Obstacle problem for the fractional Laplacian and Boltzmann equation: Difference between pages

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The obstacle problem for the fractional Laplacian refers to the particular case of the [[obstacle problem]] when the elliptic operator $L$ is given by the [[fractional Laplacian]]: $L = -(-\Delta)^s$ for some $s \in (0,1)$. Given some smooth function $\varphi$, the equation reads
{{stub}}
\begin{align}
u &\geq \varphi \qquad \text{everywhere}\\
(-\Delta)^s u &\geq 0 \qquad \text{everywhere}\\
(-\Delta)^s u &= 0 \qquad \text{wherever } u > \varphi.
\end{align}


The equation is derived from an [[optimal stopping problem]] when considering $\alpha$-stable Levy processes. It serves as the simplest model for other optimal stopping problems with purely jump processes and therefore its understanding is relevant for applications to [[financial mathematics]].
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"/>


== Existence and uniqueness ==
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"/>.
The equation can be studied from either a variational or a non-variational point of view, and with or without boundary conditions.


As a variational inequality the equation emerges as the minimizer of the homoegeneous $\dot H^s$ norm from all functions $u$ such that $u \geq \varphi$. In the case when the domain is the full space $\mathbb R^d$, a decay at infinity $u(x) \to 0$ as $|x| \to \infty$ is usually assumed. Note that in low dimensions $\dot H^s$ is not embedded in $L^p$ for any $p<\infty$ and therefore the boundary condition at infinity cannot be assured. In low dimensions one can overcome this inconvenience by minimizing the full $H^s$ norm and therefore obtaining the equation with an extra term of zeroth order:
== The classical Boltzmann equation ==
\begin{align}
u &\geq \varphi \qquad \text{everywhere}\\
(-\Delta)^s u + u &\geq 0 \qquad \text{everywhere}\\
(-\Delta)^s u + u &= 0 \qquad \text{wherever } u > \varphi.
\end{align}
This extra zeroth order term does not affect any regularity consideration for the solution.


From a non variational point of view, the solution $u$ can be obtained as the smallest $s$-superharmonic function (i.e. $(-\Delta)^s u \geq 0$ such that $u \geq \varphi$. In low dimensions one cannot assure the boundary condition at infinity because of the impossibility of constructing barriers (this is related to the fact that the fundamental solutions $|x|^{-n+2s}$ fail to decay to zero at infinity if $2s \geq n$). This can be overcome with the addition of the zeroth order term or by the study of the problem in a bounded domain with Dirichlet boundary conditions in the complement.
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


== Regularity considerations ==
\begin{equation*}
=== Regularity of the solution ===
\int_A f(x,v,t)dxdy.
Assuming that the obstacle $\varphi$ is smooth, the optimal regularity of the solution is $C^{1,s}$.
\end{equation*}


The regularity $C^{1,s}$ coincides with $C^{1,1}$ when $s=1$, which is the optimal regularity in the classical case of the Laplacian. However, adapting the ideas of the classical proof to the fractional case suggests that the optimal regularity should be only $C^{2s}$. The optimal regularity in the case $s<1$ is better than the order of the equation and cannot be justified by any simple scaling argument.
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


<div style="background:#EEEEEE;">
\begin{equation}\label{eqn: Cauchy problem}\tag{1}
'''Outline of the proof.'''
\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}


The proof consists of the following steps that we sketch below.
where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by


*'''Almost $C^{2s}$ regularity'''
\begin{equation*}
This first step of the proof is the simplest and it is the only step which is an adaptation of the classical case $s=1$.
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*}


From the statement of the equation we have $(-\Delta)^s u \geq 0$.
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


Since the average of two $s$-superharmonic function is also $s$-superharmonic, one can see that for any $h \in \mathbb R^d$, the function $v(x):=(u(x+h)+u(x-h))/2 + C|h|^2$ is $s$ superharmonic and $v \geq \varphi$ if $C = ||D^2 \varphi||_{L^\infty}$. By the comparison principle $v \geq u$. This means that $u$ is semiconvex: $D^2 u \geq -C I$.
\begin{align*}
v'  & = v-(v-v_*,e)e\\
v'_* & = v_*+(v-v_*,e)e
\end{align*}


Interpolating the semiconvexity and $L^\infty$ boundedness of $u$, we obtain that $(-\Delta)^s u \leq C$ for some constant $C$.
and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.


The boundedness of $(-\Delta)^s u$ implies that $u \in C^{2s}$ if $s\neq\frac12$.
== Collision Invariants ==


*'''$C^{2s+\alpha}$ regularity, for some small $\alpha>0$'''
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
Let $w(x) = (-\Delta)^s u(x)$. A key observation is that the function $w$ satisfies the equation
\begin{align}
(-\Delta)^{1-s} w &= -\Delta \varphi \qquad \text{in } \{u=\varphi\}, \\
w &= 0 \qquad \text{outside } \{u=\varphi\}.
\end{align}


This is a Dirichlet problem for the conjugate fractional Laplacian. However there are two difficulties. First of all we need to prove that $w$ is continuous on the boundary $\partial \{u=\varphi\}$. Second, this boundary can be highly irregular a priori so we cannot expect to obtain any H\"older continuity of $w$ from the Dirichlet problem alone.
\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*}


