Cancellation properties and unconditional well-posedness for the fifth order KdV type equations with periodic boundary condition

We consider the fifth order KdV type equations and prove the unconditional well-posedness in \(H^s({\mathbb T})\) $H^s(\mathbb{T})$ for \(s \ge 1\) $s\ge 1$ . It is optimal in the sense that the nonlinear terms can not be defined in the space-time distribution framework for \(s<1\) $s<1$ . The main idea is to employ the normal form reduction and a kinds of cancellation properties to deal with the derivative losses.