https://dispersivewiki.org/DispersiveWiki/index.php?title=Linear-derivative_nonlinear_wave_equations&feed=atom&action=historyLinear-derivative nonlinear wave equations - Revision history2024-03-28T17:30:43ZRevision history for this page on the wikiMediaWiki 1.39.3https://dispersivewiki.org/DispersiveWiki/index.php?title=Linear-derivative_nonlinear_wave_equations&diff=4081&oldid=prevTao at 23:10, 14 August 20062006-08-14T23:10:37Z<p></p>
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<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 23:10, 14 August 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</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>This equation has the same scaling as [[cubic NLW]], but is more difficult technically because of the derivative term ''uDu''. </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>This equation has the same scaling as [[cubic NLW]], but is more difficult technically because of the derivative term ''uDu''. </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>Important examples of this type of equation include the [[MKG|Maxwell-Klein-Gordon]] and [[YM|Yang-Mills]] equations (in the <del style="font-weight: bold; text-decoration: none;">Lorentz </del>gauge, at least), as well as the simplified model equations for these equations. The [[YMH|Yang-Mills-Higgs]] equation is formed by coupling equations of this type to a semi-linear wave equation. The most interesting dimensions are 3 (for physical applications) and 4 (since the energy regularity is then critical). The two-dimensional case appears to be somewhat under-explored. The Yang-Mills and Maxwell-Klein-Gordon equations behave very similarly, but from a technical standpoint the latter is slightly easier to analyze.</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>Important examples of this type of equation include the [[MKG|Maxwell-Klein-Gordon]] and [[YM|Yang-Mills]] equations (in the <ins style="font-weight: bold; text-decoration: none;">[[Lorenz </ins>gauge<ins style="font-weight: bold; text-decoration: none;">]]</ins>, at least), as well as the simplified model equations for these equations. The [[YMH|Yang-Mills-Higgs]] equation is formed by coupling equations of this type to a semi-linear wave equation. The most interesting dimensions are 3 (for physical applications) and 4 (since the energy regularity is then critical). The two-dimensional case appears to be somewhat under-explored. The Yang-Mills and Maxwell-Klein-Gordon equations behave very similarly, but from a technical standpoint the latter is slightly easier to analyze.</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>In d dimensions, the critical regularity for this equation is <math>s_c = d/2 - 1</math>. However, there are no instances of Quadratic DNLW for which any sort of well-posedness is known at this critical regularity (except for those special cases where the equation can be algebraically transformed into a simpler equation such as NLW or the free wave equation).</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 d dimensions, the <ins style="font-weight: bold; text-decoration: none;">[[</ins>critical<ins style="font-weight: bold; text-decoration: none;">]] </ins>regularity for this equation is <math>s_c = d/2 - 1</math>. However, there are no instances of Quadratic DNLW for which any sort of well-posedness is known at this critical regularity (except for those special cases where the equation can be algebraically transformed into a simpler equation such as <ins style="font-weight: bold; text-decoration: none;">[[</ins>NLW<ins style="font-weight: bold; text-decoration: none;">]] </ins>or the <ins style="font-weight: bold; text-decoration: none;">[[</ins>free wave equation<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;"><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>Energy estimates give local well-posedness for <math>s > s_c + 1</math>. Using Strichartz estimates this can be improved to <math>s > s_c + 3/4</math> in two dimensions and <math>s > s_c + 1/2</math> in three and higher dimensions [[PoSi1993]]; the point is that these regularity assumptions together with Strichartz allow one to put <math>f</math> into <math>L^2_t L^{\infty}_x</math>, hence in <math>L^1_t L^{\infty}_x</math>, so that one can then use the energy method.</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>Energy estimates give local well-posedness for <math>s > s_c + 1</math>. Using <ins style="font-weight: bold; text-decoration: none;">[[</ins>Strichartz estimates<ins style="font-weight: bold; text-decoration: none;">]] </ins>this can be improved to <math>s > s_c + 3/4</math> in two dimensions and <math>s > s_c + 1/2</math> in three and higher dimensions [[PoSi1993]]; the point is that these regularity assumptions together with Strichartz allow one to put <math>f</math> into <math>L^2_t L^{\infty}_x</math>, hence in <math>L^1_t L^{\infty}_x</math>, so that one can then use the energy method.</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>Using <math>X^{s,\theta}</math> estimates [[FcKl2000]] instead of Strichartz estimates, one can improve this further to <math>d > s_c + 1/4</math> in four dimensions and to the near-optimal <math>s > s_c</math> in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space <math>B^{s_c}_{2,1}</math> [[Stz-p3]], and local well-posedness for large Besov data.In four dimensions one has a similar result if one imposes one additional angular derivative of regularity [[Stz-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>Using <math>X^{s,\theta}</math> estimates [[FcKl2000]] instead of Strichartz estimates, one can improve this further to <math>d > s_c + 1/4</math> in four dimensions and to the near-optimal <math>s > s_c</math> in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space <math>B^{s_c}_{2,1}</math> [[Stz-p3]], and local well-posedness for large Besov data.In four dimensions one has a similar result if one imposes one additional angular derivative of regularity [[Stz-p2]].</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</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>Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in <math>H^1</math> [[Lb1993]], although one can recover well-posedness in the Besov space B^1_{2,1} [[Na1999]], or when an epsilon of radial regularity is imposed [[MacNkrNaOz-p]]. It would be interesting to determine what the situation is in the other low dimensions.