Regularity of the 3D Navier-Stokes equations in the largest critical space and related problems

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The regularity of solutions of the three-dimensional incompressible Navier Stokes equations (NSE) remains a significant problem. Even thought it is far from been solved, numerous regularity criteria have been proved since the work of Leray in the 1930s. The research funded with this award will extend classical regularity and uniqueness results, in particular, the recently established Escauriaza-Seregin-Sverak criterion, to wider classes of Besov spaces. The methods to be used will combine harmonic analysis tools and classical techaniques for the Navier-Stokes equations. In addition, it is planned to study global attractors of the 3D NSE and related models, such as the dyadic (a.k.a. shell) models of turbulence. The study of the structure of the global attractor is essential for a better understanding of turbulent phenomena. Due to the lack of a uniqueness proof, it is not known whether the 3D NSE possesses a semigroup of solution operators, and consequently a classical theory of semiflows cannot be used for this system. It is proposed to continue developing a theory of global attractors for an evolutionary system, which is a generalization of a dynamical system that can be applied in this situation.This award will support research on some 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, the existence and uniqueness of (classical) solutions is still not known. An answer to this question is expected to shed light on fundamental issues related to turbulence. Turbulence, often referred to as the last unsolved problem in classical physics, is a fundamental phenomenon occurring in fluid flows around airplane bodies, vehicles, ships, and blades of turbines. A better mathematical understanding of turbulence will result in improvements in the design of these objects.