1.1 The relativistic Boltzmann equation

Consider a collection of neutral particles in Minkowski spacetime. Let the signature of the metric be (− ,+, +,+ ). In this section we assume that all the particles have rest mass m = 1, and we normalize the speed of light c to one. We point out that in Section 2 on the Einstein–Vlasov system, the dependence on the rest mass and the speed of light will be included in the formulation of the system. The four-momentum of a particle is denoted by pa, a = 0,1,2,3. Since all particles have equal rest mass, the four-momentum for each particle is restricted to the mass shell, papa = − m2 = − 1. Thus, by denoting the three-momentum by 3 p ∈ ℝ, a p may be written a 0 p = (p ,p), where 0 ∘ ------2- p = 1 + |p| is the energy of a particle with three-momentum p, and |p | is the usual Euclidean length of p. The relativistic velocity of a particle with momentum p is denoted by ˆp and is given by
----p----- ˆp = ∘ 1 + |p|2. (1 )
Note that |ˆp| < 1 = c. The relativistic Boltzmann equation models the spacetime behavior of the one-particle distribution function f = f(t,x,p), and it has the form
( ) ∂t + p-⋅ ∇x f = Q (f,f), (2 ) p0
where the relativistic collision operator Q (f,g) is defined by
∫ ∫ Q (f,g) = k (p, q,ω)[f(p + a(p,q,ω)ω )g(q − a(p,q,ω)ω ) − f (p)g (q)]dωdp. (3 ) ℝ3 𝕊2
Note that g = f in Equation (2View Equation). Here dω is the element of surface area on 𝕊2 and k(p,q,ω ) is the scattering kernel, which depends on the differential cross-section in the interaction process. We refer to [178Jump To The Next Citation Point], [54Jump To The Next Citation Point] and [68Jump To The Next Citation Point] for examples of differential cross-sections in the relativistic case. The function a(p,q,ω ) results from the collision mechanics. If two particles, with momentum p and q respectively, collide elastically with scattering angle ω ∈ 𝕊2, their momenta will change, i.e., p → p′ and q → q ′. The relation between p,q and p′,q′ is given by
p′ = p + a(p,q,ω)ω, q′ = q − a(p,q,ω )ω, (4 )
2(p0 +-q0)p0q0(ω-⋅-(qˆ−-ˆp))- a(p,q,ω ) = (p0 + q0)2 − (ω ⋅ (p + q))2 . (5 )
This relation is a consequence of four-momentum conservation,
a a a′ a′ p + q = p + q ,

or equivalently

p0 + q0 = p0′ + q0′, (6 ) ′ ′ p + q = p + q. (7 )
These are the conservation equations for relativistic particle dynamics. In the classical case the corresponding conservation equations read
|p |2 + |q|2 = |p′|2 + |q′|2, (8 ) p + q = p′ + q′. (9 )
The function a(p,q,ω) gives the distance between p and p′ (q and q′) in momentum space, and the analogue function in the non-relativistic, Newtonian, classical case has the form
acl(p,q,ω) = ω ⋅ (q − p). (10 )
By inserting acl in place of a in Equation (3View Equation) we obtain the classical Boltzmann collision operator (disregarding the scattering kernel, which is also different). We point out that there are other representations of the collision operator (3View Equation), cf. [179].

In [44] and [178Jump To The Next Citation Point] classical solutions to the relativistic Boltzmann equations are studied as c → ∞, and it is proven that the limit as c → ∞ of these solutions satisfies the classical Boltzmann equation. The former work is more general since general initial data is considered, whereas the latter is concerned with data near vacuum. The latter result is stronger in the sense that the limit, as c → ∞, is shown to be uniform in time.

