  The initial value problem for two-dimensional Zakharov-Kuznetsov equation is
shown to be globally well-posed in $H^s({\mathbb{R}^2})$ for all
$\frac{5}{7}<s<1$ via using $I$-method in the context of atomic spaces. By
means of the increment of modified energy, the exsitence of global attractor
for weakly damped, forced Zakharov-Kuznetsov equation is also established in
$H^s({\mathbb{R}^2})$ for $\frac{10}{11}<s<1$.
