Boltzmann equation - Revision history

imported>Luis: /* Typical collision kernels */

2016-08-30T16:07:41Z

Typical collision kernels

← Older revision		Revision as of 11:07, 30 August 2016
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	The collision kernel $B(\|v_-v\|,\theta)$ should depend only on the distance $\|v-v_\|$ and the deviation angle $\theta \in [-\pi,\pi]$ given by		The collision kernel $B(\|v_-v\|,\theta)$ should depend only on the distance $\|v-v_\|$ and the deviation angle $\theta \in [-\pi,\pi]$ given by
	\[ \cos \theta = \frac{(v - v_) \cdot \sigma }{\|v-v_\|}.\]		\[ \cos \theta = \frac{(v - v_) \cdot \sigma }{\|v-v_\|}.\]

	~~The case $B(r,\theta) = Kr$ is known as Maxwell molecules.~~

	From particle models interacting by inverse $s$-power force, the collision kernel has the form		From particle models interacting by inverse $s$-power force, the collision kernel has the form

2015-05-20T21:49:29Z

The classical Boltzmann equation

← Older revision		Revision as of 16:49, 20 May 2015
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	\begin{equation*}		\begin{equation*}
	Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_, \theta) \bigg( f(v'_)f(v') - f(v_)f(v) \bigg) d\sigma dv_.		Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(\|v-v_\|, \theta) \bigg( f(v'_)f(v') - f(v_)f(v) \bigg) d\sigma dv_.
	\end{equation*}		\end{equation*}

2015-05-20T21:48:55Z

Typical collision kernels

← Older revision		Revision as of 16:48, 20 May 2015
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	== Typical collision kernels ==		== Typical collision kernels ==

	The collision kernel $B(\|v_-v\|,\theta)$ should depend only on the distance $\|v-v_\|$ and the deviation angle $\theta$ given by		The collision kernel $B(\|v_-v\|,\theta)$ should depend only on the distance $\|v-v_\|$ and the deviation angle $\theta \in [-\pi,\pi]$ given by
	\[ \cos \theta = \frac{(v - v_) \cdot \sigma }{\|v-v_\|}.\]		\[ \cos \theta = \frac{(v - v_) \cdot \sigma }{\|v-v_\|}.\]

2015-05-20T21:47:55Z

Typical collision kernels

← Older revision		Revision as of 16:47, 20 May 2015
Line 58:		Line 58:
	== Typical collision kernels ==		== Typical collision kernels ==

	The collision kernel $B(v_-v,e)$ should depend only on the distance $\|v-v_\|$ and the deviation angle $\theta$ given by		The collision kernel $B(\|v_-v\|,\theta)$ should depend only on the distance $\|v-v_\|$ and the deviation angle $\theta$ given by
	\[ \cos \theta = \frac{(v - v_) \cdot \sigma }{\|v-v_\|}.\]		\[ \cos \theta = \frac{(v - v_) \cdot \sigma }{\|v-v_\|}.\]
	~~We abuse notation by writing $B(v_-v,e) = B(\|v_-v\|,\theta)$.~~

	The case $B(r,\theta) = Kr$ is known as Maxwell molecules.		The case $B(r,\theta) = Kr$ is known as Maxwell molecules.

2015-05-20T21:46:52Z

The classical Boltzmann equation

← Older revision		Revision as of 16:46, 20 May 2015
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	\begin{equation*}		\begin{equation*}
	Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_,e) \bigg( f(v'_)f(v') - f(v_)f(v) \bigg) d\sigma~~(e)~~ dv_.		Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_, \theta) \bigg( f(v'_)f(v') - f(v_)f(v) \bigg) d\sigma dv_.
	\end{equation*}		\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		here $d\sigma$ denotes the differential of surface on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $\sigma \in \mathbb{S}^{d-1}$ we write

	\begin{align*}		\begin{align*}
	v' & = v-(v-v_*,~~e)e~~\\		v' & = \frac{v+v_}2 + \frac{\|v-v_\|}2 \sigma, \\
	v'_* & = v_+(v-v_~~,e)e~~		v'_* & =\frac{v+v_}2 - \frac{\|v-v_\|}2 \sigma.
	\end{align*}		\end{align*}

	and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.		and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions. The angle $\theta$ corresponds to $\cos \theta = \sigma \cdot (v-v_) / \|v-v_\|$.

