# Boltzmann equation

(Difference between revisions)
 Revision as of 17:06, 21 November 2012 (view source)Nestor (Talk | contribs) (→The Landau Equation)← Older edit Revision as of 00:42, 22 November 2012 (view source)Nestor (Talk | contribs) Newer edit → Line 7: Line 7: \end{equation*} \end{equation*} - then $f(x,v,t)$ solves the non-local equation + then if $f_0$ denotes the initial density, the function $f(x,v,t)$ solves the Cauchy problem - \begin{equation*} + \begin{equation}\label{eqn: Cauchy problem}\tag{1} - \partial_t f + v \cdot \nabla_x f = Q(f,f) + \left \{ \begin{array}{rll} - \end{equation*} + \partial_t f + v \cdot \nabla_x f & = Q(f,f) & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}_+\\ + f  & = f_0 & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \{ 0 \} + \end{array}\right. + - where $Q(f,f)$ is the Boltzmann collision operator, given by + where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by \begin{equation*} \begin{equation*} Line 28: Line 31: 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. + == Collision Invariants == - == Conservation laws == + The Cauchy problem \ref{eqn: Cauchy problem} enjoy several conservation laws, which in the Boltzmann literature are known as collision invariants. - + == The Landau Equation == == The Landau Equation ==

## Revision as of 00:42, 22 November 2012

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 if $f_0$ denotes the initial density, the function $f(x,v,t)$ solves the Cauchy problem

$$\tag{1} \left \{ \begin{array}{rll} \partial_t f + v \cdot \nabla_x f & = Q(f,f) & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}_+\\ f & = f_0 & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \{ 0 \} \end{array}\right.$$

where $Q(f,f)$ is the Boltzmann collision operator, a non-local 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.

## Collision Invariants

The Cauchy problem (1) enjoy several conservation laws, which in the Boltzmann literature are known as collision invariants.

## The Landau Equation

For Coulumb interactions, the corresponding collision kernel $B$ always diverges, instead in this case, one uses an equation (which is an asymptotic limit of Boltzmann equation) first derived by Landau,

\begin{equation*} f_t + x\cdot \nabla_y f = Q_{L}(f,f) \end{equation*}

where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as

\begin{equation*} Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2 \end{equation*}

where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.

Note that when $f$ is independent of $x$ the above equation becomes second-order parabolic equation where the coefficients depend non-locally on $f$, in particular, one has an apriori estimate for all higher derivatives of $f$ in terms of its $L^\infty$ and $L^1$ norms (via a bootstrapping argument).