Hele-Shaw: Difference between revisions
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\partial_t w &= a\partial_n w \text{ on } \{x_n=0\} | \partial_t w &= a\partial_n w \text{ on } \{x_n=0\} | ||
\end{align*} | \end{align*} | ||
Or in terms of the half-laplacian | Or in terms of the half-laplacian we can say that $w$ when restricted to $\mathbb R^{n-1} = \{x_n=0\}$ satisfies the heat equation | ||
\[ | \[ | ||
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w | \partial_t w = a(t)\Delta_{\mathbb R^{n-1}}^{1/2} w. | ||
\] | \] | ||
Notice the space invariance of the linearized problem implies Holder estimates for any spatial derivative of $w$ meanwhile the time regularity depends on the regularity of the coefficient $a(t)$. This approach allows to prove interior Holder estimates in space and time for the normal vector of the free boundary from a flatness hypothesis<ref name="2016arXiv160507591C"/>. | |||
== References == | == References == | ||
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<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> | <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> | ||
<ref name="2016arXiv160507591C">{{Citation | last1=Chang-Lara | first1= H.A. | last2=Guillen | first2= N. | title=From the free boundary condition for Hele-Shaw to a fractional parabolic equation | journal=ArXiv e-prints | year=2016}}</ref> | |||
}} | }} | ||
Latest revision as of 06:45, 1 August 2016
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The Hele-Shaw model describes the evolution of an incompressible flow lying between two nearby horizontal plates[1]. The following equations are given for a non-negative pressure $u=u(x,t)$, 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 $u = P + \varepsilon v$ where $P$ is a planar profile. 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 we can say that $w$ when restricted to $\mathbb R^{n-1} = \{x_n=0\}$ satisfies the heat equation \[ \partial_t w = a(t)\Delta_{\mathbb R^{n-1}}^{1/2} w. \]
Notice the space invariance of the linearized problem implies Holder estimates for any spatial derivative of $w$ meanwhile the time regularity depends on the regularity of the coefficient $a(t)$. This approach allows to prove interior Holder estimates in space and time for the normal vector of the free boundary from a flatness hypothesis[2].
References
- ↑ 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
- ↑ Chang-Lara, H.A.; Guillen, N. (2016), "From the free boundary condition for Hele-Shaw to a fractional parabolic equation", ArXiv e-prints