Regularity properties of solutions to the 3D Navier-Stokes equations

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Cheskidov DMS-1517583 The investigator studies several fundamental open questions concerning the equations governing the motion of fluids, such as liquids or gases. These equations can explain and predict the pattern of circulation of the oceans, atmosphere, or blood in the cardiovascular system; predict changes in weather; describe the motion of a boat or an aircraft. They 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 solutions are still not known. A mathematical proof of existence of solutions would justify the equations definitively. A mathematical proof of a loss of uniqueness of solutions would identify limitations of the equations. The project is also expected to shed light on certain fundamental issues related to turbulence. Turbulence, sometimes referred to as the last unsolved problem in classical physics, is a crucial phenomenon occurring in many fluid flows, for instance those around airplane bodies, vehicles, ships, and blades of turbines. A better mathematical understanding of turbulence would lead to improvements in the design of these objects. Eighty years ago, Leray suggested that turbulence was due to formation of singularities in the flow and introduced a notion of weak solutions to the three-dimensional Navier-Stokes equations. The notion became a basic concept in the theory of partial differential equations. The aim of this proposal is to study regularity properties of weak solutions to the three-dimensional Navier-Stokes equations and their relation to turbulence. One direction of this work is to construct weak solutions with various abnormal behaviors, such as norm inflation, discontinuity, or norm discontinuity. Such phenomena may be a result of an instantaneous blow-up from the right, which is much easier to control than a blow-up of a smooth solution. When the forcing term in the equations is large, the long-time behavior of solutions to the three-dimensional Navier-Stokes equations becomes very complicated and often chaotic. For fluid flows this phenomenon is known as turbulence. Even when a velocity field in a turbulent flow is chaotic, experimental and numerical evidence show that the averaged velocity still displays some regular structure. The investigator studies Leray-Hopf weak solutions to the Navier-Stokes equations and proves some of the empirical laws of turbulence, using the Littlewood-Paley and atomic decompositions of the velocity vector field.