PreprintarXiv licencearXiv
Well-posedness of Boussinesq Equation with Linear Restoring Force
Abstract
We establish a local well-posedness theory for the n−dimensional Boussinesq equation with a linear restoring force in a modified Sobolev spaces Hωs and Bourgain-type spaces Xωs,b, which are adapted to its modified dispersion relation. The restoring force introduces a new phase function whose behavior in the low-frequencies is different from ones of the classical Boussinesq equation, that in turn, affects the associated energy functional. We prove bilinear and trilinear estimates for quadratic and cubic nonlinearities, respectively, which in turn imply local well-posedness under suitable conditions on s and ω. To our knowledge, this is the first well-posedness result for this model.
Related
- Endpoint well-posedness for the Boussinesq equation
- Long-time validity of the Korteweg-de Vries approximation for the Boussinesq equation
- The asymptotic stability of the stable solitons of the generalized good Boussinesq equation
- On the long wave limit of the good Boussinesq equation
- Initial and initial-boundary value problems for a cubic sixth-order Boussinesq equation
BibTeX
@misc{esfahani2026wellposedness,
title = {{Well-posedness of Boussinesq Equation with Linear Restoring Force}},
author = {Amin Esfahani and Nazerke Zhardemova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33358},
eprint = {2609.33358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33358},
}
RIS
TY - UNPB
TI - Well-posedness of Boussinesq Equation with Linear Restoring Force
AU - Amin Esfahani
AU - Nazerke Zhardemova
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33358
UR - https://arxiv.org/abs/2609.33358
AB - We establish a local well-posedness theory for the n−dimensional Boussinesq equation with a linear restoring force in a modified Sobolev spaces H_ω^s and Bourgain-type spaces X_ω^s,b, which are adapted to its modified dispersion relation. The restoring force introduces a new phase function whose behavior in the low-frequencies is different from ones of the classical Boussinesq equation, that in turn, affects the associated energy functional. We prove bilinear and trilinear estimates for quadratic and cubic nonlinearities, respectively, which in turn imply local well-posedness under suitable conditions on s and ω. To our knowledge, this is the first well-posedness result for this model.
ER -