The Cauchy Problem for the CH2 System

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.