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On the well-posedness of the hyperelastic rod equation

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posted on 2012-03-10, 00:00 authored by David Karapetyan
It is shown that the data-to-solution map for the hyperelastic rod equation is not uniformly continuous on bounded sets of Sobolev spaces with exponent greater than 3/2 in the periodic case and non-periodic cases. The proof is based on the method of approximate solutions and well-posedness estimates for the solution and its lifespan. Building upon this work, we also prove that the data-to-solution map for the hyperelastic rod equation is H'older continuous from bounded sets of Sobolev spaces with exponent s > 3/2 measured in a weaker Sobolev norm with exponent r < s in both the periodic and non-periodic cases. The proof is based on energy estimates coupled with a delicate commutator estimate and multiplier estimate.

History

Date Modified

2017-06-05

Defense Date

2012-03-01

Research Director(s)

Alex Himonas

Committee Members

Bei Hu Yongtao Zhang Gerard Misiolek

Degree

  • Doctor of Philosophy

Degree Level

  • Doctoral Dissertation

Language

  • English

Alternate Identifier

etd-03102012-222916

Publisher

University of Notre Dame

Program Name

  • Mathematics

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