Regular and Singular Incompressible Fluid Flows

A variety of systems in natural sciences are described through physically measurable quantities which depend on each other. For instance, we routinely measure the pressure and the temperature of the air in the Earth’s atmosphere, and such measurements depend upon the time and the location of the device used. Several fundamental laws discovered by scientists during the last three centuries give relations among the rates of change of such physical quantities and the resulting mathematical objects, called partial differential equations, are therefore ubiquitous in modern science and engineering. The partial differential equations describing the motion of incompressible, viscous and ideal fluids date back to the seventeenth and eighteenth centuries. Nonetheless, a rigorous mathematical understanding of many properties of their solutions is still lacking and some reverberates in a poor understanding of certain fundamental phenomena. A pivotal example is the apparent incompatibility of the classical mathematical treatment of these equations with the basic observation in the theory of fully developed turbulence that, in the limit of the viscosity of the fluid tending to zero, turbulent flows dissipate kinetic energy. In fact, regular solutions of the equations describing the zero-viscosity limit can be proved to conserve the kinetic energy and are therefore at odds with the latter phenomenon. Starting from the latter problem as a pivotal one, this project aims to advance our understanding of other basic properties of solutions, such as regularity, uniqueness, and stability. The project provides research training opportunities for graduate students and supports the engagement of the principal investigator in popularizing mathematics to the general public. The project investigates two fundamental questions in incompressible fluid dynamics. The first goal is to find rigorous examples of the so-called "zero law of fully developed turbulence", namely the presence of anomalous dissipation in the zero-viscosity limit. The ideal solution of the latter problem is to give a proof of existence of a sequence of solutions to the incompressible Navier-Stokes equations with vanishing viscosity for which the dissipation rate of kinetic energy stays positive in the limit, without the introduction of spurious oscillations in the initial data. The second is the investigation of blow-up scenarios for smooth solutions of the Navier-Stokes and Euler equations. Both problems are formidable, and they have defied the efforts of mathematicians for decades. Given the size of the challenge, some effort will be dedicated to the investigation of simpler situations. An important example in the case of anomalous dissipation is the effect of forcing terms in the equations. An example in the case of the blow-up problem is understanding suitable deformations of the Navier-Stokes equations which embeds them in a higher parameter family of equations. 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.