is the corresponding invariant. Similarly, the invariant corresponding to rotational symmetry is
ℱ𝛼 B𝜕𝜑¤ℒ𝛼 =𝑟
𝑣𝜑+𝑞𝛼𝐴tot𝜑 .
Note that in the formulae for ℱ𝛼 (the “canonical angular momentum”) and 𝒢𝛼, components of the so-called “canonical momentum”
𝑝𝛼 =𝑣+𝑞𝛼𝐴tot appear. In the variables 𝑥, 𝑝𝛼
, the particle energy ℰ𝛼 B 𝑣0𝛼+𝑞𝛼𝜙=
q 𝑚𝛼2+
𝑝𝛼−𝑞𝛼𝐴tot
2+𝑞𝛼𝜙
is the (in general time-dependent) Hamiltonian governing the motion of the particles of the𝛼-th species. Assuming that the electromagnetic potentials are independent of time,ℰ𝛼is also independent of time and thus another invariant, the one corresponding to time symmetry.
3.3 Steady states—definition and ansatz
The preceding considerations about symmetry motivate the definition of what we call a (confined) steady state in our set-up. Before that we collect our symmetry assumptions.
Definition and Remark 3.3.1. (a) A function𝑓:Ω×R3→R/ a function𝜙:Ω→R / a vector field𝐴:Ω→R3is called
3.3 Steady states—definition and ansatz 115 (i) independent of𝑥3if𝜕𝑥3𝑓 =0 /𝜕𝑥3𝜙=0 /𝜕𝑥3𝐴=0;
(ii) axially symmetric if 𝑓(𝑅𝑥, 𝑅𝑣)= 𝑓(𝑥, 𝑣)for any𝑥∈ Ω,𝑣 ∈R3, and rotation 𝑅 ∈ R3×3 about the 𝑥3-axis / 𝜙(𝑅𝑥) = 𝜙(𝑥) for any 𝑥 ∈ Ω and rotation 𝑅 ∈ R3×3 about the 𝑥3-axis /𝐴(𝑅𝑥) = 𝑅𝐴(𝑥) for any𝑥 ∈ Ω and rotation 𝑅∈R3×3about the𝑥3-axis.
(b) With these two symmetries, the functions 𝜙,𝐴𝑟,𝐴𝜑, and𝐴3 only depend on𝑟. Accordingly, we will often view them as functions on[0, 𝑅0].
(c) An axially symmetric vector field𝐴automatically satisfies𝐴1(𝑥)= 𝐴2(𝑥)=0 if 𝑥1 =𝑥2 =0, i.e., if𝑥lies on the𝑥3-axis.
Remark 3.3.2. From a geometric point of view, the main idea of the setting and the symmetry assumptions is the following: The confinement device Ωis a coordinate surface with respect to a suitable orthogonal curvilinear coordinate system (here,𝑟= const.in cylindrical coordinates) and in these coordinates the potentials only depend on one variable, namely, on the coordinate which is constant on𝜕Ω. The symmetry assumption about the magnetic potential thus implies that the magnetic field lies in the tangent space of the submanifold 𝜕Ω, and it carries over to the electromagnetic fields, which in particular means that the magnetic field is invariant under parallel transport around closed loops on𝜕Ω. Thus, with this approach confinement devices whose boundaries have nontrivial curvature (such as a ball) are a priori excluded in order to allow nontrivial magnetic fields. Conversely, an infinitely long cylinder or (the interior of) a torus are consistent with this approach since their boundaries are coordinate surfaces of a suitable orthogonal curvilinear coordinate system and are flat.
We proceed with an assumption about the external potential, which is supposed to hold henceforth.
Condition 3.3.3. The external potential𝐴ext:Ω→Ris independent of𝑥3and axially symmetric such that 𝐴ext𝑟 =0 and𝐴ext
𝜑 , 𝐴ext
3 ∈ 𝐶1([0, 𝑅0])(viewed as functions of𝑟) with𝐴ext
𝜑 (0)=𝐴ext
3 (0)= 𝐴ext
3
0( 0)=0.
Note that 𝐴ext
3 (0) = 0 can be assumed—for simplicity—without loss of generality since adding a constant to𝐴ext
3 does not affect𝐵extbecause of curl𝑥𝑒3=0 (as opposed to this, this invariance under adding constants does not hold for𝐴ext
𝜑 , as curl𝑥𝑒𝜑 ≠0).
We first prove some technicalities.
