Decay of small energy solutions in the ABCD Boussinesq model under the influence of an uneven bottom
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| Format: | Preprint |
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2025
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| _version_ | 1866915433063383040 |
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| author | Maulén, Christopher Muñoz, Claudio Poblete, Felipe |
| author_facet | Maulén, Christopher Muñoz, Claudio Poblete, Felipe |
| contents | The $abcd$ Boussinesq system, introduced by Bona, Chen, and Saut, describes a four-parameter $(a,b,c,d)$ family of models formulated on the time-space domain $\mathbb{R}_t \times \mathbb{R}_x$. It serves as a first-order two-wave approximation to the two-dimensional incompressible, irrotational water wave equations in shallow water, inspired by Boussinesq's classical derivation. Within the different parameter regimes, the generic regime is described by $b,d>0$ and $a,c<0$ while the system becomes Hamiltonian when $b=d$. Previously, sharp local in space $H^1\times H^1$ decay properties were proved in the case of a large class of $abcd$ model under the small data assumption. In this paper, we generalize [C. Kwak, \emph{et. al.}, \emph{The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space}. J. Math. Pures Appl. (9) 127 (2019), 121--159] by considering the small data $abcd$ decay problem in the physically relevant \emph{variable bottom regime} described by M. Chen. The nontrivial bathymetry is represented by a smooth space-time dependent function $h=h(t,x)$, which obeys integrability in time and smallness in space. We prove first the existence of small global solutions in $H^1\times H^1$. Then, for a sharp set of dispersive $abcd$ systems (characterized only in terms of parameters $a, b$ and $c$), every $H^1\times H^1$ small solution must converges to zero inside of the light cone $|x|\leq |t|$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decay of small energy solutions in the ABCD Boussinesq model under the influence of an uneven bottom Maulén, Christopher Muñoz, Claudio Poblete, Felipe Analysis of PDEs The $abcd$ Boussinesq system, introduced by Bona, Chen, and Saut, describes a four-parameter $(a,b,c,d)$ family of models formulated on the time-space domain $\mathbb{R}_t \times \mathbb{R}_x$. It serves as a first-order two-wave approximation to the two-dimensional incompressible, irrotational water wave equations in shallow water, inspired by Boussinesq's classical derivation. Within the different parameter regimes, the generic regime is described by $b,d>0$ and $a,c<0$ while the system becomes Hamiltonian when $b=d$. Previously, sharp local in space $H^1\times H^1$ decay properties were proved in the case of a large class of $abcd$ model under the small data assumption. In this paper, we generalize [C. Kwak, \emph{et. al.}, \emph{The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space}. J. Math. Pures Appl. (9) 127 (2019), 121--159] by considering the small data $abcd$ decay problem in the physically relevant \emph{variable bottom regime} described by M. Chen. The nontrivial bathymetry is represented by a smooth space-time dependent function $h=h(t,x)$, which obeys integrability in time and smallness in space. We prove first the existence of small global solutions in $H^1\times H^1$. Then, for a sharp set of dispersive $abcd$ systems (characterized only in terms of parameters $a, b$ and $c$), every $H^1\times H^1$ small solution must converges to zero inside of the light cone $|x|\leq |t|$. |
| title | Decay of small energy solutions in the ABCD Boussinesq model under the influence of an uneven bottom |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.06487 |