The Vlasov–Maxwell and Vlasov

1.2 The Vlasov–Maxwell and Vlasov–Poisson systems

Let us consider a collisionless plasma, which is a collection of particles for which collisions are relatively rare and the interaction is through their charges. We assume below that the plasma consists only of one type of particle, whereas the results below also hold for plasmas with several particle species. The particle rest mass is normalized to one. In the kinetic framework, the most general set of equations for modelling a collisionless 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,

where $c$ is the speed of light. The charge density $ρ$ and current $j$ are given by

Equation (12

) is the relativistic Vlasov equation and Equations (13

, 14

) 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

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 equation. Another special case of interest is the classical limit, obtained by letting $c → ∞$ in Equations (12

, 13

, 14

), yielding:

where $β = 1$ . See Schaeffer [105 ] for a rigorous derivation of this result. This is the (nonrelativistic) Vlasov–Poisson equation, and $β = 1$ corresponds to repulsive forces (the plasma case). Taking $β = − 1$ means attractive forces and the Vlasov–Poisson equation 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 collisionless 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 (see [42 , 47 , 12 , 13 ]). 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

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 (see [46

, 54 , 14

, 42

]). For the full three-dimensional relativistic Vlasov–Maxwell system, this important problem of establishing whether or not solutions will remain smooth for all time is open. There has been an increased activity during the last years aiming at a solution of this problem. Two new methods of proofs of the same theorem as in [46

] are given in [60 , 17 ]. A different sufficient criterion for global existence is given by Pallard in [77 ] and he also shows a new bound for the electromagnetic field in terms of $Q (t)$ in [78 ]. In two space and three momentum dimensions, Glassey and Schaeffer [43 ] have shown that $Q (t)$ can be controlled, which thus yields global existence of smooth solutions in that case (see also [44

]).

The relativistic and nonrelativistic 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 nonrelativistic case Batt [14 ] gave an affirmative solution 1977 in the case of spherically symmetric data. Pfaffelmoser [80 ] (see also Schaeffer [107 ]) was the first one to give a proof for general smooth data. He obtained the bound

Q (t) ≤ C (1 + t)(51+δ)∕11,

where $δ > 0$ could be taken 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 [55 ] and reads

Q(t) ≤ C (1 + t) log (2 + t).

In the case of repulsive forces ( $β = 1$ ) Rein [83 ] has found the sharpest estimate by using a new identity for the Vlasov–Poisson equation, discovered independently by Illner and Rein [57 ] and by Perthame [79 ]. Rein’s estimate reads

Q (t) ≤ C(1 + t)2∕3.

Independently and about the same time as Pfaffelmoser gave his proof, Lions and Perthame [66 ] used a different method for proving global existence. To some extent their method seems to be more generally applicable to attack problems similar to the Vlasov–Poisson equation but which are still quite different (see [3 , 61 ]). 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 [42 ] 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 [45 ]. If $β = − 1$ it was also shown in [42 ] that blow-up occurs in finite time for spherically symmetric data with negative total energy. This system, however, is unphysical in the sense that it is not a special case of the Einstein–Vlasov system. Quite surprisingly, for general smooth initial data none of the techniques discussed above for the nonrelativistic Vlasov–Poisson equation apply in the relativistic case. This fact is annoying since it has been suggested that an understanding of this equation may be necessary for understanding the three-dimensional relativistic Vlasov–Maxwell equation. However, the relativistic Vlasov–Poisson equation lacks the Lorentz invariance; it is a hybrid of a classical Galilei invariant field equation and a relativistic transport equation (17 ). Only for spherical symmetric data is the equation a fundamental physical equation. The classical Vlasov–Poisson equation is on the other hand Galilean invariant. In [1 ] a different equation for the field is introduced that is observer independent among Lorentz observers. By coupling this equation for the field to the relativistic Vlasov equation, the function $Q (t)$ may be controlled as shown in [1 ]. This is an indication that the transformation properties are important in studying existence of smooth solutions (the situation is less subtle for weak solutions, where energy estimates and averaging are the main tools (see [56 , 35 , 86 ]). Hence, it is unclear whether or not the relativistic Vlasov–Poisson equation will play a central role in the understanding of the Lorentz invariant relativistic Vlasov–Maxwell equation.

We refer to the book by Glassey [41 ] for more information on the relativistic Vlasov–Maxwell system and the Vlasov–Poisson equation.