Well-posedness And Ill-posedness Of Some Nonlinear Dispersive Equations

This dissertation is devoted to the study of the initial value problems of some non-linear dispersive equations of KdV and Camassa-Holm types. These equations stem fromthe study of water waves, nonlinear optics, laser and plasma physics etc., and possess im-portant physical properties. The main goal is to discuss well-posedness and ill-posednessof these equations.In Chapter 2, we consider the Cauchy problem associated with the following Kawa-hara equationBy using the I-method, we prove that the problem is globally well-posed in Hs(R) withs > 6538. In order to obtain lower regularity index for global well-posedness, we introducehomogeneous Bourgain spaces. We also prove some ill-posedness results.In Chapter 3, we consider the Cauchy problem for the modified Kawahara equationBy using the I-method, we prove that the problem is globally well-posed in Hs(R) withs > -2/32. We also prove that the problem is ill-posed when s < -4/1.In Chapter 4, we study the Cauchy problem for the higher order modified Camassa-Holm equationBy using the Fourier restriction norm method, we prove that the problem is locally well-posed in Hs(R) with s > -n + 54. As a consequence of H1(R) conservation law, theproblem is globally well-posed in H1(R). By finding a new conservation law, we showthat it possesses a Hamiltonian structure. We also obtain some ill-posedness results.In Chapter 5, we study the Cauchy problem for the following equationsWe prove that it possesses a Hamiltonian structure. We also investigate the weak rotationlimit problems ofα→0 andγ→0.In Chapter 6, we study the Cauchy problem for the mKdV equation with higherdispersion By using the Fourier restriction norm method, we prove that the problem is locally well-posed in Hs(R) with s≥?n2 + 43 and is ill-posed in Hs(R) with s <-n/2 + 43.In Chapter 7, we study the Cauchy problem for the modified Camassa-Holm equationBy using the dyadic bilinear estimates, we prove that the above problem is well-posed inH1/4(R) and is ill-posed in Hs(R) when s < 0.In Chapter 8, we study the Cauchy problem for Ostrovsky-Stepanyams-Tsimringequation(OST)u(x,0) = u0(x).whereη> 0 is the dissipative coe?cient. We prove that OST is locally well-posed inHs(R) with s > -3/2 and is ill-posed in Hs(R) with s <-2/3. Thus s = ?32 is the criticalregularity index.