Parabolic Harnack inequalities were proved by Moser for linear equation with bounded and measurable coefficients. In the case of the parabolic p-Laplacean such kind of estimates cannot hold (as proved by Trudinger). In the nineties DiBenedetto introduced the so called intrinsic Harnack inequalities for the protype equation. His original proof requiries the maximum principle and the existence of suitable subsolutions. Therefore the proof for general equations (with bounded and measurable coefficient) was missing. In some recent papers, in collaboration with DiBendetto and Gianazza, we proved intrinsic Harnack inequalities for general degenerate and singular operators. We show, via suitable counterexamples, that such estimates are sharp. Moreover we proved that when p is approaching to 2, our estimates tend to the classical Moser estimates.

The Primitive Equations are a fundamental model describing large scale oceanic and atmospheric processes. They are derived from the fully compressible Navier-Stokes equations on a combined basis of scale analysis and meteorological data. While an extensive body of mathematical literature exists in the study of these systems, very little is known in the stochastic setting. In this talk we discuss recent joint work with M. Ziane concerning existence and uniqueness of solutions for the 2-D equations in the presence of multiplicative noise terms.

The two-layer beta-plane quasi-geostrophic (QG) model plays a central role in theoretical studies of atmospheric and oceanic dynamics. It is a pair of coupled non-linear partial differential equations involving functions of two space variables and one time variable (streamfunctions for coupled two-dimensional flows). Solutions represent flows in a sense intermediate between 2-d and 3-d flows: they have a mild form of the ``vortex stretching'' process, absent in 2-d flows, that is at the heart of the difficulty in proving the long-time existence of classical solutions to the
3-d Navier-Stokes equations.

Numerical solutions to these QG equations display analogues of important features of atmospheric and oceanic flow, some of which I will illustrate. As is true of climate models, many interesting features are revealed only by long time averaging of the numerical solutions. The results I will present, on long-time existence of regular solutions and on dissipativity,
are part of an effort to provide a rigorous justification for this averaging, something beyond our reach in the case of the vastly more complicated climate models.

The talk will place the model in the context of other QG models, point out a useful formal similarity to the Kuramoto-Sivishinsky equation, and sketch proofs of the main results. The work is joint with C. Foias, C. Onica, E. Titi, and M. Ziane.

Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust
analysis is proposed, and new criteria for when calculations should be terminated are given. Most of the analysis is on a $512\times 128 \times 2048$ mesh, with new analysis on a just completed $1024\times 256\times 2048$ used to confirm trends. The qualitative conclusions of Kerr (1993) are supported, but most of the proposed scaling laws will have to be modified. Assume enstrophy growth like $\Omega\sim (T_c-t)^{-\gamma_\Omega}$ and vorticity growth like $||\omega||_\infty \sim (T_c-t)^{-\gamma}$. Present results would support $\gamma_\Omega\rightarrow 1/4-1/2$ and $\gamma>$. The results are not conclusive since they require higher resolution calculations (work in progress) to further confirm the trends.

Classical ideal fluid motion is described by Euler and Navier-Stokes equations. For real fluids, their motions are more complicated and governed by Euler and Navier-Stokes equations coupled with various constitutive equations. We study viscoelastic models whose motions are
carried out by the competition between the kinetic energies and internal elastic energies. The deformation tensor plays an essential role in our studies. We will present how to use the heuristics coming from the special
structure of the deformation tensor to establish the global well-posedness results for several viscoelastic models, but will focus on a 2D Strain-Rotation model.