GKdV-3 equation: Difference between revisions

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{{equation
| name = Quartic gKdV
| equation = <math>u_t + u_{xxx} = \pm u^3 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>\dot H^{-1/6}(\R)</math>
| criticality = mass-subcritical, energy-subcritical
| covariance = -
| lwp = <math>H^s(\R)</math> for <math>s \geq -1/6</math>
| gwp = <math>H^s(\R)</math> for <math>s \geq -1/6</math>, small norm
| parent = [[gKdV]]
| special = -
| related = -
}}
== Non-periodic theory ==
== Non-periodic theory ==


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


* Scaling is <span class="SpellE">s_c</span> = -1/6.
* Scaling is s_c = -1/6.
* LWP in <span class="SpellE">H^s</span> for s > -1/6 [Gr-p3]
* LWP for s >= -1/6 [[Ta2007]]
** Was shown for s>=1/12 [[references.html#KnPoVe1993 KnPoVe1993]]
** For s > -1/6 this is in [[Gr-p3]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
** Was shown for s>=1/12 [[KnPoVe1993]]
** The result s >= 1/12 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..
** Was shown for s>3/2 in [[GiTs1989]]
* GWP in <span class="SpellE">H^s</span> for s >= 0 [Gr-p3]
** The result s >= 1/12 has also been established for the half-line [[CoKn-p]], assuming boundary data is in H^{(s+1)/3} of course.
** For s>=1 this is in [[references.html#KnPoVe1993 KnPoVe1993]]
* GWP in H^s for s >= 0 [[Gr-p3]]
** Presumably one can use either the Fourier truncation method or the "I-method" to go below L^2. Even though the equation is not completely <span class="SpellE">integrable</span>, the one-dimensional nature of the equation suggests that "correction term" techniques will also be quite effective.
** For s>=1 this is in [[KnPoVe1993]]
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{5/4}, assuming compatibility and small L^2 norm [<span class="SpellE">CoKe</span>-p]
** Presumably one can use either the Fourier truncation method or the [[I-method]] to go below L^2. Even though the equation is not [[completely integrable]], the one-dimensional nature of the equation suggests that [[correction term]] techniques will also be quite effective.
* <span class="SpellE">Solitons</span> are H^1-stable [[references.html#CaLo1982 CaLo1982]], [[references.html#Ws1986 Ws1986]], [[references.html#BnSouSr1987 BnSouSr1987]] and asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p]
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{5/4}, assuming compatibility and small L^2 norm [[CoKn-p]]
* [[Solitons]] are H^1-stable [[CaLo1982]], [[Ws1986]], [[BnSouSr1987]] and asymptotically H^1 stable [[MtMe-p3]], [[MtMe-p]]
** If one also assumes the error is small in the critical space <math>\dot H^{-1/6}(\R)</math> then one has asymptotic stability [[Ta2007]]


== Periodic theory ==
== Periodic theory ==
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The local and global [[well-posedness]] theory for the quartic [[generalized Korteweg-de Vries equation]] on the line and half-line is as follows.
The local and global [[well-posedness]] theory for the quartic [[generalized Korteweg-de Vries equation]] on the line and half-line is as follows.


* Scaling is <span class="SpellE">s_c</span> = -1/6.
* Scaling is s_c = -1/6.
* 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]]
** One has analytic ill-<span class="SpellE">posedness</span> for s<1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]] by a modification of the example in [[references.html#KnPoVe1996 KnPoVe1996]].
** One has analytic ill-posedness for s<1/2 [[CoKeStTkTa-p3]] by a modification of the example in [[KnPoVe1996]].
* GWP in <span class="SpellE">H^s</span> for s>5/6 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
* GWP in H^s for s>5/6 [[CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[references.html#St1997c St1997c]]
** Was shown for s >= 1 in [[St1997c]]
** This result may well be improvable by the "damping correction term" method in<span class="GramE"> [</span>[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]].
** This result may well be improvable by the [[correction term]] method.
* ''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 P(u).


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

Latest revision as of 22:26, 4 March 2007

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


Non-periodic theory

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

  • Scaling is s_c = -1/6.
  • LWP for s >= -1/6 Ta2007
    • For s > -1/6 this is in Gr-p3
    • Was shown for s>=1/12 KnPoVe1993
    • Was shown for s>3/2 in GiTs1989
    • The result s >= 1/12 has also been established for the half-line CoKn-p, assuming boundary data is in H^{(s+1)/3} of course.
  • GWP in H^s for s >= 0 Gr-p3
    • For s>=1 this is in KnPoVe1993
    • Presumably one can use either the Fourier truncation method or the I-method to go below L^2. Even though the equation is not completely integrable, the one-dimensional nature of the equation suggests that correction term techniques will also be quite effective.
    • On the half-line GWP is known when s >= 1 and the boundary data is in H^{5/4}, assuming compatibility and small L^2 norm CoKn-p
  • Solitons are H^1-stable CaLo1982, Ws1986, BnSouSr1987 and asymptotically H^1 stable MtMe-p3, MtMe-p
    • If one also assumes the error is small in the critical space then one has asymptotic stability Ta2007

Periodic theory

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

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