- 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

# Gluing Solutions

# Gluing Solutions

- Chapter:
- 12 Gluing Solutions
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
### Philip Isett

- Publisher:
- Princeton University Press

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 necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

*Keywords:*
continuous solution, Galilean transformation, momentum, velocity, pressure, Euler-Reynolds equations, smooth stress tensor

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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