## 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

### Citations:

Zbl 0217.25004; Zbl 0977.35125
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