On the local regularity of the Kadomtsev-Petviashvili-II equation (Q2708377)

The authors consider the initial-value problem NEWLINE\[NEWLINE(u_t+u_{xxx}+uu_x)_x+u_{yy}=0,\quad (t,x,y)\in\mathbb{R}^3,\quad u(0,x,y)=\Phi(x,y)\tag{1}NEWLINE\]NEWLINE where the initial data, \(\Phi(x,y)\), belongs to an anisotropic Sobolev space \(H^{s_1,s_2}_{x,y}(\mathbb{R}^2) =\{\phi\in S'(\mathbb{R}^2)\); \(\|\phi\|_{H^{s_1,s_2}_{x,y}}= \|((1-\partial^2_x)^{s_1/2}(1-\partial^2_y)^{s_2/2}\Phi\|_{L_2} < \infty\}\) representing the Cauchy problem for the Kadomtsev-Petviashvili-II (KP-II) equation.NEWLINENEWLINENEWLINEThe Kadomtsev-Petviashvili (KP) equations are 2-dimensional extensions of the Korteweg-de Vries equations and was obtained by \textit{B. B. Kadomtsev} and \textit{V. I. Petviashvili} [Sov. Phys. Dokl. 15, 539-541, (1970; Zbl 0217.25004)] when studying the stability of the solitary waves of KdV-equations as universal models for the propagation of weakly nonlinear dispersive long waves that are essentially one-directional with weak transverse effects.NEWLINENEWLINENEWLINEIn this paper, as a suitable combination of individual results obtained by \textit{H. Takaoka} [Adv. Differ. Equ. 5, No. 10-12, 1421-1443 (2000)]; respectively \textit{N. Tzvetkov} [Differ. Integral Equ. 13, 1289-1320 (2000; Zbl 0977.35125)] the two authors prove that (1) is locally well-posed for data in the anisotropic Sobolev spaces \(H^{s_1,s_2}_{x,y}(\mathbb{R}^2)\), \(s_1 > -\frac 13\), \(s_2\geq 0\).NEWLINENEWLINENEWLINEThe result seems to be the optimal one for the initial data in \(H^{s_1,s_2}_{x,y}(\mathbb{R}^2)\), because of the counter examples constructed in the second part of the paper.