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

Description

A new threshold for the existence of weak solutions to 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. 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