Some regularity properties for two equations arising from flows

Abstract

In this thesis, which consists of two parts, we study properties of certain solutions to equations arising from two different flow equations. In the first part of this thesis, we prove that the Ricci flow starting from any asymptotically flat manifold with its scale-invariant integral norm of curvature small relative to the inverse of its Sobolev constant exists for all positive times and converges to flat Euclidean space. To do this we show that a uniform scalar curvature-weighted Sobolev inequality holds along such a Ricci flow. In the second part of this thesis, we prove a Caffarelli–Kohn–Nirenberg-type partial regularity result for suitable weak solutions of the five-dimensional stationary hyperdissipative Navier–Stokes equations. Specifically, using methods connected to the extension theory for the fractional Laplacian we show that the 7 − 6s-dimensional Hausdorff measure of the singular set S of such solutions is zero, where s ∈ (1,2) is the power of the fractional Laplacian (−∆)^s in the equations and S is the set of points at which a weak solution is not locally bounded.