Modified Kdv Equation And Ds Equations Posedness And Ill-posedness Of The Problem

In this dissertation, we study the initial-boundary-value problem of modified Korteweg-de Vries equation and use the tool of harmonic analysis to study the Cauchy problem of Davey-Stewarson equation.In the second chapter, We obtain some linear estimates, trilinear estimates .And through these estimates , we prove the local well-posedness of modified Korteweg-de Vries equation in a quarter plane.In the third chapter, We prove an " almost conservation law " to obtain global-in-time well-posedness for the nonlinear Davey-Stewartson equation in H~S(R~2), and S>4/7.In the fourth chapter, The nonlinear Davey-Stewartson equations on R~d, with general power nonlinearity and with both the focusing and defocuing signs, are proved to be ill-posed in the Sobolev space H~s whenever the exponent s is lower than that predicated by scaling or Galilean invariance, or when the regularity is too low to support distributional solutions .