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