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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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The Coarse Scale Flow and Commutator Estimates

The Coarse Scale Flow and Commutator Estimates

Chapter:
16 The Coarse Scale Flow and Commutator Estimates
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/9780691174822.003.0016

This chapter derives estimates for the coarse scale flow and commutator. Instead of mollifying the velocity field in the time variable, it derives a Transport equation for vsubscript Element and some estimates that will be necessary for the proof. Here the quadratic term arises from the failure of the nonlinearity to commute with the averaging. Commutator estimates are then derived. To observe cancellation in the quadratic term, the control over the higher-frequency part of v is used, and cancellation is obtained from the lower-frequency parts. It becomes clear that the commutator terms can be estimated using the control of only the derivatives of v. The chapter concludes by presenting the theorem for coarse scale flow estimates.

Keywords:   coarse scale flow, commutator estimate, Transport equation, commutator, commutator term

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