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Cubic NLS on R3 - Revision history
2024-03-28T21:21:11Z
Revision history for this page on the wiki
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Tao at 06:22, 21 July 2007
2007-07-21T06:22:25Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:22, 21 July 2007</td>
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<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>* LWP for <math>s \ge 1/2\,</math> [[CaWe1990]].</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>* LWP for <math>s \ge 1/2\,</math> [[CaWe1990]].</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>** For <math>s=1/2\,</math> the time of existence depends on the profile of the data as well as the norm.</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>** For <math>s=1/2\,</math> the time of existence depends on the profile of the data as well as the norm.</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>** For <math>s<1/2\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling.</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>** For <math>s<1/2\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the <ins style="font-weight: bold; text-decoration: none;">[[</ins>virial identity<ins style="font-weight: bold; text-decoration: none;">]] </ins>and scaling.</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>** For <math>s > 1/2\,</math> there is unconditional well-posedness [[FurPlTer2001]]</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>** For <math>s > 1/2\,</math> there is unconditional well-posedness [[FurPlTer2001]]</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>*** For <math>s >= 2/3\,</math> this is in [[Ka1995]].</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>*** For <math>s >= 2/3\,</math> this is in [[Ka1995]].</div></td></tr>
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Tao
https://dispersivewiki.org/DispersiveWiki/index.php?title=Cubic_NLS_on_R3&diff=4598&oldid=prev
Tao at 20:28, 4 March 2007
2007-03-04T20:28:31Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:28, 4 March 2007</td>
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<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>** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation. One also has scattering in this case [[GiVl1985]].</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>** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation. One also has scattering in this case [[GiVl1985]].</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>** For small <math>H^{1/2}\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is <math>L^5\,</math> in spacetime.</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>** For small <math>H^{1/2}\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is <math>L^5\,</math> in spacetime.</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>** In the focusing case one has blowup whenever the energy is negative [[Gs1977]], [[OgTs1991]], and in particular one has blowup arbitrarily close to the ground state [[BerCa1981]], [[SaSr1985]].If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms <math>H^s_{loc}\,</math> remaining bounded for all time [[Bo1996c]], [[Bo1998b]]. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component [[<del style="font-weight: bold; text-decoration: none;">Ta-p7</del>]]</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>** In the focusing case one has blowup whenever the energy is negative [[Gs1977]], [[OgTs1991]], and in particular one has blowup arbitrarily close to the ground state [[BerCa1981]], [[SaSr1985]].If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms <math>H^s_{loc}\,</math> remaining bounded for all time [[Bo1996c]], [[Bo1998b]]. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component [[<ins style="font-weight: bold; text-decoration: none;">Ta2004b</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;"><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>
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<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=Cubic_NLS_on_R3&diff=3977&oldid=prev
Pblue at 14:35, 10 August 2006
2006-08-10T14:35:49Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:35, 10 August 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l30">Line 30:</td>
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<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>** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation. One also has scattering in this case [[GiVl1985]].</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>** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation. One also has scattering in this case [[GiVl1985]].</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>** For small <math>H^{1/2}\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is <math>L^5\,</math> in spacetime.</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>** For small <math>H^{1/2}\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is <math>L^5\,</math> in spacetime.</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>** In the focusing case one has blowup whenever the energy is negative [[Gs1977]], [[OgTs1991]], and in particular one has blowup arbitrarily close to the ground state [[BerCa1981]], [[SaSr1985]].If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms <math>H^s_{loc}\,</math> remaining bounded for all time [[<del style="font-weight: bold; text-decoration: none;">Bibliography#Bo1996c|</del>Bo1996c]], [[Bo1998b]]. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component [[Ta-p7]]</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>** In the focusing case one has blowup whenever the energy is negative [[Gs1977]], [[OgTs1991]], and in particular one has blowup arbitrarily close to the ground state [[BerCa1981]], [[SaSr1985]].If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms <math>H^s_{loc}\,</math> remaining bounded for all time [[Bo1996c]], [[Bo1998b]]. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component [[Ta-p7]]</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:Schrodinger]]</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:Schrodinger]]</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=Cubic_NLS_on_R3&diff=3405&oldid=prev
