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

TL;DR

This paper establishes local and global well-posedness results for the 2D generalized Zakharov-Kuznetsov equation across various Sobolev spaces, depending on the nonlinearity degree, and identifies conditions for global solutions.

Contribution

It provides new well-posedness thresholds in Sobolev spaces for different nonlinearities and proves global existence under small initial data conditions.

Findings

Local well-posedness for s>3/4 when 2≤k≤7

Local well-posedness for s>s_k when k≥8

Global solutions for small initial data in H^1 for k≥3

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 $H^s(\mathbb{R}^2)$.