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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 <span class="SpellE">s_c</span> = -1/2.
* C^0 LWP in L^2 in the defocusing case [[KpTp-p2]]
** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [[TkTs2004]]. Note one has to gauge away a nonlinear resonance term before one can apply iteration methods. 
** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[KnPoVe1993]], [[Bo1993b]].
** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [[CtCoTa-p]].
* C^0 GWP in L^2 in the defocusing case [[KpTp-p2]].
** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> 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<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> 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).

[[Category:Integrability]]
[[Category:Airy]]
[[Category:Equations]]