Regularity Properties of the Solution Map of the Incompressible Euler Equations

Doctoral Dissertation

Abstract

We study two properties of the data-to-solution map associated to the incompressible Euler equations of fluid motion. First, we show that the maximum norm of the vorticity controls the breakdown of the solution in the Eulerian coordinates for the “little” Hölder space. As a consequence we obtain an extension result for local solutions to the incompressible Euler equations. Second, we prove the non-uniform continuity of the data-to-solution map from bounded subsets of the Besov space into the space of continuous curves landing in the Besov space. In the periodic case two sequences of bounded solutions are shown to converge at time zero and to remain apart at later times. In the non-periodic case we use the initial values of two sequences of bounded approximate solutions (which converge at time zero and remain apart at later times) to construct two sequences of exact solutions to the incompressible Euler equations. Using standard energy estimates for solutions to the incompressible Euler equations and estimates for a solution to a linear Transport equation we show that the distance between the exact and approximate solutions is negligible in the Besov norm. Finally, we show that the exact solutions initially converge and remain separated at later times.

Attributes

Attribute NameValues
Author José David Pastrana Chiclana
Contributor Gerard K. Misiołek, Research Director
Contributor Richard Hind, Committee Member
Contributor Alexander Himonas, Committee Member
Contributor David Galvin, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Banner Code
  • PHD-MATH

Submission Date 2020-04-15
Subject
  • BKM criterion

  • Non-uniform dependence on initial data

  • Approximate solutions

  • Besov spaces

  • Hadamard well-posedness

  • Hölder spaces

  • Euler equations of fluid motion

Record Visibility Public
Content License
  • All rights reserved

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