List of equations and Aggregation equation: Difference between pages

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This is a list of nonlocal equations that appear in this wiki.
The aggregation equation consists in the scalar equation


== Linear equations ==
\[\begin{array}{rll}
=== Stationary linear equations from Levy processes ===
u_t+\text{div}(uv) & = 0 & \text{ in } \mathbb{R}^d \times \mathbb{R}_+\\
\[ Lu = 0 \]
u(x,0) & = u_0\\
where $L$ is a [[linear integro-differential operator]].
v (x,t) & = -\nabla (K*u(.,t))(x) &
\end{array} \]


=== parabolic linear equations from Levy processes ===
where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate or segregate (depending on whether $-x \cdot \nabla K(x)$ is always negative or
\[ u_t = Lu \]
always positive, respectively), or whether the interaction is isotropic,  or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = |x|$.
where $L$ is a [[linear integro-differential operator]].


=== [[Drift-diffusion equations]] ===
* Note that these equations can be seen as a Wasserstein Gradient Flow
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
where $b$ is a given vector field.


== [[Semilinear equations]] ==
* Note the similarities and differences when the energy generating these dynamics is written in a [[Nonlocal Phase Field]] model
=== 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]] ===
\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0, \]
where $u = R^\perp \theta := \nabla^\perp (-\Delta)^{-1/2} \theta$.


=== Conservation laws with fractional diffusion ===
Note the close connection with the [[nonlocal porous medium equation | nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
=== 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 $u = \nabla K \ast \theta$.

Revision as of 22:16, 13 March 2012

The aggregation equation consists in the scalar equation

\[\begin{array}{rll} u_t+\text{div}(uv) & = 0 & \text{ in } \mathbb{R}^d \times \mathbb{R}_+\\ u(x,0) & = u_0\\ v (x,t) & = -\nabla (K*u(.,t))(x) & \end{array} \]

where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate or segregate (depending on whether $-x \cdot \nabla K(x)$ is always negative or always positive, respectively), or whether the interaction is isotropic, or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = |x|$.

  • Note that these equations can be seen as a Wasserstein Gradient Flow
  • Note the similarities and differences when the energy generating these dynamics is written in a Nonlocal Phase Field model

Note the close connection with the nonlocal porous media equation, the key difference being that $\Delta K$ is not locally integrable in that case.

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