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

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

Prescribed fractional order curvature equation

\[ (-\Delta)^s u = Ku^\frac{n+2s}{n-2s} \]

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.