Talk:To Do List and List of equations: Difference between pages

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imported>Luis
(Created page with "This is a list of nonlocal equations that appear in this wiki. == Linear equations == === Stationary linear equations from Levy processes === \[ Lu = 0 \] where $L$ is a [[linea...")
 
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The porous medium equation is technically a quasi-linear equation. There is now a page about [[semilinear equations]] to clarify the issue.
This is a list of nonlocal equations that appear in this wiki.


I see, by the way, how come Navier-Stokes is semilinear then? I was sure it was quasi until now.
== Linear equations ==
=== Stationary linear equations from Levy processes ===
\[ Lu = 0 \]
where $L$ is a [[linear integro-differential operator]].


---> It is the heat equation plus a first order nonlinear term.
=== parabolic linear equations from Levy processes ===
\[ u_t = Lu \]
where $L$ is a [[linear integro-differential operator]].


== Main page vs no Main page? ==
=== [[Drift-diffusion equations]] ===
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
where $b$ is a given vector field.


What is currently the "Main page" should become the  "Community portal", and we can use the Main page as the starting page. What do you guys think? -Nestor.
== [[Semilinear equations]] ==
:Right now I don't have an opinion either for or against. ([[User:Luis|Luis]] 11:03, 5 June 2011 (CDT))
=== Stationary equations with zeroth order nonlinearity ===
\[ (-\Delta)^s u = f(u). \]
=== Reaction diffusion equations ===
\[ u_t + (-\Delta)^s u = f(u). \]
=== Burgers equation with fractional diffusion ===
\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
=== [[Surface quasi-geostrophic equation]] ===
\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]
=== Conservation laws with fractional diffusion ===
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
=== Hamilton-Jacobi equation with fractional diffusion ===
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
=== [[Keller-Segel equation]] ===
\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]


I also started playing around with the use of categories (see the "Quasilinear equations" category)
== Quasilinear or [[fully nonlinear integro-differential equations]] ==
=== [[Bellman equation]] ===
\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
=== [[Isaacs equation]] ===
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
=== [[obstacle problem]] ===
For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies
\begin{align}
u &\geq \varphi \qquad \text{everywhere in the domain } D,\\
Lu &\leq 0 \qquad \text{everywhere in the domain } D,\\
Lu &= 0 \qquad \text{wherever } u > \varphi.
\end{align}


== Let us try to avoid a big bias ==
=== [[Nonlocal minimal surfaces ]] ===
The set $E$ satisfies.
\[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]
=== [[Nonlocal porous medium equation]] ===
\[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\]
Or
\[ u_t +(-\Delta)^{s}(u^m) = 0. \]


I was talking to Russell today that we have to make an effort not to have a strong bias towards Caffarelli related stuff. Otherwise, the purpose of the wiki will fail. ([[User:Luis|Luis]] 00:58, 8 June 2011 (CDT))
== Inviscid equations ==
=== [[Surface quasi-geostrophic equation|Inviscid SQG]]===
\[ \theta_t + u \cdot \nabla \theta = 0,\]
where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.


Indeed! I think so far we don't have to worry about it since we are just getting started with the pages (naturally we will write first stuff we know best). I hope it won't become a problem. Also, having Moritz will help a lot too. ([[User:Nestor|Nestor]]) (5:42 pm US Eastern Time, 8 June 2011)
=== [[Active scalar equation]] (from fluid mechanics) ===
\[ \theta_t + u \cdot \nabla \theta = 0,\]
where $u = \nabla^\perp K \ast \theta$.


== Classical potential analysis ==
=== [[Aggregation equation]] ===
I removed this comment from the page because I believe it belongs to the discussion. ([[User:Luis|Luis]] 20:11, 8 June 2011 (CDT))


* Luis C. would be very disappointed if we did not even mention topics from good old potential theory. Very importantly, [[Riesz potentials]] and the [[Poisson kernel]] for [[Fractional harmonic functions]]. Someone better start dusting off their copy of Landkof.
\[ \theta_t + \mathrm{div}(\theta \ u) = 0,\]
 
where $u = \nabla K \ast \theta$.
I put some of those topics on the page about the [[fractional Laplacian]].
 
Yeah, good call on that. I will add more stuff to those articles later ([[User:Nestor|Nestor]])
 
 
*A reminder for later (to make a full article out of this): it has become clear (only now, although in hindsight its trivial) that what physicists call "strong interactions" is the same as "very non-local". Namely, strong interactions correspond to forces that approach infinity much, much faster than say Coulomb's force as the distance between two particles goes to zero, mathematically, this means the interaction kernel $k(x,y)$ goes like $|x-y|^{-\gamma}$ for a very big positive number $\gamma$. ([[User:Nestor|Nestor]])

Revision as of 19:20, 4 March 2012

This is a list of nonlocal equations that appear in this wiki.

Linear equations

Stationary linear equations from Levy processes

\[ Lu = 0 \] where $L$ is a linear integro-differential operator.

parabolic linear equations from Levy processes

\[ u_t = Lu \] where $L$ is a linear integro-differential operator.

Drift-diffusion equations

\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\] where $b$ is a given vector field.

Semilinear equations

Stationary equations with zeroth order nonlinearity

\[ (-\Delta)^s u = f(u). \]

Reaction diffusion equations

\[ u_t + (-\Delta)^s u = f(u). \]

Burgers equation with fractional diffusion

\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]

Surface quasi-geostrophic equation

\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]

Conservation laws with fractional diffusion

\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]

Hamilton-Jacobi equation with fractional diffusion

\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]

Keller-Segel equation

\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]

Quasilinear or fully nonlinear integro-differential equations

Bellman equation

\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \] where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.

Isaacs equation

\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.

obstacle problem

For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies \begin{align} u &\geq \varphi \qquad \text{everywhere in the domain } D,\\ Lu &\leq 0 \qquad \text{everywhere in the domain } D,\\ Lu &= 0 \qquad \text{wherever } u > \varphi. \end{align}

Nonlocal minimal surfaces

The set $E$ satisfies. \[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]

Nonlocal porous medium equation

\[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\] Or \[ u_t +(-\Delta)^{s}(u^m) = 0. \]

Inviscid equations

Inviscid SQG

\[ \theta_t + u \cdot \nabla \theta = 0,\] where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.

Active scalar equation (from fluid mechanics)

\[ \theta_t + u \cdot \nabla \theta = 0,\] where $u = \nabla^\perp K \ast \theta$.

Aggregation equation

\[ \theta_t + \mathrm{div}(\theta \ u) = 0,\] where $u = \nabla K \ast \theta$.