Luis Caffarelli & Alexis Vasseur on Drift Diffusion Equations
New Hot Paper Commentary, September 2011
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Article: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation
Authors: Caffarelli, LA;Vasseur,
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Luis A. Caffarelli & Alexis F. Vasseur talks with ScienceWatch.com and answers a few questions about this month's New Hot Paper in the field of Mathematics.
Why do you think your paper is highly
cited?
The interplay between viscosity and non-linear transport is one of the central issues in fluid dynamics. It is thought that understanding the regularity or lack of it may provide some insights about the difficult problem of turbulent flows. Our paper provides such regularity results for a simple model of climatology: the Surface Quasi-Geostrophic equation.
Does it describe a new discovery, methodology, or
synthesis of knowledge?
Coauthor Alexis F. Vasseur.
Actually, our approach to this problem was motivated by ideas of De Giorgi's solution of the 19th Hilbert problem in 1957. The novelty is to apply this method in the context of strongly nonlocal operators. We developed a theory which can be used in general situations like models of transport of scalars in turbulent fluid, phase transition, or nonlocal image processing.
Would you summarize the significance of your paper
in layman's terms?
In viscous fluids or diffusive processes, disturbations, like waves or shocks, tend to be attenuated. We show how, in a climatology model, the Surface Quasi-Geostrophic equation, the presence of viscosity implies that ocean surface temperature changes continuously across the surface and in time.
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 have worked for some time both in nonlocal elliptic theory and mathematical treatment of fluid mechanics. The Surface Quasi-Geostrophic equation, because of its criticality, provided a natural, challenging case to study. The main challenges were to grasp the apparent nonlocality of the problem. We handled this by the introduction of an extra dimension. Since then, we have refined that methodology to treat rather general nonlocal variational problems.
Where do you see your research leading in the
future?
This result is part of a larger program of systematic study of nonlinear
nonlocal operators. The mathematical research in this area is growing fast.
The models involved are used in a wide set of applications in such areas as
engineering, biology, economics and finance, physics, and
others.
Luis A. Caffarelli
Professor
Department of Mathematics
Institute for Computational Engineering and Sciences
University of Texas at Austin
Austin, TX, USA
Alexis F. Vasseur
Professor
Mathematical Institute
University of Oxford
Oxford, UK
KEYWORDS: DRIFT DIFFUSION EQUATIONS, FRACTIONAL DIFFUSION, QUASI-GEOSTROPHIC EQUATION, NAVIER-STOKES EQUATIONS, GLOBAL WELL-POSEDNESS, REGULARITY, BEHAVIOR, FLOWS.