https://dispersivewiki.org/DispersiveWiki/index.php?title=Modified_Korteweg-de_Vries_on_T&feed=atom&action=history
Modified Korteweg-de Vries on T - Revision history
2024-03-29T09:18:17Z
Revision history for this page on the wiki
MediaWiki 1.39.3
https://dispersivewiki.org/DispersiveWiki/index.php?title=Modified_Korteweg-de_Vries_on_T&diff=4547&oldid=prev
M. Otani at 06:35, 15 February 2007
2007-02-15T06:35:53Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:35, 15 February 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2">Line 2:</td>
<td colspan="2" class="diff-lineno">Line 2:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Scaling is <span class="SpellE">s_c</span> = -1/2.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Scaling is <span class="SpellE">s_c</span> = -1/2.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* C^0 LWP in L^2 in the defocusing case [KpTp-p2]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* C^0 LWP in L^2 in the defocusing case <ins style="font-weight: bold; text-decoration: none;">[</ins>[KpTp-p2<ins style="font-weight: bold; text-decoration: none;">]</ins>]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [<del style="font-weight: bold; text-decoration: none;">Takaoka and <span class="SpellE">Tsutsumi</span>?</del>]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [<ins style="font-weight: bold; text-decoration: none;">[TkTs2004]</ins>]<ins style="font-weight: bold; text-decoration: none;">. </ins>Note one has to gauge away a nonlinear resonance term before one can apply iteration methods. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[KnPoVe1993]], [[Bo1993b]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[KnPoVe1993]], [[Bo1993b]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<del style="font-weight: bold; text-decoration: none;"><span class="SpellE"></del>CtCoTa<del style="font-weight: bold; text-decoration: none;"></span></del>-p]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<ins style="font-weight: bold; text-decoration: none;">[</ins>CtCoTa-p]<ins style="font-weight: bold; text-decoration: none;">].</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* C^0 GWP in L^2 in the defocusing case [KpTp-p2]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* C^0 GWP in L^2 in the defocusing case <ins style="font-weight: bold; text-decoration: none;">[</ins>[KpTp-p2]<ins style="font-weight: bold; text-decoration: none;">].</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Was proven for s >= 1 in [[KnPoVe1993]], [[Bo1993b]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Was proven for s >= 1 in [[KnPoVe1993]], [[Bo1993b]].</div></td></tr>
</table>
M. Otani
https://dispersivewiki.org/DispersiveWiki/index.php?title=Modified_Korteweg-de_Vries_on_T&diff=3957&oldid=prev
Pblue at 14:30, 10 August 2006
2006-08-10T14:30:43Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:30, 10 August 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td>
<td colspan="2" class="diff-lineno">Line 4:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 LWP in L^2 in the defocusing case [KpTp-p2]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 LWP in L^2 in the defocusing case [KpTp-p2]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [Takaoka and <span class="SpellE">Tsutsumi</span>?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [Takaoka and <span class="SpellE">Tsutsumi</span>?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[<del style="font-weight: bold; text-decoration: none;">Bibliography#KnPoVe1993|</del>KnPoVe1993]], [[<del style="font-weight: bold; text-decoration: none;">Bibliography#Bo1993b|</del>Bo1993b]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[KnPoVe1993]], [[Bo1993b]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[<del style="font-weight: bold; text-decoration: none;">Bibliography#KnPoVe1996|</del>KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<span class="SpellE">CtCoTa</span>-p]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<span class="SpellE">CtCoTa</span>-p]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 GWP in L^2 in the defocusing case [KpTp-p2]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 GWP in L^2 in the defocusing case [KpTp-p2]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** Was proven for s >= 1 in [[<del style="font-weight: bold; text-decoration: none;">Bibliography#KnPoVe1993|</del>KnPoVe1993]], [[<del style="font-weight: bold; text-decoration: none;">Bibliography#Bo1993b|</del>Bo1993b]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** Was proven for s >= 1 in [[KnPoVe1993]], [[Bo1993b]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[<del style="font-weight: bold; text-decoration: none;">Bibliography#Bo1995c|</del>Bo1995c]]. Indeed one has an invariant measure. Note that such data barely fails to be in H<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[Bo1995c]]. Indeed one has an invariant measure. Note that such data barely fails to be in H<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Integrability]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Integrability]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Airy]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Airy]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Equations]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Equations]]</div></td></tr>
</table>
Pblue
https://dispersivewiki.org/DispersiveWiki/index.php?title=Modified_Korteweg-de_Vries_on_T&diff=2360&oldid=prev
Pblue: More bib cleaning
2006-07-31T16:10:23Z
<p>More bib cleaning</p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:10, 31 July 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td>
