Electron. J. Differential Equations, Vol. 2022 (2022), No. 06, pp. 1-22.

Mathematical analysis of a Dupuit-Richards model Safaa Al Nazer, Carole Rosier, Munkhgerel Tsegmid

Abstract:
This article concerns an alternative model to the 3D-Richards equation to describe the flow of water in shallow aquifers. The model couples the two dominant types of flow existing in the aquifer. The first is described by the classic Richards problem in the upper capillary fringe. The second results from Dupuit's approximation after vertical integration of the conservation laws between the bottom of the aquifer and the saturation interface. The final model consists of a strongly coupled system of parabolic-type partial differential equations that are defined in a time-dependent domain. First, we show how taking the low compressibility of the fluid into account eliminates the nonlinearity in the time derivative of the Richards equation. Then, the general framework of parabolic equations is used in non-cylindrical domains to give a global in time existence result to this problem.

Submitted May 5, 2021. Published January 17, 2022.
Math Subject Classifications: 35A01, 35D30, 35K59, 35Q86, 35R35.
Key Words: Dupuit-Richards equations; free boundary problems; global solution; weak solution; fluid flow modeling.
DOI: https://doi.org/10.58997/ejde.2022.06

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 Safaa Al Nazer Université du Littoral Côte d'Opale, UR 2597, LMPA Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville F-62100 Calais, France email: safaaalnazer9@gmail.com Carole Rosier Université du Littoral Côte d'Opale, UR 2597, LMPA Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville F-62100 Calais, France email: Carole.Rosier@univ-littoral.fr Munkhgerel Tsegmid School of Applied Sciences Mongolian University of Science and Technology Ulaanbaatar, Mongolia email: tsemuugii@must.edu.mn