As the last term on the r.h.s. is further estimated by 2
R2k∂vϕk∞(kf◦k1+ 1)2,
we can chooseRlarge so that this second term becomes less than/6. Using the operator TeBRR
0
introduced in Lemma 2.4.10, the first term in our estimate for|Jn2|may be rewritten
as
In this expression we estimate using H¨older’s inequality to obtain the upper bound C(ϕ, p)kfn◦k∞
Z T
0
kTBR
eR0 (ρn(t)−ρ0(t)kpdt. (2.24) Choosing pproperly we can use the compactness of the operator TeBRR
0
to see that
n→∞lim kTBR
eR0 (ρn(t)−ρ0(t))kp = 0 ∀t∈[0, T]
so that we can pass to the limit in (2.24) using the dominated convergence theorem. Hence we have shown that |Jn2|< /3 forn large enough.
Finally, term|Jn3|is estimated along the same lines as |Jn2|. Similar techniques apply
also for In00 and the proof is complete. 2
2.5 Appendix
It is the first goal of this appendix to prove the relation (2.21). To do so we start with two Lemmas. We define ρ◦n(x) :=R Using this claim we can construct a sequence (Rk)k∈Nsuch that
Z
We can now construct a monotone smooth functionG: [0,∞[→]0,∞[ which satisfies
The last condition can be fulfilled because G(Rk+1)−G(Rk) Proof of Lemma 2.5.2. Let G be the function from Lemma 2.5.1. We compute using (2.13)
Next we prove Eq. (2.21). From Z
So by energy conservation and Lemma 2.5.2 we get Z
and the claim follows. 2
Next we prove (2.22). DefineT(R) :=
q
G(R/√
2) whereGis the function from Lemma 2.5.1. We estimate as follows
Z
where (2.25) was used. Eq. (2.22) now follows easily. 2
3 Global classical solutions of the Vlasov-Darwin system for small initial data
A global-in-time existence theorem for classical solutions of the Vlasov-Darwin system is given under the assumption of smallness of the initial data. Furthermore, it is shown that in case of spherical symmetry the system degenerates to the relativistic Vlasov-Poisson system. The results of this section have been published by the author in the article [43].
3.1 Introduction
Kinetic models play an increasingly important role in todays plasma physics. On the one hand much effort is used to deepen our analytical understanding of some problems where no other description seems to be adequate. On the other hand progress has also been achieved especially with numerical simulations (see, e.g., [42]).
In the kinetic picture, the particle distribution of a one-species plasma is described by a time dependent density function f(t, x, p) on phase space. If collisions of the particles are neglected and a relativistic model is used, then f is subject to the transport equation
∂tf+v(p)· ∇xf+K(t, x)· ∇pf = 0 (3.1) with force termK =E+v×B. HereE andB denote the electric and the magnetic field respectively and the relativistic velocity is given by
v(p) = p
p1 +|p|2. (3.2)
Note that all physical constants such as the speed of light or the rest mass of the particles have been set equal to unity.
Eq. (3.1) is usually called the Vlasov equation. Expressions for the charge and current densities ρ and j in terms of the phase space densityf are given by
ρ(t, x) = Z
f(t, x, p)dp, j(t, x) = Z
f(t, x, p)v(p)dp. (3.3) To obtain a self consistent closed system one has to take into account how the ensemble modeled by the density f creates the fields E and B. Usually this is done with the full system of Maxwell’s equations, but numerical difficulties of simulations of that system have stimulated a search for alternatives (compare [7]). The present chapter deals with what is known as the Darwin approximation. Here the electric field is split into a transverse and a longitudinal component as follows:
E =EL+ET, ∇ ×EL= 0, ∇ ·ET = 0. (3.4)
In the evolution part of the Maxwell equations the transverse part of the electric field is neglected, resulting in
∂tEL− ∇ ×B =−j, ∇ ·EL=ρ (3.5)
∂tB+∇ ×ET = 0, ∇ ·B = 0. (3.6)
The system consisting of Eqns. (3.1) – (3.6) is called theVlasov-Darwin system. The main feature of this system is that the field equations are elliptic which in particular facilitates a numerical treatment since a time integration step, which is needed to solve the Maxwell system can be avoided here ([42]). The justification of the model seems possible in case the particle velocities are not too fast when compared to the speed of light.
Up to now there are only few mathematical results known for this system. In 2003 Benachour et al. [6] proved an existence theorem for small initial data: this assumption implies global-in-time existence of weak solutions of the Cauchy problem. Later Pallard [36] removed the smallness assumption and added a result about solvability of the Cauchy problem in a classical sense: To a given initial datumf0 ∈Cc2(R6) there existsT >0 and a classical solutionf: [0, T[×R6 →Rof the Vlasov-Darwin system satisfyingf(0) =f0.
In the main part of this chapter we present a result which is well known for the Vlasov-Poisson system (VP), the Relativistic Vlasov-Maxwell system (RVM), and other related systems such as the relativistic Vlasov-Poisson system or the spherically symmet-ric Vlasov-Einstein system (cf. [2, 3, 23, 41]) but seems to be new for the Vlasov-Darwin system: we consider classical solutions of the Cauchy problem and show that these exist for all times if the initial data are chosen sufficiently small. The precise statement of our result is contained in the next section, where we also formulate three propositions which are used to prove the theorem. Sections 3.3–3.6 are devoted to proofs. In the final Section 3.7 we take a look at spherically symmetric solutions. First it is shown that any symmetry of the initial datum with respect to an orthogonal transformation is preserved for all times. This allows the conclusion that in case of spherical symmetry the VD system reduces to the well known relativistic Vlasov-Poisson system. So in this case the solutions are global-in-time as well [21].