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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.

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