On Mixing and Resonances in Fluid Systems

This habilitation thesis is concerned with the long-time stability of partial differential equations modeling coupled fluid systems. A particular focus lies on the role of mixing in the Euler, Navier-Stokes and Boussinesq equations.
While the corresponding linearized equations exhibit stability in Sobolev regularity, this is not reflected in the nonlinear dynamics. Instead, nonlinear resonances give rise to norm inflation and instability.
We introduce ``traveling waves'' as a new tool to capture these resonances and the effects of mixing.