GKdV-4 equation
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Description | |
---|---|
Equation | |
Fields | |
Data class | |
Basic characteristics | |
Structure | Hamiltonian |
Nonlinearity | semilinear with derivatives |
Linear component | Airy |
Critical regularity | |
Criticality | mass-critical, energy-subcritical |
Covariance | - |
Theoretical results | |
LWP | for |
GWP | for , small |
Related equations | |
Parent class | gKdV |
Special cases | - |
Other related | - |
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
- 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 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 Pu.