- 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

# Checking Frequency Energy Levels for the Velocity and Pressure

# Checking Frequency Energy Levels for the Velocity and Pressure

- Chapter:
- 24 Checking Frequency Energy Levels for the Velocity and Pressure
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
### Philip Isett

- Publisher:
- Princeton University Press

This chapter checks frequency energy levels for the velocity and pressure. It begins by comparing the different estimates obtained for the corrections to the velocity and the pressure with the Main Lemma. It then considers bounds that will be established for a particular constant *C* once the constant *B*subscript Greek Small Letter Lamda has been chosen. It also checks whether the frequency and energy levels of the new velocity and pressure are consistent with the claims of the Main Lemma (10.1). To complete the proof of the Main Lemma (10.1), it now only remains to choose a constant *B*subscript Greek Small Letter Lamda so that (243) and (244) can be verified for the new energy levels. This choice of *B*λ is the last step of the proof.

*Keywords:*
frequency energy levels, velocity, pressure, correction, Main Lemma

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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