Advection and Enhanced Diffusion in an Active Scalar Problem

We start by demonstrating that the interplay between advection and diffusion in the incompressible porous media equation with diffusion -- a dissipative version of the classical active scalar problem -- can lead to the enhanced dissipation. Subsequently, we derive a scaling limit that perfectly balances these two physical mechanisms. The high degeneracy of the limiting equation prevents us from proving existence of weak solutions in the distributional form. To address this challenge, we use the gradient flow structure of the equation to define weak solutions within a robust "geometric" framework and show that the solution space is compact.

The talk is based on the joint work with Felix Otto and Bian Wu.