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Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation. (English) Zbl 1216.35119
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 $H^s(\bbfR^2)$.

35Q53KdV-like (Korteweg-de Vries) equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35A02Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness
Full Text: DOI arXiv
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