A toy model for damped water waves

2024/06/01 by Gary Moon · 1 voice · 1 citation
Engineering · Computer Science · Earth and Planetary Sciences · #Stability and Controllability of Differential Equations #Advanced Mathematical Modeling in Engineering #Arctic and Antarctic ice dynamics

paper · doi:10.1142/s0219891624500097

openalex publication_date 2024/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21

Abstract

We consider a toy model for a damped water waves system in a domain [Formula: see text]. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J. 158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the [Formula: see text] water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys. 205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

Citations

• The Cauchy problem for quasi-linear symmetric hyperbolic systems
• Nonstationary flows of viscous and ideal fluids in R3
• Well-posedness in Sobolev spaces of the full water wave problem in 2-D

Cited by

• Interior energy estimates for the linear wave equation

Discussions

• In “A toy model for damped water waves,” Gary Moon studies a damped water waves toy model using paradifferential formulation and modified sponge-layer damping, proving quadratic lifespan for small Cau [bsky, 0 points, 0 comments]