Cubic NLS on T

The theory of the cubic NLS on the circle is as follows.

• LWP for $s\geq 0\,$ Bo1993.
  • For $s<0\,$ one has failure of uniform local well-posedness CtCoTa-p, BuGdTz-p. In fact, the solution map is not even continuous from $H^{s}\,$ to $H^{\sigma }\,$ for any $\sigma$, even for small times and small data CtCoTa-p3.
• GWP for $s\geq 0\,$ thanks to $L^{2}\,$ conservation Bo1993.
  • One also has GWP for random data whose Fourier coefficients decay like $1/|k|\,$ (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
• If the cubic non-linearity is of ${\underline {uuu}}\,$ type (instead of $|u|^{2}u\,$) then one can obtain LWP for $s>-1/3\,$ Gr-p2
• Remark: This equation is completely integrable AbMa1981; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.