| Abstract: |
| This talk concerns the Muskat problem with surface tension, modeling the filtration of two incompressible immiscible fluids in porous media. This non-local and non-linear partial differential equation is a basic mathematical model in petroleum engineering; it was formulated by the petroleum engineer M. Muskat in 1934 to describe the mixture of water into an oil sand. Given its origins and its equivalence with Hele-Shaw flows, the Muskat problem has received a lot of attention from the physics community.
We consider the case in which the fluids have different constant densities together with different constant viscosities. The Rayleigh-Taylor condition cannot hold for a closed curve, which makes this situation unstable. In this case the equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available to bound the amplitude and the slopes of the interface. We prove global in time existence and uniqueness results for medium size initial stable data in critical functional spaces. In particular we prove for the first time the global in time stability of star shaped bubbles influenced by Gravity. This is joint work with Gancedo, Garcia-Juarez, and Patel. |
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