An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation

The purpose of kinetic equations is the description of dilute particle 
gases at an intermediate scale between the microscopic scale and the 
hydrodynamical scale. By dilute gases, one has to understand a system with 
a large number of particles, for which a description of the position and of the 
velocity of each particle is irrelevant, but for which the decription cannot be 
reduced to the computation of an average velocity at any time 
$t\in \mathbb R$ and any 
position $x\in \mathbb R^d:$ one wants to take into account more than one possible velocity 
at each point, and the description has therefore to be done at the level of 
the phase space – at a statistical level – by a distribution function $f(t, x, v)$. 
 
This course is intended to make an introductory review of the literature on 
kinetic equations. Only the most important ideas of the proofs will be given. 
The two main examples we shall use are the Vlasov-Poisson system and the 
Boltzmann equation in the whole space.


The distribution function
The main object of kinetic theory is the distribution function f(t;x;v) which is a nonnegative function depending on the time: t2IR, the position: x2IR d , the velocity: v 2IR d or the impulsion ).A basic requirement is that f(t;:;:) belongs to L 1 loc (IR d IR d ) and from a physical point of view f(t;x;v)dxdv represents \the probability of nding particles in an element of volume dxdv, at time t, at the point (x;v) in the (one-particle) phase space".f describes the statistical evolution of the system of particles: f has to be constant along the characteristics (X(t);V (t)) in the phase space given by Newton's law: _ X = dX dt = V ; _ V = dV dt = F(t;X(t))=?@x U(t;X) if F derives from a potntial U. 0 = d dt f t;X(t);V (t) = @ t f +V (t) @ x f +F t;X(t) @ v f and satis es therefore the transport equation: @ t f +v:@ x f +F (t;x) @ v f = 0 (1.1) with the notations: @ t f = @f @ , @ x f = r x f = @f @x1 , @f @x1 ;::: @f @x d , @ v f = r v f = @f @v1 ; @f @v2 ;::: @f @v d .

Mean eld approximation and collisions
A mean eld approximation corresponds to the case where the force itself depends on some average of the distribution function, for instance F(t;x)=(@ x V 0 x )(t;x) ; (t;x) = Z I R d f(t;x;v)dv : The Vlasov-Poisson system is given by V 0 (z) = 1  4 jzj (in dimension d=3), or div x F = in general.
Another limit corresponds to short range two-body potentials, for which the e ects of the interaction can be considered as a collision: it occurs at a xed time t for a given position x and acts only on the velocities (in the thermodynamical limit).For dilute gases,no more than two particles are involved in a collision.The fundamental example is the Boltzmann equation: @ t f +v:@ x f = Q(f;f) (t;x;v) 2IR IR d IR d (BE) where the collision kernel takes the form Q(f;f) = Z Z I R d S d?1 B(v?v ;!)(f 0 f 0 ?ff ) dv d! ; (1.2) f = f(t;x;v) ; f = f(t;x;v ) ; f 0 = f(t;x;v 0 ) ; f 0 = f(t;x;v 0 ) ; v and v are the velocities of the incoming particles (before collision), v 0 and v 0 are the velocities of the outgoing particles (after collision) and are given in terms of v and v by v 0 = v? (v ?v ):! ! ; v 0 = v + (v ?v ):! ! ; for some !2S d?1 which parametrizes the set of admissible outgoing velocities under the constraints v+v = v 0 +v 0 and jvj 2 +jv j 2 = jv 0 j 2 +jv 0 j 2 and B(v? v ;!) is the di erential cross-section, which measures the probability of the collision process (v;v ) 7 !(v 0;v 0 ) = T !(v;v ).Note that the collision operator is local in (t;x) and has two parts: \the incoming part": Q ?(f;f) = R R B(v?v ;!)ff dv d!, \the outgoing part": Q + (f;f) = R R B(v?v ;!)f 0 f 0 dv d!, and we may write: Q(f;f)=Q + (f;f)?Q ?(f;f).

Conservation of mass
Consider a solution f(t;x;v) of the linear transport equation (1.1) or of the Boltzmann equation (BE) and formally perform an integration w.r.t.v: if the mass ux is de ned by then one obtains: @ t (t;x)+div x (j(t;x)) = 0 since the force term is in divergence in v form or since R Q(f;f)dv = 0 (of course, one has to assume a su cient decay of f to justify this computation).This expresses the local conservation of mass (or of the number of the particles).
If the problem is stated in the whole space (x 2 = IR d ), performing one more integration w.r.t.x and provided f has a su cient decay in x too, then: d dt This relation is the global conservation of the mass.