Lemma 3.3.4. Let𝜙, 𝐴𝜑, 𝐴3∈ 𝐶1([0, 𝑅0])with 𝜙0(0)=𝐴𝜑(0)=𝐴0
3(0)=0 (3.3.1)
and assume𝐴𝑟 =0. Then:
(i) The potentials𝜙=𝜙(𝑥)and𝐴=𝐴(𝑥)are continuously differentiable onΩ. Thus, the electromagnetic fields
𝐸=−𝜕𝑥𝜙=−𝜙0𝑒𝑟, 𝐵=curl𝑥𝐴=−𝐴0
3𝑒𝜑+1
𝑟 𝑟𝐴𝜑0𝑒
3 (3.3.2) are continuous onΩ. Moreover,div𝑥𝐴=0onΩ.
(ii) If𝜙, 𝐴3∈ 𝐶2([0, 𝑅0]), they are even twice continuously differentiable onΩwith respect to𝑥. Accordingly,𝐸is of class𝐶1onΩ. If moreover𝐴𝜑 ∈ 𝐶2(]0, 𝑅0])such that
𝐴0
𝜑(𝑟) − 𝐴𝜑(𝑟)
𝑟 =𝒪(𝑟), 𝐴00
𝜑(𝑟)=𝒪(1) for𝑟→0, (3.3.3) then𝐴∈𝑊2,∞ Ω;R3∩𝐶2
Ω\R𝑒3;R3
. Accordingly,𝐵is of class𝑊1,∞onΩand of class𝐶1onΩ\R𝑒3.
Proof. We easily see that the maps𝑥↦→𝜙(𝑥)and𝑥 ↦→𝐴3(𝑥)𝑒3are (twice) continuously differentiable on Ω if the maps 𝑟 ↦→ 𝜙(𝑟)and 𝑟 ↦→ 𝐴3(𝑟) are (twice) continuously differentiable on [0, 𝑅0] since 𝜙0(0) = 𝐴0
3(0) = 0. There remains to take care of 𝑥 ↦→ 𝐴𝜑(𝑥)𝑒𝜑(𝑥), in particular at 𝑟 = 0. Indeed, this map can be continuously extended to wholeΩbecause of𝐴𝜑(0)=0 and is differentiable for𝑟 >0 with
𝜕𝑥 𝐴𝜑𝑒𝜑 𝑟, 𝜑
=
©
«
−sin𝜑cos𝜑
𝐴0𝜑(𝑟) − 𝐴𝜑𝑟(𝑟)
−sin2𝜑
𝐴0𝜑(𝑟) − 𝐴𝜑𝑟(𝑟)
− 𝐴𝜑𝑟(𝑟) 0 cos2𝜑
𝐴0
𝜑(𝑟) −𝐴𝜑𝑟(𝑟)
+ 𝐴𝜑𝑟(𝑟) sin𝜑cos𝜑 𝐴0
𝜑(𝑟) − 𝐴𝜑𝑟(𝑟) 0
0 0 0
ª
®
®
® (3.3.4)¬ where all entries have a limit as𝑟→0. Hence, also𝐴𝜑𝑒𝜑is continuously differentiable onΩ. Furthermore,𝐴is divergence free with respect to𝑥, as was already observed in Section 3.2 because of (3.2.1). Thus, part 3.3.4.(i) is proved. If moreover the assumptions about𝐴𝜑in part 3.3.4.(ii) are satisfied, all second order derivatives (with respect to𝑥) of𝐴𝜑𝑒𝜑are bounded for𝑟→0, since we see by differentiating the entries of (3.3.4) once more that these second order derivatives are expressions in sin𝜑, cos𝜑,
1𝑟
𝐴0
𝜑(𝑟) − 𝐴𝜑𝑟(𝑟)
, and𝐴00
𝜑(𝑟), and thus bounded by assumption. Therefore, all second order derivatives exist onΩin the weak sense, coincide with the classical derivatives almost everywhere, and are bounded. This proves the remaining part of 3.3.4.(ii).
Note that this lemma yields that under Condition 3.3.3 the external potential𝐴extis continuously differentiable onΩand divergence free, and that the external magnetic field𝐵ext=curl𝑥𝐴extis continuous onΩ.
Remark 3.3.5. In Lemma 3.3.4.(ii), we cannot expect that𝐴∈ 𝐶2 Ω;R3
in general if 𝐴𝜑 ∈𝐶2([0, 𝑅0])and (3.3.3) holds, as the example𝐴𝜑(𝑟)=𝑟2shows since
Δ𝑥 𝐴𝜑𝑒𝜑
1=−Δ𝑥 𝑟2sin𝜑
=−3 sin𝜑
3.3 Steady states—definition and ansatz 117 has no limit for𝑟→0.
We proceed with a basic definition.
