摘要
In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt + umx + 2ux m = 0, m =(1- δx^2)^2u,u(x, 0) = u0(x) ∈ H^s(R), x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s(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.
In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt + umx + 2ux m = 0, m =(1- δx^2)^2u,u(x, 0) = u0(x) ∈ H^s(R), x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s(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.
基金
supported by the National Natural Science Foundation of China(11226159)