On Periodic KP-I Type Equations

Abstract:

We analyze bilinear estimates related to periodic KP-I type equations. We prove that the Cauchy problem for the fifth order KP-I equation, posed on ?×ℝ, is globally well-posed in the energy space. Despite the loss of a smoothing relation (which is essential to recuperate the derivative loss), used in the context of the KdV and KP-II type equations, we are able to gain a moothing thanks to some specific properties of the set of Fourier modes where the smoothing relation fails. Counterexamples in the context of KP-I type equations posed on ?×? are given. These examples indicate that contrary to the KP-II case it would be difficult to obtain the local well-posedness for the periodic KP-I type equations by an iteration scheme using Bourgain spaces.