A priori energy estimates
Consider a solution of @ t f + v:@ x f ?@ x U:@ v f = 0 : (1.The Vlasov equation is now nonlinear (quadratic) and nonlocal: @ t f +v:@ x f ?@x (K ):@ v f = 0 and this can also be seen at the level of the energy: exactly as before, combining (1.
Note here the factor 1 2 in front of the potential energy term.

Velocity averaging lemmas
Velocity averaging lemmas are a basic tool to obtain some compactness in the framework of kinetic equations with distribution functions in C( 0;T ];L 1 (IR d IR d )).We follow the presentation given in 21] and 7], but the basic reference is 23] and also more recent papers by Lions and al.These results together with the notion of renormalization are two crucial steps in the construction of the renormalized solutions for the Boltzmann equation by DiPerna and Lions.
Lemma 1.1 Let f 2L

Interpolation lemmas
In the two following lemmas, the relations between the norms and the exponents are easily recovered using scalings in x and v.The rst lemma can be found for instance in 31, 32].The second one is a generalization of the rst lemma to higher moments.These lemmas are related to the estimates used by B. Proof: Assume to simplify that p<+1 and consider the integral de ning : The L q -norm of is now bounded and using H older's inequality, we obtain the result for a convenient choice of the exponents.In this section we consider the Vlasov-Poisson system @ t f +v:@ x f ?@x U:@ v f = 0 t>0; x;v 2IR d (2.1) in dimension d (with d=3 unless it is speci ed; for d=2, see 15]) and with = ?1 (plasma physics or eletrostatic case) or = +1 (gravitational case).The global existence of weak solutions goes back to Arsen'ev 3] and is now known under weak assumptions like: Here we will rather focuse on strong solutions { solutions for which the characteristics are de ned in a classical sense { or even classical C 1 solutions for which each of the terms makes sense as a continuous function (and @ x U as a Lispchitz function).For stationary solutions, see 4], 15], 16], 19].

Classical solutions and characteristics
We present here in dimension d=3 a result which has been established rst by K. Pfa elmoser 35] and then improved by several authors, in the version given by R. Glassey in 22] (initially given by Schae er in 37]).The main ingredient of this approach is to start with a solution which is initially compactly supported and to control the growth of the size of the support.Let Q(t)=1+sup n jvj:9(t;x)2(0;t) IR 3 s:t: f(t;x;v)6 = 0 o Theorem 2.1 Let f 0 be a non negative C 1 compactly supported function.Then the Cauchy problem for (2.1) has a unique C 1 solution and Q(t) C p (1+t) p with p> 33  17 : Note that the rate of growth has been improved but its optimal value is still unknown.
Proof: The proof relies on the iteration scheme 8 < : @ x f n+1 +v:@ x f n+1 ?@x U n :@ v f n+1 = 0 which is solved at each step by the characteristics method.Passing to the limit is easy after proving the right uniform bounds (energy estimates, bounds on the eld and its derivatives, bounds on the derivatives of f) which are easily obtained as soon as one has a uniform estimate of the size of the support of f (whatever it is).
To simplify the notation, we shall forget the index n and work directly with a solution.The main step to estimate Q(t) is then to compute for any t>0, 2]0;t the quantity: f(s;y;v) j X(s)?X(s;t;y;v)j 2 dydv using the fact that the map (x;v) 7 !(X(s;t;x;v);V (s;t;x;v)) given by dX ds = V ; dV ds = ?@x U(s;X); (X;V )(t;t;x;v) = (x;v) is measure preversing (here X(s)) denotes any xed given characteristics): it is indeed deriving from the ow of an hamiltonian system.
The nexlastt step is to split the integral in (2.3) into the integral over three sets (usually called the \good", the \bad" and the \ugly") and to optimize on the parameters de ning these sets, thus obtaining The proof is valid in the gravitational case as well as in the plasma physics case since both parts of the energy (kinetic and self consistent potential parts) are uniformly bounded (for some xed term interval 0;T ]) even in the gravitational case, where they enter in the energy with opposite signs.The reason is the following.
L P (I R N ) C L q (I R N ) ; with 0 < 1 p = 1 q + N ?1, we can control jj@ x Ujj L 2 by Then, using H older's inequality, we get and the L 5=3 -norm is controlled by the following interpolation identity (which is a limit case of Lemma 1.3): 2 dxdv and P(t)= 1 2 R j@ x U(t;x)j 2 dx are the kinetic energy and the potential energy respectively, then the total energy is Const = K(t)?P(t) K ?CK 10=12 proving therefore that K and also P are uniformly bounded (in t) in terms of f 0 .

