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Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time$
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Philip Isett

Print publication date: 2017

Print ISBN-13: 9780691174822

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691174822.001.0001

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Bounds for Coefficients from the Stress Equation

Bounds for Coefficients from the Stress Equation

Chapter:
20 Bounds for Coefficients from the Stress Equation
Source:
Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
Author(s):

Philip Isett

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691174822.003.0020

This chapter estimates the bounds for coefficients from the Stress equation. It starts by considering the coefficients γ‎subscript I and the equation that implicitly defines it. It then estimates the derivatives of γ‎subscript I by differentiating the equation. The first transport derivative always costs a factor Ξ‎eᵥ½ in the estimates, and each spatial derivative costs a factor of Ξ‎ until the total order of differentiation exceeds L, at which point one obtains a larger cost of Nsuperscript 1/LΞ‎ per derivative. The chapter also considers the bounds satisfied by the coefficients γ‎subscript I and shows that the final bound for the coefficients γ‎subscript I is exactly the same quality as the corresponding bound for ε‎.

Keywords:   coefficient, Stress equation, transport derivative, spatial derivative

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