# Mathematics Colloquia and Seminars

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### On regularity for the Muskat problem

**PDE and Applied Math Seminar**

Speaker: | Huy Nguyen, Brown |

Location: | 2112 MSB |

Start time: | Thu, Nov 7 2019, 4:30PM |

The Muskat problem models the evolution of two immiscible

fluids of varying density in a porous medium. The free interface

between the two fluids obeys a quasilinear parabolic equation, which

to leading order has a natural scaling. We will discuss our recent

results on local and global well-posedness respectively in all

subcritical spaces and certain critical spaces, allowing for

curvature singularities of the initial interface. We employ a

paradifferential calculus approach which is robust enough to

incorporate various features in the problem such as varying viscosity

and physical boundaries. We also obtain results on the vanishing

surface tension limit and the infinite depth limit.