The supercritical generalized KdV equation: Global well-posedness in the energy space and below

We consider the generalized Korteweg-de Vries (gKdV) equation ${\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x^{3}u+\mu {\mathrm{\partial }}_x(u^{k+1})=0$, where $k\ge 5$ is an integer number and $\mu =\pm 1$. In the focusing case ($\mu =1$), we show that if the initial data $u_0$ belongs to $H^1(\mathbb{R})$ and satisfies $E(u_0)^{s_k} M(u_0)^{1-s_k} < E(Q)^{s_k} M(Q)^{1-s_k}$,$E(u_0)\geq0$,and$|\partial_x u_0|{L^2}^{s_k}|u_0|{L^2}^{1-s_k} < |\partial_x Q|{L^2}^{s_k}|Q|{L^2}^{1-s_k}$,where$M(u)$and$E(u)$ are the mass and energy, then the corresponding solution is global in $H^1(\mathbb{R})$. Here, $s_k=\frac{(k-4)}{2k}$ and $Q$ is the ground state solution corresponding to the gKdV equation. In the defocusing case ($\mu =-1$), if $k$ is even, we prove that the Cauchy problem is globally well-posed in the Sobolev spaces $H^s(\mathbb{R})$, $s>\frac{4(k-1)}{5k}$.