Nonlinear Dynamics and PDE Mini-Conference

Fall, 2007

Schedule

Abstracts

Alp Eden, Department of Mathematics, Bogazici University:

In this talk I will review some of the work done jointly with C. Babaoglu, S. Erbay, H. Erbay, G. Muslu, I. A. Topaloglu and E. Kuz on the generalized Davey-Stewartson equations in the purely elliptic case. I will cast these equations as a Nonlinear Schroedinger Equation (NLS) with cubic like non-local non-linearity in 2D and discuss some of the conserved quantities and their relation to the global existence of solutions; virial identity and its relation with blow-up of solutions ; the pseudoconformal invariance and its relation with the time asymptotics of solutions. I will also review the on-going work on the existence and uniqueness of solutions for NLS equations with non-local terms on various Sobolev spaces in the spirit of Kato's[1987] work for NLS.


Luan Thach Hoang, School of Mathematics, University of Minnesota:

We study the asymptotic behavior of the statistical solutions to the 3D Navier-Stokes equations using a normalization map developed by Foias and Saut. It is then applied to the study of mean energy, mean dissipation rate of energy, and mean helicity of the spatial periodic flows driven by potential body forces. The statistical distribution of the asymptotic Beltrami flows are also investigated. We connect our mathematical analysis with the empirical theory of decaying turbulence. With appropriate mathematically defined ensemble averages, the Kolmogorov universal features are shown to be transient in time. We provide an estimate for the time interval in which those features may still be present.

Igor Kukavica, Department of Mathematics, USC:

TBA


Pablo Padilla, Department of Mathematics, University of Mexico:

We consider dynamical systems generated by ordinary and partial differential equations. By studying area functionals in phase space, we formulate variational principles in order to determine the existence of particular invariant sets. These can also be implemented numerically and several applications to heteroclinic and periodic orbits are presented.


Edriss Titi, Department of Mathematics, U.C. Irvine:

The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, the so-called the ``Primitive Equations'', is often prohibitively expensive computationally, and hard to study analytically. In this talk I will survey the main obstacles in proving the global regularity for the three-dimensional Navier--Stokes equations and their geophysical counterparts. Even though the Primitive Equations look as if they are more difficult to study analytically than the three-dimensional Navier--Stokes equations I will show in this talk that they have a unique global (in time) regular solution for all initial data.