Non-uniform dependence on initial data for the modified Camassa-Holm equation on the line

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt+umx+2uxm=0,  m=(1-∂x2)2u,u(x,0)=u0(x)∈Hs(ℝ),    x∈ℝ, t>0,u(x,0)=u0(x)∈Hs(ℝ),    x∈ℝ, t>0,and show that the solution map is not uniformly continuous in Sobolev spaces HS(ℝ) for s>7/2. Compared with the periodic problem, the non-periodic problem is more difficult, e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.