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Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations. (English) Zbl 1157.35094

Summary: We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space \(H^{s_1,s_2}(\mathbb{R}^2)\) with \(s_1>-\tfrac 12\) and \(s_2 \geq 0\). On the \(H^{s_1,0}(\mathbb{R}^2)\) scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation: \[ (u_t-|D_x|^\alpha u_x+ (u^2)_x)_x+u_{yy}=0,\quad u(0)=u_0, \] for \(\tfrac 43<\alpha\leq 6\), \(s_1>\max(1-\tfrac 34\alpha,\tfrac 14-\tfrac 38 \alpha)\) \(s_2\geq 0\) and \(u_0\in H^{s_1,s_2} (\mathbb{R}^2)\). We deduce global well-posedness for \(s_1\geq 0\), \(s_2=0\) and real valued initial data.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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