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About this book

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.

Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract
We present the context of the partial regularity theory of the three-dimensional incompressible Navier-Stokes equations due to Caffarelli, Kohn and Nirenberg.
Wojciech S. Ożański

Chapter 2. The Caffarelli–Kohn–Nirenberg Theorem

Abstract
In this chapter, we prove the Caffarelli–Kohn–Nirenberg theorem (Theorem 1.7), where we will also assume that \(\nu =1\).
Wojciech S. Ożański

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

Abstract
In this chapter, we construct the first of Scheffer’s counterexamples that is a weak solution of the Navier–Stokes inequality that blows up in finite time at a point.
Wojciech S. Ożański

Chapter 4. Weak Solution of the Navier–Stokes Inequality with a Blow-Up on a Cantor Set

Abstract
In this chapter, we construct Scheffer’s counterexample (Theorem 1.8), that is, a weak solution of the Navier–Stokes inequality that blows up on a Cantor set \(S\times \{ T_0 \}\) with \(d_H (S)\ge \xi \) for any preassigned \(\xi \in (0,1)\).
Wojciech S. Ożański

Backmatter

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