The problems of blow-up for the Navier-Stokes and Euler equations have been extensively studied for decades using different techniques. This research is aimed at developing a new unified approach to the blow-up problem for the equations of incompressible fluid motion. Motivated by Kolmogorov's theory of turbulence, the project introduces a time-dependent dissipation wavenumber that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. This frequency separation will be used to obtain new regularity criteria, uniqueness conditions, local wellposedness results, conditions preventing type I blow-up. The developed technique will also be applied to related equations. The methods to be used will combine harmonic analysis tools and classical techniques for the Navier-Stokes and Euler equations.
This research is devoted to several fundamental open questions concerning the equations of fluid motion. The equations were introduced almost two centuries ago but are still not well understood mathematically. Even though the equations are broadly used by physicists and engineers for real-life applications, and are widely believed to be an accurate representation of the physical phenomena involved, the existence and uniqueness of solutions is still not known. A mathematical proof of existence of solutions would justify the equations definitively. The proposed research is also expected to shed light on certain fundamental issues related to turbulence. Turbulence, often referred to as the last unsolved problem in classical physics, is a crucial phenomenon occurring in fluid flows around airplane bodies, vehicles, ships, and blades of turbines. A better mathematical understanding of turbulence will lead to improvements in the design of these objects.
