Scaling bounds on dissipation in turbulent flows

We present a new rigorous method for estimating statistical quantities in fluid dynamics such as the (average) energy dissipation rate directly from the equations of motion. The method is tested on shear flow, channel flow, Rayleigh--B\'enard convection and porous medium convection

A quantitative theory for the continuity equation

In this work, we provide stability estimates for the continuity equation with Sobolev vector fields. The results are inferred from contraction estimates for certain logarithmic Kantorovich--Rubinstein distances. As a by-product, we obtain a new proof of uniqueness in the DiPerna--Lions setting. The novelty in the proof lies in the fact that it is not based on the theory of renormalized solutions.Comment: Final version, includes optimality result. Accepted for publication in Annales IH

Diffusion limited mixing rates in passive scalar advection

We are concerned with flow enhanced mixing of passive scalars in the presence of diffusion. Under the assumption that the velocity gradient is suitably integrable, we provide upper bounds on the exponential rates of mixing and of enhanced dissipation. Our results suggest that there is a crossover from advection dominated to diffusion dominated mixing, and we observe a slow down in the exponential decay rates by (some power of) a logarithm of the diffusivity.Comment: Generalized resul