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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - FLIRT (Fluid Flows and Irregular Transport)

Teaser

Summary

\"Several physical phenomena can be modelled and described in terms of mathematical objects, frequently using so-called \"\"partial differential equations\"\" (PDEs). These are relations that correlate variations (in time and in space) of an unknown function, that represents the physical quantity the evolution of which we want to study. The mathematical theory of PDEs analyses existence, uniqueness, and further properties of solutions to such equations. From the point of view of physics, a solid mathematical result guarantees a better understanding of the phenomena under investigation and the possibility of making previsions on the future development of the system.
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.\"

Work performed

\"In the first half of the time-span of the project we have published several scientific articles with the findings of our research, disseminated our results at conferences and on the occasion of invited talks, organised conferences as a platform for scientific exchange, and supported the education of young talented mathematicians working on topics of interest for the project. Among the most relevant mathematical results we can single out the following ones.
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.\"

Final results

This project has offered and will offer until its end several progresses beyond the state of the art. Our research is in pure mathematics, and one general objective is the design of new mathematical tools, that we will apply for the study of our specific questions, but also develop as methods that can be borrowed by other researchers and applied to other research questions in different contexts. We will turn our attention specifically to problems with a physical motivation, trying to combine the creation of a beautiful mathematical theory with a progress in the understanding of the involved physics.