Formation of small scales in nonlinear PDEs

Dissipation Enhancement, Mixing, and Blow-up Suppression

Gautam Iyer

Carnegie Mellon University

We study the energy decay of solutions to the advection diffusion with an incompressible drift. We quantitatively estimate the rate at solutions dissipate energy in terms of the mixing rate of the advecting velocity field. Moreover, in the discrete time setting, we show that for a large class of toral automorphisms the $L^2$ energy decays double exponentially in time. We also study certain non-linear PDEs (e.g. Keller-Segel) and show how singularities in them can be avoided by the addition of an (initial data dependent) convective term that sufficiently enhances dissipation. We provide simple examples of such convective terms using a fine and strong enough cellular flows.