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September  2010, 3(3): 409-427. doi: 10.3934/dcdss.2010.3.409

The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics

1. 

Department of Mathematics, University of Texas at Austin, 1 University Station – C1200, Austin, TX 78712-0257, United States, United States

Received  February 2009 Revised  March 2010 Published  May 2010

This paper is dedicated to the application of the De Giorgi-Nash-Moser kind of techniques to regularity issues in fluid mechanics. In a first section, we recall the original method introduced by De Giorgi to prove $C^\alpha$-regularity of solutions to elliptic problems with rough coefficients. In a second part, we give the main ideas to apply those techniques in the case of parabolic equations with fractional Laplacian. This allows, in particular, to show the global regularity of the Surface Quasi-Geostrophic equation in the critical case. Finally, a last section is dedicated to the application of this method to the 3D Navier-Stokes equation.
Citation: Luis A. Caffarelli, Alexis F. Vasseur. The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 409-427. doi: 10.3934/dcdss.2010.3.409
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