A two-phase mother body and a Muskat problem

יום ב', 20/06/2016 - 15:05

A Muskat problem describes an evolution of the interface  $\Gamma (t)\subset{\mathbb R}^{2}$  between two immiscible fluids, occupying regions $\Omega _1$ and $\Omega _2$ in a Hele-Shaw cell. The interface evolves due to the presence of sinks and sources located in $\Omega _j$, $j=1,2$.
The case where one of the fluids is effectively inviscid, that is, it  has a constant pressure, is called
one-phase problem.  This case has been studied extensively. Much less progress has been made for the two-phase problem, the Muskat problem.
The main difficulty of the two-phase problem is the fact that the pressure on the interface, separating the fluids, is unknown. In this talk we introduce a notion of a two-phase mother body (the terminology comes from the potential theory) as a union of two distributions $\mu _j$  with integrable densities of sinks and sources, allowing to control the evolution of the interface, such that $\rm{supp}\, \mu _j \subset\Omega _j$. We use the Schwarz function approach and the introduced two-phase mother body to find the evolution of the curve $\Gamma (t)$  as well as two harmonic functions $p_j$, the pressures,  defined almost everywhere in $\Omega_j$ and satisfied prescribed boundary conditions on $\Gamma (t)$.