Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov–Kuznetsov equation, namely,{ut+∂xΔu+ukux=0,(x,y)∈R2,t>0,u(x,y,0)=u0(x,y). For 2⩽k⩽72⩽k⩽7, the IVP above is shown to be locally well posed for data in Hs(R2)Hs(R2), s>3/4s>3/4. For k⩾8k⩾8, local well-posedness is shown to hold for data in Hs(R2)Hs(R2), s>sks>sk, where sk=1−3/(2k−4)sk=1−3/(2k−4). Furthermore, for k⩾3k⩾3, if u0∈H1(R2)u0∈H1(R2) and satisfies ‖u0‖H1≪1‖u0‖H1≪1, then the solution is shown to be global in H1(R2)H1(R2). For k=2k=2, if u0∈Hs(R2)u0∈Hs(R2), s>53/63s>53/63, and satisfies ‖u0‖L2<3‖φ‖L2, where φ   is the corresponding ground state solution, then the solution is shown to be global in Hs(R2)Hs(R2).