Time Regularity for Nonlocal Parabolic Equations and Hele-Shaw: Difference between pages

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One of the phenomena that are exclusive to nonlocal parabolic equations is how the boundary data, posed in the complement of a given domain might drastically affect the regularity of the solution. Consider the fractional heat equation of order $\sigma\in(0,2)$
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{alignat*}{3}
\begin{align*}
u_t &= \Delta^{\sigma/2} u \quad &&\text{ in } \quad &&B_1\times\mathbb R\\
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
u &= g \quad &&\text{ on } \quad &&(\mathbb R^n \setminus B_1)\times\mathbb R
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \partial \{u>0\}\cap \Omega
\end{alignat*}
\end{align*}
If $g$ has a sudden discontinuity in time then it is expected that the nonlocal effect, transmitted into the equation by $\Delta^\sigma$, makes $u_t$ discontinuous in time. A specific example was presented by Chang-Lara and Dávila<ref name="MR3148110"/>.
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
\[
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad  A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
\]


For fully nonlinear, nonlocal parabolic equations it was established by Chang-Lara and Kriventsov<ref name="2015arXiv150507889C"/> that $u_t$ is Holder continuous provided that the boundary is Holder continuous in time. Under the assumption that $g$ is merely bounded, it was also proven that $u$ is Holder continuous in time for every exponent $\beta \in(0,1)$ with an estimate that degenerates as $\beta$ approaches 1. It remains open whether Lipschitz regularity in time also holds under the previous hypothesis.
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.


One application of the result in <ref name="2015arXiv150507889C"/> was to extend the Evans-Krylov estimate for parabolic equations under a mild continuity hypothesis for the boundary data.
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 ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="MR3148110">{{Citation | last1=Chang-Lara | first1= Héctor | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | url=http://dx.doi.org/10.1007/s00526-012-0576-2 | journal=Calc. Var. Partial Differential Equations | issn=0944-2669 | year=2014 | volume=49 | pages=139--172 | doi=10.1007/s00526-012-0576-2}}</ref>


<ref name="2015arXiv150507889C">{{Citation | last1=Chang-Lara | first1= Héctor | last2=Kriventsov | first2= Dennis | title=Further Time Regularity for Non-Local, Fully Non-Linear Parabolic Equations | journal=ArXiv e-prints | year=2015}}</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>
 
}}
}}

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 
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