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

Abstract

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, \{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x Δu+u^ku_x = 0,}\qquad (x,y) \in \mathbb{R}^2, \,\,\,\, t>0, {\displaystyle u(x,y,0)=u_0(x,y)}. \{array} \right. \{equation*} For $2\le k\le 7$, the IVP above is shown to be locally well-posed for data in $H^s({\mathbb{R}}^2)$, $s>3/4$. For $k\ge 8$, local well-posedness is shown to hold for data in $H^s({\mathbb{R}}^2)$, $s>s_k$, where $s_k=1-3/(2k-4)$. Furthermore, for $k\ge 3$, if $u_0\in H^1({\mathbb{R}}^2)$ and satisfies $\Vert u_0{\Vert }_{H^1}\ll 1$, then the solution is shown to be global in $H^1({\mathbb{R}}^2)$. For $k=2$, if $u_0\in H^s({\mathbb{R}}^2)$, $s>53/63$, and satisfies $\Vert u_0{\Vert }_{L^2}<\sqrt{3}\Vert \varphi {\Vert }_{L^2}$, where $\varphi$ is the corresponding ground state solution, then the solution is shown to be global in $H^s({\mathbb{R}}^2)$.