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?
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5$\begingroup$ Could you change your title to something more descriptive, As it is now, your title is useless, and would describe a non-trivial amount of the questions here. Imagine a world where every fifth question here is titled "Help in understanding a statement in a paper" $\endgroup$– user35370May 27, 2015 at 15:56
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3$\begingroup$ I took a stab at a better title. Feel free to change it if there is something else you prefer. I also corrected the spelling of "Schrödinger". $\endgroup$– Nate EldredgeMay 27, 2015 at 16:35
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What you need is "persistence of regularity".
As you said, data in $H^1$ lead to global solutions that remain bounded in $H^1$ by conservation of mass and energy.
If you take data in $H^2$, say, you just need to check that the solution does not blow up its $H^2$-norm in finite time. (Of course, it may be the case that $\lim_{t\to\infty} \|u(t)\|_{H^2} = \infty$.)
To do this, you can run a bootstrap-type argument using Strichartz estimates. For example, on an interval $[0,T]$ you could use the $L_t^2 L_x^{6/5}$ endpoint and Sobolev embedding to estimate
\begin{align*} \|u\|_{L_t^\infty H_x^2} \lesssim \|u_0\|_{H^2} + T^{1/2}\|u\|_{L_t^\infty H_x^1}^2 \|u\|_{L_t^\infty H_x^2}. \end{align*}
Choosing $T$ sufficiently small depending on $\|u\|_{L_t^\infty H_x^1}$ (which is bounded purely in terms of the mass and energy), you can deduce that $$ \|u\|_{L_t^\infty H_x^2([0,T]\times\mathbb{R}^3)} \lesssim 2\|u_0\|_{H^2}. $$ In particular, $$ \|u(T)\|_{H^2} \lesssim 2 \|u_0\|_{H^2}. $$
Now you can run the same argument on $[T,2T]$. In particular, you see that the $H^2$-norm at most doubles on intervals of length $T$. Thus $u$ remains in $H^2$ throughout its lifespan (although the $H^2$-norm may blow up in "infinite time").
Modifications of this type of argument also give continuity in $H^2$.
