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Zbl 1216.35119
Linares, Felipe; Pastor, Ademir
Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation. (English)
[J] J. Funct. Anal. 260, No. 4, 1060-1085 (2011). ISSN 0022-1236

Summary: This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, $$\cases u_t+ \partial_x\Delta u+ u^k u_x= 0, &(x,y)\in\bbfR^2,\ t> 0,\\ u(x,y,0)= u_0(x,y). \endcases$$ For $2\le k\le 7$, the IVP above is shown to be locally well-posed for data in $H^s(\bbfR^2)$, $s> 3/4$. For $k\ge 8$, local well-posedness is shown to hold for data in $H^s(\bbfR^2)$, $s> s_k$, where $s_k= 1- 3/(2k- 4)$. Furthermore, for $k\ge 3$, if $u_0\in H^1(\bbfR^2)$ and satisfies $\Vert u_0\Vert_{H^1}\ll 1$, then the solution is shown to be global in $H^1(\bbfR^2)$. For $k= 2$, if $u_0\in H^s(\bbfR^2)$, $s> 53/63$, 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 $H^s(\bbfR^2)$.

MSC 2000:: *35Q53 KdV-like equations
35A01
35A02

Keywords: Zakharov-Kuznetsov equation; local well-posedness; global well-posedness

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