The PDE system

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–

Maxwell system. Collisions among the plasma particles can be neglected if the plasma is sufficiently rarefied or hot. The particles only interact through electromagnetic fields created collectively. We consider the following setting: There are𝑁species of particles, all of which are located in a containerΩ⊂R³, which is a bounded domain, for example, a fusion reactor. Thus, boundary conditions on𝜕Ωhave to be imposed.

In the exterior ofΩ, there are external currents, for example, in electric coils, that may serve as a control of the plasma if adjusted suitably. In order to model objects that are placed somewhere in space, for example, the reactor wall, electric coils, and (almost perfect) superconductors, we consider the permittivity𝜀and permeability𝜇, which are functions of the space coordinate, take values in the set of symmetric, positive definite matrices of dimension three, and do not depend on time, as given. With this assumption we can model linear, possibly anisotropic materials that stay fixed in time.

We should mention that in reality𝜀and𝜇will on the one hand additionally depend on the particle density inside Ω and on the other hand additionally locally on the electromagnetic fields, typically via their frequencies (maybe even nonlocally because of hysteresis). However, this would cause further nonlinearities which we avoid in this work.

The unknowns are on the one hand the particle densities 𝑓^𝛼 = 𝑓^𝛼(𝑡, 𝑥, 𝑣), 𝛼 = 1, . . . , 𝑁, which are functions of time 𝑡 ≥ 0, the space coordinate 𝑥 ∈ Ω, and the momentum coordinate 𝑣 ∈ R³. Roughly speaking, 𝑓^𝛼(𝑡, 𝑥, 𝑣)indicates how many particles of the 𝛼-th species are at time𝑡 at position 𝑥 with momentum𝑣. On the other hand there are the electromagnetic fields𝐸=𝐸(𝑡, 𝑥),𝐻=𝐻(𝑡, 𝑥), which depend on time𝑡and space coordinate𝑥 ∈R³. The𝐷- and𝐵-fields are computed from𝐸and 𝐻by the linear constitutive equations𝐷=𝜀𝐸and𝐵=𝜇𝐻. We will only view𝐸and 𝐻as unknowns in the following.

The Vlasov part, which is to hold for each𝛼, reads as follows:

𝜕^𝑡𝑓^𝛼+

b^𝑣𝛼·𝜕^𝑥𝑓^𝛼+𝑞_𝛼(𝐸+

b^𝑣𝛼×𝐻) ·𝜕^𝑣𝑓^𝛼=0, (0.1a) 1

𝑓₋^𝛼=𝒦_𝛼𝑓₊^𝛼+𝑔^𝛼, (0.1b) 𝑓^𝛼(0)= 𝑓˚𝛼. (0.1c) Here, (0.1a) is the Vlasov equation equipped with the boundary condition (0.1b) on

𝜕Ωand the initial condition (0.1c) for𝑡=0. In (0.1c), 𝑓^𝛼(0)denotes the evaluation of 𝑓^𝛼at time𝑡=0, that is to say, the function 𝑓^𝛼(0,·,·). We will use this notation often, also similarly for the electromagnetic fields and other functions.

Note that throughout this work we use modified Gaussian units such that the speed of light (in vacuum) is normalized to unity and all rest masses𝑚_𝛼of a particle of the respective species are at least 1. In (0.1a),𝑞_𝛼 is the charge of the𝛼-th particle species andb^𝑣𝛼the velocity, which is computed from the momentum𝑣via

b^𝑣𝛼= 𝑣 q

𝑚²_𝛼+ |𝑣|² according to special relativity. Clearly, |

b^𝑣𝛼| < 1, that is, the velocities are bounded by the speed of light. Moreover, we assume that𝜀 = 𝜇 = Id onΩ, Id denoting the 3×3-identity matrix. Thus, the speed of light is constant inΩand𝐵=𝐻onΩ.

