A Toy Model for Damped Water Waves

We consider a toy model for a damped water waves system in a domain $\Omega_t \subset \mathbb{T} \times \mathbb{R}$. The toy model is based on the paradifferential water waves equation derived in the work of Alazard-Burq-Zuily. The form of damping we utilize we utilize is a modified sponge layer proposed for the three-dimensional water waves system by Clamond, et. al. 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 the energy being constructed from appropriately chosen vector fields.