Difference between revisions of "Modified Korteweg-de Vries on T"

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* Scaling is <span class="SpellE">s_c</span> = -1/2.
 
* Scaling is <span class="SpellE">s_c</span> = -1/2.
* C^0 LWP in L^2 in the defocusing case [KpTp-p2]
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* 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 [Takaoka and <span class="SpellE">Tsutsumi</span>?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.
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** 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]].
 
** 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> [<span class="SpellE">CtCoTa</span>-p]
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** 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]
+
* 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.
 
** 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]].
 
** Was proven for s >= 1 in [[KnPoVe1993]], [[Bo1993b]].

Latest revision as of 06:35, 15 February 2007

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).