Nonlocal Evans-Krylov theorem and Interacting Particle Systems: Difference between pages

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The classical [[Evans-Krylov theorem]] <ref name="E"/> <ref name="K"/> says that convex or concave fully nonlinear elliptic equations have $C^{2,\alpha}$ (therefore classical) solutions. This type of equations can be written as a Hamilton-Jacobi-Bellman equation.
The (second order) integro-differential equation
\[ \sup_\beta a_{ij}^\beta \partial_{ij} u = f \]
\[ \begin{array}{rl}
for a family of uniformly elliptic coefficients $a_{ij}^\alpha$.
\partial_t \rho &= \text{div} \left( D(\rho) \nabla \rho+\sigma(\rho) \nabla V[\rho]\right )\\
V[\rho] & = J * \rho
\end{array}\]
describes at the macroscopic scale the phase segregation in  a gas whose particles are interacting at long ranges, as shown by Giacomin and Lebowitz <ref name="GL97"/>. This equation not only arises as the limit of the microscopic system but the approximation is good enough to capture both qualitative and quantitative phenomena of the microscopic system <ref name="GL97"/>. More concretely, the above equation arises as the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics. Giacomin and Lebowitz also note that unlike the standard Cahn-Hilliard equation, the above integro-differential equation has been shown rigorously to arise as the macroscopic limit of a microscopic model of interacting particles <ref name="GL97"/>.


A purely integro-differential version of this theorem<ref name="CS3"/> says that solutions of an integro-differential [[Bellman equation]] of the form
\[ \sup_\beta \int_{\R^n} (u(x+y) - u(x)) K_\beta (y) \mathrm d y = 0 \qquad \text{in } B_1\]
are $C^{s+\alpha}(B_{1/2})$ (which implies that they are classical) if the kernels satisfy the following assumptions
\begin{align*}
\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{uniform ellipticity of order $s$} \\
D^2 K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s+2}} && \text{Decay of the tails of $K$ in $C^2$} \\
K(y) &= K(-y) && \text{symmetry}
\end{align*}


The $C^{s+\alpha}$ estimate '''does not blow up as $s \to 2$'''. Thus, the result is a true generalization of Evans-Krylov theorem.
== The interacting particle system ==


Note that the result is relevant only if $s>1$, otherwise it is a weaker result compared to the [[differentiability estimates|C^{1,\alpha} estimates]].
At the microscopic level, the system is described by a function
 
\[ \eta  : \Lambda_\gamma \to \{ 0,1\} \]
 
where  $\gamma>0$ taken very small represents the spatial scale and  $\Lambda_\gamma $ denotes the finite $d$-dimensional lattice
 
\[ \Lambda_\gamma = \{ 1,2,...,[\gamma^{-1}]\}^d\]
 
in other words, a cube inside $\mathbb{Z}^d$ with sides given by $[\gamma^{-1}]$, as $\gamma \to 0$, this exhausts all of $\mathbb{Z}^d$. The set of all posible configurations $\eta$ will be denoted by $\Omega_\gamma$, this is the state space where the (microscopic scale) dynamics takes place. As $\gamma \to 0$ we expect to recover the above as a limiting dynamical system the integro-differential equation listed above, of course first we have to describe the microscopic dynamics.
 
Given any initial condition $\eta_0 : \Lambda_\gamma \to \{0,1\}$, we consider a stochastic Poisson jump process with values in $\Lambda_\gamma$ generated by the operator
 
\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]
 
where $\eta^{x,y}$ denotes the state $\eta$ where the values at $x$ and $y$ have been interchanged and the kernel $c_\gamma(x,y;\eta)$ is defined as
 
\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl}
\Phi \left ( \beta\left [ H(\eta^{x,y}-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\
0 & \text{ otherwise }
  \end{array}\right.\]


The hypothesis above are most probably not optimal. But unlike the [[differentiability estimates|C^{1,\alpha} estimates]], no extension of this result has been done.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="CS3">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=The Evans-Krylov theorem for non local fully non linear equations | year=to appear | journal=[[Annals of Mathematics]] | issn=0003-486X}}</ref>
 
<ref name="E">{{Citation | last1=Evans | first1=Lawrence C. | title=Classical solutions of fully nonlinear, convex, second-order elliptic equations | url=http://dx.doi.org/10.1002/cpa.3160350303 | doi=10.1002/cpa.3160350303 | year=1982 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=35 | issue=3 | pages=333–363}}</ref>
<ref name="GL97"> {{Citation | last1=Lebowitz | first1=Joel | last2=Giacomin | first2=Giambattista | title=Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits | doi=10.1007/BF02181479 | year=1997 | journal=Journal of Statistical Physics | issn=0022-4715 | volume=87 | issue=1 | pages=37–61}} </ref>
<ref name="K">{{Citation | last1=Krylov | first1=N. V. | title=Boundedly inhomogeneous elliptic and parabolic equations | year=1982 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=46 | issue=3 | pages=487–523}}</ref>
 
}}
}}

Revision as of 00:35, 1 February 2012

The (second order) integro-differential equation \[ \begin{array}{rl} \partial_t \rho &= \text{div} \left( D(\rho) \nabla \rho+\sigma(\rho) \nabla V[\rho]\right )\\ V[\rho] & = J * \rho \end{array}\] describes at the macroscopic scale the phase segregation in a gas whose particles are interacting at long ranges, as shown by Giacomin and Lebowitz [1]. This equation not only arises as the limit of the microscopic system but the approximation is good enough to capture both qualitative and quantitative phenomena of the microscopic system [1]. More concretely, the above equation arises as the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics. Giacomin and Lebowitz also note that unlike the standard Cahn-Hilliard equation, the above integro-differential equation has been shown rigorously to arise as the macroscopic limit of a microscopic model of interacting particles [1].


The interacting particle system

At the microscopic level, the system is described by a function

\[ \eta : \Lambda_\gamma \to \{ 0,1\} \]

where $\gamma>0$ taken very small represents the spatial scale and $\Lambda_\gamma $ denotes the finite $d$-dimensional lattice

\[ \Lambda_\gamma = \{ 1,2,...,[\gamma^{-1}]\}^d\]

in other words, a cube inside $\mathbb{Z}^d$ with sides given by $[\gamma^{-1}]$, as $\gamma \to 0$, this exhausts all of $\mathbb{Z}^d$. The set of all posible configurations $\eta$ will be denoted by $\Omega_\gamma$, this is the state space where the (microscopic scale) dynamics takes place. As $\gamma \to 0$ we expect to recover the above as a limiting dynamical system the integro-differential equation listed above, of course first we have to describe the microscopic dynamics.

Given any initial condition $\eta_0 : \Lambda_\gamma \to \{0,1\}$, we consider a stochastic Poisson jump process with values in $\Lambda_\gamma$ generated by the operator

\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]

where $\eta^{x,y}$ denotes the state $\eta$ where the values at $x$ and $y$ have been interchanged and the kernel $c_\gamma(x,y;\eta)$ is defined as

\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl} \Phi \left ( \beta\left [ H(\eta^{x,y}-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\ 0 & \text{ otherwise } \end{array}\right.\]


References

  1. 1.0 1.1 1.2 Lebowitz, Joel; Giacomin, Giambattista (1997), "Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits", Journal of Statistical Physics 87 (1): 37–61, doi:10.1007/BF02181479, ISSN 0022-4715