NLS wellposedness

In order to establish the well-posedness of the NLS in $H^{s}$ one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume

$p$ is an odd integer, or $p>[s]+1.$

With this assumption, one has LWP for $s\geq 0,s_{c}\,$, CaWe1990; see also Ts1987; for the case $s=1\,,$ see GiVl1979. In the $L^{2}\,$-subcritical cases one has GWP for all $s\geq 0\,$ by $L^{2}\,$ conservation; in all other cases one has GWP and scattering for small data in $H^{s}\,$, $s\,\geq s_{c}.\,$ 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 $p\,$ is either an odd integer or larger than $[s]+1\,.$ Then some of the above results are still known to hold:

• In the $H^{1}\,$ subcritical case one has GWP in $H^{1}\,,$ assuming the nonlinearity is smooth near the origin Ka1986
  • In $R^{6}\,$ one also has Lipschitz well-posedness BuGdTz-p5

In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for $s>0,s_{c}\,$ if $p>1\,$, with the endpoint $s=0\,$ being attained when $1\leq p\leq 4\,$ Bo1993. In particular one has GWP in $L^{2}\,$ when $p<4\,,$ or when $p=4\,$ and the data is small norm.For $6>p\geq 4\,$ one also has GWP for random data whose Fourier coefficients decay like $1/|k|\,$ (times a Gaussian random variable) Bo1995c. (For $p=6\,$ one needs to impose a smallness condition on the $L^{2}\,$ norm or assume defocusing; for $p>6\,$ one needs to assume defocusing).

• For the defocussing case, one has GWP in the $H^{1}\,$-subcritical case if the data is in $H^{1}\,.$

Many of the global results for $H^{s}\,$ also hold true for $L^{2}(|x|^{2s}dx)\,$. 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.