A local well-posed result for the fifth order KP-II initial value problem

The well-posedness of the real valued fifth order KP-II equation in Sobolev spaces Hs1,s2(R2) has been studied by several authors. It was known to be locally well-posed in Hs,0(R2) for s>−5/4 (see Hadac (2008) [3], Isaza et al. (2006) [5]). We are more interested in the case s=−5/4. We obtain the local well-posedness for s1≥−5/4,s2≥0, which improves previous results by Saut and Tzvetkov (2000) [15] and by Isaza, López and Mejía (2006) [5]. Moreover our main contribution is that we set up the global in time Strichartz estimates for the fifth order KP equation (no matter KP-I or KP-II equation) on dyadic pieces. And especially, on the low frequency part, we obtain a BS→L4 estimate, in which we obtain 3/8 regularity which comes from the effect of ∂x−1∂y2 term.