The Cauchy Problem for the CH2 System

Doctoral Dissertation


For Sobolev exponent s > 5=2, it is shown that the data-to-solution map for the2-component Camassa-Holm system is continuous from Hs x Hs􀀀-1 into C([0; T];Hs x Hs􀀀-1) but not uniformly continuous. The proof of non-uniform dependence on the initial data is based on the method of approximate solutions, delicate commutator and multiplier estimates, and well-posedness results for the solution and its lifespan. Also, the solution map is Holder continuous if the Hs x Hs-􀀀1 norm is replaced by an Hr x Hr􀀀-1 norm for 0 r < s.


Attribute NameValues
  • etd-03162015-154155

Author Ryan Cole Thompson
Advisor Alex Himonas
Contributor Alex Himonas, Committee Chair
Contributor Gerard Misiolek, Committee Member
Contributor Yongtao Zhang, Committee Member
Contributor Alan Lindsay, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name PhD
Defense Date
  • 2015-03-03

Submission Date 2015-03-16
  • United States of America

  • approximate solutions

  • well-posedness

  • integrability

  • initial value problem

  • Sobolev spaces

  • Holder continuity

  • non-uniform dependence on initial data

  • multiplier estimate.

  • commutator estimate

  • peakons

  • continuity properties

  • 2-component Camassa-Holm system equation

  • University of Notre Dame

  • English

Record Visibility Public
Content License
  • All rights reserved

Departments and Units


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