A fundamental model in mathematical fluid dynamics is that of a passive scalar subject to both molecular diffusion, as well as advection due to the motion of some fluid. In particular, one expects the interplay of the advection and diffusion terms to result in a rate of convergence to the equilibrium for the passive scalar that is far quicker than the natural dissipative one, a phenomenon known as enhanced dissipation. This effect is particularly pronounced when the fluid behaves in a disorderly and turbulent way. A particularly simple model for such fluid motion is given by stochastic transport noise, which manages to model the chaotic and seemingly random nature of turbulence very effectively.

In this talk, we will discuss the heuristics behind why certain random flows can be seen to enhance dissipation, and derive a spectral criterion for when a given SPDE driven by transport noise achieves enhanced dissipation. Furthermore, we will obtain explicit rates for enhanced dissipation by transport noise, as well as the precise hypoelliptic regularisation it enjoys, in the special case when this noise generates a shear flow.