Since the seminal paper by Perona and Malik nonlinear diffusions have successfully been used for various image processing tasks. They also have attracted steadfast interest in the mathematical community. In this talk we will give an historical overview of the developments on the Perona-Malik equation and describe two new nonlinear diffusions which resolve the main mathematical shortcomings of Perona-Malik without sacrificing but rather enhancing the cherished practical qualities of the Perona-Malik model. The new equations, while
well-posed and purely diffusive, exhibit a non trivial dynamical behavior, which makes them mathematically interesting and practically
effective.

The question of global regularity for the 3D Navier-Stokes equations is a major open problem in applied analysis. It is well-known that the existence and uniqueness of strong solutions could be obtain under suitable additional assumptions. In this talk I will review some old and new results about the sufficient conditions for the global regularity to the 3D Navier-Stokes equations.

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.