Quintic NLS on T

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

• This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
• Scaling is $s_{c}=0\,$.
• LWP is known for $s>0\,$ Bo1993.
  • For $s<0\,$ the solution map is not uniformly continuous from $C^{k}\,$ to $C^{-k}\,$ for any $k\,$ CtCoTa-p3.
• GWP is known in the defocusing case for $s>4/9\,$ (De Silva, Pavlovic, Staffilani, Tzirakis)
  • For $s>2/3\,$ this is commented upon in Bo-p2 and is a minor modification of CoKeStTkTa-p.
  • For $s\geq 1\,$ one has GWP in the defocusing case, or in the focusing case with small $L^{2}\,$ norm, by Hamiltonian conservation.
    • In the defocusing case one has GWP for random data whose Fourier coefficients decay like $1/|k|\,$ (times a Gaussian random variable) Bo1995c; this is roughly of the regularity of $H^{1/2}\,$. Indeed one has an invariant measure. In the focusing case the same result holds assuming the $L^{2}\,$ norm is sufficiently small.