</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>Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in <math>H^1</math> [[Lb1993]], although one can recover well-posedness in the Besov space B^1_{2,1} [[Na1999]], or when an epsilon of radial regularity is imposed [[MacNkrNaOz-p]]. It would be interesting to determine what the situation is in the other low dimensions.</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>If the non-linearity <math><del style="font-weight: bold; text-decoration: none;">fDf</del></math> has a null structure then one can improve upon the previous results. For instance, the model equations for the Yang-Mills and Maxwell-Klein-Gordon equations are locally well-posed for <math>s > s_c</math> in three [[KlMa1997]] and higher [[KlTt1999]] dimensions. It would be interesting to determine what happens in two dimensions for these equations (probably one can get down to <math>s > s_c + 1/4)</math>. In one dimension the model equation trivially collapses to the free wave equation.</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>If the non-linearity <math><ins style="font-weight: bold; text-decoration: none;">\phi D\phi</ins></math> has a null structure then one can improve upon the previous results. For instance, the model equations for the Yang-Mills and Maxwell-Klein-Gordon equations are locally well-posed for <math>s > s_c</math> in three [[KlMa1997]] and higher [[KlTt1999]] dimensions. It would be interesting to determine what happens in two dimensions for these equations (probably one can get down to <math>s > s_c + 1/4)</math>. In one dimension the model equation trivially collapses to the free wave equation.</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:Wave]]</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:Wave]]</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>Taohttps://dispersivewiki.org/DispersiveWiki/index.php?title=Linear-derivative_nonlinear_wave_equations&diff=2170&oldid=prevTao at 21:11, 30 July 20062006-07-30T21:11:11Z<p></p>
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<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 21:11, 30 July 2006</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>A '''linear-derivative nonlinear wave equation''' <del style="font-weight: bold; text-decoration: none;">takes </del>the schematic form</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>A '''linear-derivative nonlinear wave equation''' <ins style="font-weight: bold; text-decoration: none;">is a [[DNLW|derivative nonlinear wave equation]] of with </ins>the schematic form</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><center><math>\Box u = F(u) Du + G(u)</math></center></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><center><math>\Box u = F(u) Du + G(u)</math></center></div></td></tr>
</table>Taohttps://dispersivewiki.org/DispersiveWiki/index.php?title=Linear-derivative_nonlinear_wave_equations&diff=2169&oldid=prevTao at 21:10, 30 July 20062006-07-30T21:10:43Z<p></p>
<p><b>New page</b></p><div>A '''linear-derivative nonlinear wave equation''' takes the schematic form<br />
<br />
<center><math>\Box u = F(u) Du + G(u)</math></center><br />
<br />
An important subclass of such equations are the '''Yang-Mills-type equations''' of the form<br />
<br />
<center><math>\Box u = u D u + u^3 </math></center><br />
<br />
This equation has the same scaling as [[cubic NLW]], but is more difficult technically because of the derivative term ''uDu''. <br />
<br />
Important examples of this type of equation include the [[MKG|Maxwell-Klein-Gordon]] and [[YM|Yang-Mills]] equations (in the Lorentz gauge, at least), as well as the simplified model equations for these equations. The [[YMH|Yang-Mills-Higgs]] equation is formed by coupling equations of this type to a semi-linear wave equation. The most interesting dimensions are 3 (for physical applications) and 4 (since the energy regularity is then critical). The two-dimensional case appears to be somewhat under-explored. The Yang-Mills and Maxwell-Klein-Gordon equations behave very similarly, but from a technical standpoint the latter is slightly easier to analyze.<br />
<br />
In d dimensions, the critical regularity for this equation is <math>s_c = d/2 - 1</math>. However, there are no instances of Quadratic DNLW for which any sort of well-posedness is known at this critical regularity (except for those special cases where the equation can be algebraically transformed into a simpler equation such as NLW or the free wave equation).<br />
<br />
Energy estimates give local well-posedness for <math>s > s_c + 1</math>. Using Strichartz estimates this can be improved to <math>s > s_c + 3/4</math> in two dimensions and <math>s > s_c + 1/2</math> in three and higher dimensions [[PoSi1993]]; the point is that these regularity assumptions together with Strichartz allow one to put <math>f</math> into <math>L^2_t L^{\infty}_x</math>, hence in <math>L^1_t L^{\infty}_x</math>, so that one can then use the energy method.<br />
<br />
Using <math>X^{s,\theta}</math> estimates [[FcKl2000]] instead of Strichartz estimates, one can improve this further to <math>d > s_c + 1/4</math> in four dimensions and to the near-optimal <math>s > s_c</math> in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space <math>B^{s_c}_{2,1}</math> [[Stz-p3]], and local well-posedness for large Besov data.In four dimensions one has a similar result if one imposes one additional angular derivative of regularity [[Stz-p2]].<br />
<br />
Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in <math>H^1</math> [[Lb1993]], although one can recover well-posedness in the Besov space B^1_{2,1} [[Na1999]], or when an epsilon of radial regularity is imposed [[MacNkrNaOz-p]]. It would be interesting to determine what the situation is in the other low dimensions.<br />
<br />
If the non-linearity <math>fDf</math> has a null structure then one can improve upon the previous results. For instance, the model equations for the Yang-Mills and Maxwell-Klein-Gordon equations are locally well-posed for <math>s > s_c</math> in three [[KlMa1997]] and higher [[KlTt1999]] dimensions. It would be interesting to determine what happens in two dimensions for these equations (probably one can get down to <math>s > s_c + 1/4)</math>. In one dimension the model equation trivially collapses to the free wave equation.<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Tao