The Lions and Perthame approach for strong solutions
An alternating approach to nd strong solutions in dimension d=3 when the initial data is not compactly supported has been developped by Lions and Perthame in 32].It is mainly based on a priori estimates for the eld @ x U and for moments of order m>3.

2.3 Time-dependent rescalings and dispersion
In this section, we introduce as in 20] the time-dependent rescalings for kinetic equations on the example of the Vlasov- A 2 (t) : Here _ always denotes derivative with respect to t.Let F be the rescaled distribution function: f(t;x;v)=G(t)F( ; ; ).The aim is to choose this transformation in such a way that the rescaled Vlasov equation is still a transport equation on the phase space and contains a given, external force and a friction term.If the rescaled potential is given by We want F to be a conservation law on ( ; )-space (preservation of the L 1norm), so we require _ 2d (up to a multiplicative constant) and the Vlasov equation becomes @ F + @ F +div Next we require that the external force in the above Vlasov equation becomes time independent and that there is no time-dependent factor in front of the nonlinear term.We therefore require where c 0 >0 is an arbitrary constant.Thus we get A=R d=4 ; G=R d?4 2 d and R has to solve R=? c 0 R 1?d : Without any restriction, we may asume that c 0 = 1, R(0)=1 and _ R(0)=0: F( = 0; ; ) = f(t=0; ; ) = f 0 ( ; ) : By considering for F the derivative of the energy E( ) = 1 2 Z Z j j 2 +W ( ; )? j j 2 F( ; ; )d d : with respect to : is decreasing for d=2, 3, 4.
In dimension d=3, if = ?1,R(t) behaves as t!1 as t, which essentially proves that R R f(t;x;v)jx?vtj 2 dxdv = O(t).By an interpolation between this moment and the L 1 -norm of f, Perthame and Illner & Rein (see 34] and 27]) proved the following decay estimate.Corollary 2.5 Consider a solution of the Vlasov-Poisson system in the electrostatic case ( = ?1)corresponding to a nonnegative initial data f 0 2L 1 \ L 1 (IR 3 IR 3 ) such that R R f 0 (x;v) jxj 2 +jxj 2 ] dxdv is bounded.Then k (t;:)k L 5=3 (I R 3 ) = O(t ?3=5 ) : Further estimates on @ x U for instance can also be obtained.The method introduced in 20] provides re ned estimates and explain how to obtain Lyapunov functionals using time-dependent rescalings in various related systems of uid dynamics or quantum mechanics, and what is the relation with the pseudo-conformal law.

Introduction to the Boltzmann equation
For Sections 3.1 and 3.3, we essentially follow the presentation of B. Perhame in 9].For a detailed study of the hard spheres case we shall refer to 7] and for a more classical theory of perturbations, to 22].The results on the homogeneous case (and the limit of grazing collisions) are directly collected from the original papers.The dispersion results for renormalized solutions are new results.For the moment, there is no book covering all the mathematical aspects of the Boltzmann equation, the most complete at this time beeing probably the book by Cercignani, Illner and Pulvirenti 7] (hard sphere case only).

The Boltzmann equation
The non homogeneous Boltzmann equation (BE) in IR d describes a cloud of particles expanding in the vacuum.It is an integro-di erential equation where the integral part is the Boltzmann collision operator is given by (1.2).We are assuming that the particles have the same mass and are a ected only by (binary) elastic collisions, so that the conservation of the impulsion and of the kinetic energy respectively give v 0 +v 0 = v+v ; (3.1) jv 0 j 2 +jv 0 j 2 = jvj 2 +jv j 2 ; (3.2) where v and v are the incoming velocities, v 0 and v 0 the outgoing velocities.These relations can be solved into for any !2S d?1 .We denote by T ! the operator acting on IR d IR d such that (v 0 ;v 0 ) = T !(v;v ) The di erential cross-section B is a measurement of the probability of a collision corresponding to a given !. Physical considerations (microreversibility, galilean invariance) allow to consider B depending only on jv?v j and (v ?v ) !, and further formal considerations show that for power-like two body potentials u 0 (r) = k r 1?s ; B takes the form B(z;!)=jzj: (cos ) with cos = z jzj ! and = s?5 s?1 .has a singularity for = 2 : (cos ) 2 ?
Unless it is explicitely speci ed, we shall assume that d=3.In the case of a power-law interaction, the rst two assumptions respectively mean: >?3 (or s>2) and b2L 1 (S 2 ) and <2 or s>1)

