GKdV-4 equation: Difference between revisions

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{{equation
| name = Quintic gKdV
| equation = <math>u_t + u_{xxx} = \pm u^4 u_x</math>
| fields = <math>u: \R \times \R \to \R</math>
| data = <math>u(0) \in H^s(\R)</math>
| hamiltonian = [[Hamiltonian]]
| linear = [[Airy equation|Airy]]
| nonlinear = [[semilinear|semilinear with derivatives]]
| critical = <math>L^2(\R)</math>
| criticality = mass-critical, energy-subcritical
| covariance = -
| lwp = <math>H^s(\R)</math> for <math>s \geq 0</math>
| gwp = <math>H^s(\R)</math> for <math>s \geq 0</math>, small
| parent = [[gKdV]]
| special = -
| related = -
}}
== 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].
(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.  
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 <span class="SpellE">s_c</span> = 0 (i.e. L^2-critical).
* Scaling is s_c = 0 (i.e. L^2-critical).
* LWP in <span class="SpellE">H^s</span> for s >= 0 [[references.html#KnPoVe1993 KnPoVe1993]]
* LWP in H^s for s >= 0 [[KnPoVe1993]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
** Was shown for s>3/2 in [[GiTs1989]]
** The same result s >= 0 has also been established for the half-line [<span class="SpellE">CoKe</span>-p], assuming boundary data is in H<span class="GramE">^{</span>(s+1)/3} of course..
** 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 <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 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]]
** 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], [[references.html#MtMe2001 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.
** 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 [<span class="SpellE">BnWe</span>-p] but these are not in L^2.
** 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 <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 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 [<span class="SpellE">CoKe</span>-p]
** 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]]
* <span class="SpellE">Solitons</span> are H^1-unstable [[references.html#MtMe2001 MtMe2001]]. However, small H^1 perturbations of a <span class="SpellE">soliton</span> must asymptotically converge weakly to some rescaled <span class="SpellE">soliton</span> shape provided that the H^1 norm stays comparable to 1 [[references.html#MtMe-p <span class="SpellE">MtMe</span>-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 ==
== Periodic theory ==
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The local and global [[well-posedness]] theory for the quintic [[generalized Korteweg-de Vries equation]] on the torus is as follows.  
The local and global [[well-posedness]] theory for the quintic [[generalized Korteweg-de Vries equation]] on the torus is as follows.  


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


[[Category:Equations]]
[[Category:Equations]]
[[Category:Airy]]

Latest revision as of 22:03, 4 March 2007

Quintic gKdV
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
    • 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 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.