A priori estimates for the 2D one-phase Muskat problem with contact points
Venue: Edoardo Bocchi - Aula Buzano (DISMA Politecnico di Torino)
We address the dynamics of a viscous and incompressible free-surface fluid in a Hele-Shaw cell or, equivalently, in a 2D bounded region of a porous medium. The contact between the fluid surface and the boundary of the domain occurs at vertical lateral walls. In order to close a scheme of a priori estimates, following the approach of Guo and Tice for the Stokes problem, we bootstrap from energy-dissipation control of the time derivatives to higher spatial regularity via elliptic estimates. In particular, despite the presence of corners in the stationary domain that might create singularities in the velocity potential (and its time derivative), we gain the necessary $H^2$-regularity avoiding weights and restrictions on the angles. This talk is based on a joint work with Ángel Castro and Francisco Gancedo.
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