Conservation laws and H-theorem
For any functions f, ' such that all the involved quantities are well de ned, As a consequence, we have the Lemma 3.1 (i) Conservation of mass: These identities are easily proved by applying (3.3) with '=1, v, jvj @ t +@ x ( u) = 0 @ t ( u)+@ x (p+ u u) = 0 @ t h p( + 1 These equations and the equation of state (3.4) provide 6 scalar equations for 13 unknowns and we need to impose \constitutive equations" to relate those quantities and to close the system.We may for instance consider the following cases: Euler equations for ideal uids: p ij (t;x) = p(t;x) ij ; q i = 0 Navier Stokes equations for viscous uids : p ij (t;x) = p(t;x) ij ?@u j @x i + @u i @x j ?(@ x :u) ij ; q i = ?k@T @x i : Grad hierarchy: f(t;x;v)=M ;u;T P(v) where M ;u;T is a local Maxwellian having the same moments in 1;v;jvj 2 as f, and P is a well choosen polynomial.However, this system is not hyperbolic (see 8]).
Levermore hierarchy: f(t;x;v)=e pt;x(v) .The closure of this hierarchy is not explicit, but the rst nontrivial system (with 17 moments) is hyperbolic (see 28]).Note that we may derive a macroscopic entropy inequality @ @ t (S(t;x))+div( (t;x)) 0 which is fundamental to describe the shocks in the uid limit.

Stability, existence of renormalized solutions to the Boltzmann equation
In this section, we consider the Boltzmann equation under the weak (Grad) angular cut-o , the mild growth condition and the positivity (and symmetry) assumptions of Section 3.1.The global existence of solution to the Cauchy problem for arbitrarily large initial data has been proved by DiPerna and Lions in 13] and 14] (see also 21] and 29]) in the framework of the renormalized solutions.We assume that the initial data f 0 is a nonnegative L 1 (IR d IR d ) function such that (x;v) 7 !f0 (x;v)(jxj 2 +logjvj 2 +jlogf 0 j)dxdv belongs to L 1 (IR d IR d ): ( As a consequence of the a priori estimate which holds because the Boltzmann collision kernel is local in (t;x), and because of the H-theorem: S(t) is bounded from below.To prove it, we may use Jensen's inequality (jvj 2 +jx?tvj 2 ) dxdv !>?1 8t>0: The main di culty of the Boltzmann equation is to give a sense to the products f(t;x;v)f(t;x;v ) and to f(t;x;v 0 )f(t;x;v 0 ) when f is only a L 1 function.Even for a bounded collision kernel B, if we write the simplest possible estimate: we can see that (t;x) 7 !R I R d f(t;x;v)dv 2 still does not make much sense.
The main idea of renormalized solutions is to replace the equation by a renormalized equation and write that Q + (f;f) 1+f belongs to L 1 (R + ;L 1 (IR d K)) for any compact set K in IR d v .A nonnegative distribution function f is said to be a renormalized solution of the Boltzmann equation if f 2C 0 (IR + ;L 1 (IR d IR d )) is such that Z Z I R d I R d f(t;x;v) 1+jvj 2 +jx?tvj 2 +jlog f(t;x;v) j dxdv <+1 for any t>0, and if for any 2C 1 (IR + ;IR + ) such that 0 (t)(1+t) is bounded in IR + , @ @t +v:@ x (f) = 0 (f)Q(f;f) in D 0 (IR + IR This result is obtained through compactness arguments and appropriate regularization, so that an almost equivalent result is the following stability result.The boundary problem has been studied by Hamdache 26], Arkeryd and Cercignani 1].One should also mention the study of the large time asymptotics by Desvillettes 11] and Cercignani 6] (in a bounded domain).For the theory of small classical solutions or perturbations of a stationary solution, we refer to 9] and 22].An overview of the results in the homogeneous case will be given the next Section.Let us nally mention the existence results recently given by Arkeryd and Nouri in 2] for stationary solutions in a bounded domain.

