In the first part of the talk I will provide rigorous asymptotic models for the free boundary Darcy problem under the assumption of weak nonlinear interaction, in a regime in which the steepness parameter of the interface is considered to be very small. The second part of the talk will be devoted to prove some rigorous result of well-posedness in critical spaces for the asymptotic model. Joint work with R. Granero-Belinchón. 

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