Whether the 3D incompressible Euler equation can develop a finite time
singualrity from smooth initial data has been an outstanding open problem.
It has been believed that a finite singularity of the 3D Euler equation
could be the onset of turbulence. Here we review some existing computational
and theoretical work on possible finite blow-up of the 3D Euler equation.
Further, we show that there is a sharp relationship between the geometric
properties of the vortex filament and the maximum vortex stretching. By
exploring this geometric property of the vorticity field, we have obtained
a global existence of the 3D incompressible Euler equation provided that
the normalized unit vorticity vector has certain mild regularity property
in a very localized region containing the maximum vorticity. Our assumption
on the local geometric regularity of the vorticity field is consistent
with recent numerical experiments. Further, we discuss how viscosity may
help preventing singularity formation for the 3D Navier-Stokes equation,
and present a new result on the global existence of the viscous Boussinesq
equation.

This talk outlines recent work by Feldman, Ha, and Slemrod on the dynamics of the sheath boundary layer which occurs in a plasma consisting of ions and electrons. The equations for the motion are derived from the classical Euler- Poisson equations. Of particular interest is that the boundary layer interface moves via motion by mean curvature where the acceleration of the front (not the velocity) is proportional to the mean curvature of the front.

In this talk, he will discuss the regularization effect of the noise in ordinary and partial differential equations. The main results are the existence and uniqueness of strong solutions for nonlinear equations when the drift coefficient is not Lipschitz. The proofs of these results are based on the Girsanov transformation of measure. Some recent regularization results by fractional noise will be also presented.

The NLS equation describes solitonic transmission in
fiber optic communication and is generically encountered in propagation through nonlinear media. One of its most important aspects is its modulational instability: regular wavetrains are unstable to modulation and break up to more complicated structures.

The IVP for the NLS equation is solvable by the method of inverse scattering. The initial spectral data of the Zakharov Shabhat (ZS) operator, a particular linear operator having the solution to NLS as the potential, are calculated from the initial data of the NLS; they evolve in a simple way as a result of the integrability of the problem, and produce the solution through the inverse spectral transformation.

In collaboration with A. Tovbis, we have developed
a one parameter family of initial data for which the derivation of the spectral data is explicit. Then, in collaboration with A. Tovbis and X. Zhou, we have obtained the following results:

1) We prove the existence and basic properties of the
first breaking curve (curve in space-time above which the character
of the solution changes by the emergence of a new
oscillatory phase) and show that for pure radiation
no further breaks occur.

2) We construct the solution beyond the first break-time.

3) We derive a rigorous estimate of the error.

4) We derive rigorous asymptotics for the large
time behavior of the system in the pure radiation case.