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## The Euler-Reynolds System

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter provides a background on the Euler-Reynolds system, starting with some of the underlying philosophy behind the argument. It describes low frequency parts and ensemble averages of Euler ... More

## The Main Iteration Lemma

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0010
- Subject:
- Mathematics, Computational Mathematics / Optimization

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 ... More

## Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

*Philip Isett*

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.001.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization

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 ... More

## Structure of the Book

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0002
- Subject:
- Mathematics, Computational Mathematics / Optimization

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 ... More

## Basic Technical Outline

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0003
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter provides a more technical outline of the construction, starting with a solution to the Euler-Reynolds system and a correction v₁ = v + V, p₁ = p + P. The correction V is a divergence ... More

## A Main Lemma for Continuous Solutions

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0005
- Subject:
- Mathematics, Computational Mathematics / Optimization

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 ... More

## Frequency and Energy Levels

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0009
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter shows how to measure the Hölder regularity of the weak solutions that are constructed when the scheme is executed more carefully. For this aspect of the convex integration scheme, a ... More

## Gluing Solutions

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0012
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if ... More

## On Onsager's Conjecture

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0013
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with Onsager's conjecture, which would be implied by a stronger form of Lemma (10.1). It considers what could be proven assuming Conjecture (10.1) by turning to Theorem 13.1, which ... More

## The Divergence Equation

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0006
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter introduces the divergence equation. A key ingredient in the proof of the Main Lemma for continuous solutions is to find special solutions to this divergence equation, which includes a ... More

## Constructing the Correction

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0007
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter explains how the correction is constructed, first by considering the transportation of the phase functions. A solution (v, p, R) to the Euler-Reynolds equations is fixed and a correction ... More

## Main Lemma Implies the Main Theorem

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0011
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter shows that the Main Lemma implies the main theorem. It proves Theorem (10.1) by inductively applying the Main Lemma in order to construct a sequence of solutions of the Euler-Reynolds ... More

## Cellular Homology

*Graham Ellis*

### in An Invitation to Computational Homotopy

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0002
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology

This chapter introduces more basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: chain complex, chain mapping, ... More

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