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

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, {ut + partial derivative(x)Delta u+ u(k)u(x) = 0, (x, y) is an element of R(2), t > 0, u(x, y, 0) = u(0)(x, y). For 2 <= k <= 7, the IVP above is shown to be locally well posed for data in H(s) (R(2)), s > 3/4. For k >= 8, local well-posedness is shown to hold for data in H(s) (R(2)), s > s(k), where s(k) = 1 - 3/(2k - 4). Furthermore, for k >= 3, if u(0) is an element of H(1) (R(2)) and satisfies parallel to u(0)parallel to(H1) << 1, then the solution is shown to be global in H(1)(R(2)). For k = 2, if u(0) is an element of H(s)(R(2)), s > 53/63, and satisfies parallel to u(0)parallel to(L2) < root 3 parallel to phi parallel to(L2), where phi is the corresponding ground state solution, then the solution is shown to be global in H(s)(R(2)). (C) 2010 Elsevier Inc. All rights reserved.260410601085Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq [152234/2007-1