Well-Posedness of the Kadomtsev–Petviashvili-II in the Negative Sobolev Space with Respect to y Direction

In this paper, we first establish some new dyadic bilinear estimates of KP-II equation. Then following some ideas in [ 7 ], we consider the Cauchy problem of the 2-D KP-II equation in negative Sobolev space with respect to y direction ∂ _t u + ∂ _xxxu+∂ _x^-1(∂ _yy )u +∂ _x (u^2) =0, (x,y,t) ∈ℝ^3. It follows that the 2D-KP-II is locally well-posed in space H^(s_0,s_1) with s_0=0, s_1≥ -41/360 ( or s_0≥ -1/3 and s_1≥ -1/360 ) although s_1≥ -41/360 is not critical case.