### On the dam problem with two fluids governed by a nonlinear Darcy's law

#### Abstract

We consider the problem of two fluids flow through a porous medium governed by a nonlinear law. We prove the existence of a weak solution, establish the local Lipschitz continuity of this solution in the zone above the lower fluid, and prove the continuity of the upper free boundary. In the rectangular case, we prove the existence of a monotone solution with respect to the vertical variable, and the continuity of the lower free boundary. Finally, we prove the uniqueness of a monotone solution with respect to $x$ and $y$, when the dam is rectangular and the flow is governed by the linear Darcy law.

#### Article information

Source
Adv. Differential Equations, Volume 11, Number 8 (2006), 841-892.

Dates
First available in Project Euclid: 18 December 2012

Mathematical Reviews number (MathSciNet)
MR2261733

Zentralblatt MATH identifier
1167.35550

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35D05 76S05: Flows in porous media; filtration; seepage

#### Citation

Challal, S.; Lyaghfouri, A. On the dam problem with two fluids governed by a nonlinear Darcy's law. Adv. Differential Equations 11 (2006), no. 8, 841--892. https://projecteuclid.org/euclid.ade/1355867618