H-Principle and Fluid Dynamics

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Fluid dynamics is the branch of physics that studies the motion of fluids, that are liquids and gases, and it uses sophisticated mathematics to describe the motions, make useful predictions and find efficient ways to solve practical problems in science and engineering. While much of the theory behind such descriptions are rooted in the physics of the nineteenth century, due to the complexity of the mathematical objects involved in them many aspects are still poorly understood. Two problems are of primary interest in this research project. (1) In theory the mathematical models should determine the future evolution of the fluid when the forces acting on it and its initial status are known; nonetheless a rigorous mathematical proof of this expectation is missing for the simplest and most popular models. In layman terms the reason is that we do not understand "how and why" fluid motions which are apparently "smooth and calm" evolve into "chaotic and turbulent" ones. (2) Even if we expect the complete predictability of the models, nonetheless the computations are often so complex that become practically impossible. For this reason scientists have developed statistical descriptions of the motions, which allow to infer their typical behavior without resorting to heavy computations. The rigorous justification of these statistical descriptions from the primary mathematical models describing the motions is still a big challenge of modern science.This project is concerned with some of the most popular systems of partial differential equations used in the mathematics of fluid dynamics, the Euler and Navier-Stokes equations. The three main themes of the project are the following. (A) The convergence of solutions of the Navier-Stokes equations to the Euler equations when the viscosity of the fluid goes to zero. A primary concern is a tenet of the theory of turbulence that the rate of energy dissipation is, for fluid motions of a fixed macroscopic scale, independent of the viscosity. The recent resolution of a well-known conjecture of Lars Onsager on the energy dissipation for low regularity solutions of Euler could pave the way to a mathematical proof of the existence of such anomalous dissipation. (B) Regularity and singularity of the solutions. One of the celebrated Millennium prize problems asks whether smooth solutions of the Navier-Stokes equations develop singularities in finite time. The Navier-Stokes equations can be "embedded" in a one-parameter family of models where the parameter describes the regularizing strength of the viscous term. While some of the known results concerning the Navier-Stokes equations have been extended to the latter family, there is still quite a few which lack a suitable counterpart. (C) Rigidity and flexibility of the isometric embeddings of Riemannian manifolds. The recent resolution of the Onsager conjecture mentioned at point (A) is based on the discovery of the project leader and Laszlo Szekelyhidi Jr. of unexpected connections with a classical problem in Riemannian geometry, solved by John Nash in the fifties. The counterpart of the Onsager conjecture in the geometric framework is however not yet fully solved.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.