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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

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

Pages: 409 - 427, Volume 3, Issue 3, September 2010      doi:10.3934/dcdss.2010.3.409

 
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Luis A. Caffarelli - Department of Mathematics, University of Texas at Austin, 1 University Station – C1200, Austin, TX 78712-0257, United States (email)
Alexis F. Vasseur - Department of Mathematics, University of Texas at Austin, 1 University Station – C1200, Austin, TX 78712-0257, United States (email)

Abstract: 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.

Keywords:  De Giorgi's method, fluid mechanic, regularity, Navier-Stokes equation, Quasi-geostrophic equation.
Mathematics Subject Classification:  Primary: 76D03, 35Q35, 35B65.

Received: February 2009;      Revised: March 2010;      Available Online: May 2010.