In this paper, we consider the evolution dam problem (P) related to a compressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet boundary conditions on some part of the boundary. We establish existence of a solution for this problem. We choose a convenient regularized problem (P) for which we prove the existence and uniqueness of solution using the comparison Lemma 2.1 and the Schauder fixed-point theorem. Then, we pass to the limit, when goes to 0, to get a solution for our problem. Moreover, we will see another approach for the incompressible case where we pass to the limit in (P), when goes to 0, to get a solution.
We consider a class of parabolic free boundary problems with heterogeneous coefficients including, from a physical point of view, the evolutionary dam problem. We establish existence of a solution for this problem. We use a regularized problem for which we prove existence of a solution by applying the Tychonoff fixed point theorem. Then we pass to the limit to get a solution of our problem. We also give a regularity result of the solutions.
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