Sharp Well-posedness Of The Cauchy Problem For The Higher-order Dispersive Equation

This dissertation is to study the Cauchy problem for high-order dispersive equation ut + (?)x2n+1u = (?)x(u(?)xnu)+(?)xn-1(ux2), n ? 2, n ? N+which contains our main results:(1) We prove that the problem is locally well-posed in modified Sobolev space H(s,1/2n)with s < -n/2 + 3/4.(2) The flow map is not C2 at the origin if we assume that the problem is well-posed in H(s,w) with 0 < w<1/2n) for any s ?R.(3) By using the Besov-type spaces, the Cauchy problem for the high-order dispersive equation is locally well-posed in H(-n/2+3/4,-1/2n)(R). When n is evel, we also prove that the Cauchy problem for the higher order dispersive equation is ill-posed in H(s,a)(R) with s < -n/2 + 3/4 and all ? ? R.(4) The highlight and difficulty here is to establish some new dyadic bilinear estimates.The detailed outline is stated as:In Chapter one, we introduce the background and significance of the KdV equation firstly, then show the main problems and results.In Chapter two, based on some useful lemmas, we derived that the Cauchy problem of high order dispersive equation is locally well-posed in modified Sobolev space H(s,1/2n)with s < -n/2 + 3/4.In Chapter three, by using Besov-type spaces, we proved that the Cauchy problem for the high-order dispersive equation is locally well-posed in H(-n/2+3/4,-1/2n)(R).