List of equations and Hele-Shaw: Difference between pages

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This is a list of nonlocal equations that appear in this wiki.
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== Linear equations ==
The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates<ref name="MR0097227"/>. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain,
=== Stationary linear equations from Levy processes ===
\begin{align*}
\[ Lu = 0 \]
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
where $L$ is a [[linear integro-differential operator]].
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega
\end{align*}
The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side).
Particular solutions are given for instance by the planar profiles
\[
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad  A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
\]


=== parabolic linear equations from Levy processes ===
The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.
\[ u_t = Lu \]
where $L$ is a [[linear integro-differential operator]].


=== [[Drift-diffusion equations]] ===  
Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
\[
where $b$ is a given vector field.
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
\]
By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies
\begin{align*}
\Delta w &= 0 \text{ in } \{x_n>0\}\\
\partial_t w &= a\partial_n w \text{ on } \{x_n=0\}
\end{align*}
Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$,
\[
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w
\]


== [[Semilinear equations]] ==
== References ==
=== Stationary equations with zeroth order nonlinearity ===
{{reflist|refs=
\[ (-\Delta)^s u = f(u). \]
=== 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 ===
<ref name="MR0097227">{{Citation | last1=Saffman | first1= P. G. | last2=Taylor | first2= Geoffrey | title=The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid | journal=Proc. Roy. Soc. London. Ser. A | issn=0962-8444 | year=1958 | volume=245 | pages=312--329. (2 plates)}}</ref>
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
=== 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 ==
=== [[Surface quasi-geostrophic equation|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.

Revision as of 12:23, 29 July 2016

This article is a stub. You can help this nonlocal wiki by expanding it.

The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates[1]. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain, \begin{align*} \Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\ \frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega \end{align*} The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side). Particular solutions are given for instance by the planar profiles \[ P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0 \]

The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.

Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives \[ \frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2 \] By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies \begin{align*} \Delta w &= 0 \text{ in } \{x_n>0\}\\ \partial_t w &= a\partial_n w \text{ on } \{x_n=0\} \end{align*} Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$, \[ \partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w \]

References

  1. ↑ Saffman, P. G.; Taylor, Geoffrey (1958), "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid", Proc. Roy. Soc. London. Ser. A 245: 312--329. (2 plates), ISSN 0962-8444 
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