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 [[ | ** Was shown for s>=1/12 [[Bibliography#KnPoVe1993|KnPoVe1993]] | ||
** Was shown for s>3/2 in [[ | ** Was shown for s>3/2 in [[Bibliography#GiTs1989|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 [[ | ** For s>=1 this is in [[Bibliography#KnPoVe1993|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 [[ | * <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] | ||
== Periodic theory == | == Periodic theory == | ||
| Line 19: | Line 19: | ||
* 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 [[ | * LWP in <span class="SpellE">H^s</span> for s>=1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]] | ||
** Was shown for s >= 1 in [[ | ** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]] | ||
** One has analytic ill-<span class="SpellE">posedness</span> for s<1/2 [[ | ** 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]]. | ||
* GWP in <span class="SpellE">H^s</span> for s>5/6 [[ | * GWP in <span class="SpellE">H^s</span> for s>5/6 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]] | ||
** Was shown for s >= 1 in [[ | ** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]] | ||
** This result may well be improvable by the "damping correction term" method in<span class="GramE"> [</span>[ | ** 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). | ||
[[Category:Equations]] | [[Category:Equations]] | ||
[[Category:Airy]] | [[Category:Airy]] | ||
Revision as of 16:13, 31 July 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
- 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 "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).
