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| This is a list of nonlocal equations that appear in this wiki.
| | The Monge-Ampere equation refers to |
| | \[ \det D^2 u = f(x). \] |
| | The right hand side $f$ is always nonnegative. The unknown function $u$ should be convex. |
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| == Linear equations ==
| | The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator |
| === Stationary linear equations from Levy processes === | | \[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\] |
| \[ Lu = 0 \] | | is concave. |
| where $L$ is a [[linear integro-differential operator]].
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| === parabolic linear equations from Levy processes ===
| | {{stub}} |
| \[ u_t = Lu \]
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| where $L$ is a [[linear integro-differential operator]].
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| === [[Drift-diffusion equations]] ===
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| \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]
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| where $b$ is a given vector field.
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| == [[Semilinear equations]] ==
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| === Stationary equations with zeroth order nonlinearity ===
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| \[ (-\Delta)^s u = f(u). \]
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| === Reaction diffusion equations ===
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| \[ u_t + (-\Delta)^s u = f(u). \]
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| === Burgers equation with fractional diffusion ===
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| \[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
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| === [[Surface quasi-geostrophic equation]] ===
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| \[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0, \]
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| where $u = R^\perp \theta := \nabla^\perp (-\Delta)^{-1/2} \theta$.
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| === Conservation laws with fractional diffusion ===
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| \[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
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| === Hamilton-Jacobi equation with fractional diffusion ===
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| \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
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| === [[Keller-Segel equation]] ===
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| \[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]
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| == Quasilinear or [[fully nonlinear integro-differential equations]] ==
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| === [[Bellman equation]] ===
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| \[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
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| where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
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| === [[Isaacs equation]] ===
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| \[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
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| where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
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| === [[obstacle problem]] ===
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| For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies
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| \begin{align}
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| u &\geq \varphi \qquad \text{everywhere in the domain } D,\\
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| Lu &\leq 0 \qquad \text{everywhere in the domain } D,\\
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| Lu &= 0 \qquad \text{wherever } u > \varphi.
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| \end{align}
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| === [[Nonlocal minimal surfaces ]] ===
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| The set $E$ satisfies.
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| \[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]
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| === [[Nonlocal porous medium equation]] ===
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| \[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\]
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| Or
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| \[ u_t +(-\Delta)^{s}(u^m) = 0. \]
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| == Inviscid equations ==
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| === [[Surface quasi-geostrophic equation|Inviscid SQG]]===
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| \[ \theta_t + u \cdot \nabla \theta = 0,\]
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| where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.
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| === [[Active scalar equation]] (from fluid mechanics) ===
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| \[ \theta_t + u \cdot \nabla \theta = 0,\]
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| where $u = \nabla^\perp K \ast \theta$.
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| === [[Aggregation equation]] ===
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| \[ \theta_t + \mathrm{div}(\theta \ u) = 0,\]
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| where $u = \nabla K \ast \theta$.
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