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Non-Uniform Dependence for the Periodic CH Equation |
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| Authors: | A. Alexandrou Himonas Carlos Kenig Gerard Misiołek |
| Affiliation: | 1. Department of Mathematics , University of Notre Dame , Notre Dame, Indiana, USA himonas.1@nd.edu;3. Department of Mathematics , The University of Chicago , Chicago, Illinois, USA;4. Department of Mathematics , University of Notre Dame , Notre Dame, Indiana, USA |
| Abstract: | We show that the solution map of the periodic CH equation is not uniformly continuous in Sobolev spaces with exponent greater than 3/2. This extends earlier results to the whole range of Sobolev exponents for which local well-posedness of CH is known. The crucial technical tools used in the proof of this result are a sharp commutator estimate and a multiplier estimate in Sobolev spaces of negative index. |
| Keywords: | Approximate solutions CH equation Commutator estimate Multiplier estimate Non-uniform dependence on initial data Periodic Cauchy problem Sobolev spaces Well-posedness |