Intermittent Convex Integration for the 3D Euler Equations

Tristan Buckmaster, Nader Masmoudi, Matthew Novack, Vlad Vicol

• Publisher: Princeton University Press
• Year: 2023
• Print ISBN: 9780691249544, 9780691249551
• eBook ISBN: 9780691249568
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

A new threshold for the existence of weak solutions to the incompressible Euler equations

To gain insight into the nature of turbulent fluids, mathematicians start from experimental facts, translate them into mathematical properties for solutions of the fundamental fluids PDEs, and construct solutions to these PDEs that exhibit turbulent properties. This book belongs to such a program, one that has brought convex integration techniques into hydrodynamics. Convex integration techniques have been used to produce solutions with precise regularity, which are necessary for the resolution of the Onsager conjecture for the 3D Euler equations, or solutions with intermittency, which are necessary for the construction of dissipative weak solutions for the Navier-Stokes equations. In this book, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions enjoy a spatial regularity index in L^2 that can be taken as close as desired to 1/2, thus lying at the threshold of all known convex integration methods. This property matches the measured intermittent nature of turbulent flows. The construction of such solutions requires technology specifically adapted to the inhomogeneities inherent in intermittent solutions. The main technical contribution of this book is to develop convex integration techniques at the local rather than global level. This localization procedure functions as an ad hoc wavelet decomposition of the solution, carrying information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates.

Table of contents

• Cover
• Contents
• 1. Introduction
• 2. Outline of the convex integration scheme
• 3. Inductive assumptions
• 4. Building blocks
• 5. Mollification
• 6. Cutoffs
• 7. From q to q + 1: breaking down the main inductive estimates
• 8. Proving the main inductive estimates
• 9. Parameters
• Appendix A: Useful Lemmas
• Bibliography
• Index