/3}$ in one dimension and of order ( β∗ 1/4 μ 1/4 h0 ε−1)2− in two dimensions, where β∗= min(|3β−1|, 1/2), assuming a non-cavitation condition with a parameter h0>0. In addition, we established well-posedness for the classical Whitham equation with either β=0 or β>1/3 on the timescale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} ε^{-1} )^{4/3}$. These results explicitly capture how the existence time depends on all the parameters involved. Our proofs rely on dispersive and Strichartz estimates combined with energy estimates.
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BibTeX
@misc{kamet2026enhanced,
title = {{Enhanced lifespan of solutions to Whitham-Boussinesq system with surface tension}},
author = {Madina Kamet and Achenef Tesfahun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23612},
eprint = {2609.23612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23612},
}
RIS
TY - UNPB
TI - Enhanced lifespan of solutions to Whitham-Boussinesq system with surface tension
AU - Madina Kamet
AU - Achenef Tesfahun
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23612
UR - https://arxiv.org/abs/2609.23612
AB - We consider a Whitham-Boussinesq type system with surface tension arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water, which is characterized by a nonlinearity parameter 0<ε≤ 1, a shallow water parameter 0<μ≤ 1 and a nonnegative surface tension parameter β. In this paper, we establish well-posedness of the associated Cauchy problem for β>1/3 on a time scale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} h_0 ε^{-1} )^{4/3}$ in one dimension and of order ( β_∗^ 1/4 μ^ 1/4 h_0 ε^−1)^2− in two dimensions, where β_∗= min(|3β−1|, 1/2), assuming a non-cavitation condition with a parameter h_0>0. In addition, we established well-posedness for the classical Whitham equation with either β=0 or β>1/3 on the timescale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} ε^{-1} )^{4/3}$. These results explicitly capture how the existence time depends on all the parameters involved. Our proofs rely on dispersive and Strichartz estimates combined with energy estimates.
ER -