Structure of the Book

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Euler Equations ◽  
Weak Solutions ◽  
Energy Levels ◽  
Main Lemma ◽  
Optimal Regularity ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Energy Part ◽  
Error Terms ◽  
Incompressible Euler

This chapter provides an overview of the book's structure. Section 3 deals with the error terms which need to be controlled, whereas Part III explains some notation of the book and presents a basic construction of the correction. The goal is to clarify how the scheme can be used to construct Hölder continuous weak solutions—continuous in space and time—to the incompressible Euler equations that fail to conserve energy. Part IV shows how to iterate the construction of Part III to obtain continuous solutions to the Euler equations. It then discusses the concept of frequency energy levels, along with the Main Lemma. It also highlights some additional difficulties which arise as one approaches the optimal regularity and illustrates how these difficulties can be overcome. Parts V–VII verify all the estimates needed for the proof of the Main Lemma.

The Main Iteration Lemma

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Euler Equations ◽  
Dimensional Analysis ◽  
Compact Support ◽  
Energy Levels ◽  
Main Lemma ◽  
Nonzero Solution ◽  
Galilean Invariance ◽  
Reynolds Equations ◽  
Scaling Symmetry ◽  
Onsager’S Conjecture

This chapter properly formalizes the Main Lemma, first by discussing the frequency energy levels for the Euler-Reynolds equations. Here the bounds are all consistent with the symmetries of the Euler equations, and the scaling symmetry is reflected by dimensional analysis. The chapter proceeds by making assumptions that are consistent with the Galilean invariance of the Euler equations and the Euler-Reynolds equations. If (v, p, R) solve the Euler-Reynolds equations, then a new solution to Euler-Reynolds with the same frequency energy levels can be obtained. The chapter also states the Main Lemma, taking into account dimensional analysis, energy regularity, and Onsager's conjecture. Finally, it introduces the main theorem (Theorem 10.1), which states that there exists a nonzero solution to the Euler equations with compact support in time.

scholarly journals Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Weak Solutions ◽  
Compact Support ◽  
Main Lemma ◽  
Three Dimensions ◽  
Nonlinear Pdes ◽  
Hölder Continuous ◽  
Nonlinear Phase ◽  
Nonuniqueness Of Solutions ◽  
Fluid Dynamics Equations ◽  
Onsager’S Conjecture

Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if their spatial regularity was below 1/3-Hölder. This book uses the method of convex integration to achieve the best-known results regarding nonuniqueness of solutions and Onsager's conjecture. Focusing on the intuition behind the method, the ideas introduced now play a pivotal role in the ongoing study of weak solutions to fluid dynamics equations. The construction itself—an intricate algorithm with hidden symmetries—mixes together transport equations, algebra, the method of nonstationary phase, underdetermined partial differential equations (PDEs), and specially designed high-frequency waves built using nonlinear phase functions. The powerful “Main Lemma”—used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem—has been extended to a broad range of applications that are surveyed in the appendix. Appropriate for students and researchers studying nonlinear PDEs, this book aims to be as robust as possible and pinpoints the main difficulties that presently stand in the way of a full solution to Onsager's conjecture.

Introduction

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Euler Equations ◽  
Ideal Fluid ◽  
Weak Solutions ◽  
Modern Language ◽  
Incompressible Euler Equations ◽  
Spatial Dimensions ◽  
Periodic Weak Solutions ◽  
Incompressible Euler ◽  
Onsager’S Conjecture

In the paper [DLS13], De Lellis and Székelyhidi introduce a method for constructing periodic weak solutions to the incompressible Euler equations{∂tv+div v⊗v+∇p=0                       div v=0in three spatial dimensions that are continuous but do not conserve energy. The motivation for constructing such solutions comes from a conjecture of Lars Onsager [Ons49] on the theory of turbulence in an ideal fluid. In the modern language of PDE, Onsager's conjecture can be translated as follows....

A Main Lemma for Continuous Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Euler Equations ◽  
Compact Support ◽  
Finite Time ◽  
Main Lemma ◽  
Time Interval ◽  
Finite Time Interval ◽  
Continuous Solutions ◽  
Reynolds Equations ◽  
Uniformly Continuous ◽  
Uniformly Bounded

This chapter introduces the Main Lemma that implies the existence of continuous solutions. According to this lemma, there exist constants K and C such that the following holds: Let ϵ‎ > 0, and suppose that (v, p, R) are uniformly continuous solutions to the Euler-Reynolds equations on ℝ x ³, with v uniformly bounded⁷ and suppR ⊆ I x ³ for some time interval. The Main Lemma implies the following theorem: There exist continuous solutions (v, p) to the Euler equations that are nontrivial and have compact support in time. To establish this theorem, one repeatedly applies the Main Lemma to produce a sequence of solutions to the Euler-Reynolds equations. To make sure the solutions constructed in this way are nontrivial and compactly supported, the lemma is applied with e(t) chosen to be any sequence of non-negative functions whose supports are all contained in some finite time interval.

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