Well-posedness Of The Two Dimensional Benjamin Equations

Two dimensional Bnnjamin equation (?) is an extension of the Benjamin equation in high dimensional space and it is model-ing weak spatial variations transverse to the direction of propagation.2D Benjamin equations has not any scaling invariant property or uniform t-1 decay of the linear solution according to the frequency.But the t-1 decay is very important in setting up bilinear estimates.It seems that the global well-posedness problem is very interesting and challenging.In the paper we set up well-posedness of the 2D Benjamin equation of Third order and Fifth order.1.we set up the global well-posedness in the energy space for the Cauchy problem of the 2D Benjamin equation (?) First we rescale the problem into small initial data problem.And then set up Strichartz estimates on dyadic blocks if the scaling number ? and the size of the block 2k are not equivalence.If they are approximately equal.which is absolutely different with the KP-1 equation,as the main contribution in the paper,we introduce a paraproduct and give a.direct proof.And then we set up bilinear estimates,nonlinear estimates and energy estimates.In the end,witout Picard iteration method,we prove the main theorem by the skill introduced by A.Ioneseu,C.Kenig and D.Tataru.2.We concern with the local well-posedness for the C'auchy problem of the fifth order 2D Benjamin equation (?) and obtain its local well-posedness in Hs.0 with s?-5/4.and global well-posednessss in Hs1,s2(R2)with s1?0.s2?0.First we set up Strichartz estimates on dyadic blocks {(?)-N} global in time and obtained Bs-L4 estimates for N?1.For the case of N<1,we turn the modulation into regularity by the method introduced by J.Li and S,Shi,Shi,and get the Bs-L4 estimates.And then set trilinear estimates on blocks {(?)-N}.Finally,we prove the well-posdedness results by Picard iteration method via bilinear estimates.