Open problems and Hele-Shaw: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Luis
(Created page with "== Well posedness of the supercritical surface quasi-geostrophic equation == Let $\theta_0 : \R^2 \to \R$ be a smooth function either with compact support or periodic. Let $s...")
 
imported>Hector
No edit summary
 
Line 1: Line 1:
== Well posedness of the supercritical [[surface quasi-geostrophic equation]] ==
{{stub}}
Let $\theta_0 : \R^2 \to \R$ be a smooth function either with compact support or periodic. Let $s \in (0,1/2)$. Is there a global classical solution $\theta :\R^2 \to \R$ for the SQG equation?
 
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*}
\begin{align*}
\theta(x,0) &= \theta_0(x) \\
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
\theta_t + u \cdot \nabla \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \partial \{u>0\}\cap \Omega
\end{align*}
\end{align*}
where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
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
\[
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad  A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
\]


This is a very difficult open problem. It is believed that a solution would be a major step towards the understanding of Navier-Stokes equation. In the supercritical regime $s\in (0,1/2)$, the effect if the drift term is larger than the diffusion in small scales. Therefore, it seems unlikely that a proof of well posedness could be achieved with the methods currently known and listed in this wiki.
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 if the relation between $u$ and $\theta$ was changed by $u = R\theta$, then the equation is ill posed. This suggests that the divergence free nature of $u$ must play an important role, unlike the critical and subcritical cases $s \geq 1/2$.
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 $\mathbb R^{n-1} = \{x_n=0\}$,
\[
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w
\]


== Regularity of [[nonlocal minimal surfaces]] ==
== References ==
{{reflist|refs=


A nonlocal minimal surface that is sufficiently flat is known to be smooth. The possibility of singularities in the general case reduces to the analysis of a possible existence of nonlocal minimal cones. The problem can be stated as follows.
<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>


For any $s \in (0,1)$, and any natural number $n$, is there any set $A \in \R^n$, other than a half space, such that
}}
# $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
# If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds
\[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]
 
When $s$ is sufficiently close to one, such set does not exist if $n < 8$.
 
== An integral ABP estimate ==
 
The nonlocal version of the [[Alexadroff-Bakelman-Pucci estimate]] holds either for a right hand side in $L^\infty$ (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels. Would the following result be true?
 
Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
\[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
Assume that the kernels $K$ satisfy symmetry and a uniform ellipticity condition
\begin{align*}
K(x,y) &= K(x,-y) \\
\lambda |y|^{-n-s} \leq K(x,y) &\leq \Lambda |y|^{-n-s} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2).
\end{align*}
If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?

Revision as of 12:20, 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 } \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 
Retrieved from "https://web.ma.utexas.edu/mediawiki/index.php?title=Open_problems&oldid=1341"