### On a Free Boundary Problem for a Strongly Degenerate Quasilinear Parabolic Equation with an Application to a Model of Pressure Filtration

Raimund Burger, Hermano Frid, and Kenneth H. Karlsen

Abstract: We consider a free boundary problem of a quasilinear strongly degenerate parabolic equation arising from a model of pressure filtration of flocculated suspensions. We provide definitions of generalized solutions of the free boundary problem in the framework of $L^2$ divergence-measure fields. The formulation of boundary conditions is based on a Gauss-Green theorem for divergence-measure fields on bounded domains with Lipschitz deformable boundaries and avoids referring to traces of the solution. This allows to consider generalized solutions from a larger class than $BV$. Thus it is not necessary to derive the usual uniform estimates on spatial {\em and} time derivatives of the solutions of the corresponding regularized problem requires in the $BV$ approach. We first prove existence and uniqueness of the solution of the regularized parabolic free boundary problem and then apply the vanishing viscosity method to prove existence of a generalized solution to the degenerate free boundary problem.

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Author(s):
Raymund Bürger <buerger@mathematik.uni-stuttgart.de>
Hermano Frid, <hermano@impa.br>
Kenneth Hvistendahl Karlsen <kennethk@mi.uib.no>
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