## Final Report Summary - HPFLUDY (The h-Principle for Fluid Dynamics)

Turbulence is one of the major unsolved problems of classical physics, and, although there is a well-developed statistical theory, on the level of the deterministic equations very little is known. One specific mathematical problem in this context arises from the work of Lars Onsager in 1949 concerning the phenomenon of anomalous dissipation - the phenomenon, where turbulent fluids seem to be able to dissipate kinetic energy without viscosity. Onsager considers inviscid fluids, described by the Euler equations, and conjectured a precise regularity threshold below which energy dissipation is possible.

This threshold is given in terms of the spatial Hölder regularity of the velocity and is conjectured to be at exponent 1/3.

In this project we contributed towards establishing a link between the observed non-deterministic nature of turbulence and a principle from geometry and topology: the h-principle of Gromov. Athough the h-principle is not thought to apply to deterministic equations from classical physics, we show that, when interpreted correctly in terms of weak solutions, the h-principle indeed applies. This is a phenomenon, which is similar to what happens in non-linear elasticity: although one is dealing with an inherently deterministic process from classical physics, variational models for elasticity do not predict a unique solution but rather a statistical description (e.g. in terms of Young measures). In turn, exact realizations of such Young measure solutions can be constructed using the method of convex integration, a central technique for the h-principle. In the same way one may argue that certain weak solutions of the Euler equations should be seen as exact realizations of a supposed statistical description of turbulent fluids. A specific test in this programme is the conjecture of Onsager: the precise regularity exponent of Onsager is directly linked to the scaling of the energy spectrum in Kolmogorov’s K41 statistical theory and the famous Kolmogorov-Obukhov 5/3 law.

Overall we succeeded in showing that convex integration can indeed be applied to the Euler equations, in a variety of ways. We have made significant progress towards the conjecture of Onsager and the problem of anomalous dissipation, that had major impact not just in the mathematical community on PDE but also in the physics turbulence community. As a result, Onsager’s conjecture is now fully resolved. Perhaps even more importantly, our results show that these examples of weak dissipating Euler flows are not isolated singular examples, but belong to a general, essentially generic class of solutions at this low regularity.

This threshold is given in terms of the spatial Hölder regularity of the velocity and is conjectured to be at exponent 1/3.

In this project we contributed towards establishing a link between the observed non-deterministic nature of turbulence and a principle from geometry and topology: the h-principle of Gromov. Athough the h-principle is not thought to apply to deterministic equations from classical physics, we show that, when interpreted correctly in terms of weak solutions, the h-principle indeed applies. This is a phenomenon, which is similar to what happens in non-linear elasticity: although one is dealing with an inherently deterministic process from classical physics, variational models for elasticity do not predict a unique solution but rather a statistical description (e.g. in terms of Young measures). In turn, exact realizations of such Young measure solutions can be constructed using the method of convex integration, a central technique for the h-principle. In the same way one may argue that certain weak solutions of the Euler equations should be seen as exact realizations of a supposed statistical description of turbulent fluids. A specific test in this programme is the conjecture of Onsager: the precise regularity exponent of Onsager is directly linked to the scaling of the energy spectrum in Kolmogorov’s K41 statistical theory and the famous Kolmogorov-Obukhov 5/3 law.

Overall we succeeded in showing that convex integration can indeed be applied to the Euler equations, in a variety of ways. We have made significant progress towards the conjecture of Onsager and the problem of anomalous dissipation, that had major impact not just in the mathematical community on PDE but also in the physics turbulence community. As a result, Onsager’s conjecture is now fully resolved. Perhaps even more importantly, our results show that these examples of weak dissipating Euler flows are not isolated singular examples, but belong to a general, essentially generic class of solutions at this low regularity.