Small Scale Formations in the Incompressible Porous Media Equation

Explore a 27-minute lecture on small scale formations in the incompressible porous media equation, presented by Yao Yao from Georgia Institute of Technology at the Transport and Mixing in Complex and Turbulent Flows 2021 conference. Delve into the complexities of this active scalar equation, where density is transported by an incompressible velocity field given by a singular integral operator. Examine the open question of global regularity versus finite-time blow-up for smooth initial data, and investigate numerical evidence suggesting small scale formation. Learn about rigorous examples of small scale formations, including solutions exhibiting infinite-in-time growth of Sobolev norms. Understand the connection to the 2D SQG equation and gain insights into local opposedness, blowup scenarios, previous results, and associated difficulties. Follow the proof and computational aspects of the H2 norm growth, presented as part of joint work with Alexander Kiselev.

Introduction
Incompressible porous media equation
Local opposedness
Blowup
Previous results
Difficulties
Infinite time growth
Proof
Computation
The H2 norm