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

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  • © 2019

Overview

Authors:
  1. Wojciech S. Ożański

    1. Department of Mathematics, University of Southern California, Los Angeles, USA

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  • 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

Part of the book series: Advances in Mathematical Fluid Mechanics (AMFM)

Part of the book sub series: Lecture Notes in Mathematical Fluid Mechanics (LNMFM)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-vi
    Download chapter 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
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Keywords

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.

Reviews

“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)

Authors and Affiliations

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

    Wojciech S. Ożański

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