Talk:Introduction to nonlocal equations: Difference between revisions
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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. | 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. [[User:Luis|Luis]] 15:17, 14 March 2012 (CDT) | ||
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...). | 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...). | ||
Latest revision as of 15:17, 14 March 2012
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. Luis 15:17, 14 March 2012 (CDT)
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...).