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Non-uniform dependence for the periodic CH equation. (English) Zbl 1193.35189

Summary: 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B45 A priori estimates in context of PDEs
35B33 Critical exponents in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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