Extremal operators 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*}

here $d\sigma$ 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*}

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

@@ Line 1: / Line 1: @@
-Given any family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
+{{stub}}
-\begin{align*}
-M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
+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
-M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x)
-\end{align*}
+\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*}
-If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula
+here $d\sigma$ 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*}
-P^+(D^2 u) &= \Lambda \mathrm(D^2u^+) - \lambda \mathrm(D^2u^-)\\
-P^-(D^2 u) &= \lambda \mathrm(D^2u^+) - \Lambda \mathrm(D^2u^-)
-\end{align*}
-If $\mathcal{L}$ consists of all [[linear integro-differential operator|symmetric purely integro-differential operators, uniformly elliptic of order $s$]], then the extremal operators have the formula<ref name="S"/>
 \begin{align*}
-M^+\, u &= \int_{\R^n} \left( \Lambda \delta u(x,y)^+ - \lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y \\
+v'  & = v-(v-v_*,e)e\\
-M^-\, u &= \int_{\R^n} \left( \lambda \delta u(x,y)^+ - \Lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y
+v'_* & = v_*+(v-v_*,e)e
 \end{align*}
-where $\delta u(x,y) = (u(x+y) + u(x-y) - 2u(x))$. These two extremal operator are sometimes called "the ''monster'' Pucci operators".
-== References ==
+and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.
-{{reflist|refs=
-<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
-}}

Extremal operators and Boltzmann equation: Difference between pages

Revision as of 11:05, 20 November 2012

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