global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS):
$$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t),  \ (x, t\in \mathbb R),$$
$$\phi (x,0) = \phi_{0}(x)\in H^{s}(\mathbb R);$$
where $H^{s}(\mathbb R)$ is usual Sobolev space.
In 1978  Giniberg-Velo have shown that the above NLS is globally wellposed: that is, for the initial data in $\phi_{0}\in H^{1}(\mathbb R)$, the NLS has  a uniqe solution in $C(\mathbb R, H^{1}(\mathbb R)).$

My Question: Let  $\phi_{0}\in H^{s}(\mathbb R), (0<s<1).$  Then what can we say about the 
  local and global well posedness of the above NLS ? (If it is well know just the proper refeence will be o.k for me) 

Thanks,
 A: For the NLS in 1D, quintic is $L^2$ critical. So you are quite comfortably in the subcritical regime. Indeed, you have the result of 
Tsutsumi, Yoshio; "$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups"
which implies for $H^s$ (and hence $L^2$) initial data, you have global existence of solutions in the space $C(\mathbb{R}; L^2(\mathbb{R}))$ intersected with some Strichartz space. 
For local solutions in $H^s$, in the sense of having solutions in the space $C((T_{min}, T_{max}); H^s(\mathbb{R}))$, you can look at 
Kato, Tosio; "On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness"
(Remark, the nonlinearities considered in Kato's paper allow more general forms than that of Tsutsumi's.)
The main problem here is that we have no direct conservation law on the level of $H^s$, and so global well-posedness cannot follow, as in the $L^2$ case, from local well-posedness. On the other hand, the Strichartz type estimates used in Kato's work is sufficient to allow a direct proof of global well posedness provided that the initial data in $H^s$ is sufficiently small. 
(Remark: for general results concerning NLS, a good reference book is Cazenave's Semilinear Schroedinger equations,)
However, global wellposedness for the cubic NLS is true in 1 spatial dimension. Here we can use the fact that we are looking at an algebraic nonlinearity: that the right hand side is $\pm \phi\bar{\phi} \phi$. In particular, for this equation we can prove persistence of regularity. For this equation the statement is
Proposition (Prop. 3.11, p132, Tao, Nonlinear dispersive equations)
Let $u\in C((t_1,t_2); H^s(\mathbb{R}))$ be a (local) solution for $s\geq 0$, with $t_1 < 0 < t_2$. Then there exists a constant $C$ depending on $s$ such that
$$ \|u\|_{L^\infty((t_1,t_2); H^s(\mathbb{R}))} \leq \|u(0)\|_{H^s} \exp (C \|u\|^3_{L^2((t_1,t_2); L^\infty(\mathbb{R}))}) $$
provided that the right hand side of the estimate is finite. 
It turns out that for the 1D Schrodinger equation you can control $L^4_t L^\infty_x$ using Strichartz estimates from $L^2$ of initial data, and so restricting to finite intervals you also pick up $L^2_t L^\infty_x$ which then allows you to use the $L^2$ global well-posedness result above to obtain global wellposedness in any $H^s$ for $s > 0$; however, we lose control over the $H^s$ norm over time. If you run the argument using the above proposition, you get that $\|u(t)\|_{H^s(\mathbb{R})}$ is allowed to grow exponentially in $t$. For more details see Section 3.4 in Tao, op. cit.
