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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 the Corrections

Bounds for the Corrections

Chapter:
(p.143) 22 Bounds for the Corrections
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/9780691174822.003.0022

This chapter derives the bounds for the correction terms, starting with bounds for the velocity correction. Based on V of the form V = Δ‎ x W, it introduces a proposition for estimating the spatial derivatives of W. Since the number of Wsubscript I supported at any given region of ℝ x ³ is bounded by a universal constant, it suffices to estimate Wsubscript I uniformly in I. For an individual wave, it is easy to see that the estimate will hold. During repeated differentiation, the derivative hits either the oscillatory factor, the phase direction, or the amplitude wsubscript I or one of its derivatives. In any case, the largest cost happens when differentiating the phase function. The chapter also gives estimates for derivatives of the coarse scale material derivative of W and concludes with bounds for the pressure correction.

Keywords:   correction term, velocity correction, spatial derivative, oscillatory factor, phase direction, amplitude, phase function, pressure correction

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