Abstract
From the mathematical formulation of a one-dimensional flow through a partially saturated porous medium, we arrive at a nonlinear free boundary problem, the boundary being between the saturated and the unsaturated regions in the medium. In particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part.
Existence, uniqueness, a maximum principle and regularity properties are proved for weak solutions of a Cauchy-Dirichlet problem in the cylinder {(x,t): 0≦x≦1, t≧0} and the nature, in particular the regularity, of the free boundary is discussed.
Finally, it is shown that solutions of a large class of Cauchy-Dirichlet problems converge towards a stationary solution as t → ∞ and estimates are given for the rate of convergence.
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Communicated by J. B. McLeod
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Van Duyn, C.J., Peletier, L.A. Nonstationary filtration in partially saturated porous media. Arch. Rational Mech. Anal. 78, 173–198 (1982). https://doi.org/10.1007/BF00250838
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DOI: https://doi.org/10.1007/BF00250838