Periodic Reporting for period 3 - FLIRT (Fluid Flows and Irregular Transport)
Reporting period: 2019-06-01 to 2020-11-30
In the ERC Starting Grant ""FLIRT - Fluid Flows and Irregular Transport"" we investigate a vast class of PDEs which show up in the study of the dynamics of fluids. Due to the chaotic and turbulent behaviour of a fluid, these PDEs often display very irregular properties, either in their structure or for what concerns their solutions. We aim at obtaining a better understanding of these irregular PDEs. Our approach consists of (1) the use of geometric measure theory as a powerful tool to understand irregularities, and (2) the search for quantitative results, which provide more robust and significant information in contrast to merely qualitative statements. The main objectives are (1) a deep insight into the structure of nonlinear PDEs, which are the most complicated but also the most physically significant ones, and (2) rigorous bounds in mixing phenomena, in which we investigate the intertwining on small scales of different fluids present in the same physical region."
The PI in collaboration with Alberti and Mazzucato has finalised a paper (currently in press on the prestigious Journal of the AMS) in which examples of fluid evolutions are shown, which saturate previously known bounds on the irregularity of the solution, thus showing that no better quantitative statements can be proved. The same authors have recently applied these constructions to build striking examples of irregularity for solutions of linear problems.
The PI in collaboration with Colombo and Spinolo is studying some equations originating from traffic modelling. These equations can be either ""nonlocal"" (the propagation speed depends on the distribution of cars in the vicinity of the point under consideration), or ""local"" (the propagation speed depends on the punctual value of the distribution of cars only). It was conjectured (based on some numerical simulations) that in suitable situations the nonlocal equations would be an approximation of the local ones. We could actually show by means of examples that this is not the case in general, and provide an explanation for the numerical observation, based on the fact that the numerical schemes retain a small amount of ""diffusivity"", numerically essential as a stabilising tool, but not present in the actual physical model.
A further achievement has been made by a PostDoc involved in the project, Dr. Paolo Bonicatto, in collaboration with Bianchini. They are currently finalising a seminal paper, most of which was written during the PhD studies of Bonicatto, in which they prove a celebrated conjecture by Bressan on the compactness of solutions to linear equations under bounds that are naturally available in the study of associated nonlinear problems."