	The notation $f' = f(v')$, $f_\ast = f(v_\ast)$ and $f'_\ast = f(v'_\ast)$ is customary.		The notation $f' = f(v')$, $f_\ast = f(v_\ast)$ and $f'_\ast = f(v'_\ast)$ is customary.

2013-12-30T17:46:05Z

Entropy

← Older revision		Revision as of 12:46, 30 December 2013
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	\[ D(f) = -\frac{\mathrm d H(f)}{\mathrm d t} = \frac 14 \iint_{\R^{2d}} \int_{S^{d-1}} B(v-v_\ast,\sigma) (f' f'_\ast - f f_\ast) \left( \log (f'f'_\ast) - \log(f f_\ast) \right) \, \mathrm d \sigma \, \mathrm d v \, \mathrm d v_\ast \geq 0. \]		\[ D(f) = -\frac{\mathrm d H(f)}{\mathrm d t} = \frac 14 \iint_{\R^{2d}} \int_{S^{d-1}} B(v-v_\ast,\sigma) (f' f'_\ast - f f_\ast) \left( \log (f'f'_\ast) - \log(f f_\ast) \right) \, \mathrm d \sigma \, \mathrm d v \, \mathrm d v_\ast \geq 0. \]

	Without Grad's angular cut-off assumption, and for a rather general family of cross-sections $B$, the entropy dissipation is bounded below by a fractional Sobolev norm minus a small correction <ref name="ADVW2000" />.		Without Grad's angular cut-off assumption, and for a rather general family of cross-sections $B$, the entropy dissipation is bounded below by a fractional Sobolev norm in terms of the velocity variable minus a small correction <ref name="ADVW2000" />.
	\[ D(f) \geq c_1 \\|\sqrt f\\|_{H^{\nu/2}}^2 - c_2 \\|f\\|_{L^1_2}^2,\]		\[ D(f) \geq c_1 \\|\sqrt f\\|_{H^{\nu/2}_{v}}^2 - c_2 \\|f\\|_{L^1_2}^2,\]
	where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$.		where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$ at $x$. In the space homogeneous case, this estimate shows a regularization effect by the Boltzmann equation.

	== The Landau Equation ==		== The Landau Equation ==

2013-12-26T18:08:25Z

The Landau Equation

2013-12-26T18:07:53Z

Entropy

← Older revision		Revision as of 13:07, 26 December 2013
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	Without Grad's angular cut-off assumption, and for a rather general family of cross-sections $B$, the entropy dissipation is bounded below by a fractional Sobolev norm minus a small correction <ref name="ADVW2000" />.		Without Grad's angular cut-off assumption, and for a rather general family of cross-sections $B$, the entropy dissipation is bounded below by a fractional Sobolev norm minus a small correction <ref name="ADVW2000" />.
	\[ D(f) \geq c_1 \\|f\\|_{H^{\nu/2}}^2 - c_2 \\|f\\|_{L^1_2}^2,\]		\[ D(f) \geq c_1 \\|\sqrt f\\|_{H^{\nu/2}}^2 - c_2 \\|f\\|_{L^1_2}^2,\]
	where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$.		where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$.

2013-12-26T18:06:54Z

The classical Boltzmann equation

← Older revision		Revision as of 13:06, 26 December 2013
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	and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.		and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.

			The notation $f' = f(v')$, $f_\ast = f(v_\ast)$ and $f'_\ast = f(v'_\ast)$ is customary.

	== Collision Invariants ==		== Collision Invariants ==

2013-12-26T18:05:16Z

The classical Boltzmann equation