Linear integro-differential operator and Boltzmann equation: Difference between pages

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The linear integro-differential operators that we consider ''in this wiki'' are the generators of [[Levy processes]]. According to the Levy-Kintchine formula, they have the general form
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 <ref name="Bal2009"/>


\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
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 <ref name="Vil2002"/>. A basic reference is also <ref name="CerIllPul1994"/>.
where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]


The above definition is very general. Many theorems, and in particular regularity theorems, require extra assumptions in the kernels $K$. These assumptions restrict the study to certain sub-classes of linear operators. The simplest of all is the [[fractional Laplacian]]. We list below several extra assumptions that are usually made.
== The classical Boltzmann equation ==


== Absolutely continuous measure ==
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


In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
\begin{equation*}
\int_A f(x,v,t) \, \mathrm d x \mathrm dv.
\end{equation*}


We keep this assumption in all the examples below.
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


== Purely integro-differential operator ==
\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}


In this case we neglect the local part of the operator
where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]


== Symmetric kernels ==
\begin{equation*}
If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.
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*}


In the purely integro-differentiable case, it reads as
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
\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]


The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
\begin{align*}
 
v'  & = v-(v-v_*,e)e\\
== Translation invariant operators ==
v'_* & = v_*+(v-v_*,e)e
In this case, all coefficients are independent of $x$.
\end{align*}
\[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]
 
== The fractional Laplacian ==
 
The [[fractional Laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.
 
\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]


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


These are the operators whose kernel is homogeneous in $y$
The notation $f' = f(v')$, $f_\ast = f(v_\ast)$ and $f'_\ast = f(v'_\ast)$ is customary.
\[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\]
They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure.


== Uniformly elliptic of order $s$ ==
== Collision Invariants ==  


This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order.
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
\[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]


The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
\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*}


An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.
and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have


== Smoothness class $k$ of order $s$ ==
\begin{equation*}
This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded
\frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0
\[ |\partial_y^r K(x,y)| \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]
\end{equation*}


== Order strictly below one ==
according to what $\phi$ we pick this equation corresponds to  conservation of mass, conservation momentum or conservation of energy.


If a non symmetric kernel $K$ satisfies the extra local integrability assumption
== Typical collision kernels ==
\[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
then the extra gradient term is not necessary in order to define the operator.


\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]
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 modification in the integro-differential part of the operator becomes an extra drift term.
The case $B(r,\theta) = Kr$ is known as Maxwell molecules.


A uniformly elliptic operator of order $s<1$ satisfies this condition.
From particle models interacting by inverse $s$-power force, the collision kernel has the form
 
\[ B(r,\theta) = r^\gamma \theta^{-d+1-\nu} b(\theta),\]
== Order strictly above one ==
where
 
If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
\[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
then the gradient term in the integral can be taken global instead of being cut off in the unit ball.
 
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]
 
The modification in the integro-differential part of the operator becomes an extra drift term.
 
A uniformly elliptic operator of order $s>1$ satisfies this condition.
 
== More singular/irregular kernels ==
 
The concept of uniform ellipticity can be relaxed in various ways. The following family of operators was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />. The integro-differential operators have the form
\begin{align*}
\begin{align*}
\int_{\R^d} \big(u(x+y) - u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\
\gamma &= \frac{s - (2d - 1)}{s-1}, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\
\nu &= \frac 2 {s-1}, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha > 1.
\end{align*}
\end{align*}
$b(\theta)$ is some positive bounded function which is not known explicitly.


For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \Omega \times \R^d \to \R$ is assumed to satisfy the following assumptions for all $x \in \Omega$,
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 $\nu \in (0,2)$.
* $K(x,h) \geq 0$ for all $h\in\R^d$.
This is a basic assumption for the integral operator to be a legit diffusion operator.


* For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(x,h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]
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''.
This assumption is more general than $K(x,h) \leq (2-\alpha) \Lambda |h|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.


* For every $r>0$, there exists a set $A_r$ such that
=== Grad's angular cutoff assumption ===
** $A_r \subset B_{2r} \setminus B_r$.
** $A_r$ is symmetric in the sense that $A_r = -A_r$.
** $|A_r| \geq \mu |B_{2r} \setminus B_r|$.
** $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.
Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(x,-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_r|.
\]


This assumption says that the lower bound $K(x,h) \geq (2-\alpha) \lambda |h|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.
This assumption consists in using a collision kernel that is integrable in the angular variable $\theta$. That is
\[ \int_{S^{d-1}} B(r,e) \mathrm d \sigma(e) < +\infty \text{ for all values of } r. \]


* For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}. \]
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.


This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$.
== Stationary solutions ==


== Indexed by a matrix ==
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
In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
\[ f(t,x,v) = a e^{-b|v-v_0|^2}, \]
\[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
is a solution. In fact, for this function one can check that the integrand in the definition of $Q$ is identically zero since $f' \, f'_* = f \, f_*$.
This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$.


