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 [[KnPoVe1993]] | ** Was shown for s>=1/12 [[KnPoVe1993]] | ||
** Was shown for s>3/2 in [[GiTs1989]] | ** Was shown for s>3/2 in [[GiTs1989]] | ||
** The result s >= 1/12 has also been established for the half-line [ | ** The result s >= 1/12 has also been established for the half-line [[CoKn-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 [[KnPoVe1993]] | ** For s>=1 this is in [[KnPoVe1993]] | ||
** Presumably one can use either the Fourier truncation method or the | ** 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 [ | ** 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]] | ||
== Periodic theory == | == Periodic theory == | ||
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* GWP in <span class="SpellE">H^s</span> for s>5/6 [[CoKeStTkTa-p3]] | * GWP in <span class="SpellE">H^s</span> for s>5/6 [[CoKeStTkTa-p3]] | ||
** Was shown for s >= 1 in [[St1997c]] | ** Was shown for s >= 1 in [[St1997c]] | ||
** This result may well be improvable by the | ** 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''<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). | ||
[[Category:Equations]] | [[Category:Equations]] | ||
[[Category:Airy]] | [[Category:Airy]] | ||
Revision as of 22:56, 14 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 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
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
- Was shown for s >= 1 in St1997c
- One has analytic ill-posedness for s<1/2 CoKeStTkTa-p3 by a modification of the example in KnPoVe1996.
- 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 correction term method.
- Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).
