List of equations

(Difference between revisions)
 Revision as of 00:20, 5 March 2012 (view source)Luis (Talk | contribs) (Created page with "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 [[linea...")← Older edit Latest revision as of 19:31, 15 May 2015 (view source)Xavi (Talk | contribs) m (→obstacle problem) (3 intermediate revisions not shown) Line 22: Line 22: $u_t + u \ u_x + (-\Delta)^s u = 0$ $u_t + u \ u_x + (-\Delta)^s u = 0$ === [[Surface quasi-geostrophic equation]] === === [[Surface quasi-geostrophic equation]] === - $\theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0$ + $\theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0,$ + where $u = R^\perp \theta := \nabla^\perp (-\Delta)^{-1/2} \theta$. + === Conservation laws with fractional diffusion === === Conservation laws with fractional diffusion === $u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.$ $u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.$ Line 29: Line 31: === [[Keller-Segel equation]] === === [[Keller-Segel equation]] === $u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.$ $u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.$ + + === [[Prescribed fractional order curvature equation]] === + $(-\Delta)^s u = Ku^\frac{n+2s}{n-2s}$ == Quasilinear or [[fully nonlinear integro-differential equations]] == == Quasilinear or [[fully nonlinear integro-differential equations]] == Line 37: Line 42: $\sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x),$ $\sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x),$ where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$. 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]] === + === [[Obstacle problem]] === For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies \begin{align} \begin{align} Line 64: Line 69: === [[Aggregation equation]] === === [[Aggregation equation]] === - $\theta_t + \mathrm{div}(\theta \ u) = 0,$ + $u_t + \mathrm{div}(u \;v) = 0,$ - where $u = \nabla K \ast \theta$. + where $v = -\nabla K \ast u$, $K$ typically being a radially symmetric positive kernel such that $\Delta K$ is locally integrable.

Latest revision as of 19:31, 15 May 2015

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.