Definition 3.3.6. Let Condition 3.3.3 hold.
(a) A tuple 𝑓𝛼
𝛼,𝜙, 𝐴
is called an axially symmetric steady state of the two and one-half dimensional relativistic Vlasov–Maxwell system on Ω with external potential𝐴ext (hereafter abbreviated as steady state) if the following conditions are satisfied:
(i) For each𝛼=1, . . . , 𝑁, the functions 𝑓𝛼:Ω×R3 → [0,∞[are continuously differentiable satisfying 𝑓𝛼(𝑥,·) ∈𝐿1 R3
for each𝑥 ∈Ω. (ii) The potentials satisfy
𝜙∈𝐶2 Ω
, 𝐴∈𝐶1 Ω;R3
∩𝐶2
Ω\R𝑒3;R3
∩𝑊2,∞ Ω;R3. (This condition is motivated in view of Lemma 3.3.4.)
(iii) Any 𝑓𝛼 and𝜙,𝐴are independent of𝑥3and axially symmetric.
(iv) The equations
b𝑣𝛼·𝜕𝑥𝑓𝛼+𝑞𝛼 𝐸+
b𝑣𝛼×𝐵tot·
𝜕𝑣𝑓𝛼=0 onΩ×R3, (3.3.5a) 𝑓𝛼(𝑥, 𝑣−2𝑣𝑟𝑒𝑟)= 𝑓𝛼(𝑥, 𝑣), 𝑥 ∈𝜕Ω, 𝑣∈R3, 𝑣𝑟 <0,
(3.3.5b)
−Δ𝑥𝜙=4𝜋𝜌, −Δ𝑥𝐴=4𝜋𝑗, div𝑥𝐴=0 onΩ, (3.3.5c) are satisfied. Here,𝑒𝑟 =𝑒𝑟(𝑥),𝑣𝑟 =𝑣·𝑒𝑟, and
𝐸=−𝜕𝑥𝜙, 𝐵tot=curl𝑥 𝐴+𝐴ext, 𝜌=
𝑁
Õ
𝛼=1
𝑞𝛼
∫
R3
𝑓𝛼𝑑𝑣, 𝑗=
𝑁
Õ
𝛼=1
𝑞𝛼
∫
R3
b𝑣𝛼𝑓𝛼𝑑𝑣.
(b) A steady state 𝑓𝛼
𝛼,𝜙, 𝐴
is said to (i) have finite charge if
∫
𝐵𝑅 0
∫
R3
𝑓𝛼𝑑𝑣𝑑(𝑥1, 𝑥2)<∞
for each𝛼=1, . . . , 𝑁;
(ii) be compactly supported with respect to𝑣if there is𝑆>0 such that𝑓𝛼(𝑥, 𝑣)= 0 for each𝛼=1, . . . , 𝑁,𝑥 ∈Ω,|𝑣| ≥𝑆;
(iii) be nontrivial if 𝑓𝛼 .0 for each𝛼=1, . . . , 𝑁;
(iv) be confined with radius at most𝑅if 0< 𝑅 < 𝑅0 such that 𝑓𝛼(𝑥, 𝑣) =0 for each𝛼=1, . . . , 𝑁,𝑥∈Ωwith|(𝑥1, 𝑥2)| ≥𝑅, and𝑣 ∈R3.
Note that perfect conductor boundary conditions are automatically satisfied due to symmetry, as was already observed in Section 3.2, and are thus omitted in (3.3.5).
Remark 3.3.7. A physically reasonable steady state should have finite charge, which usually means 𝑓𝛼 ∈ 𝐿1 Ω×R3
for each𝛼 = 1, . . . , 𝑁. However, this is impossible in our setting (unless all 𝑓𝛼 vanish identically) by 𝑓𝛼 being independent of𝑥3. Thus, here we have to modify this definition suitably as above.
According to [Deg90], the natural ansatz for 𝑓𝛼is that
𝑓𝛼=𝜂𝛼(ℰ𝛼,ℱ𝛼,𝒢𝛼) (3.3.6) is a function of the three invariants obtained in Section 3.2. We collect some basic assumptions about the ansatz functions𝜂𝛼.
Condition 3.3.8. For each𝛼=1, . . . , 𝑁it holds that:
(i) 𝜂𝛼 ∈𝐶1 R3;[0,∞[
. (ii) There exists𝜂𝛼∗ ∈𝐿1 R2
such that
∫
R2
ℰ𝜂𝛼∗(ℰ,𝒢)
𝑑(ℰ,𝒢)<∞ and
𝜂𝛼(ℰ,ℱ,𝒢)
≤𝜂𝛼∗(ℰ,𝒢) for all(ℰ,ℱ,𝒢) ∈R3.
