Dirichlet form and Boltzmann equation: Difference between pages

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The Boltzmann equation is an evolution equation used to describe the configuration of particles in a gas, but only statistically. Specifically, if the probability that a particle in the gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by

\begin{equation*} \int_A f(x,v,t)dxdy \end{equation*}

then $f(x,v,t)$ solves the non-local equation

\begin{equation*} \partial_t f + v \cdot \nabla_x f = Q(f,f) \end{equation*}

where $Q(f,f)$ is the Boltzmann collision operator, given by

\begin{equation*} Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_* \end{equation*}

where $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write

\begin{align*} v' & = v-(v-v_*,e)e\\ v'_* & = v_*+(v-v_*,e)e \end{align*}

where $B$, known as the Boltzmann collision kernel, measures the strength of collisions in different directions.

@@ Line 1: / Line 1: @@
-$$
+{{stub}}
-\newcommand{\dd}{\mathrm{d}}
-\newcommand{\R}{\mathbb{R}}
+The Boltzmann equation is an evolution equation used to describe the configuration of particles in a gas, but only statistically. Specifically, if the probability that a particle in the gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by
-$$
+\begin{equation*}
+\int_A f(x,v,t)dxdy
+\end{equation*}
-A Dirichlet form refers to a quadratic functional defined by an integral of the form
+then $f(x,v,t)$ solves the non-local equation
-\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]
-for some nonnegative kernel $K$.
-If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$. If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared.
+\begin{equation*}
+\partial_t f + v \cdot \nabla_x f = Q(f,f)
+\end{equation*}
-Dirichlet forms are natural generalizations to fractional order of the Dirichlet integrals
+where $Q(f,f)$ is the Boltzmann collision operator, given by
-\[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \]
-where $a_{ij}$ is elliptic.
-The Euler-Lagrange equation of a Dirichlet form is a fractional order version of elliptic equations in divergence form. They are studied using variational methods and they are expected to satisfy similar properties <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.
+\begin{equation*}
+Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_*
+\end{equation*}
-== References ==
+where $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write
-(There should be a lot more references here)
-{{reflist|refs=
-<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
-<ref name="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</ref>
-<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
-}}
+\begin{align*}
+v'  & = v-(v-v_*,e)e\\
+v'_* & = v_*+(v-v_*,e)e
+\end{align*}
-{{stub}}
+where $B$, known as the Boltzmann collision kernel, measures the strength of collisions in different directions.

Dirichlet form and Boltzmann equation: Difference between pages

Revision as of 10:42, 20 November 2012

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