<td colspan="2" class="diff-lineno">Line 4:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 LWP in L^2 in the defocusing case [KpTp-p2]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 LWP in L^2 in the defocusing case [KpTp-p2]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [Takaoka and <span class="SpellE">Tsutsumi</span>?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [Takaoka and <span class="SpellE">Tsutsumi</span>?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[<del style="font-weight: bold; text-decoration: none;">references.html</del>#KnPoVe1993 KnPoVe1993]], [[<del style="font-weight: bold; text-decoration: none;">references.html</del>#Bo1993b Bo1993b]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[<ins style="font-weight: bold; text-decoration: none;">Bibliography</ins>#KnPoVe1993<ins style="font-weight: bold; text-decoration: none;">|</ins>KnPoVe1993]], [[<ins style="font-weight: bold; text-decoration: none;">Bibliography</ins>#Bo1993b<ins style="font-weight: bold; text-decoration: none;">|</ins>Bo1993b]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[<del style="font-weight: bold; text-decoration: none;">references.html</del>#KnPoVe1996 KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<span class="SpellE">CtCoTa</span>-p]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[<ins style="font-weight: bold; text-decoration: none;">Bibliography</ins>#KnPoVe1996<ins style="font-weight: bold; text-decoration: none;">|</ins>KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<span class="SpellE">CtCoTa</span>-p]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 GWP in L^2 in the defocusing case [KpTp-p2]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* C^0 GWP in L^2 in the defocusing case [KpTp-p2]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[<del style="font-weight: bold; text-decoration: none;">references.html</del>#CoKeStTaTk-p2 CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[<ins style="font-weight: bold; text-decoration: none;">Bibliography</ins>#CoKeStTaTk-p2 <ins style="font-weight: bold; text-decoration: none;">|</ins>CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** Was proven for s >= 1 in [[<del style="font-weight: bold; text-decoration: none;">references.html</del>#KnPoVe1993 KnPoVe1993]], [[<del style="font-weight: bold; text-decoration: none;">references.html</del>#Bo1993b Bo1993b]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** Was proven for s >= 1 in [[<ins style="font-weight: bold; text-decoration: none;">Bibliography</ins>#KnPoVe1993<ins style="font-weight: bold; text-decoration: none;">|</ins>KnPoVe1993]], [[<ins style="font-weight: bold; text-decoration: none;">Bibliography</ins>#Bo1993b<ins style="font-weight: bold; text-decoration: none;">|</ins>Bo1993b]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[<del style="font-weight: bold; text-decoration: none;">references.html</del>#Bo1995c Bo1995c]]. Indeed one has an invariant measure. Note that such data barely fails to be in H<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[<ins style="font-weight: bold; text-decoration: none;">Bibliography</ins>#Bo1995c<ins style="font-weight: bold; text-decoration: none;">|</ins>Bo1995c]]. Indeed one has an invariant measure. Note that such data barely fails to be in H<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Integrability]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Integrability]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Airy]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Airy]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Equations]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Equations]]</div></td></tr>
<!-- diff cache key DispersiveWiki_db-dw_:diff::1.12:old-2333:rev-2360 -->
</table>
Pblue
https://dispersivewiki.org/DispersiveWiki/index.php?title=Modified_Korteweg-de_Vries_on_T&diff=2333&oldid=prev
Tao at 07:41, 31 July 2006
2006-07-31T07:41:10Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:41, 31 July 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references.html#Bo1995c Bo1995c]]. Indeed one has an invariant measure. Note that such data barely fails to be in H<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references.html#Bo1995c Bo1995c]]. Indeed one has an invariant measure. Note that such data barely fails to be in H<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Integrability]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Airy]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Equations]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Equations]]</div></td></tr>
</table>
Tao
https://dispersivewiki.org/DispersiveWiki/index.php?title=Modified_Korteweg-de_Vries_on_T&diff=1681&oldid=prev
Tao at 05:39, 28 July 2006
2006-07-28T05:39:51Z
<p></p>
<p><b>New page</b></p><div>The local and global [[well-posedness]] theory for the [[modified Korteweg-de Vries equation]] on the torus is as follows.<br />
<br />
* Scaling is <span class="SpellE">s_c</span> = -1/2.<br />
* C^0 LWP in L^2 in the defocusing case [KpTp-p2]<br />
** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [Takaoka and <span class="SpellE">Tsutsumi</span>?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.<br />
** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[references.html#KnPoVe1993 KnPoVe1993]], [[references.html#Bo1993b Bo1993b]].<br />
** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[references.html#KnPoVe1996 KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<span class="SpellE">CtCoTa</span>-p]<br />
* C^0 GWP in L^2 in the defocusing case [KpTp-p2]<br />
** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.<br />
** Was proven for s >= 1 in [[references.html#KnPoVe1993 KnPoVe1993]], [[references.html#Bo1993b Bo1993b]].<br />
** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references.html#Bo1995c Bo1995c]]. Indeed one has an invariant measure. Note that such data barely fails to be in H<span class="GramE">^{</span>1/2}, however one can modify the local well-<span class="SpellE">posedness</span> theory to go below H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the Fourier coefficients (which is indeed the case almost surely).<br />
<br />
[[Category:Equations]]</div>
Tao