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(ut−β∂x3u+∂x(u2))+β′∂y2u−γu=0 in the anisotropic Sobolev spaces Hs1,s2(R2). When β>0 and γ>0,β′<0, we show that the Cauchy problem is locally well-posed in Hs1,s2(R2) with s1>−12 and s2≥0. The main difficulty in establishing bilinear estimates related to nonlinear term of RMKP type equation is that the resonant function|3βξξ1ξ2−γ(ξ12−ξ1ξ2+ξ22)ξξ1ξ2−β′ξ1ξ2ξ(μ1ξ1−μ2ξ2)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 Hs1,0(R2) with s1<−12 in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not C3.