Quintic NLS on T
From DispersiveWiki
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
.
- LWP is known for
Bo1993.
- For
the solution map is not uniformly continuous from 
to 
for any 
CtCoTa-p3.
- For
- GWP is known in the defocusing case for
(De Silva, Pavlovic, Staffilani, Tzirakis)
- For
this is commented upon in Bo-p2 and is a minor modification of CoKeStTkTa-p.
- For
one has GWP in the defocusing case, or in the focusing case with small 
norm, by Hamiltonian conservation.
- In the defocusing case one has GWP for random data whose Fourier coefficients decay like
(times a Gaussian random variable) Bo1995c; this is roughly of the regularity of 
. Indeed one has an invariant measure. In the focusing case the same result holds assuming the norm is sufficiently small.
- In the defocusing case one has GWP for random data whose Fourier coefficients decay like
- For
