On recent results regarding the mathematical model of turbulent fluid mixing via convex integration

2022. 10. 13. 14:00
H306 terem, online
Kolumbán József

On recent results regarding the mathematical modelling of turbulent fluid mixing via convex integration

Since the pioneering works of De Lellis and Székelyhidi Jr. regarding the deterministic mathematical modelling of turbulent fluids via convex integration from the late 2000s, many pertinent new results were published using similar techniques for different models in continuum mechanics. One subclass of particular interest is that of fluid mixing models. Indeed, in such situations often there is a sharp interface between two fluids which becomes unstable and leads to turbulent mixing. This not only leads to mathematical ill-posedness, but also to numerical simulations and real life experiments producing different results every time they are carried out, due to the instability of the interface. In this talk we will study in particular the Rayleigh-Taylor instability, where a heavier fluid is layered on top of a lighter fluid, subject to gravity. Using methods of convex integration, we show that there exist infinitely many weak solutions for the model, and hence there is a need to find other pertinent quantities to study instead of the usual fluid density and velocity. One such option is the averaged flow, which corresponds to the notion of subsolution in convex integration. We then look at different selection criteria for subsolutions, such as maximizing initial energy dissipation (in which case one reproduces the quadratic growth rate for the turbulent zone, which until now was only observed in experiments and simulations), and in the last part of the talk we try to establish a least action principle of the level of the averaged flow. The talk is based on various papers of the speaker from the last couple of years co-authored with B. Gebhard (Madrid), J. Hirsch (Leipzig), L. Székelyhidi Jr. (Leipzig).


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