https://web.ma.utexas.edu/mediawiki/index.php?action=history&feed=atom&title=Talk:Introduction_to_nonlocal_equationsTalk:Introduction to nonlocal equations - Revision history2024-03-29T14:30:40ZRevision history for this page on the wikiMediaWiki 1.40.1https://web.ma.utexas.edu/mediawiki/index.php?title=Talk:Introduction_to_nonlocal_equations&diff=1128&oldid=previmported>Luis at 20:17, 14 March 20122012-03-14T20:17:57Z<p></p>
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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>One thought that keeps me from starting to write this page is that it has to be very general, but at the same time mention some important particular cases. It is tempting to start writing about the equations from stochastic control with Levy processes. But we cannot leave out other nonlocal equations like active scalars, aggregation or even Boltzmann.</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 thought that keeps me from starting to write this page is that it has to be very general, but at the same time mention some important particular cases. It is tempting to start writing about the equations from stochastic control with Levy processes. But we cannot leave out other nonlocal equations like active scalars, aggregation or even Boltzmann. <ins style="font-weight: bold; text-decoration: none;">[[User:Luis|Luis]] 15:17, 14 March 2012 (CDT)</ins></div></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;"><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>Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, etc...).</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>Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, etc...).</div></td></tr>
</table>imported>Luishttps://web.ma.utexas.edu/mediawiki/index.php?title=Talk:Introduction_to_nonlocal_equations&diff=1127&oldid=previmported>Luis at 20:17, 14 March 20122012-03-14T20:17:38Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:17, 14 March 2012</td>
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<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;">One thought that keeps me from starting to write this page is that it has to be very general, but at the same time mention some important particular cases. It is tempting to start writing about the equations from stochastic control with Levy processes. But we cannot leave out other nonlocal equations like active scalars, aggregation or even Boltzmann.</ins></div></td></tr>
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<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 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>Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, etc...).</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>Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, etc...).</div></td></tr>
</table>imported>Luishttps://web.ma.utexas.edu/mediawiki/index.php?title=Talk:Introduction_to_nonlocal_equations&diff=1126&oldid=previmported>Russell at 03:13, 14 March 20122012-03-14T03:13:54Z<p></p>
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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>Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, )</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>Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, <ins style="font-weight: bold; text-decoration: none;"> etc...</ins>)<ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
</table>imported>Russellhttps://web.ma.utexas.edu/mediawiki/index.php?title=Talk:Introduction_to_nonlocal_equations&diff=1125&oldid=previmported>Russell: Created page with "Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This c..."2012-03-14T03:01:24Z<p>Created page with "Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This c..."</p>
<p><b>New page</b></p><div>Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, )</div>imported>Russell