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The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics. (English) Zbl 1210.76039

Summary: 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.

MSC:

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
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