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On the local regularity of the Kadomtsev-Petviashvili-II equation. (English) Zbl 0977.35126

The authors consider the initial-value problem \[ (u_t+u_{xxx}+uu_x)_x+u_{yy}=0,\quad (t,x,y)\in\mathbb{R}^3,\quad u(0,x,y)=\Phi(x,y)\tag{1} \] 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.
The Kadomtsev-Petviashvili (KP) equations are 2-dimensional extensions of the Korteweg-de Vries equations and was obtained by B. B. Kadomtsev and 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.
In this paper, as a suitable combination of individual results obtained by H. Takaoka [Adv. Differ. Equ. 5, No. 10-12, 1421-1443 (2000)]; respectively 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 \(H^{s_1,s_2}_{x,y}(\mathbb{R}^2)\), \(s_1 > -\frac 13\), \(s_2\geq 0\).
The 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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