NLS wellposedness
From DispersiveWiki
In order to establish the well-posedness of the NLS in Hs one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume
With this assumption, one has LWP for 
, CaWe1990; see also Ts1987; for the case 
see GiVl1979. In the 
-subcritical cases one has GWP for all 
by conservation; in all other cases one has GWP and scattering for small data in 
, 
These results apply in both the focussing and defocussing cases. At the critical exponent one can prove Besov space refinements Pl2000, Pl-p4. This can then be used to obtain self-similar solutions, see CaWe1998, CaWe1998b, RiYou1998, MiaZg-p1, MiaZgZgx-p, MiaZgZgx-p2, Fur2001.
Now suppose we remove the regularity assumption that 
is either an odd integer or larger than ![[s]+1\,.](/DispersiveWiki/images/math/0/b/6/0b606cd6be75dc01fa40f69ed9a14f22.png)
Then some of the above results are still known to hold:
- In the
subcritical case one has GWP in 
assuming the nonlinearity is smooth near the origin Ka1986
- In
one also has Lipschitz well-posedness BuGdTz2003
- In
In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for 
if 
, with the endpoint 
being attained when 
Bo1993. In particular one has GWP in when 
or when 
and the data is small norm.For 
one also has GWP for random data whose Fourier coefficients decay like 
(times a Gaussian random variable) Bo1995c. (For 
one needs to impose a smallness condition on the norm or assume defocusing; for 
one needs to assume defocusing).
- For the defocussing case, one has GWP in the -subcritical case if the data is in
Many of the global results for also hold true for 
. Heuristically this follows from the pseudo-conformal transformation, although making this rigorous is sometimes difficult. Sample results are in CaWe1992, GiOzVl1994, Ka1995, NkrOz1997, NkrOz-p. See NaOz2002 for further discussion.
