1.2 The Vlasov–Maxwell and Vlasov–Poisson systems

Let us consider a collision-less plasma, which is a collection of particles for which collisions are relatively rare and the interaction is through their charges. For simplicity we assume that the plasma consists of one type of particle, although the results below hold for plasmas with several particle species. The particle rest mass and the particle charge are normalized to one. In the kinetic framework, the most general set of equations for modeling a collision-less plasma is the relativistic Vlasov–Maxwell system:
∂tf + ˆv ⋅ ∇xf + (E (t,x) + vˆ× B (t,x)) ⋅ ∇vf = 0 (12 ) ∂tE + j = c∇ × B, ∇ ⋅ E = ρ, (13 ) ∂tB = − c∇ × E, ∇ ⋅ B = 0. (14 )
The notation follows the one already introduced with the exception that the momenta are now denoted by v instead of p. This has become a standard notation in this field. E and B are the electric and magnetic fields, and ˆv is the relativistic velocity,
ˆv = ∘-----v------, (15 ) 1 + |v|2∕c2
where c is the speed of light. The charge density ρ and current j are given by
∫ ∫ ρ = 3 fdv, j = 3 ˆvfdv. (16 ) ℝ ℝ
Equation (12View Equation) is the relativistic Vlasov equation and Equations (13View Equation, 14View Equation) are the Maxwell equations.

A special case in three dimensions is obtained by considering spherically-symmetric initial data. For such data it can be shown that the solution will also be spherically symmetric, and that the magnetic field has to be constant. The Maxwell equation ∇ × E = − ∂tB then implies that the electric field is the gradient of a potential ϕ. Hence, in the spherically-symmetric case the relativistic Vlasov–Maxwell system takes the form

∂tf + ˆv ⋅ ∇xf + βE (t,x) ⋅ ∇vf = 0, (17 ) E = ∇ ϕ, Δ ϕ = ρ. (18 )
Here β = 1, and the constant magnetic field has been set to zero, since a constant field has no significance in this discussion. This system makes sense for any initial data, without symmetry constraints, and is called the relativistic Vlasov–Poisson system. Another special case of interest is the classical limit, obtained by letting c → ∞ in Equations (12View Equation, 13View Equation, 14View Equation), yielding:
∂tf + v ⋅ ∇xf + βE (t,x ) ⋅ ∇vf = 0, (19 ) E = ∇ϕ, Δ ϕ = ρ, (20 )
where β = 1. We refer to Schaeffer [166] for a rigorous derivation of this result. This is the Vlasov–Poisson system, and β = 1 corresponds to repulsive forces (the plasma case). Taking β = − 1 means attractive forces and the Vlasov–Poisson system is then a model for a Newtonian self-gravitating system.

One of the fundamental problems in kinetic theory is to find out whether or not spontaneous shock formations will develop in a collision-less gas, i.e., whether solutions to any of the equations above will remain smooth for all time, given smooth initial data.

If the initial data are small this problem has an affirmative solution in all cases considered above [81Jump To The Next Citation Point, 86, 32Jump To The Next Citation Point, 33]. For initial data unrestricted in size the picture is more involved. In order to obtain smooth solutions globally in time, the main issue is to control the support of the momenta

Q(t) := sup {|v| : ∃ (s, x) ∈ [0,t] × ℝ3 such that f(s,x,v) ⁄= 0}, (21 )
i.e., to bound Q (t) by a continuous function so that Q(t) will not blow up in finite time. That such a control is sufficient for obtaining global existence of smooth solutions follows from well-known results in the different cases, cf. [85Jump To The Next Citation Point, 104, 39, 96, 34Jump To The Next Citation Point, 81Jump To The Next Citation Point]. For the full three-dimensional relativistic Vlasov–Maxwell system, the problem of establishing whether or not solutions will remain smooth for all time is open. A different sufficient criterion for global existence in this case is given by Pallard in [129], and he also shows a new bound for the electromagnetic field in terms of Q (t) in [130]. In two space and three momentum dimensions, Glassey and Schaeffer [82, 83] have shown that Q (t) can be controlled for the relativistic Vlasov–Maxwell system, which thus yields global existence of smooth solutions in that case.

The relativistic and non-relativistic Vlasov–Poisson equations are very similar in form. In particular, the equation for the field is identical in the two cases. However, the mathematical results concerning the two systems are very different. In the non-relativistic case, Batt [34] gave an affirmative solution in 1977 in the case of spherically-symmetric data. Pfaffelmoser [133] was the first one to give a proof for general smooth data. A simplified version of the proof is given by Schaeffer in [168]. Pfaffelmoser obtained the bound

(51+ δ)∕11 Q(t) ≤ C (1 + t) ,

where δ > 0 can be taken as arbitrarily small. This bound was later improved by different authors. The sharpest bound valid for β = 1 and β = − 1 has been given by Horst [97] and reads

Q(t) ≤ C (1 + t) log (2 + t).

In the case of repulsive forces (β = 1) Rein [137] has found a better estimate by using a new identity for the Vlasov–Poisson system, discovered independently by Illner and Rein [98] and by Perthame [132]. Rein’s estimate reads

2∕3 Q (t) ≤ C(1 + t) .

Independently, and at about the same time as Pfaffelmoser gave his proof, Lions and Perthame [113] used a different method for proving global existence. Their method is more generally applicable, and the two studies [5] and [105] are examples of problems in related systems, where their method has been successful. On the other hand, their method does not give such strong growth estimates on Q (t) as described above. For the relativistic Vlasov–Poisson equation, Glassey and Schaeffer [81Jump To The Next Citation Point] showed in the case β = 1 that if the data are spherically symmetric, Q(t) can be controlled, which is analogous to the result by Batt mentioned above. Also in the case of cylindrical symmetry they are able to control Q (t); see [84]. If β = − 1 it was shown in [81] that blow-up occurs in finite time for spherically-symmetric data with negative total energy. More recently, Lemou et al. [111Jump To The Next Citation Point] have investigated the structure of the blow-up solution. They show that the blow-up is determined by the self-similar solution of the ultra-relativistic gravitational Vlasov–Poisson system. It should be pointed out that the relativistic Vlasov–Poisson system is unphysical since it lacks the Lorentz invariance; it is a hybrid of a classical Galilei invariant field equation and a relativistic transport equation (17View Equation), cf. [3]. In particular, in the case β = − 1, it is not a special case of the Einstein–Vlasov system. Only for spherically-symmetric data, in the case β = 1, is the equation a fundamental physical equation. The results mentioned above all concern classical solutions. The situation for weak solutions is different, in particular the existence of weak solutions to the relativistic Vlasov–Maxwell system is known [70, 139].

We also mention that models, which take into account both collisions and the electric and magnetic fields generated by the particles have been investigated. Classical solutions near a Maxwellian for the Vlasov–Maxwell–Boltzmann system are constructed by Guo in [90]. A similar result for the Vlasov–Maxwell–Landau system near a Jüttner solution is shown by Guo and Strain in [180].

We refer to the book by Glassey [79] and the review article by Rein [141Jump To The Next Citation Point] for more information on the relativistic Vlasov–Maxwell system and the Vlasov–Poisson system.


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