The homogeneous Boltzmann equation
In the case where the distribution function does not depend on x, the situation is much simpler and better results have been proved for already a long time.
The general framework is given by L 1 -spaces with weights: consider L 1 s and LlogL such that The hard spheres collision kernel is We follow here the presentation of 7].Theorem 3.5 Let f 0 0 be an initial data such that f 0 2L 1 4 \LlogL.Then there exists a unique solution f in C 0 (IR + ;L 1 (IR 3 )).Moreover, f 2L 1 4 (IR 3 ) and

Dispersion for the renormalized solutions
We conclude this introduction to the Boltzmann equation by giving a dispersion result for the renormalized solutions.A preliminary result has been obtained by B. Perthame in 34], but we follow here the approach of 17] based on Jensen's inequality.Theorem 3.8 Under the same assumptions as in Section 3.4, consider a renormalized solution f 2C 0 (IR + ;L 1 (IR 3 IR 3 )) corresponding to an initial datum f 0 2L 1 (IR 3 IR 3 ) such that f 0 (jxj 2 +jvj 2 +jlogf 0 j) is bounded in L 1 (IR 3 IR 3 ).We now use (3.10) to obtain a dispersion relation via an interpolation which is in a sense the limit case (see Section 1.6) as p!1 of an interpolation between moments and an L p -norm.The result is obtained using several times Jensen's inequality: if f and g are two nonnegative L 1 ( ) solutions such that f(jlogfj+ jloggj) belongs to L 1 ( ), Jensen's inequality applied to t7 !tlogt=s(t) with the measure d (y) = g(y)dy R g(y)dy gives R f log( f g ) dy R g(y)dy = Z s( f g ) d (y) s Z f g d (y) = R f(y)dy R g(y)dy log R f(y)dy R g(y)dy : (3.11) Applying rst this inequality to g = e ?(1+t 2 )jv? t 1+t 2 xj 2 with y = v, = IR 3 , and then integrating w.r.t.x, we get 2 f(x;v)jvj k dv : If we optimize on R, then we get (x) C(d;p;k) Z I R d jf(x;v)j p dv k d(p?1)+kp Z I R d f(x;v)jvj k dv d(p?1) d(p?1)+kp :
d IR d ) (RBE) Theorem 3.3 (DiPerna & Lions) Under the above assumptions, there exists a global in time renormalized solution to the Boltzmann equation.

Theorem 3 . 4 (
DiPerna & Lions) Consider a sequence of initial data f n 0 converging in L 1 (IR d IR d ) to some f 0 such that (3.5) is uniformly satis ed.Then the corresponding renormalized solutions f n converge up to the extraction of a subsequence to a renormalized solution to the Cauchy problem associated to f 0 .These results are uncomplete from several points of view: the conservation of the energy is not established, the H-theorem holds as an inequality and the question of the uniqueness is open.
) and Z I R d jf(v)log(f(v))jdv <+1: Consider now the Cauchy problem for the homogeneous Boltzmann equation 1 L 1 theory for the hard spheres case (d=3)

3. 5 . 3 2 (
Gain of moments and regularizing e ects for collision kernel without cut-o sConsider the case of hard potentials and assume that the initial data is bounded in L 1 2+ with >0.According to Povzner inequality see 12], 33]).In other words, f 2L 1 ( 0;T ]; L 1 2+ + ) for any >0.Thus by iteration any moment becomes nite for any positive time.More interesting probably are the regularization properties of the Boltzmann collision kernel.For forces with an in nite range, and especially for inverse power laws, the weak angular cut-o assumption is not satis ed: if B(z;!) = jzj ( ), then has a singularity of order s+1 s?1 = 1+ .P.-L.Lions proved in 30] properties of Q + .Recently, further results have been given by Villani, and Desvillettes and Wennberg.
a positive constant C = C(f 0 ) which depends only on f 0 such that for any r >0,
Poisson system (2.1)-(2.2) (see also 17], 18]) and show in dimension d=3 how this provides the dispersion estimates found independently by Perthame and Illner & Rein (see 34] and 27]).Consider the Vlasov-Poisson system and compute the transformation of vari- 1(if we write d! = sin d d ).?' allow to give a sense to the weak formulation whenever the right hand side in(3.6)isreplaced by expression (3.8).
N.B. the right hand side in(3.6)has to be understood in the sense (3.7).'