This project is concerned with solutions of the Navier-Stokes equations, which are the principal model for flow of a viscous incompressible fluid. The investigator addresses questions regarding regularity, qualitative behavior of solutions (size, complexity, decay), qualitative description of attractors, determination of solutions by observables, and minimal scale. Similar questions are addressed for the Euler equations, Kuramoto-Sivashinsky equation, and the complex Ginzburg-Landau equation.
The understanding of dynamics of a fluid flow is important in many applied fields, such as oceanography, atmospheric science, aerodynamics, and turbulence theory. The Navier-Stokes equations are the main model for study of the viscous fluid flow. The equations have a rich history and have been a subject of intense study due to their immense challenges and their direct connections with applications. The investigator studies regularity questions and qualitative behavior of solutions. Potential applications include better understanding of small structures of turbulent flows (vortices, oscillations), reconstructing dynamics from measurements, and a rigorous interpretation of numerical simulations of a fluid motion.
