GKdV-4 equation: Difference between revisions

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Revision as of 05:03, 29 July 2006

Non-periodic theory

(Thanks to Felipe Linares for help with the references here - Ed.)A good survey for the results here is in [Tz-p2].

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

  • Scaling is s_c = 0 (i.e. L^2-critical).
  • LWP in H^s for s >= 0 references.html#KnPoVe1993 KnPoVe1993
    • Was shown for s>3/2 in references.html#GiTs1989 GiTs1989
    • The same result s >= 0 has also been established for the half-line [CoKe-p], assuming boundary data is in H^{(s+1)/3} of course..
  • GWP in H^s for s > 3/4 in both the focusing and defocusing cases, though one must of course have smaller L^2 mass than the ground state in the focusing case [FoLiPo-p].
    • For s >= 1 and the defocusing case this is in references.html#KnPoVe1993 KnPoVe1993
    • Blowup has recently been shown for the focussing case for data close to a ground state with negative energy [Me-p]. In such a case the blowup profile must approach the ground state (modulo scalings and translations), see [MtMe-p4], references.html#MtMe2001 MtMe2001. Also, the blow up rate in H^1 must be strictly faster than t^{-1/3} [MtMe-p4], which is the rate suggested by scaling.
    • Explicit self-similar blow-up solutions have been constructed [BnWe-p] but these are not in L^2.
    • GWP for small L^2 data in either case references.html#KnPoVe1993 KnPoVe1993. In the focussing case we have GWP whenever the L^2 norm is strictly smaller than that of the ground state Q (thanks to Weinstein's sharp Gagliardo-Nirenberg inequality). It seems like a reasonable (but difficult) conjecture to have GWP for large L^2 data in the defocusing case.
    • 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]
  • Solitons are H^1-unstable references.html#MtMe2001 MtMe2001. However, small H^1 perturbations of a soliton must asymptotically converge weakly to some rescaled soliton shape provided that the H^1 norm stays comparable to 1 [[references.html#MtMe-p MtMe-p]].

Periodic theory

The local and global well-posedness theory for the quintic generalized Korteweg-de Vries equation on the torus is as follows.