We study two properties of the datatosolution 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 nonuniform continuity of the datatosolution 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 nonperiodic 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.
Regularity Properties of the Solution Map of the Incompressible Euler Equations
Doctoral Dissertation
Abstract
Attribute Name  Values 

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 
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Submission Date  20200415 
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Record Visibility  Public 
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