Alexander Kiselev & Fedor Nazarov on the Quasi-Geostrophic (SQG) Equation
Fast Moving Front Commentary, November 2010
 Alexander Kiselev |
ArticlE: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
Authors: Kiselev, A;Nazarov, F;Volberg,
A |
Alexander Kiselev & Fedor Nazarov talk with ScienceWatch.com and answer a few questions about this month's Fast Moving Fronts paper in the field of Mathematics.
Why do you think your paper is highly
cited?
In this paper we introduced a new technique for studying solutions of the surface quasi-geostrophic (SQG) equation. This equation comes from atmospheric science and describes phenomena such as temperature front formation. The new technique allowed us to understand the behavior of solutions in an important particular case, and turned out to be useful in other problems as well.
Does it describe a new discovery, methodology, or
synthesis of knowledge?
The paper introduces a new idea useful for studying solutions of a class of equations in fluid mechanics.
Would you summarize the significance of your paper in layman's terms?
Fluid mechanics equations describe a wide range of phenomena in nature and engineering. They are some of the trickiest equations to study, with solutions that may be quite unstable. The flow of water in a pipe or in a river, fronts in the atmosphere, plane wake, and nuclear combustion in stars are all processes described by these equations, and give an idea of the complexity solutions can exhibit.
Coauthor Fedor Nazarov
An especially interesting phenomena is formation of singularities in solutions. Singularities of solutions are important, since, depending on the situation, they may describe catastrophic events, suggest transition to a different regime (like turbulence), or indicate a failure of the model.
Some of the biggest questions in mathematical fluid mechanics focus on singularities and their role (including one of the millennium problems by the Clay Institute). We ruled out singularity formation in one particular equation (critical SQG), which models the temperature dynamics on Earth's surface.
How did you become involved in this research, and
how would you describe the particular challenges, setbacks, and
successes that you've encountered along the way?
We learned about this problem from others (Peter Constantin and Diego Cordoba). We liked this problem since it combines ideas from different parts of mathematics and addresses a physically relevant question. Initially, we worked on a simpler problem but then realized that the technology generalizes to the SQG equation.
Where do you see your research leading in the
future?
We are investigating whether the ideas of this paper can be useful in studying wider classes of equations, such as Euler equations, which describe a flow of any incompressible fluid. Better understanding of the solutions of basic fluid mechanics equations is likely to improve our understanding of many processes, such as transition to turbulence, small-scale structures, and stability. Such knowledge helps to produce better modeling for weather forecasting and engineering problems involving fluids.
Do you foresee any social or political
implications for your research?
Perhaps we can get some extra funds to create jobs and hire postdocs and
graduate assistants, since there are really many questions around the
techniques introduced in this paper to think about. On a more serious note,
we do not anticipate any implications of that sort, at least in the visible
future.
Professor Alexander Kiselev
Department of Mathematics
University of Wiscosin, Madison
Madison, WI, USA
Professor Fedor Nazarov
Department of Mathematics
University of Wisconsin, Madison
Madison, WI, USA
Professor Alexander Volberg
Department of Mathematics
Michigan State University
East Lansing, MI, USA
KEYWORDS: quasi-geostrophic equation; Besov space; Littlewood-Paley decomposition; well-posedness.