Cubic NLS on T
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The theory of the cubic NLS on the circle is as follows.
- LWP for Bo1993.
- GWP for thanks to conservation Bo1993.
- One also has GWP for random data whose Fourier coefficients decay like (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
- If the cubic non-linearity is of type (instead of ) then one can obtain LWP for 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.