On the well-posedness of the hyperelastic rod equation

Doctoral Dissertation

Abstract

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.

Attributes

Attribute NameValues
URN
  • etd-03102012-222916

Author David Karapetyan
Advisor Alex Himonas
Contributor Bei Hu, Committee Member
Contributor Alex Himonas, Committee Chair
Contributor Yongtao Zhang, Committee Member
Contributor Gerard Misiolek, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name PhD
Defense Date
  • 2012-03-01

Submission Date 2012-03-10
Country
  • United States of America

Subject
  • energy estimates

  • Sobolev spaces

  • Hyperelastic rod equation

  • periodic

  • continuity of solution map

  • non-periodic

  • approximate solutions

  • well-posedness

  • initial value problem

Publisher
  • University of Notre Dame

Language
  • English

Record Visibility Public
Content License
  • All rights reserved

Departments and Units

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