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

2025/07/09 by Christopher Maulén, Claudio Muñoz, Maulén, Christopher +3 · 1 citation
Earth and Planetary Sciences · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2507.06487

openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℝ_t × 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 H¹× H¹ 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, et. al., 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 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 H¹× H¹. Then, for a sharp set of dispersive abcd systems (characterized only in terms of parameters a, b and c), every H¹× H¹ small solution must converges to zero inside of the light cone |x|≤ |t|.

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• Ill-posedness issues on (abcd)-Boussinesq system