From the semiconvexity of $u$ we have $-\Delta u \leq C$, and therefore we derive the extra condition $(-\Delta)^{1-s} w \leq C$ in the full space $\R^d$ (in particular across the boundary $\partial \{u=\varphi\}$). Moreover, we also know that $w \geq 0$ everywhere.
and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have


The $C^\alpha$ Holder continuity of $w$ on the boundary $\partial \{u=\varphi\}$ is obtained from an [[iterative improvement of oscillation]] procedure. Since $w \geq 0$ and $(-\Delta)^{1-s} u \leq C$, for any $x_0$ on $\partial \{u=\varphi\}$ we can show that $\max_{B_r(x_0)} w$ decays provided that $\{u > \varphi\} \cap B_r$ is sufficiently "thick" using the [[weak Harnack inequality]]. We cannot rule out the case in which $\{u > \varphi\} \cap B_r$ has a very small measure. However, in the case that $\{u > \varphi\} \cap B_r$ is too small in measure, we can prove that $u$ separates very slowly from $\varphi$. This slow separation is used to prove that $w$ must also improve its oscillation and this step is particularly tricky <ref name="S"/>.
\begin{equation*}
\frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0
\end{equation*}


Once we know that $w(x) = (-\Delta)^s u(x)$ is $C^\alpha$, this implies that $u \in C^{2s+\alpha}$ by classical potential analysis theory.
according to what $\phi$ we pick this equation corresponds to  conservation of mass, conservation momentum or conservation of energy.


*'''$C^{1,s}$ regularity'''
== The Landau Equation ==
If the contact set $\{u=\varphi\}$ is convex or at least has an exterior ball condition, a fairly simple barrier function can be constructed to show that $w$ must be $C^{1-s}$ on the boundary $\partial \{u=\varphi\}$. This is the generic boundary regularity for solutions of fractional Laplace equations in smooth domains.


Without assuming anything on the contact set $\{u=\varphi\}$, one can still obtain that $w \in C^\alpha$ for every $\alpha < 1-s$ through an iterative use of barrier functions <ref name="S"/>. The sharp $w \in C^{1-s}$ regularity in full generality was obtained rewriting the equation as a [[thin obstacle problem]] using the [[extension technique]] and then applying blowup techniques, the Almgren monotonicity formula and classification of global solutions <ref name="CSS"/>.
A closely related evolution equation is 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,


The $C^{1-s}$ regularity of $w$ implies that $u \in C^{1,s}$ by classical potential analysis.
\begin{equation*}
</div>
f_t + x\cdot \nabla_y f = Q_{L}(f,f)
\end{equation*}


=== Regularity of the free boundary ===
where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as
A ''regular point'' of the free boundary is where the solution $u$ is exactly $C^{1,s}$ an no better. This is classified explicitly in terms of the limits of the Almgren frequency formula <ref name="CSS"/>. Around any regular point, the free boundary is a smooth $C^{1,\alpha}$ surface <ref name="CSS"/>.


A ''singular point'' is defined as a point on the free boundary where the measure of the contact set has vanishing density. More precisely, if
\begin{equation*}
\[ \lim_{r \to 0} \frac{|\{u=\varphi\} \cap B_r|}{r^n} = 0.\]
Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2
\end{equation*}


In the case $s=1/2$, it was shown by Nicola Garofalo and Arshak Petrosyan <ref name="GP"/> that the singular points of the free boundary are contained  inside a differentiable surface. The proof is done in the context of the [[thin obstacle problem]] and presumably can be extended to other powers of the Laplacian using the [[extension technique]].
where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.  


It is important to notice that the definitions of regular and singular points of the free boundary are mutually exclusive but they do not exhaust all possible free boundary points. It is an interesting open problem to understand what other type of free boundary points are possible if any.


== The parabolic version ==
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).


The parabolic version of the fractional obstacle problem was studied by Caffarelli and Figalli <ref name="CF"/>. They concluded that the solution $u$ has the following regularity estimates.
== References ==
\begin{align}
{{reflist|refs=
u_t, (-\Delta)^s u \in LogLip_t C_x^{1-s}, \text{ if } s\leq 1/3,\\
u_t, (-\Delta)^s u \in C_{t,x}^{\frac{1-s}{2s},{1-s}}, \text{ if } s > 1/3.
\end{align}


It turns out that it is crucial to consider solutions $u$ to be non decreasing in time (which is assured by taking the initial value coinciding with the obstacle). Otherwise the regularity of the solution is reduced to merely $C^{2s}$ in space.
<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>


The regularity of the free boundary has not been explored in the parabolic setting yet.
<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>


== References ==
<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>
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | publisher=Wiley Online Library | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
<ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
<ref name="CF">{{Citation | last1=Figalli | first1=A. | last2=Caffarelli | first2=Luis | title=Regularity of solutions to the parabolic fractional obstacle problem | year=2011 | journal=Arxiv preprint arXiv:1101.5170}}</ref>
<ref name="GP">{{Citation | last1=Petrosyan | first1=A. | last2=Garofalo | first2=N. | title=Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2009 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=177 | issue=2 | pages=415–461}}</ref>
}}
}}

Revision as of 20:11, 18 October 2013

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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) (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 \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.

The Landau Equation

A closely related evolution equation is 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).

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