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 of??0 and??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).where?> 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.