Nonlocal minimal surfaces

The Boltzmann equation is a nonlinear evolution equation first put forward by Ludwig Boltzmann to describe the configuration of particles in a gas, but only statistically. However, this equation and related equations are used in other physical situations, such as in optics. The corresponding linear inverse problem is also used in tomography

In reality, the Boltzmann equation is not a single equation but a family of equations, where one obtains different equations depending on the nature of the interaction between particles (see below). Although there has been a lot of progress in the analysis of the Cauchy problem under many circumstances, the broad understanding of the equation and the dynamics of its solutions remains largely incomplete. For an overview of the mathematical issues revolving around this equation see for instance . A basic reference is also .

The classical Boltzmann equation

As explained originally by Boltzmann in the probabilistic description of a gas, we assume that the probability that a particle in a gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by some function

\begin{equation*} \int_A f(x,v,t) \, \mathrm d x \mathrm dv. \end{equation*}

Then, under certain (natural) physical assumptions, Boltzmann derived an evolution equation for $f(x,v,t)$. In particular, if one imposes $f$ at time $t=0$ then $f$ should solve the Cauchy problem

\begin{equation}\label{eqn: 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. \end{equation}

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) \bigg( f(v'_*)f(v') - f(v_*)f(v) \bigg) 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.

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

Collision Invariants

The Cauchy problem \ref{eqn: Cauchy problem} enjoys several conservation laws, which in the Boltzmann literature are known as collision invariants. Take $\phi(v)$ to be any of the following functions

\begin{equation*} \phi(v) = 1, \;\;v,\;\; \tfrac{|v|^2}{2} \end{equation*} \begin{equation*} \text{(the first and third ones are real valued functions, the second one is vector valued)} \end{equation*}

and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have

\begin{equation*} \frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0 \end{equation*}

according to what $\phi$ we pick this equation corresponds to conservation of mass, conservation momentum or conservation of energy.

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 \[ \cos \theta = \frac{(v - v_*) \cdot \sigma }{|v-v_*|}.\] We abuse notation by writing $B(v_*-v,e) = B(|v_*-v|,\theta)$.

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

From particle models interacting by inverse $s$-power force, the collision kernel has the form \[ B(r,\theta) = r^\gamma \theta^{-d+1-\nu} b(\theta),\] where \begin{align*} \gamma &= \frac{s - (2d - 1)}{s-1}, \\ \nu &= \frac 2 {s-1}, \\ \end{align*} $b(\theta)$ is some positive bounded function which is not known explicitly.

The operator $Q(f,f)$ may not make sense even for $f$ smooth if $\gamma$ is too negative or $\nu$ is too large. The operator does make sense if $\gamma \in (-d,0)$ and $\nu \in (0,2)$.

In the case $s = 2d-1$, we obtain that the collision kernel $B(r,\theta)$ depends only on the angular variable $\theta$. This case is called Maxwellian molecules.

Grad's angular cutoff assumption

This assumption consists in using a collision kernel that is integrable in the angular variable $\theta$. That is \[ \int_{S^{d-1}} B(r,e) \mathrm d \sigma(e) < +\infty \text{ for all values of } r. \]

The purpose of this assumption is to simplify the mathematical analysis of the equation. Note that for particles interaction by inverse power forces this assumption never holds.

Stationary solutions

The Gaussian (or Maxweillian) distributions in terms of $v$, which are constant in $x$, are stationary solutions of the equation. That is, any function of the form \[ f(t,x,v) = a e^{-b|v-v_0|^2}, \] is a solution. In fact, for this function one can check that the integrand in the definition of $Q$ is identically zero since $f' \, f'_* = f \, f_*$.

Entropy

The following quantity is called the entropy and is monotone decreasing along the flow of the Boltzmann equation \[ H(f) = \iint_{\R^d_x \times \R^d_v} f \ \log f \ \mathrm d v \, \mathrm d x. \]

The derivative of the entropy is called entropy dissipation and is given by the expression \[ D(f) = -\frac{\mathrm d H(f)}{\mathrm d t} = \frac 14 \iint_{\R^{2d}} \int_{S^{d-1}} B(v-v_\ast,\sigma) (f' f'_\ast - f f_\ast) \left( \log (f'f'_\ast) - \log(f f_\ast) \right) \, \mathrm d \sigma \, \mathrm d v \, \mathrm d v_\ast \geq 0. \]

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 . \[ D(f) \geq c_1 \|f\|_{H^{\nu/2}}^2 - c_2 \|f\|_{L^1_2}^2,\] where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$.

The Landau Equation

A closely related evolution equation is the Landau equation. For Coulomb 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).^{[citation needed]}