== Second order elliptic operators as limits of purely integro-differential ones ==
== Entropy ==


Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators
The following quantity is called the entropy and is monotone decreasing along the flow of the Boltzmann equation
\[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \]
\[ H(f) = \iint_{\R^d_x \times \R^d_v} f \ \log f \ \mathrm d v \, \mathrm d x. \]
define in the limit $\sigma \to 2^-$ a second order linear elliptic operator (possibly degenerate). This can be checked for any fixed $C^2$ function $u$ by a straightforward computation using the second order Taylor expansion. We get
\[ \lim_{\sigma \to 2^-} L_\sigma u(x) = a_{ij} \partial_{ij} u(x),\]
where $a_{ij}$ is determined by the quadratic form
\[ a_{ij} e_i e_j = \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x), \qquad \text{for vectors } e \in \R^d.\]


A class of kernels that is big enough to recover all translation invariant elliptic operators of the form  $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels
The derivative of the entropy is called ''entropy dissipation'' and is given by the expression
\[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\]
\[ D(f) = -\frac{\mathrm d H(f)}{\mathrm d t} = \frac 14 \iint_{\R^{2d}} \int_{S^{d-1}} B(v-v_\ast,\sigma) (f' f'_\ast - f f_\ast) \left( \log (f'f'_\ast) - \log(f f_\ast) \right) \, \mathrm d \sigma \, \mathrm d v \, \mathrm d v_\ast \geq 0. \]
where $A$ is an invertible symmetric matrix.


Conversely, the condition
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" />.
\[ \lambda |e|^2 \leq \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x) \leq \Lambda |e|^2, \qquad \text{for all vectors } e \in \R^d,\]
\[ D(f) \geq c_1 \|f\|_{H^{\nu/2}}^2 - c_2 \|f\|_{L^1_2}^2,\]
defines the largest class of stable operators that may be considered uniformly elliptic. Indeed, this is the condition that ensures regularity of solutions to translation invariant integro-differential equations.<ref name="ros2014regularity" /> If we let the operator depend on $x$, it is an outstanding [[Open problems#Hölder estimates for singular integro-differential equations|open problem]] whether this condition alone (for every value of $x$) ensures [[Holder estimates]] for the integro-differential equation.
where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$.


== Characterization via global maximum principle ==
== The Landau Equation ==


A bounded linear operator
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,


\[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \]
\begin{equation*}
f_t + x\cdot \nabla_y f = Q_{L}(f,f)
\end{equation*}


is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have
where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as


\[ (Lu)(x_0) \leq 0 \]
\begin{equation*}
Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2
\end{equation*}


It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège <ref name="C65"/> <ref name="C64"/>: if $L$ satisfies the global maximum principle then it has the form
where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.


\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]


where again $A(x)$ is a nonnegative matrix for all $x$, $c(x)\leq 0$ and $\mu_x$ is a nonnegative measure for all $x$ satisfying
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}}


\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
== References ==
{{reflist|refs=


and $A(x),c(x)$ and $b(x)$ are bounded.
<ref name="Bal2009">{{Citation | last1=Bal | first1=G. | title=Inverse transport theory and applications | publisher=IOP Publishing | year=2009 | journal=Inverse Problems | volume=25 | issue=5 | pages=053001}}</ref>


<ref name="Vil2002">{{Citation | last1=Villani | first1=C. | title=A review of mathematical topics in collisional kinetic theory | publisher=[[Elsevier]] | year=2002 | journal=Handbook of mathematical fluid dynamics | volume=1 | pages=71–74}}</ref>


== See also ==
<ref name="CerIllPul1994">{{Citation | last1=Cercignani | first1=Carlo | last2=Illner | first2=R. | last3=Pulvirenti | first3=M. | title=The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences vol 106) | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1994}}</ref>


* [[Fractional Laplacian]]
<ref name="ADVW2000">{{Citation | last1=Alexandre | first1= R. | last2=Desvillettes | first2= L. | last3=Villani | first3= C. | last4=Wennberg | first4= B. | title=Entropy dissipation and long-range interactions | url=http://dx.doi.org/10.1007/s002050000083 | journal=Arch. Ration. Mech. Anal. | issn=0003-9527 | year=2000 | volume=152 | pages=327--355 | doi=10.1007/s002050000083}}</ref>
* [[Levy processes]]
* [[Dirichlet form]]
 
 
== References ==
{{reflist|refs=
<ref name="C64">{{Citation | last1=Courrège | first1=Philippe | title=Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine | year=1964 | journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=88 | pages=3–30}}</ref>
<ref name="C65">{{Citation | last1=Courrège | first1=P. | title=Sur la forme intégro-différentielle des opéateurs de  $C_k^\infty(\mathbb{R}^n)$  dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum | journal=Sém. Théorie du potentiel (1965/66) Exposé | volume=2}}</ref>
<ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref>
<ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}}</ref>
}}
}}

Revision as of 13:06, 26 December 2013

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 [1]

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 [2]. A basic reference is also [3].

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 [4]. \[ D(f) \geq c_1 \|f\|_{H^{\nu/2}}^2 - c_2 \|f\|_{L^1_2}^2,\] 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]

References

  1. Bal, G. (2009), "Inverse transport theory and applications", Inverse Problems (IOP Publishing) 25 (5): 053001 
  2. Villani, C. (2002), "A review of mathematical topics in collisional kinetic theory", Handbook of mathematical fluid dynamics (Elsevier) 1: 71–74 
  3. Cercignani, Carlo; Illner, R.; Pulvirenti, M. (1994), The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences vol 106), Berlin, New York: Springer-Verlag 
  4. Alexandre, R.; Desvillettes, L.; Villani, C.; Wennberg, B. (2000), "Entropy dissipation and long-range interactions", Arch. Ration. Mech. Anal. 152: 327--355, doi:10.1007/s002050000083, ISSN 0003-9527, http://dx.doi.org/10.1007/s002050000083