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On the Cauchy problem for Kadomtsev-Petviashvili equation. (English) Zbl 0934.35161

Local solvability and uniqueness of the Cauchy problem for the Kadomtsev-Petviashvili-II (KP-II) equation in the case of two and three space dimensions are studied. The functional spaces used for this are (non-isotropic) modified Sobolev spaces and the associated Bourgain type spaces. In the case of two space dimensions they are \[ \widetilde H^{b,s_1,s_2} (\mathbb{R}^3)=\biggl\{u; \bigl(1+|\tau |\bigr)^b\bigl(1 +|\xi |\bigr)^{s_1} \bigl(1+ |\eta |\bigr)^{s_2} \bigl(1+|\xi|^{-1} \bigr)\widehat u(\tau, \xi,\eta)\in L^2(\mathbb{R}^3) \biggr\} \] and \(B^{b,s_1s_2} (\mathbb{R}^3)=\{u;U(-t)u\in \widetilde H^{b,s_1,s_2} (\mathbb{R}^3)\}\), where \(U\) is the unitary group which defines the free evolution associated with the KP-II equation. The main result obtained in the paper is the following:
Theorem (the case of two space dimensions). Let \(s_1>-1/4\) and \(s_2\geq 0\). Then there exists \(b>1/2\) such that if the initial value of the Cauchy problem for KP-II equation belongs to \(\widetilde H^{s_1,s_2}(\mathbb{R}^2)\) then there exists a positive \(T\) (depending on the initial value) and a unique solution of the Cauchy problem in the time interval \([-T,T]\) which belongs to \(C([-T,T]; \widetilde H^{s_1,s_2} (\mathbb{R}^2))\cap B^{b,s_1,s_2} (\mathbb{R}^3)\).
In case of three space dimensions a similar result holds with \(s=s_1=s_2= s_3>3/2\) and \(b=1/2\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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