The main result concerning the existence of solutions to the classical Boltzmann equation is a theorem by DiPerna and Lions [71] that proves existence, but not uniqueness, of renormalized solutions. An analogous result holds in the relativistic case, as was shown by Dudyński and Ekiel-Jeżewska [72], cf. also [102]. Regarding classical solutions, Illner and Shinbrot [99Jump To The Next Citation Point] have shown global existence of solutions to the non-relativistic Boltzmann equation for initial data close to vacuum. Glassey showed global existence for data near vacuum in the relativistic case in a technical work [80Jump To The Next Citation Point]. He only requires decay and integrability conditions on the differential cross-section, although these are not fully satisfactory from a physics point of view. By imposing more restrictive cut-off assumptions on the differential cross-section, Strain [178] gives a different proof, which is more related to the proof in the non-relativistic case [99] than [80] is. For the homogeneous relativistic Boltzmann equation, global existence for small initial data has been shown in [126] under the assumption of a bounded differential cross-section. For initial data close to equilibrium, global existence of classical solutions has been proven by Glassey and Strauss [87Jump To The Next Citation Point] using assumptions on the differential cross-section, which fall into the regime “hard potentials”, whereas Strain [177Jump To The Next Citation Point] has shown existence in the case of soft potentials. Rates of the convergence to equilibrium are given in both [87Jump To The Next Citation Point] and [177]. In the non-relativistic case, we refer to [189, 172, 119] for analogous results.

The collision operator Q (f, g) may be written in an obvious way as

Q(f,g ) = Q+ (f, g) − Q − (f, g),

where Q+ and Q − are called the gain and loss term, respectively. If the loss term is deleted the gain-term-only Boltzmann equation is obtained. It is interesting to note that the methods of proof for the small data results mentioned above concentrate on gain-term-only equations, and once that is solved it is easy to include the loss term. In [14] it is shown that the gain-term-only classical and relativistic Boltzmann equations blow up for initial data not restricted to a small neighborhood of trivial data. Thus, if a global existence proof of classical solutions for unrestricted data will be given, it will necessarily use the full collision operator.

The gain term has a nice regularizing property in the momentum variable. In [4Jump To The Next Citation Point] it is proven that given f ∈ L2 (ℝ3) and g ∈ L1 (ℝ3) with f,g ≥ 0, then

+ ∥Q (f,g)∥H1(ℝ3p) ≤ C ∥f ∥L2(ℝ3p)∥g∥L1(ℝ3p), (11 )
under some technical requirements on the scattering kernel. Here s H is the usual Sobolev space. This regularizing result was first proven by Lions [112] in the classical situation. The proof relies on the theory of Fourier integral operators and on the method of stationary phase, and requires a careful analysis of the collision geometry, which is very different in the relativistic case. Simplified proofs in the classical and relativistic case are given in [193, 194].

The regularizing theorem has many applications. An important application is to prove that solutions tend to equilibrium for large times. More precisely, Lions used the regularizing theorem to prove that solutions to the classical Boltzmann equation, with periodic boundary conditions, converge in L1 to a global Maxwellian,

− α|p|2+β⋅p+γ 3 M = e with α, γ ∈ R, α > 0, β ∈ ℝ ,

as time goes to infinity. This result was first obtained by Arkeryd [29] by using non-standard analysis. It should be pointed out that the convergence takes place through a sequence of times tending to infinity and it is not known whether the limit is unique or depends on the sequence. In the relativistic situation, the analogous question of convergence to a relativistic Maxwellian, or a Jüttner equilibrium solution,

√ ----2 J = e−α 1+|p| +β⋅p+γ, α,β, and γ as above, with α > |β|,

was studied by Glassey and Strauss [87, 88]. In the periodic case, they proved convergence in a variety of function spaces for initial data close to a Jüttner solution. Having obtained the regularizing theorem for the relativistic gain term, it is a straightforward task to follow the method of Lions and prove convergence to a global Jüttner solution for arbitrary initial data (satisfying the natural bounds of finite energy and entropy), which are periodic in the space variables, cf. [4]. We also mention that in the non-relativistic case Desvillettes and Villani [69] have studied the convergence rate to equilibrium in detail. A similar study in the relativistic case has not yet been achieved.

For more information on the relativistic Boltzmann equation on Minkowski space we refer to [54, 68, 181, 79Jump To The Next Citation Point] and in the non-relativistic case we refer to [190, 79Jump To The Next Citation Point, 53].

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