The Partial Regularity Theory of Caffarelli, Kohn, and Nirenberg and its Sharpness

This monograph focuses on the partial regularity theorem, as developed by Caffarelli, Kohn, and Nirenberg (CKN), and offers a proof of the upper bound on the Hausdorff dimension of the singular set of weak solutions of the Navier-Stokes inequality, while also providing a clear and insightful presentation of Scheffer's constructions showing their bound cannot be improved. A short, complete, and self-contained proof of CKN is presented in the second chapter, allowing the remainder of the book to be fully dedicated to a topic of central importance: the sharpness result of Scheffer. Chapters three and four contain a highly readable proof of this result, featuring new improvements as well. Researchers in mathematical fluid mechanics, as well as those working in partial differential equations more generally, will find this monograph invaluable.

Provides a simple proof of the classical Caffarelli-Kohn-Nirenberg theorem with brevity and completeness Promotes understanding of Scheffer's constructions by providing streamlined proofs based on his arguments Explains the geometric building blocks of the constructions by presenting numerous helpful figures

Zusammenfassung

"This is a well written, and this makes it easy to read, mathematical text. ... Essentially self-contained, the book can be used as a straightforward introduction to the topic of regularity of solutions of the Navier-Stokes equations." (Florin Catrina, zbMATH 1441.35004, 2020)

Inhalt
1 Introduction.- 2 The Caffarelli-Kohn-Nirenberg theorem.- 3 Point blow-up.- 4. Blow-up on a Cantor set.