Mathematical study of some evolving and stationary interfaces.

Summary This thesis gathers existence and uniqueness results for evolving and stationary free boundary problems from biology, oil extraction or fluid mechanics. One of the main features of these models is that they exclude any presence of surface tension. The first part deals with a class of free boundary problems including the so-called muskat and quasi-stationary stefan problems. In all these cases, we show by an analysis of the nonlinear and nonlocal terms, the existence of a unique classical local solution whose regularity is the same in time and space. In the second part, we study a class of stationary transport equations with a non-local and positive lower order term. This type of equation comes directly from the free boundary problem considered in part 3, but can also be studied as such. A pseudo-differential analysis gives the existence of a unique solution in the sobolev spaces. In part 3, we consider an incompressible three-dimensional fluid subject to gravity flowing along an inclined plane, in the absence of surface tension. In this context, we prove that there exists a unique solution in sobolev spaces with weights by using the results of the second part, and at the cost of a nash-moser theorem.