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 H s 1 , s 2 ( R 2 ) has been studied by several authors. It was known to be locally well-posed in H s , 0 ( R 2 ) 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 s 1 ≥ − 5 / 4 , s 2 ≥ 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 B S → L 4 estimate, in which we obtain 3 / 8 regularity which comes from the effect of ∂ x − 1 ∂ y 2 term.