- Sponsor
- UIUC Math Department
- Speaker
- Andrej Zlatos
- Contact
- Jeremy Tyson
- Originating Calendar
- General Events - Department of Mathematics
Title: Fluid Dynamics in Porous Media
Abstract: The motion of fluids is modeled by various partial differential equations, depending on the properties of the fluid, the medium through which it flows, or the space-time scale on which this flow is studied. I will start by discussing two of the most fundamental such models: the 2D Euler equations modeling motions of ideal fluids, and the incompressible porous media equation (IPM) for fluids in porous media such as oil or ground water in an aquifer. These are also some of the simplest-looking fluid models because they can each be restated as a single scalar transport equation for either the vorticity or the density of the fluid. Nevertheless, their analysis is far from elementary due to non-local dependence of the transporting velocity on the transported quantity. In the second part of the talk I will present a recent result that demonstrates development of finite time singularities for the two-fluid IPM. It is based on showing that for a large class of initial data, certain physically relevant quantities - the maximal slope of the interface between the two fluids as well as the potential energy of the system - always decrease in time.