The Cauchy problem for the rotation-modified Kadomtsev-Petviashvili type equation

Abstract This paper is devoted to studying the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) type equation ∂ x ( u t − β ∂ x 3 u + ∂ x ( u 2 ) ) + β ′ ∂ y 2 u − γ u = 0 in the anisotropic Sobolev spaces H s 1 , s 2 ( R 2 ) . When β > 0 and γ > 0 , β ′ 0 , we show that the Cauchy problem is locally well-posed in H s 1 , s 2 ( R 2 ) with s 1 > − 1 2 and s 2 ≥ 0 . The main difficulty in establishing bilinear estimates related to nonlinear term of RMKP type equation is that the resonant function | 3 β ξ ξ 1 ξ 2 − γ ( ξ 1 2 − ξ 1 ξ 2 + ξ 2 2 ) ξ ξ 1 ξ 2 − β ′ ξ 1 ξ 2 ξ ( μ 1 ξ 1 − μ 2 ξ 2 ) 2 | may tend to zero since β > 0 , γ > 0 and β ′ 0 . When β > 0 and γ > 0 and β ′ 0 , we also prove that the Cauchy problem for RMKP equation is ill-posed in H s 1 , 0 ( R 2 ) with s 1 − 1 2 in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not C 3 .