Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation (1010.5404v1)

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 \Delta u+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 $2\leq k \leq 7$, the IVP above is shown to be locally well-posed for data in $H^s(\mathbb{R}^2)$, $s>3/4$. For $k\geq8$, 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 $k\geq3$, if $u_0\in H^1(\mathbb{R}^2)$ and satisfies $|u_0|{H^1}\ll1$, 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 $|u_0|{L^2}<\sqrt3 \, |\phi|_{L^2}$, where $\phi$ is the corresponding ground state solution, then the solution is shown to be global in $H^s(\mathbb{R}^2)$.