List of equations - Revision history

imported>Xavi: /* obstacle problem */

2015-05-15T19:31:50Z

obstacle problem

← Older revision		Revision as of 14:31, 15 May 2015
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	\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]		\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
	where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.		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]] ===		=== [[Obstacle problem]] ===
	For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies		For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies
	\begin{align}		\begin{align}

imported>Tianling: /* Semilinear equations */

2013-09-23T21:56:02Z

Semilinear equations

← Older revision		Revision as of 16:56, 23 September 2013
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	=== [[Keller-Segel equation]] ===		=== [[Keller-Segel equation]] ===
	\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]		\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]

			=== [[Prescribed fractional order curvature equation]] ===
			\[ (-\Delta)^s u = Ku^\frac{n+2s}{n-2s} \]

	== Quasilinear or [[fully nonlinear integro-differential equations]] ==		== Quasilinear or [[fully nonlinear integro-differential equations]] ==

imported>Nestor: /* Aggregation equation */

2012-03-13T00:45:25Z

Aggregation equation

← Older revision		Revision as of 19:45, 12 March 2012
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	=== [[Aggregation equation]] ===		=== [[Aggregation equation]] ===

	\[ ~~\theta_t~~ + \mathrm{div}(\~~theta \ u~~) = 0,\]		\[ u_t + \mathrm{div}(u \;v) = 0,\]
	where $u = \nabla K \ast \~~theta~~$.		where $v = -\nabla K \ast u$, $K$ typically being a radially symmetric positive kernel such that $\Delta K$ is locally integrable.

imported>Luis: /* Surface quasi-geostrophic equation */

2012-03-06T14:48:10Z

Surface quasi-geostrophic equation

← Older revision		Revision as of 09:48, 6 March 2012
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	\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]		\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
	=== [[Surface quasi-geostrophic equation]] ===		=== [[Surface quasi-geostrophic equation]] ===
	\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]		\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0, \]
			where $u = R^\perp \theta := \nabla^\perp (-\Delta)^{-1/2} \theta$.

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

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..."

2012-03-05T00:20:12Z

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..."

New page

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 ==
=== [[Surface quasi-geostrophic equation|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 $u = \nabla K \ast \theta$.