Modified Korteweg-de Vries on R: Difference between revisions
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*** This has been extended to the [[defocusing]] case in [[CtCoTa-p]], by a high-frequency approximation of [[mKdV]] by [[Cubic NLS on R|cubic NLS]]. (This high frequency approximation has also been utilized in [[Sch1998]]). | *** This has been extended to the [[defocusing]] case in [[CtCoTa-p]], 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 [[ | ** 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]] | ||
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** Was proven for s>3/5 in [[FoLiPo1999]] | ** Was proven for s>3/5 in [[FoLiPo1999]] | ||
** Is implicit for s >= 1 from [[KnPoVe1993]] | ** 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 [[ | ** 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]]. | ** GWP for smooth data can also be achieved from inverse scattering methods [[BdmFsShp-p]]; the same approach also works on an interval [[BdmShp-p]]. | ||
** [[Soliton]]s are asymptotically H^1 stable [[MtMe-p3]], [[MtMe-p]] | ** [[Soliton]]s are asymptotically H^1 stable [[MtMe-p3]], [[MtMe-p]] | ||
Revision as of 22:54, 14 August 2006
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 KnPoVe-p, in the sense that the solution map is no longer uniformly continuous for s < 1/4.
- 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 <xi>^s hat(u_0) lies in L^{r\u2019} for 4/3 < r <= 2 and s >= ½ - 1/2r Gr-p4
- 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
