Isaacs equation and Hele-Shaw: Difference between pages

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The Isaacs equation is the equality
{{stub}}
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.


The equation appears naturally in zero sum stochastic games with [[Levy processes]].
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,
\begin{align*}
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \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 equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.
Non-local aspects of the equation can be appreciated by noticing that a given 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.


Note that any second order fully nonlinear uniformly elliptic PDE $F(D^2 u)=0$ can be written as an Isaacs equation by the following two steps:
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
# $F(X)$ is Lipschitz with constant $\Lambda$, so it is the infimum of all cones $C_{X_0}(x) = F(X_0) + \Lambda|X-X_0|$.
\[
# Each cone $C(X)$ is the supremum of all linear functions of the form $L(X) = F(X_0) + \mathrm{tr} \, A \cdot (X-X_0)$ for $||A||\leq \Lambda$.
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
\]
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 $\mathbb R^{n-1} = \{x_n=0\}$,
\[
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w
\]


A more general second order fully nonlinear uniformly elliptic PDE $F(D^2 u, Du, u, x)=0$ can also be written as an Isaacs equation if it is linear with respect to all parameters.
== References ==
{{reflist|refs=
 
<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>
 
}}

Revision as of 12:20, 29 July 2016

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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 } \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 \[ P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0 \]

Non-local aspects of the equation can be appreciated by noticing that a given 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