The 2017 DiPerna Lecture was given by Herbert Koch (Bonn) on March 9, 2017, 4PM in 60 Evans Hall.
Regularity and asymptotics of solutions to the porous medium equation
Abstract: The porous medium equation is one of the simplest equations describing the propagation of gas in a porous medium. Its most striking feature is the propagation of the front, which is the boundary of the area filled by gas. Many questions about its regularity are open.
In this lecture I will explain old and new results and open questions. Caffarelli devised a scheme for proving regularity of 'flat' solutions to free boundary problems: Flat solutions are regular if one is able to show that flatness improves on smaller scales. This scheme can be used for the porous medium equation. The lecture will conclude with consequences for the asymptotics for solutions with compactly supported initial data.