May 15 Vortex sheets are asymptotic models for the motion of infinitely thin shear layers in slightly viscous liquids. For two dimensional motion, D.W. Moore's pioneering work gives rise to the interpretation that branch point singularities in the sheet location are present in the complex plane of the circulation variable. These singularites move towards and reach the real axis in finite time. A question arises on the origin of such singularites when the initial data is analytic. By continuing the evolution equations into the complex plane, we can see how these singularities form immediately at certain points in the complex plane.
Recent work indicates such singularities are also present in axi-symmetric flow.
The presence of these singularities makes numerical work very difficult. One needs a highly accurate method to generate the tail of the Fourier spectrum, yet one must deal with the inherent ill-posed nature of the motion. In this talk, will be presented the existence of a compact global attractor for the Navier-Stokes equations of compressible flow in one space dimension, as well as an estimate the number of its determining modes in terms of the force and a scale invariant of the system. June 5 We will review recent work on stochastic Navier-Stokes, stochastic Burgers and stochastic passive scalar equations, with the objective of understanding the statistical aspects of turbulence. We will discuss the issues of existence and uniqueness of invariant measures, characterization of solutions supported on the invariant measures, scaling of structure functions, probability density functions, etc.