Small scale formations in the incompressible porous media equation

Yao Yao
Georgia Institute of Technology

The incompressible porous media (IPM) equation is an active scalar equation where the density is transported by an incompressible velocity field given by a singular integral operator, which is analogous to the 2D SQG equation. The question of global regularity vs finite-time blow-up remains open for smooth initial data, although numerical evidences suggest that small scale formation can happen. In this talk, I will discuss rigorous examples of small scale formations in the IPM equation: we construct solutions to IPM that exhibit infinite-in-time growth of Sobolev norms, provided that they remain globally smooth in time. This is a joint work with Alexander Kiselev.

Presentation (PDF File)

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