Analysis of a Family of Shallow Water Waves

Doctoral Dissertation


We study a family of shallow water wave equations called the b-family equation. Known for having multi-peakon solutions, this family includes the Camassa-Holm equation and the Degasperis-Procesi equation as its most notable and only integrable members (using the perturbative symmetry definition). We show that the periodic and non-periodic Cauchy problem for the b-family equation is well-posed in Sobolev spaces with exponent greater than 3/2. Moreover, we find that the corresponding data-to-solution map is continuous on Sobolev spaces but not uniformly continuous. We prove that this map is not uniformly continuous using approximate solutions together with delicate commutator and multiplier estimates. The novelty of the proof lies in the fact that it makes no use of conserved quantities. Lastly, given a weaker topology, we show that the data-to-solution map is Holder continuous.


Attribute NameValues
  • etd-11092012-124733

Author Katelyn Jean Grayshan
Advisor Richard Hind
Contributor Gerard Misiolek, Committee Member
Contributor Alex Himonas, Committee Member
Contributor Bei Hu, Committee Member
Contributor Richard Hind, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Defense Date
  • 2012-11-06

Submission Date 2012-11-09
  • United States of America

  • peakon

  • continuity

  • Cauchy problem

  • sobolev space

  • b-family equation

  • initial value problem

  • soliton

  • well-posedness

  • partial differential equations

  • data-to-solution map

  • University of Notre Dame

  • English

Record Visibility Public
Content License
  • All rights reserved

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