# University of Minnesota

Aerospace Engineering and Mechanics

**Spring 2001 Seminar Series**

*On self-similar singular solutions of the incompressible Navier-Stokes
equations and the Complex Ginzburg-Landau equations*

*Abstract*

### The topic of this talk is motivated by the open problem of regularity of
solutions of the incompressible Navier-Stokes equations (NSE). It is well-known
that, in three-dimensions, the non-linearity in the NSE is super-critical. This
means that - unless some hidden and yet undiscovered restrictions on solutions
are taken into account - the energy dissipation may by itself be insufficient
to prevent a breakdown of solutions which are initially smooth. An example of a
simpler PDE where one can study this phenomena is the Complex Ginzburg Landau
Equation (CGL). The simplest conceivable singularities for both NSE and CGL
which would be compatible with energy dissipation are the self-similar
singularities. For the NSE, the question about the existence of such
singularities was raised already in 1934 by J. Leray. During the last few years
a significant progress has been made in our understanding of these questions,
and in the talk I will explain these new results.

### Friday, January 26, 2001

209 Akerman
Hall

2:30-3:30 p.m.

### Refreshments served after the seminar in
227 Akerman Hall.

Disability accomodations provided upon request.

Contact Kristal Belisle, Senior
Secretary, 625-8000.