Title:Cauchy problem for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities

Time: Fri, June 17, 2016, PM 2:30-3:30

Speaker:Prof. Chunlai Mu （ Chongqing University）

Location:Room 4318, Building No.4, Wushan Campus

Abstract:In this talk, we study the Cauchy problem for a generalized cross-coupled Camassa-Holm system with peakons and higher-order nonlinearities. By the transport equation theory and the classical Friedrichs regularization method, we obtain the local well-posedness of solutions for the system in nonhomogeneous Besov spaces B^s_{p,r} × B^s_{p,r} with 1 ≤ p, r ≤ +∞ and s > max{2 + 1/p , 5/2}. Moreover, we construct the local well-posedness in the critical Besov space B^{5/2}_{2,1} × B^{5/2}_{2,1} and the blow-up criteria. We also consider the well-posedness problem in the sense of Hadamard, non-uniform dependence, and H¨older continuity of the data-to-solution map for the system on both the periodic and the non-periodic case. In light of a Galerkin-type approximation scheme, the system is shown well-posed in the Sobolev spaces H^s×H^s, s > 5/2 in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous. Furthermore, we prove the H¨older continuity in the H^r×H^r topology when 0 ≤ r < s with H¨older exponent α depending on both s and r.