On an asymptotic model for free boundary darcy flow in porous media

We provide a rigorous mathematical study of an asymptotic model describing Darcy flow with free boundary in a small amplitude/large wavelength approximation. In particular, we prove several well-posedness results in critical spaces. Furthermore, we also study how the solution decays towards the flat equilibrium.The work of the first author was supported by the Ministerio de Ciencia, Innovacion y Universidades MICIU through the project ``Mathematical Analysis of Fluids and Applications,"" grant PID2019-109348GA-I00 and acronym ``MAFyA."" The work of the second author was supported by the Basque Government through the BERC 2018-2021 program and the Spanish Ministry of Economy and Competitiveness (MINECO) through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by AEI/FEDER, UE, and acronym ``DESFLU.