GKdV-3 equation: Difference between revisions

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* Scaling is <span class="SpellE">s_c</span> = -1/6.
* Scaling is <span class="SpellE">s_c</span> = -1/6.
* LWP in <span class="SpellE">H^s</span> for s > -1/6 [Gr-p3]
* LWP in <span class="SpellE">H^s</span> for s > -1/6 [Gr-p3]
** Was shown for s>=1/12 [[Bibliography#KnPoVe1993|KnPoVe1993]]
** Was shown for s>=1/12 [[KnPoVe1993]]
** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]]
** Was shown for s>3/2 in [[GiTs1989]]
** 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..
** 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..
* GWP in <span class="SpellE">H^s</span> for s >= 0 [Gr-p3]
* GWP in <span class="SpellE">H^s</span> for s >= 0 [Gr-p3]
** For s>=1 this is in [[Bibliography#KnPoVe1993|KnPoVe1993]]
** 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 <span class="SpellE">integrable</span>, the one-dimensional nature of the equation suggests that "correction term" techniques will also be quite effective.
** 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.
** 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]
** 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]
* <span class="SpellE">Solitons</span> are H^1-stable [[Bibliography#CaLo1982|CaLo1982]], [[Bibliography#Ws1986|Ws1986]], [[Bibliography#BnSouSr1987|BnSouSr1987]] and asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p]
* <span class="SpellE">Solitons</span> are H^1-stable [[CaLo1982]], [[Ws1986]], [[BnSouSr1987]] and asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p]


== Periodic theory ==
== Periodic theory ==
Line 20: Line 20:
* Scaling is <span class="SpellE">s_c</span> = -1/6.
* Scaling is <span class="SpellE">s_c</span> = -1/6.
* LWP in <span class="SpellE">H^s</span> for s>=1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]]
* LWP in <span class="SpellE">H^s</span> for s>=1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]]
** Was shown for s >= 1 in [[St1997c]]
** One has analytic ill-<span class="SpellE">posedness</span> for s<1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]] by a modification of the example in [[Bibliography#KnPoVe1996|KnPoVe1996]].
** One has analytic ill-<span class="SpellE">posedness</span> for s<1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]] by a modification of the example in [[KnPoVe1996]].
* GWP in <span class="SpellE">H^s</span> for s>5/6 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]]
* GWP in <span class="SpellE">H^s</span> for s>5/6 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[Bibliography#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>[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]].
** This result may well be improvable by the "damping correction term" method in<span class="GramE"> [</span>[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]].
* ''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''<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).

Revision as of 14:30, 10 August 2006

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 in H^s for s > -1/6 [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 [CoKe-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 [CoKe-p]
  • Solitons are H^1-stable CaLo1982, Ws1986, BnSouSr1987 and asymptotically H^1 stable [MtMe-p3], [MtMe-p]

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 in H^s for s>=1/2 CoKeStTkTa-p3
  • GWP in H^s for s>5/6 CoKeStTkTa-p3
    • Was shown for s >= 1 in St1997c
    • This result may well be improvable by the "damping correction term" method in [[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]].
  • Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).