We 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](http://en.wikipedia.org/wiki/Sobolev_space).


In 1978 [Giniberg-Velo](https://eudml.org/doc/75982) 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,



global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$
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"](http://www.ams.org/mathscinet-getitem?mr=MR0915266)


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"](http://www.ams.org/mathscinet-getitem?mr=1383498)


(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.*