(iii) There exists𝜂𝛼#:R2→Rsuch that
∀𝑑 ∈R:𝜂𝛼#,|ℰ |𝜂𝛼# ∈𝐿1(]𝑑,∞[ ×R) and
∇𝜂𝛼(ℰ,ℱ,𝒢)
≤𝜂#𝛼(ℰ,𝒢) for all(ℰ,ℱ,𝒢) ∈R3.
We first prove that the ansatz (3.3.6) already ensures (3.3.5a) and (3.3.5b). Here and in the following, we will always write𝐴tot=𝐴+𝐴ext.
Lemma 3.3.9. Let Conditions 3.3.3 and 3.3.8.(i) hold and let𝜙, 𝐴𝜑, 𝐴3∈𝐶1([0, 𝑅0])with 𝜙0(0)=𝐴𝜑(0)=𝐴0
3(0)=0.
3.3 Steady states—definition and ansatz 119 Then, for each𝛼=1, . . . , 𝑁,
𝑓𝛼:Ω×R3 →R, 𝑓𝛼(𝑥, 𝑣)=𝜂𝛼(ℰ𝛼(𝑥, 𝑣),ℱ𝛼(𝑥, 𝑣),𝒢𝛼(𝑥, 𝑣))
=𝜂𝛼 𝑣0
𝛼+𝑞𝛼𝜙(𝑟), 𝑟
𝑣𝜑+𝑞𝛼𝐴tot
𝜑 (𝑟)
, 𝑣3+𝑞𝛼𝐴tot
3 (𝑟) (3.3.7) is continuously differentiable, independent of 𝑥3, axially symmetric, and satisfies (3.3.5a) and(3.3.5b).
Proof. We first note that 𝑓𝛼is continuously differentiable because of𝑟𝑣𝜑=𝑥1𝑣2−𝑥2𝑣1 and 𝜙0(0)=
𝑟𝐴tot𝜑 0
(0)= 𝐴tot
3
0(
0) =0. Clearly, 𝑓𝛼 is independent of𝑥3and axially symmetric. Furthermore, it is easy to see that (3.3.5b) holds sinceℰ𝛼is even in𝑣𝑟and ℱ𝛼, 𝒢𝛼 do not depend on𝑣𝑟. To ensure (3.3.5a) for 𝑓𝛼 it suffices to prove that ℰ𝛼, ℱ𝛼, and 𝒢𝛼 themselves satisfy (3.3.5a)—this clearly holds, as they are invariants of the motion; for the sake of completeness, we carry out the computation. Since they are of class𝐶1 onΩ×R3, this only needs to be verified for𝑟 >0. In the following, have (3.3.2) in mind. Firstly,
b𝑣𝛼·𝜕𝑥ℰ𝛼+𝑞𝛼 𝐸+
b𝑣𝛼×𝐵tot·
𝜕𝑣ℰ𝛼 =−𝑞𝛼
b𝑣𝛼·𝐸+𝑞𝛼 𝐸+
b𝑣𝛼×𝐵tot· b𝑣𝛼 =0. Secondly,
b𝑣𝛼·𝜕𝑥ℱ𝛼+𝑞𝛼 𝐸+
b𝑣𝛼×𝐵tot·
𝜕𝑣ℱ𝛼
=b𝑣𝛼·
𝑣𝜑+𝑞𝛼𝐴tot
𝜑
𝑒𝑟−
b𝑣𝛼·𝑣𝑟𝑒𝜑+𝑞𝛼 b𝑣𝛼·𝑟
𝐴tot
𝜑
0
𝑒𝑟+𝑞𝛼 𝐸+
b𝑣𝛼×𝐵tot·𝑟𝑒
𝜑
=𝑞𝛼 b𝑣𝛼·𝑒𝑟
𝐴tot𝜑 +𝑟 𝐴tot𝜑 0
−𝑟·1 𝑟
𝑟𝐴tot𝜑 0
=0. Thirdly,
b𝑣𝛼·𝜕𝑥𝒢𝛼+𝑞𝛼 𝐸+
b𝑣𝛼×𝐵tot·𝜕
𝑣𝒢𝛼=𝑞𝛼
b𝑣𝛼· 𝐴tot
3
0𝑒
𝑟+𝑞𝛼 𝐸+
b𝑣𝛼×𝐵tot·𝑒
3=0. Thus, (3.3.5a) holds for 𝑓𝛼by the chain rule.
The ansatz (3.3.6) in turn can be inserted into the definition of𝜌 and 𝑗 to derive representations of these densities in terms of the potentials.
