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Continuity Properties of the Data-To-Solution Map for the Hunter-Saxton Equation

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posted on 2011-04-15, 00:00 authored by Curtis Alan Holliman
It is proved that the data-to-solution map for the Hunter-Saxton (HS) is continuous but not uniformly continuous on bounded subsets. To demonstrate this sharpness of continuity, two sequences of bounded solutions to the HS equation are constructed whose distance at the initial time converges to zero and whose distance at any later time is bounded from below by a positive constant. To achieve this result, approximate solutions that satisfy this property are chosen, after which actual solutions are found by solving the HS Cauchy problem with initial data taken to be the value of approximate solutions at time zero. Then, using well-posedness estimates, it is shown that the difference between solutions and approximate solutions is negligible.

History

Date Modified

2017-06-05

Defense Date

2011-04-04

Research Director(s)

Alex Himonas

Committee Members

Alex Himonas

Degree

  • Doctor of Philosophy

Degree Level

  • Doctoral Dissertation

Language

  • English

Alternate Identifier

etd-04152011-140328

Publisher

University of Notre Dame

Program Name

  • Mathematics

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