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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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Constructing Continuous Solutions

Constructing Continuous Solutions

Chapter:
8 Constructing Continuous Solutions
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/9780691174822.003.0008

This chapter demonstrates how the preceding construction, combined with a few estimates from Part V, can be used to prove the Main Lemma for continuous solutions. The first step is to mollify the velocity, followed by mollification of the stress. The lifespan is then chosen, preferring a small parameter to ensure that the first term in the parametrix for the High–High term is controlled. The chapter proceeds by discussing the bounds for the new stress and solving the divergence equation, along with the bounds for the corrections and finally, control of the energy increment. The equation for the energy increment includes a smooth vector field and involves bounding the error term.

Keywords:   mollification, Main Lemma, continuous solution, velocity, stress, divergence equation, correction, energy increment, smooth vector field

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