Lemma 3.3.10. Let 𝜙:[0, 𝑅0] → R,𝐴:[0, 𝑅0] → R3, Condition 3.3.8.(ii) hold, and 𝑓𝛼 be defined as in (3.3.7)for each 𝛼 = 1, . . . , 𝑁. Then, 𝑓𝛼(𝑥,·) ∈ 𝐿1 R3
for each 𝑥 ∈ Ω. Furthermore,𝜌and 𝑗are independent of𝑥3and axially symmetric, and we have
4𝜋𝜌(𝑟)=𝑔1
𝑟,𝜙(𝑟), 𝐴tot𝜑 (𝑟), 𝐴tot
3 (𝑟)
, (3.3.8a)
𝑗𝑟(𝑟)=0, (3.3.8b)
4𝜋𝑗𝜑(𝑟)=𝑔2
𝑟,𝜙(𝑟), 𝐴tot𝜑 (𝑟), 𝐴tot
3 (𝑟)
, (3.3.8c)
4𝜋𝑗3(𝑟)=𝑔3
are continuous functions. Moreover, Proof. At least formally we have
∫
3.3 Steady states—definition and ansatz 121 where we introduced polar coordinates in the (𝑣1, 𝑣2)-plane with basis 𝑒𝑟, 𝑒𝜑
and then substituted firstlyℰ=
q𝑚2𝛼+𝑢2+𝑣2
3+𝑞𝛼𝜙(𝑟)and secondly𝒢=𝑣3+𝑞𝛼𝐴tot
3 (𝑟). Note that the integral in the second line vanishes after substituting𝑦=sin𝜃. Due to Condition 3.3.8.(ii), the modulus of the integrand in the first line can be estimated by
|ℰ | + and is hence integrable. Because of|
b𝑣𝛼| < 1 also the other integrals exist. Thus, the above calculation is justified. Multiplying these identities with𝑞𝛼and summing over 𝛼yields the representation. The above estimate on the integrands also implies that𝑔𝑖 is continuous,𝑖=1,2,3. Finally, (3.3.10) is also a consequence of|
b𝑣𝛼|<1.
Remark 3.3.11. The proof of the preceding lemma additionally shows that any steady state obtained in the following sections has finite charge. Indeed, for this it is sufficient that𝑟𝜙is integrable over[0, 𝑅0], which is of course the case when𝜙is continuous.
According to Lemma 3.3.10, after integrating (3.2.2) and using the representation (3.3.8), the problem of finding a steady state with the ansatz (3.3.6) reduces to finding 𝜙,𝐴3 ∈𝐶2([0, 𝑅0]),𝐴𝜑 ∈𝐶2(]0, 𝑅0]) ∩𝐶1([0, 𝑅0])satisfying (3.3.1), (3.3.3), and for𝑟 > 0 in view of Lemmas 3.3.4 and 3.3.9; note that we could prescribe arbitrary values for𝜙and𝐴3at𝑟=0, and we choose both of these values to be zero. Therefore, it is convenient to introduce the map
ℳ:𝐶 [0, 𝑅0];R3→𝐶 [
The following lemma shows that indeedℳis well-defined (with the obvious inter-pretationℳ 𝜙, 𝐴𝜑, 𝐴𝑟(
0)=(0,0,0)) and that it suffices to search for fixed points of ℳ.
Lemma 3.3.12. Assume Conditions 3.3.3, 3.3.8.(i), and 3.3.8.(ii).
(i) For any 𝜙, 𝐴𝜑, 𝐴3∈𝐶 [
Furthermore,
𝜙˜,𝐴˜𝜑,𝐴˜3
satisfies(3.3.1)and(3.3.3).
(ii) If 𝜙, 𝐴𝜑, 𝐴3 ∈ 𝐶 [
0, 𝑅0];R3
is a fixed point ofℳ, then 𝑓𝛼
𝛼,𝜙, 𝐴
is a steady state, where the 𝑓𝛼are defined via the ansatz(3.3.6).
Proof. Due to Lemma 3.3.10, the functions 𝑔˜𝑖:[0, 𝑅0] →R, 𝑔˜𝑖(𝜎)=𝑔𝑖 for𝑟→0. Furthermore, the ‘tilde’-potentials are twice continuously differentiable on ]0, 𝑅0]with
they are continuously differentiable on[0, 𝑅0]with vanishing derivative at𝑟=0, and moreover𝐴˜0𝜑(𝑟)=𝒪(𝑟)for𝑟→0. Furthermore, by l’Hôpital’s rule we have
3.4 Existence of steady states 123