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Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation. (English) Zbl 1216.35119

Summary: This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely,

u t + x Δu+u k u x =0,(x,y) 2 ,t>0,u(x,y,0)=u 0 (x,y)·

For 2k7, the IVP above is shown to be locally well-posed for data in H s ( 2 ), s>3/4. For k8, local well-posedness is shown to hold for data in H s ( 2 ), s>s k , where s k =1-3/(2k-4). Furthermore, for k3, if u 0 H 1 ( 2 ) and satisfies u 0 H 1 1, then the solution is shown to be global in H 1 ( 2 ). For k=2, if u 0 H s ( 2 ), s>53/63, and satisfies u 0 L 2 <3φ L 2 , where φ is the corresponding ground state solution, then the solution is shown to be global in H s ( 2 ).

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35A02Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness
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