Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations

Abstract

We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H s 1 ,s 2(R 2 ) with s 1 > -j and s 2 > 0. On the H s 1 ,0 (R 2 ) scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation: (u t -|D x | α u x + (u 2 ) x ) x + u yy = 0, U(0) = u 0 , for 4 3 max(1 - 3 4α, 1 4 - 3 8α), s 2 > 0 and u 0 ∈ H s 1 ,s 2(R 2 ). We deduce global well-posedness for s 1 > 0, s 2 = 0 and real valued initial data.