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

1. Front Matter

   Pages i-vi

   PDF

2. Introduction

   • Wojciech S. Ożański

   Pages 1-11

3. The Caffarelli–Kohn–Nirenberg Theorem

   • Wojciech S. Ożański

   Pages 13-36

4. Weak Solution of the Navier–Stokes Inequality with a Point Blow-Up

   • Wojciech S. Ożański

   Pages 37-102

5. Weak Solution of the Navier–Stokes Inequality with a Blow-Up on a Cantor Set

   • Wojciech S. Ożański

   Pages 103-131

6. Back Matter

   Pages 133-138

   PDF

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.

• Cafarelli-Kohn-Nirenberg partial regularity theorem
• Caffarelli-Kohn-Nirenberg book
• Caffarelli-Kohn-Nirenberg theorem simple
• Caffarelli-Kohn-Nirenberg inequalities
• Caffarelli-Kohn-Nirenberg
• Scheffer constructions
• Partial regularity theory
• Navier-Stokes equations
• Weak solutions to the Navier-Stokes inequality
• Vladimir Scheffer
• Leray-Hopf weak solutions
• Local energy inequality
• partial differential equations
• fluid- and aerodynamics

“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)

• Department of Mathematics, University of Southern California, Los Angeles, USA

  Wojciech S. Ożański

Policies and ethics