I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Shrodinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the following NLS $$i\partial_t\phi(x,t)+\Delta\phi(x,t)=\vert\phi(x,t)\vert^2\phi(x,t)$$ with initial datum $\phi_0\in H^s(\mathbb{R}^3)$. At the biginning of the paper they state that conservation laws and the local-in-time theory immediately yield global-in-time well-posedness for $s\geq 1$. I understand the case $s=1$ just using the conservation of the energy. But now, let's consider for instance the case $s=2$. In particular an $H^2$-solution is an $H^1$-solution, but how to control that the solution doesn't blow-up in the $H^2$-norm?

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the following NLS $$i\partial_t\phi(x,t)+\Delta\phi(x,t)=\vert\phi(x,t)\vert^2\phi(x,t)$$ with initial datum $\phi_0\in H^s(\mathbb{R}^3)$. At the beginning of the paper they state that conservation laws and the local-in-time theory immediately yield global-in-time well-posedness for $s\geq 1$. I understand the case $s=1$ just using the conservation of the energy. But now, let's consider for instance the case $s=2$. In particular an $H^2$-solution is an $H^1$-solution, but how to control that the solution doesn't blow-up in the $H^2$-norm?