## Seminar:

Fall 2009, Thursdays, Milner 216, 4:00-4:50 PM
__ Date:__ ** September 24 **

**" The Navier-Stokes Equations. A Bit of Theory and Numerics "**

**Abner Salgado**

__Abstract__ The flow of a viscous incompressible fluid can be fully described if we know the density, velocity and pressure at every point. A mathematical description of their interaction is given by the Navier-Stokes equations, which is a system of partial differential equations that presents hyperbolic, parabolic and elliptic features. Although these equations have been known since the XVIII century, from the mathematical point of view we still know very little about them.

In this talk I will scratch the surface of the theory on these equations in the constant density case, listing the main results that have been obtained so far and trying to describe (in simple terms) the main open problems and, to the best of my understanding, why do these have proved so difficult.

As a second part, I will describe one of the most popular methods for the numerical solution of these equations. Namely, the so-called projection methods. I will describe the very simple idea that lies behind them and present the most relevant results and open problems in the numerical analysis of these methods.

Finally, if time allows, I will describe the case when the density is variable and some of the work that has been recently done in the numerical analysis of this system.

This talk should appeal not only to people that are interested in PDE's or numerical analysis. After all, what I will be talking about is real-world phenomena. If this is not convincing enough, I promise to make the talk more entertaining by showing several movies of numerical simulations that I have carried out.