In the first half of this project, important advances in the ab initio mathematical understanding of turbulence have been achieved: weak solutions of the 3D Euler equation for an incompressible, inviscid fluid, related to K41 solutions and exhibiting anomalous dissipation in line with Onsager’s theory, have been constructed. In particular, Onsager’s conjecture, one of the headline goals in the proposal, is now fully resolved.
On the other hand, as is well known, in real turbulence K41 scale invariance is broken and one observes multifractal (intermittent) scaling. Thus, in the second half of the project considerable effort was invested to study this broken invariance in the context of weak solutions. The path chosen was motivated by analytical approaches to intermittency via Littlewood-Paley theory. This amounts to the construction of weak solutions of the Euler equations in certain Besov spaces. We have been able to develop a general technique, based on concentration/oscillation, which is suitable to attack this question. A further central direction of research was developing a general framework for the macroscopic evolution of fluid interfaces near instabilities. This was implemented first for the unstable Muskat problem, describing the unstable interface between two fluids of different densities in a porous medium, the subsequently for the vortex-sheet problem, describing the 2D motion of vortex lines (Kelvin-Helmholtz instability), and finally for the unstable interface between two incompressible fluids of different densities (Rayleigh-Taylor instability) at very high Atwood number.
Finally, we have opened a new strand of research concerning MHD turbulence, where new mathematical challenges, compared to the pure hydrodynamic case appear. Indeed, an important question is to understand mechanisms leading to magnetic relaxation, where a small-scale oscillations in velocity may lead to anomalous dissipation of magnetic energy.
As part of the ERC project dissemination of the scientific results played an important role at various levels. Both the PI and members of the group were invited speakers at various international conferences, in Austria, Czech Republic, Slovakia, France, USA, Japan, China, Taiwan, to name a few. The PI gave several public lectures, both for a general scientific audience (“Rudolph Kalman Lecture” in Budapest 2018) and for a general non-scientific audience (“Gauss Lecture” in Göttingen 2019). An International Conference on "Fluids and Variational Methods" was organized and funded in Budapest in June 2019.
