Modified Korteweg-de Vries on T

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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 TkTs2004. 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 KnPoVe1993, Bo1993b.
    • This is sharp in the sense of analytic well-posedness 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 [[Bibliography#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 KnPoVe1993, Bo1993b.
    • One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) 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).