## 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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