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.
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