# Difference between revisions of "Modified Korteweg-de Vries on R"

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** Was shown for s>3/2 in [[GiTs1989]] | ** Was shown for s>3/2 in [[GiTs1989]] | ||

** This is sharp in the [[focusing]] case [[KnPoVe2001]], in the sense that the solution map is no longer uniformly continuous for s < 1/4. | ** This is sharp in the [[focusing]] case [[KnPoVe2001]], in the sense that the solution map is no longer uniformly continuous for s < 1/4. | ||

− | *** This has been extended to the [[defocusing]] case in [[ | + | *** This has been extended to the [[defocusing]] case in [[CtCoTa2003]], by a high-frequency approximation of [[mKdV]] by [[Cubic NLS on R|cubic NLS]]. (This high frequency approximation has also been utilized in [[Sch1998]]). |

*** Below 1/4 the solution map was known to not be C^3 in [[Bo1993b]], [[Bo1997]]. | *** Below 1/4 the solution map was known to not be C^3 in [[Bo1993b]], [[Bo1997]]. | ||

** The same result has also been established for the half-line [[CoKn-p]], assuming boundary data is in H<span class="GramE">^{</span>(s+1)/3} of course. | ** The same result has also been established for the half-line [[CoKn-p]], assuming boundary data is in H<span class="GramE">^{</span>(s+1)/3} of course. | ||

** Global weak solutions in L^2 were constructed in [[Ka1983]]. Thus in L^2 one has global existence but no uniform continuity. | ** Global weak solutions in L^2 were constructed in [[Ka1983]]. Thus in L^2 one has global existence but no uniform continuity. | ||

** Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 [[GiTs1989]] | ** Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 [[GiTs1989]] | ||

− | ** LWP has also been demonstrated when <math><\xi> ^s \ | + | ** LWP has also been demonstrated when <math><\xi> ^s \widehat{u_0}</math> lies in <math>L^{r/(r-1)}</math> for 4/3 < r <= 2 and s >= ½ - 1/2r [[Gr2004]] |

* GWP in <span class="SpellE">H^s</span> for s > 1/4 [[CoKeStTkTa2003]], 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. | * GWP in <span class="SpellE">H^s</span> for s > 1/4 [[CoKeStTkTa2003]], 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>3/5 in [[FoLiPo1999]] | ** Was proven for s>3/5 in [[FoLiPo1999]] |

## Latest revision as of 01:27, 17 March 2007

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

- Scaling is s_c = -1/2.
- LWP in H^s for s >= 1/4 KnPoVe1993
- Was shown for s>3/2 in GiTs1989
- This is sharp in the focusing case KnPoVe2001, in the sense that the solution map is no longer uniformly continuous for s < 1/4.
- This has been extended to the defocusing case in CtCoTa2003, by a high-frequency approximation of mKdV by cubic NLS. (This high frequency approximation has also been utilized in Sch1998).
- Below 1/4 the solution map was known to not be C^3 in Bo1993b, Bo1997.

- The same result has also been established for the half-line CoKn-p, assuming boundary data is in H^{(s+1)/3} of course.
- Global weak solutions in L^2 were constructed in Ka1983. Thus in L^2 one has global existence but no uniform continuity.
- Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 GiTs1989
- LWP has also been demonstrated when lies in for 4/3 < r <= 2 and s >= ½ - 1/2r Gr2004

- GWP in H^s for s > 1/4 CoKeStTkTa2003, via the KdV theory and the Miura transform, for both the focussing and defocussing cases.
- Was proven for s>3/5 in FoLiPo1999
- Is implicit for s >= 1 from KnPoVe1993
- On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm CoKn-p
- GWP for smooth data can also be achieved from inverse scattering methods BdmFsShp-p; the same approach also works on an interval BdmShp-p.
- Solitons are asymptotically H^1 stable MtMe-p3, MtMe-p