Intermittent Convex Integration for the 3D Euler Equations : : (AMS-217) /

Intermittent Convex Integration for the 3D Euler Equations : : (AMS-217) / / Matthew Novack, Vlad Vicol, Nader Masmoudi, Tristan Buckmaster.

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

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Bibliographic Details
Superior document:Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2023 English
VerfasserIn:
Buckmaster, Tristan,
Masmoudi, Nader,
Novack, Matthew,
Place / Publishing House:Princeton, NJ : : Princeton University Press, , [2023]
©2023
Year of Publication:2023
Language:English
Series:Annals of Mathematics Studies ; 217
Online Access:
Physical Description:1 online resource (256 p.) :; 11 b/w illus.
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Table of Contents:
  • Frontmatter
  • Contents
  • 1. Introduction
  • 1.1 Context and motivation
  • 1.2 Ideas and difficulties
  • 1.3 Organization of the book
  • 1.4 Acknowledgments
  • 2. Outline of the convex integration scheme
  • 2.1 A Guide to the Parameters
  • 2.2 Inductive Assumptions
  • 2.3 Intermittent Pipe Flows
  • 2.4 Higher Order Stresses
  • 2.5 Cutoff Functions
  • 2.6 The Perturbation
  • 2.7 The Reynolds Stress Error and Heuristic Estimates
  • 3. Inductive Assumptions
  • 3.1 General Notations
  • 3.2 Inductive Estimates
  • 3.3 Main Inductive Proposition
  • 3.4 Proof of Theorem 1.1
  • 4. Building Blocks
  • 4.1 A Careful Construction of Intermittent Pipe Flows
  • 4.2 Deformed Pipe Flows and Curved Axes
  • 4.3 Placements Via Relative Intermittency
  • 5. Mollification
  • 6. Cutoffs
  • 6.1 Definition of the Velocity Cutoff Functions
  • 6.2 Properties of the Velocity Cutoff Functions
  • 6.3 Definition of the Temporal Cutoff Functions
  • 6.4 Estimates on Flow Maps
  • 6.5 Stress Estimates on the Support of the New Velocity Cutoff Functions
  • 6.6 Definition of the Stress Cutoff Functions
  • 6.7 Properties of the Stress Cutoff Functions
  • 6.8 Definition and Properties of the Checkerboard Cutoff Functions
  • 6.9 Definition of the Cumulative Cutoff Function
  • 7. From q to q + 1: Breaking Down the Main Inductive Estimates
  • 7.1 Induction on Q
  • 7.2 Notations
  • 7.3 Induction on ñ
  • 8. Proving the Main Inductive Estimates
  • 8. 1 Definition of R̊q, n̄, p̄ and Wq+1, n̄, p̄
  • 8. 2 Estimates For Wq+1, n̄, p̄
  • 8.3 Identification of Error Terms
  • 8.4 Transport Errors
  • 8.5 Nash Errors
  • 8.6 Type 1 Oscillation Errors
  • 8.7 Type 2 Oscillation Errors
  • 8.8 Divergence Corrector Errors
  • 8.9 Time Support of Perturbations and Stresses
  • 9. Parameters
  • 9.1 Definitions and Hierarchy of the Parameters
  • 9.2 Definitions of the Q-Dependent Parameters
  • 9.3 Inequalities and Consequences of the Parameter Definitions
  • 9.4 Mollifiers and Fourier Projectors
  • 9.5 Notations
  • Appendix A: Useful Lemmas
  • Introduction
  • A.1 Transport Estimates
  • A.2 Proof of Lemma 6.2
  • A.3 Lp Decorrelation
  • A.4 Sobolev Inequality with Cutoffs
  • A.5 Consequences of the Faà di Bruno Formula
  • A.6 Bounds for Sums and Iterates of Operators
  • A.7 Commutators with Material Derivatives
  • A.8 Intermittency-Friendly Inversion of the Divergence
  • Bibliography
  • Index

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