- Title Pages
- Preface
- Introduction
- 1 The Euler-Reynolds System
- Part II General Considerations of the Scheme
- 2 Structure of the Book
- 3 Basic Technical Outline
- 4 Notation
- 5 A Main Lemma for Continuous Solutions
- 6 The Divergence Equation
- 7 Constructing the Correction
- 8 Constructing Continuous Solutions
- 9 Frequency and Energy Levels
- 10 The Main Iteration Lemma
- 11 Main Lemma Implies the Main Theorem
- 12 Gluing Solutions
- 13 On Onsager's Conjecture
- 14 Preparatory Lemmas
- 15 The Coarse Scale Velocity
- 16 The Coarse Scale Flow and Commutator Estimates
- 17 Transport Estimates
- 18 Mollification along the Coarse Scale Flow
- 19 Accounting for the Parameters and the Problem with the High-High Term
- Part VI Construction of Regular Weak Solutions: Estimating the Correction
- 20 Bounds for Coefficients from the Stress Equation
- 21 Bounds for the Vector Amplitudes
- 22 Bounds for the Corrections
- 23 Energy Approximation
- 24 Checking Frequency Energy Levels for the Velocity and Pressure
- Part VII Construction of Regular Weak Solutions: Estimating the New Stress
- 25 Stress Terms Not Involving Solving the Divergence Equation
- 26 Terms Involving the Divergence Equation
- 27 Transport-Elliptic Estimates
- Appendices
- References
- Index

# Frequency and Energy Levels

# Frequency and Energy Levels

- Chapter:
- 9 Frequency and Energy Levels
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
### Philip Isett

- Publisher:
- Princeton University Press

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 notion of frequency energy levels is introduced. This notion is meant to accurately record the bounds which apply to the (*v*, *p*, *R*) coming from the previous stage of the construction. The chapter presents an example of a candidate definition for frequency and energy levels. Based on this definition, the effect of one iteration of the convex integration procedure can be summarized in a single lemma, which states that there is a solution to the Euler-Reynolds equations with new frequency and energy levels. The chapter also considers the High–Low Interaction term and the Transport term.

*Keywords:*
convex integration, Hölder regularity, weak solution, frequency energy levels, Euler-Reynolds equations, High–Low Interaction term, Transport term

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- Title Pages
- Preface
- Introduction
- 1 The Euler-Reynolds System
- Part II General Considerations of the Scheme
- 2 Structure of the Book
- 3 Basic Technical Outline
- 4 Notation
- 5 A Main Lemma for Continuous Solutions
- 6 The Divergence Equation
- 7 Constructing the Correction
- 8 Constructing Continuous Solutions
- 9 Frequency and Energy Levels
- 10 The Main Iteration Lemma
- 11 Main Lemma Implies the Main Theorem
- 12 Gluing Solutions
- 13 On Onsager's Conjecture
- 14 Preparatory Lemmas
- 15 The Coarse Scale Velocity
- 16 The Coarse Scale Flow and Commutator Estimates
- 17 Transport Estimates
- 18 Mollification along the Coarse Scale Flow
- 19 Accounting for the Parameters and the Problem with the High-High Term
- Part VI Construction of Regular Weak Solutions: Estimating the Correction
- 20 Bounds for Coefficients from the Stress Equation
- 21 Bounds for the Vector Amplitudes
- 22 Bounds for the Corrections
- 23 Energy Approximation
- 24 Checking Frequency Energy Levels for the Velocity and Pressure
- Part VII Construction of Regular Weak Solutions: Estimating the New Stress
- 25 Stress Terms Not Involving Solving the Divergence Equation
- 26 Terms Involving the Divergence Equation
- 27 Transport-Elliptic Estimates
- Appendices
- References
- Index