List of results that are fundamentally different to the local case and Boltzmann equation: Difference between pages

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In this page we collect some results in nonlocal equations when things behave very differently compared to the local counterpart. A result makes it to this list if it is somewhat suprising or counterintuitive.
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The list is very incomplete right now. Please help expand it by editing it.
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"/>


=== Traveling fronts in Fisher-KPP equations with fractional diffusion have exponential speed ===
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"/>.
Let us consider the reaction diffusion equation
\[ u_t + (-\Delta)^s u = f(u), \]
with a Fisher-KPP type of nonlinearity (for example $f(u) = u(1-u)$). In the local diffusion case, the stable state $u=1$ invades the unstable state $u=0$ at a constant speed. In the nonlocal case (any $s<1$), the invasion holds at an exponential rate.


The explanation of the difference can be understood intuitively from the fact that the fat tails in the fractional heat kernels make diffusion happen at a much faster rate <ref name="CR"/>.
== The classical Boltzmann equation ==  


=== The optimal regularity for the fractional obstacle problem exceeds the scaling of the 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
Given a function $\varphi$, the obstacle problem consists in the solution to an equation of the form
\[ \min((-\Delta)^s u , u-\varphi) = 0.\]


If $\varphi$ is smooth enough, the solution $u$ to the obstacle problem will be $C^{1,s}$ and no better. There is a big difference between the case $s=1$ and $s<1$ which makes the proof fundamentally different. In the classical case $s=1$, the optimal regularity matches the scaling of the equation. The classical proof of optimal regularity is to show an upper bound in the separation of $u$ from the obstacle in the unit ball and then just scale it. In the fractional case $s<1$, this method only gives $C^{2s}$ regularity, which matches the scaling of the equation. It is somewhat surprising that a better regularity result holds and it requires a different method for the proof.
\begin{equation*}
\int_A f(x,v,t)dxdy.
\end{equation*}


The intuitive explanation is that $(-\Delta)^s u$ satisfies an extra elliptic equation in terms of its Laplacian to the power $1-s$, and that equation provides the extra regularity <ref name="S"/>.
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


=== Nonlocal elliptic equations can have interior maximums ===
\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}


A solution to a [[fully nonlinear integro-differential equation]] satisfies a ''nonlocal'' maximum principle: they cannot have a ''global'' maximum or minumum in the interior of the domain of the equation. Local extrema are possible.
where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by


This is related to the fact that Dirichlet boundary conditions have to be given in the whole complement of the Domain and not only on its boundary. It is also related to the failure in general of the classical [[Harnack inequality]] unless the possitivity of the function is assumed in the full space <ref name="K"/>.
\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*}


=== Viscosity solutions can be evaluated at points ===
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


The concept of [[viscosity solutions]] is developed in order to make sense of an elliptic equation even for continuous functions for which the equation cannot be evaluated classically at points. The idea is to use test functions whose graphs are tangent to the graphs of the weak solution at some point, and then evaluate the equation on that test function. The ellipticity property tells us that the value of the equation for that test function at the point of contact must have certain sign, and this is the condition that a viscosity solution fulfill.
\begin{align*}
v'  & = v-(v-v_*,e)e\\
v'_* & = v_*+(v-v_*,e)e
\end{align*}


It turns out that for a large class of [[fully nonlinear integro-differential equations]], every time a viscosity solution can be touched by a smooth test function at a point, then the equation can be evaluated classically for '''the original function''' at that point <ref name="CS"/>.
and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.


=== Viscosity solutions to fully nonlinear integro-differential equations can be approximated with classical solutions ===
== Collision Invariants ==  
It is a very classical trick that if we have a weak solution to a linear PDE with constant coefficients, we can approximate it with a smooth solution via a simple mollification. For nonlinear equations this trick is no longer available and we are always forced to deal with the technical difficulties of viscosity solutions. This is an apparent difficulty for example when proving [[regularity estimates]], since in general we cannot derive them an a priori estimate for a classical solution. On the other hand, [[viscosity solutions]] to [[fully nonlinear integro-differential equations]] can be approximated by $C^2$ solutions to approximate equations <ref name="CS3"/>.


This procedure can also be carried out to approximate viscosity solutions to fully nonlinear elliptic equations with smooth solutions of an approximate equation. However, the approximated equation is integro-differential <ref name="CS4"/>.
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).


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="CR">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire | url=http://dx.doi.org/10.1016/j.crma.2009.10.012 | doi=10.1016/j.crma.2009.10.012 | year=2009 | journal=Comptes Rendus Mathématique. Académie des Sciences. Paris | issn=1631-073X | volume=347 | issue=23 | pages=1361–1366}}</ref>
 
<ref name="CR2">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=The influence of fractional diffusion in Fisher-KPP equations | url=http://arxiv.org/abs/1202.6072 | year=to appear | journal=Comm. Math. Phys.}}</ref>
<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="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
 
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=The classical Harnack inequality fails for non-local operators | year=Preprint}}</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>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
 
<ref name="CS3">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=The Evans-Krylov theorem for non local fully non linear equations | year=to appear | journal=[[Annals of Mathematics]] | issn=0003-486X}}</ref>
<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>
<ref name="CS4">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Nonlinear partial differential equations and related topics | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Amer. Math. Soc. Transl. Ser. 2 | year=2010 | volume=229 | chapter=Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations | pages=67–85}}</ref>
}}
}}

Revision as of 15:43, 24 December 2013

This article is a stub. You can help this nonlocal wiki by expanding it.

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)dxdy. \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.

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).

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