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| The fractional Laplacian $(-\Delta)^s$ is a classical operator which gives the standard Laplacian when $s=1$. One can think of $-(-\Delta)^s$ as the most basic [[elliptic linear integro-differential operator]] of order $2s$ and can be defined in several equivalent ways (listed below). A range of powers of particular interest is $s \in (0,1)$, in which case for $u \in \mathcal{S}(\mathbb{R}^d)$ we can write the operator as
| | This is a list of nonlocal equations that appear in this wiki. |
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| \[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\] | | == Linear equations == |
| | === Stationary linear equations from Levy processes === |
| | \[ Lu = 0 \] |
| | where $L$ is a [[linear integro-differential operator]]. |
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| where $c_{n,s}$ is a universal constant and $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$. This particular expression shows that in this range of $s$ the operator enjoys the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)^s u(x) \geq 0$, with equality only if $u$ is constant. From this monotonicity, a [[comparison principle]] can be derived for equations involving the fractional Laplacian.
| | === parabolic linear equations from Levy processes === |
| | \[ u_t = Lu \] |
| | where $L$ is a [[linear integro-differential operator]]. |
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| == Definitions == | | === [[Drift-diffusion equations]] === |
| | \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\] |
| | where $b$ is a given vector field. |
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| All the definitions below are equivalent.
| | == [[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 $u = R^\perp \theta := \nabla^\perp (-\Delta)^{-1/2} \theta$. |
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| === As a pseudo-differential operator === | | === Conservation laws with fractional diffusion === |
| The fractional Laplacian is the pseudo-differential operator with symbol $|\xi|^{2s}$. In other words, the following formula holds
| | \[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\] |
| \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\] | | === Hamilton-Jacobi equation with fractional diffusion === |
| for any function (or tempered distribution) for which the right hand side makes sense.
| | \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\] |
| | === [[Keller-Segel equation]] === |
| | \[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\] |
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| This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.
| | === [[Prescribed fractional order curvature equation]] === |
| | \[ (-\Delta)^s u = Ku^\frac{n+2s}{n-2s} \] |
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| === From functional calculus === | | == Quasilinear or [[fully nonlinear integro-differential equations]] == |
| Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.
| | === [[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} |
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| This definition is not as useful for practical applications, since it does not provide any explicit formula.
| | === [[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 |
| | \[ u_t +(-\Delta)^{s}(u^m) = 0. \] |
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| === As a singular integral === | | == Inviscid equations == |
| If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula
| | === [[Surface quasi-geostrophic equation|Inviscid SQG]]=== |
| \[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\] | | \[ \theta_t + u \cdot \nabla \theta = 0,\] |
| | where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$. |
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| Where $c_{n,s}$ is a constant depending on dimension and $s$.
| | === [[Active scalar equation]] (from fluid mechanics) === |
| | \[ \theta_t + u \cdot \nabla \theta = 0,\] |
| | where $u = \nabla^\perp K \ast \theta$. |
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| This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.
| | === [[Aggregation equation]] === |
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| === As a generator of a [[Levy process]] ===
| | \[ u_t + \mathrm{div}(u \;v) = 0,\] |
| The operator can be defined as the generator of $\alpha$-stable Lévy processes. More precisely, if $X_t$ is the isotropic $\alpha$-stable Lévy process starting at zero and $f$ is a smooth function, then
| | where $v = -\nabla K \ast u$, $K$ typically being a radially symmetric positive kernel such that $\Delta K$ is locally integrable. |
| \[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \] | |
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| This definition is important for applications to probability.
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| == Inverse operator ==
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| The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the ''Riesz potential'':
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| \[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\]
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| which holds as long as $u$ is regular enough for the right hand side to make sense.
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| == Heat kernel ==
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| The fractional heat kernel $p(t,x)$ is the fundamental solution to the [[fractional heat equation]]. It is the function which solves the equation
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| \begin{align*}
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| p(0,x) &= \delta_0 \\
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| p_t(t,x) + (-\Delta)^s p &= 0
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| \end{align*}
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| The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables, but the following inequalities are known to hold for some constant $C$
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| \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]
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| Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
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| \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]
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| == Poisson kernel ==
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| Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem
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| \begin{align*} | |
| u(x) &= g(x) \qquad \text{if } x \notin B_1 \\
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| (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1.
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| \end{align*}
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| The solution can be computed explicitly using the Poisson kernel
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| \[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
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| where<ref name="R"/> | |
| \[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]
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| The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')<ref name="L"/>.
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| == Regularity issues ==
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| Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This is a classical fact for pseudo-differential operators{{citation needed}}.
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| === Full space regularization of the Riesz potential ===
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| If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions ''up to $2s$ derivatives''. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of <ref name="S"/>, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$.
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| === Boundary regularity ===
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| From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel).
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| == References ==
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| {{reflist|refs=
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| <ref name="L">{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1972}}</ref>
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| <ref name="R">{{Citation | last1=Riesz | first1=M. | title=Intégrales de Riemann-Liouville et potentiels | url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article | year=1938 | journal=Acta Sci. Math. Szeged | issn=0001-6969 | volume=9 | issue=1 | pages=1–42}}</ref>
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| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
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| }}
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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.
\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]
where $b$ is a given vector field.
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
\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0, \]
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
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]
\[ (-\Delta)^s u = Ku^\frac{n+2s}{n-2s} \]
\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
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}
The set $E$ satisfies.
\[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]
\[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\]
Or
\[ u_t +(-\Delta)^{s}(u^m) = 0. \]
Inviscid equations
\[ \theta_t + u \cdot \nabla \theta = 0,\]
where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.
\[ \theta_t + u \cdot \nabla \theta = 0,\]
where $u = \nabla^\perp K \ast \theta$.
\[ u_t + \mathrm{div}(u \;v) = 0,\]
where $v = -\nabla K \ast u$, $K$ typically being a radially symmetric positive kernel such that $\Delta K$ is locally integrable.