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 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 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.