Decay of small energy solutions in the ABCD Boussinesq model under the influence of an uneven bottom

TL;DR

This paper proves decay of small energy solutions for the ABCD Boussinesq model with uneven bottom topography, extending previous results to more general variable bottom conditions and establishing global existence and decay in the energy space.

Contribution

It generalizes decay results for the ABCD Boussinesq system to include variable bottom topography with smallness and integrability conditions, covering physically relevant regimes.

Findings

Global existence of small solutions in H^1×H^1

Solutions decay to zero inside the light cone

Results apply to dispersive ABCD systems with specific parameter conditions

Abstract

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 $|x|\leq |t|$.