Difference between revisions of "GKdV-4 equation"

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== Non-periodic theory ==
 
== Non-periodic theory ==
  
(Thanks to Felipe <span class="SpellE">Linares</span> for help with the references here - Ed.)A good survey for the results here is in [Tz-p2].
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(Thanks to Felipe <span class="SpellE">Linares</span> 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.  
 
The local and global [[well-posedness]] theory for the quintic [[generalized Korteweg-de Vries equation]] on the line and half-line is as follows.  
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* GWP in <span class="SpellE">H^s</span> 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 [<span class="SpellE">FoLiPo</span>-p].
 
* GWP in <span class="SpellE">H^s</span> 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 [<span class="SpellE">FoLiPo</span>-p].
 
** For s >= 1 and the defocusing case this is in [[KnPoVe1993]]
 
** For s >= 1 and the defocusing case this is in [[KnPoVe1993]]
** Blowup has recently been shown for the <span class="SpellE">focussing</span> 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 <span class="SpellE">scalings</span> and translations), see [MtMe-p4], [[MtMe2001]]. Also, the blow up rate in H^1 must be strictly faster than t<span class="GramE">^{</span>-1/3} [MtMe-p4], which is the rate suggested by scaling.
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** Blowup has recently been shown for the <span class="SpellE">focussing</span> 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 <span class="SpellE">scalings</span> and translations), see [[MtMe-p4]], [[MtMe2001]]. Also, the blow up rate in H^1 must be strictly faster than t<span class="GramE">^{</span>-1/3} [[MtMe-p4]], which is the rate suggested by scaling.
 
** Explicit self-similar blow-up solutions have been constructed [<span class="SpellE">BnWe</span>-p] but these are not in L^2.
 
** Explicit self-similar blow-up solutions have been constructed [<span class="SpellE">BnWe</span>-p] but these are not in L^2.
 
** GWP for small L^2 data in either case [[KnPoVe1993]]. In the <span class="SpellE">focussing</span> case we have GWP whenever the L^2 norm is strictly smaller than that of the ground state Q (thanks to Weinstein's sharp <span class="SpellE">Gagliardo-Nirenberg</span> inequality). It seems like a reasonable (but difficult) conjecture to have GWP for large L^2 data in the defocusing case.
 
** GWP for small L^2 data in either case [[KnPoVe1993]]. In the <span class="SpellE">focussing</span> case we have GWP whenever the L^2 norm is strictly smaller than that of the ground state Q (thanks to Weinstein's sharp <span class="SpellE">Gagliardo-Nirenberg</span> inequality). It seems like a reasonable (but difficult) conjecture to have GWP for large L^2 data in the defocusing case.

Revision as of 23:58, 15 February 2007

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 KnPoVe1993
    • Was shown for s>3/2 in 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 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, 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 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 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 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.

  • Scaling is s_c = 0.
  • LWP in H^s for s>=1/2 CoKeStTkTa-p3
    • Was shown for s >= 1 in St1997c
    • Analytic well-posedness fails for s < 1/2; this is essentially in KnPoVe1996
  • GWP in H^s for s>=1 St1997c
    • This is almost certainly improvable by the techniques in CoKeStTkTa-p3, probably to s > 6/7. There are some low-frequency issues which may require the techniques in KeTa-p.
  • Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).