The local well-posedness in Besov spaces and non-uniform dependence on initial data for the interacting system of Camassa-Holm and Degasperis-Procesi equations

摘要

This paper deals with the Cauchy problem for the interacting system of the Camassa- Holm and Degasperis- Procesi equations mt = - 3m(2ux + vx) - mx (2u + v), nt = - 2n(2ux + vx) - nx (2u + v), where m = u - uxx and n = v - vxx. By the transport equations theory and the classical Friedrichs regularization method, the local well- posedness of solutions for this system in nonhomogeneous Besov spaces Bs p, r x Bs p, r with 1 = p, r = +8 and s max 2 + 1 p, 5 2 is obtained, and the local well- posedness in critical Besov space B 5/ 2 2,1 x B 5/ 2 2,1 is also established. Moreover, by the approach for approximate solutions and well- posedness estimates, we obtain two sequences of solution for this equation, which are bounded in the Sobolev space Hs (R) x Hs (R) with s 5/ 2, and the distance between the two sequences is lower- bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous.