Geodesic
Proof of Dupuit's Assumption for the Free Boundary Problem in an Inhomogeneous Porous Medium
Matematičeskie zametki, Tome 103 (2018) no. 1, pp. 49-64

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For free boundary problems describing steady groundwater flows, the asymptotic behavior of solutions is studied in the situation where the scale in one of the spatial directions is much less than that in the other directions. The convergence of solutions to a certain limit is proved. Properties of the limit solution agree with the assumption known as Dupuit's assumption in engineering applications, which customarily serves as a basis for constructing approximate models of groundwater flows in thin aquifers.
Keywords: variational inequality, Darcy's law, dam problem. 
@article{MZM_2018_103_1_a4,
     author = {A. Yu. Belyaev},
     title = {Proof of {Dupuit's} {Assumption} for the {Free} {Boundary} {Problem} in an {Inhomogeneous} {Porous} {Medium}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {49--64},
     publisher = {mathdoc},
     volume = {103},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_103_1_a4/}
}
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A. Yu. Belyaev. Proof of Dupuit's Assumption for the Free Boundary Problem in an Inhomogeneous Porous Medium. Matematičeskie zametki, Tome 103 (2018) no. 1, pp. 49-64. http://geodesic.mathdoc.fr/item/MZM_2018_103_1_a4/