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 + ∂x u +ukux = 0, (x, y) ∈ R2, t >0, u(x, y, 0) = u0(x, y). For 2 k 7, the IVP above is shown to be locally well posed for data in Hs (R2), s > 3/4. For k 8, local well-posedness is shown to hold for data in Hs (R2), s >sk, where sk = 1−3/(2k−4). Furthermore, for k 3, if u0 ∈ H1(R2) and satisfies u0 H1 1, then the solution is shown to be global in H1(R2). For k = 2, if u0 ∈ Hs (R2), s > 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). © 2010 Elsevier Inc. All rights reserved