Modified Korteweg-de Vries on T

The local and global well-posedness theory for the modified Korteweg-de Vries equation on the torus is as follows.

• Scaling is s_c = -1/2.
• C^0 LWP in L^2 in the defocusing case [KpTp-p2]
  • C^0 LWP in H^s for s > 3/8 [Takaoka and Tsutsumi?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.
  • Analytic LWP in H^s for s >= 1/2, in both focusing and defocusing cases references.html#KnPoVe1993 KnPoVe1993, references.html#Bo1993b Bo1993b.
  • This is sharp in the sense of analytic well-posedness references.html#KnPoVe1996 KnPoVe1996 or uniform well-posedness [CtCoTa-p]
• C^0 GWP in L^2 in the defocusing case [KpTp-p2]
  • Analytic GWP in H^s for s >= 1/2 [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]], via the KdV theory and the Miura transform, for both the focussing and defocussing cases.
  • Was proven for s >= 1 in references.html#KnPoVe1993 KnPoVe1993, references.html#Bo1993b Bo1993b.
  • One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) references.html#Bo1995c Bo1995c. Indeed one has an invariant measure. Note that such data barely fails to be in H^{1/2}, however one can modify the local well-posedness theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).