Small Scales in the Navier-Stokes Equations

University Of Southern California, Los Angeles CA

Investigators

Abstract

0072662 Kukavica This project will address definition, interplay, and rigorous estimates of various natural small scales arising in a viscous incompressible flow. In particular, we will consider those lengths that can be derived directly from solutions of the Navier-Stokes equations, which is the main model for a fluid flow. Examples of such scales are the those resulting from the Fourier spectrum of solutions, such as the inverse of the wave number at which the Fourier spectrum is cut off exponentially, and the length scales measuring complexity of level sets of solutions. Related problems will be addressed also for other dissipative partial differential equations arising in fluid dynamics, such as the Ginzburg-Landau and the Kuramoto- Sivashinsky equations. Navier-Stokes equations are perhaps the most widely studied system of nonlinear partial differential equations. They are generally believed to contain necessary ingredients to explain much of turbulence phenomena. This project will address properties of solutions which would have implications in understanding creation and properties of fine structures in a flow. Potential applications include quantifying complexity of a fluid flow, establishment of spectral properties of solutions, estimation of the size of the mesh needed to resolve a flow numerically, and information on locating observables for the flow's monitoring