Regularity of weak soutions to the Navier-Stokes equations

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Department of Mathematics

Synopsis:

The goal of the subject is to inform students about the basics of the regularity theory for the weak solutions of the Navier-Stokes equtions (NSE) for the incompressible fluid. The content of the subject: the description of NSE, the introduction of the fundamental concepts from the mathematical theory of NSE, the definition of the basic function spaces, the definition of the weak solution, a brief proof of the existence of the weak solution by the Galerkin method, structure theorem, epochs of irregularity, Hausdorff measure and dimension, parabolic measure, the size of the set of time singular points, the definition of the suitable solution, regular and singular points in spacetime, partial regularity, local regularity conditions, dimension of the set of singular points, conditional regularity, Prodi-Serrin conditions, conditional regularity in terms of one or two components of the velocity field, conditional regularity in terms of some items of the velocity gradient, conditional regularity in terms of pressure, pressure gradient, vorticity and other quantities.

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