Modified Korteweg-de Vries on R

From DispersiveWiki
Revision as of 03:13, 31 July 2006 by Pblue (talk | contribs) (Cleaning bibliographic references)
Jump to navigationJump to search
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 focussing case KnPoVe-p, in the sense that the solution map is no longer uniformly continuous for s < 1/4.
      • This has been extended to the defocussing case in [CtCoTa-p], by a high-frequency approximation of mKdV by [schrodinger.html#Cubic NLS on R 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 [CoKe-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 CoKeStTkTa-p2, 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 [CoKe-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]