Tao at 05:05, 8 August 2006
2006-08-08T05:05:52Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 05:05, 8 August 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<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>====<del style="font-weight: bold; text-decoration: none;">Cubic NLS on </del><math>R^3</math>====</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><ins style="font-weight: bold; text-decoration: none;">{{equation</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;"> | name </ins>= <ins style="font-weight: bold; text-decoration: none;">Cubic NLS on <math>\R^3</math></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;"> | equation = <math>iu_t + \Delta u = \pm |u|^2 u</math></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;"> | fields = <math>u: \R \times \R^3 \to \mathbb{C}</math></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;"> | data </ins>= <ins style="font-weight: bold; text-decoration: none;"><math>u(0) \in H^s(\R^3)</math></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;"> | hamiltonian = [[Hamiltonian]]</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;"> | linear </ins>= <ins style="font-weight: bold; text-decoration: none;">[[free Schrodinger equation|Schrodinger]]</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;"> | nonlinear = [[semilinear]]</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;"> | critical </ins>= <math><ins style="font-weight: bold; text-decoration: none;">\dot H^{1/2}(\</ins>R^3<ins style="font-weight: bold; text-decoration: none;">)</math></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;"> | criticality = mass-supercritical;<br> energy-subcritical;<br> scattering-subcritical</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;"> | covariance = [[Galilean]]</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;"> | lwp = <math>H^s(\R)</ins></math> <ins style="font-weight: bold; text-decoration: none;">for <math>s \geq 1/2</math></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;"> | gwp </ins>= <ins style="font-weight: bold; text-decoration: none;"><math>H^s(\R)</math> for <math>s \geq 4/5</math> (+)<br>or for <math>s \geq 0</math>, small norm</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;"> | parent </ins>= <ins style="font-weight: bold; text-decoration: none;">[[cubic NLS]]</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;"> | special </ins>= <ins style="font-weight: bold; text-decoration: none;">-</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;"> | related </ins>= <ins style="font-weight: bold; text-decoration: none;">-</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;">}}</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> </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;">The theory of the [[cubic NLS]] on <math>\R^3</math> is as follows.</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;"><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" 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><del style="font-weight: bold; text-decoration: none;">* Scaling is <math>s_c = 1/2\,</math>.</del></div></td><td colspan="2" class="diff-side-added"></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>* LWP for <math>s \ge 1/2\,</math> [[CaWe1990]].</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>* LWP for <math>s \ge 1/2\,</math> [[CaWe1990]].</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>** For <math>s=1/2\,</math> the time of existence depends on the profile of the data as well as the norm.</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>** For <math>s=1/2\,</math> the time of existence depends on the profile of the data as well as the norm.</div></td></tr>
</table>
Tao
https://dispersivewiki.org/DispersiveWiki/index.php?title=Cubic_NLS_on_R3&diff=3300&oldid=prev
Tao at 21:20, 5 August 2006
2006-08-05T21:20:02Z
<p></p>
<p><b>New page</b></p><div>====Cubic NLS on <math>R^3</math>====<br />
<br />
* Scaling is <math>s_c = 1/2\,</math>.<br />
* LWP for <math>s \ge 1/2\,</math> [[CaWe1990]].<br />
** For <math>s=1/2\,</math> the time of existence depends on the profile of the data as well as the norm.<br />
** For <math>s<1/2\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling.<br />
** For <math>s > 1/2\,</math> there is unconditional well-posedness [[FurPlTer2001]]<br />
*** For <math>s >= 2/3\,</math> this is in [[Ka1995]].<br />
* GWP and scattering for <math>s > 4/5\,</math> in the defocussing case [[CoKeStTkTa-p8]]<br />
** For <math>s > 5/6\,</math> GWP is in [[CoKeStTkTa2002]]<br />
** For <math>s>11/13\,</math> GWP is in [[Bo1999]]<br />
** For radial data and <math>s > 5/7\,</math> GWP and scattering is in <math>s>5/7\,</math> [[Bo1998b]], [[Bo1999]].<br />
** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation. One also has scattering in this case [[GiVl1985]].<br />
** For small <math>H^{1/2}\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is <math>L^5\,</math> in spacetime.<br />
** In the focusing case one has blowup whenever the energy is negative [[Gs1977]], [[OgTs1991]], and in particular one has blowup arbitrarily close to the ground state [[BerCa1981]], [[SaSr1985]].If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms <math>H^s_{loc}\,</math> remaining bounded for all time [[Bibliography#Bo1996c|Bo1996c]], [[Bo1998b]]. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component [[Ta-p7]]<br />
<br />
[[Category:Schrodinger]]<br />
[[Category:Equations]]</div>
Tao