Periodic Reporting for period 4 - DIFFINCL (Differential Inclusions and Fluid Mechanics)
Período documentado: 2021-10-01 hasta 2022-09-30
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, through 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 was 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 is to contribute to a more efficient and accurate parametrization based not on case-by-case modelling and approximations but directly on the underlying equations of hydrodynamics.
A central aim of this project was 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 is to contribute to 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. 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.
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.
Beyond the resolution of Onsager’s conjecture, we were able to develop a general framework for the construction of intermittent solutions, firstly for the linear transport equation and later for non-Newtonian models of incompressible viscous fluids. The former lead in particular to the surprising statement that even for Sobolev vector fields the transport/continuity equation may have non-unique and non-renormalized weak solutions. The latter, in turn, was applicable to Leray-Hopf solutions of certain models used in the applied literature, showing that the underlying PDE is actually ill-posed. These developments were of course largely motivated and enriched by the work of Buckmaster and Vicol on weak solutions of the Navier-Stokes equations, work that in turn arose from previous collaboration on Onsager’s conjecture within this ERC project.
Concerning our work on instabilities, two important advances were made: firstly, a completely new approach to the evolution of vortex sheet, which is able to combine both the advantage of the weak solution theory of Delort (general existence) with the advantage of the Birkhoff-Rott setting (which gives geometric information, but is in general ill-posed). Our techniques have since been adapted to the partially stable Muskat problem, another breaktrhough in the area. Secondly, we were able to devise a similar strategy via differential inclusions for the Rayleigh-Taylor instability in the high Atwood number case — a situation which is still lacking an accepted physical theory. Our analysis showed that the macroscopic evolution in such unstable interface problems can be understood in the general context of differential inclusions, regardless of whether there is an underlying gradient flow structure, as has been suspected previously.
In the context of isometric embeddings and Nash-Kuiper theory, we developed an induction on dimension argument via simplicial decompositions which avoids the loss of regularity in the global Nash-Kuiper scheme. Very recently we have been able to apply this techniqe to the embedding of general compact manifolds to the Whitney-critical dimension with the optimal Hölder exponent.
Finally, in the context of magnetohydrodynamic turbulence, we were able to identify a new mechanism leading to non-trivial lower bounds, analogous to magnetic helicity and asymptotic linking number, albeit in the weak setting where such topological information is not readily available. Our observation is that the natural div-curl structure in the Maxwell system can be interpreted as a weak version of magnetic helicity conservation. We then constructed weak solutions showing optimality of various Onsager-type theorems in the literature concerning magnetic helicity conservation.
Concerning our work on instabilities, two important advances were made: firstly, a completely new approach to the evolution of vortex sheet, which is able to combine both the advantage of the weak solution theory of Delort (general existence) with the advantage of the Birkhoff-Rott setting (which gives geometric information, but is in general ill-posed). Our techniques have since been adapted to the partially stable Muskat problem, another breaktrhough in the area. Secondly, we were able to devise a similar strategy via differential inclusions for the Rayleigh-Taylor instability in the high Atwood number case — a situation which is still lacking an accepted physical theory. Our analysis showed that the macroscopic evolution in such unstable interface problems can be understood in the general context of differential inclusions, regardless of whether there is an underlying gradient flow structure, as has been suspected previously.
In the context of isometric embeddings and Nash-Kuiper theory, we developed an induction on dimension argument via simplicial decompositions which avoids the loss of regularity in the global Nash-Kuiper scheme. Very recently we have been able to apply this techniqe to the embedding of general compact manifolds to the Whitney-critical dimension with the optimal Hölder exponent.
Finally, in the context of magnetohydrodynamic turbulence, we were able to identify a new mechanism leading to non-trivial lower bounds, analogous to magnetic helicity and asymptotic linking number, albeit in the weak setting where such topological information is not readily available. Our observation is that the natural div-curl structure in the Maxwell system can be interpreted as a weak version of magnetic helicity conservation. We then constructed weak solutions showing optimality of various Onsager-type theorems in the literature concerning magnetic helicity conservation.