Georgia Institute of Technology College of Sciences

In this talk, we will present some results on global classical solution to the two-dimensional compressible Navier-Stokes equations with density-dependent of viscosity, which is the shear viscosity is a positive constant and the bulk viscosity is of the type $\r^\b$ with $\b>\frac43$. This model was first studied by Kazhikhov and Vaigant who proved the global well-posedness of the classical solution in periodic case with $\b> 3$ and the initial data is away from vacuum. Here we consider the Cauchy problem and the initial data may be large and vacuum is permmited. Weighted stimates are applied to prove the main results.

I will discuss a two dimensional spatial pattern formation problem proposed by Doelman, Sandstede, Scheel, and Schneider in 2003 as a phenomenological model of convective fluid flow . In the same work the authors just mentioned use geometric singular perturbation theory to show that the coexistence of certain spatial patterns is equivalent to the existence of some heteroclinic orbits between equilibrium solutions in a four dimensional vector field. More recently Andrea Deschenes, Jean-Philippe Lessard, Jan Bouwe van den Berg and the speaker have shown, via a computer assisted argument, that these heteroclinic orbits exist. Taken together these arguments provide mathematical proof of the existence of some non-trivial patterns in the original planar PDE. I will present some of the ingredients of this computer assisted proof.

The Steenrod algebra consists of all natural transformations of cohomology over a prime field. I will present work of Milnor showing that the Steenrod algebra also has a natural coalgebra structure and giving an explicit description of the dual algebra.

In this talk, we propose moment identities for point processes. After revisiting the case of Poisson point processes, we propose a direct approach to derive (joint factorial) moment identities for point processes admitting Papangelou intensities. Applications of such identities are given to random transformations of point processes and to their distribution invariance properties.