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Differential Inclusions and Fluid Mechanics

Periodic Reporting for period 3 - DIFFINCL (Differential Inclusions and Fluid Mechanics)

Reporting period: 2020-04-01 to 2021-09-30

Many problems in science involve structures on several distinct length scales. Two typical examples are fine phase mixtures in solid-solid phase transitions and the complex mixing patterns in turbulent or multiphase flows. The microstructures in such situations influence in a crucial way the macroscopic behavior of the system, and understanding the formation, interaction and overall effect of these structures is a great scientific challenge. Although there is a large variety of models and descriptions for such phenomena, a recurring issue in the mathematical analysis is that one has to deal with very complex and highly non-smooth structures in solutions of the associated partial differential equations. A common ground is provided by the analysis of partial differential inclusions, a theory whose development was strongly influenced by the influx of ideas from two directions: Geometry, throigh the work of Gromov on partial differential relations, building on celebrated constructions of Nash for isometric immersions, and Analysis: the work of Tartar in the study of oscillation phenomena in nonlinear partial differential equations.

A central aim of this project is to develop these ideas in the context of fluid mechanics, with the ultimate goal to address important challenges on hydrodynamical turbulence: providing an analytic foundation for the statistical theory of turbulence and for the behavior of turbulent flows near instabilities and boundaries. Understanding the fine-scale structure of turbulence is of paramount importance in several scientific contexts, from engineering applications to meteorology. With current technology, the small scales in turbulent flow are not truly represented in simulations, but are modelled by certain ad-hoc parametrizations; The validity of these, however, is invariably limited. A long-term effect of this project would be a more efficient and accurate parametrization based not on case-by-case modelling and approximations but directly on the underlying equations of hydrodynamics.
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. The correct scaling exponents seem not to be universal, and are still subject to experimental research. So far we have not managed to obtain physically reasonable scaling exponents in the mathematical implementation for the full hydrodynamical equations, but we now have a road-map and were able to succeed in simpler PDE models.

A further achievement so far is the development of a general technique for calculating the macroscopic evolution of fluid interfaces near instabilities. This was implemented so far for the Muskat problem, describing the unstable interface between two fluids of different densities in a porous medium, the vortex-sheet problem, describing the 2D motion of vortex lines (Kelvin-Helmholtz instability), and the unstable interface between two incompressible fluids of different densities (Rayleigh-Taylor instability) at very high Atwood number, a situation which is still lacking an accepted physical theory.

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.
We expect to be able to reach the goal of constructing weak solutions of the Euler equations with multi-fractal scaling. This would be a major result, as it would lead to a new experimental playground via “synthetic turbulence”, rather than direct numerical simulation (DNS) that, in turn, can be explored numerically. There are various parameters in the construction that can be fine-tuned to fit experimental data (on averaged quantities). Then, one might ask (and test): how far are multifractal/beta-models agreeing with Lagrangian measurements? Also, are there other characteristics of such flows that can be checked in DNS or real experiments (e.g. flight crash events, local dissipation measure, ...)? With such applications in view, new collaborations within the applied sciences were formed, with an eye towards numerical implementation.

A further major goal of the overall project is regarding the characterization of oscillatory behaviour in sequences of solutions of nonlinear PDE. One strand is to make progress concerning Morrey’s conjecture, with ramifications in complex analysis and harmonic analysis. A second strand is to build on our experience concerning compressible fluid models and the Rayleigh-Taylor instability and understand better the relationship between mixing of density (RT) and momentum (KH), a topic that remained largely unexplored so far.

Finally, returning to the connections with geometry, our aim is to further explore flexibility phenomena in certain differential structures, such as divergence-free fields and isometric embeddings. The former is related to questions in magnetism, specifically the possibility of magnetic reversal in ideally soft ferromagnetic bodies, the latter is related to nonlinear elasticity thoery and bending of thin objects.