Well-posedness for the fifth order KP-II initial data problem in Hs,0(R×T)

The well-posed properties for the fifth order KP-II initial value problem for x∈R, y∈T are considered. It is proved to be locally well posed in Hs,0(R×T) for s≥−34 with small initial data and s>−34 with general initial data. By the L2 conservation law of KP equation, the L2 global well-posedness is also obtained. The crucial ingredient of the argument is the L2 estimates of a bilinear operator which was introduced in recent works [14] and [13]. This operator is Galilean invariant in the content of T2 and R2 but not in the content R×T.