List of equations

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

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

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

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