To derive a precise statement of the boundary condition (0.1b) and a definition of𝑓_±^𝛼, the operator𝒦_𝛼, and where (0.1b) has to hold, we have a look at typical examples at first. Most commonly, the operator𝒦_𝛼describes a specular boundary condition. For this, we assume thatΩhas a (at least piecewise)𝐶¹-boundary that is a submanifold ofR³, and denote the outer unit normal of𝜕Ωat some𝑥 ∈𝜕Ωby𝑛(𝑥). Now consider a particle moving insideΩand then hitting the surface𝜕Ωat some time𝑡at𝑥 ∈𝜕Ω. Its momentum𝑣(shortly) after the reflection satisfies𝑣·𝑛(𝑥)<0 and its momentum (shortly) before the hit is thus given by𝑣−2(𝑣·𝑛(𝑥))𝑛(𝑥). In other words, this means that the components of the momentum which are tangential to𝑛(𝑥)stay the same, and that the component which is normal to 𝑛(𝑥)changes the sign. On the level of a particle density 𝑓^𝛼, this consideration yields the condition

𝑓^𝛼(𝑡, 𝑥, 𝑣)= 𝑓^𝛼(𝑡, 𝑥, 𝑣−2(𝑣·𝑛(𝑥))𝑛(𝑥))C ^{𝐾 𝑓𝛼}_{(𝑡, 𝑥, 𝑣)}

(0.2) for𝑥 ∈𝜕Ωand𝑣·𝑛(𝑥)<0.

More generally, we can consider the case that only a portion of the particles that hit the boundary are reflected and the rest is absorbed and, additionally, more particles are added from outside. Thus, we may demand

𝑓^𝛼(𝑡, 𝑥, 𝑣)=𝑎^𝛼(𝑡, 𝑥, 𝑣) 𝐾 𝑓^𝛼(𝑡, 𝑥, 𝑣) +𝑔^𝛼(𝑡, 𝑥, 𝑣) (0.3) for𝑥 ∈ 𝜕Ωand 𝑣·𝑛(𝑥) < 0. Here, 0 ≤ 𝑎𝛼(𝑡, 𝑥, 𝑣) ≤ 1 is a coefficient; that is to say, 𝑎^𝛼(𝑡, 𝑥, 𝑣)-times the amount of the particles hitting the boundary at time𝑡at𝑥 ∈ 𝜕Ω with momentum𝑣are reflected and the rest is absorbed. Furthermore,𝑔^𝛼(𝑡, 𝑥, 𝑣) ≥0 is the source term describing how many particles are added from outside.

Since the boundary condition is to hold only if𝑣·𝑛(𝑥)<0, it is natural to decompose [0,∞[ ×𝜕Ω×R³into three parts:

𝛾⁺B (𝑡, 𝑥, 𝑣) ∈ [0,∞[ ×𝜕Ω×R^{3 |𝑣·𝑛(𝑥)}>0 ,

0.1 The PDE system 3 𝛾⁻B (𝑡, 𝑥, 𝑣) ∈ [0,∞[ ×𝜕Ω×R^{3 |𝑣·𝑛(𝑥)}<0 ,

𝛾⁰B (𝑡, 𝑥, 𝑣) ∈ [0,∞[ ×𝜕Ω×R^{3 |𝑣·𝑛(𝑥)=}0 .

Therefore, (0.3) is to hold for(𝑡, 𝑥, 𝑣) ∈ 𝛾⁻. Moreover,𝐾 can be seen as an operator mapping functions on𝛾⁺to functions on𝛾⁻. In accordance with (0.1b), we define 𝑓_±^𝛼 to be the restriction of 𝑓𝛼 to 𝛾^±. Of course, this only makes sense if we have some regularity of 𝑓^𝛼, for example, continuity on[0,∞[ ×Ω×R³. But even if a solution 𝑓^𝛼 (of a Vlasov equation) is only an𝐿^𝑝-function, it is possible to define a trace𝑓_±^𝛼of 𝑓^𝛼on 𝛾^±; see Definition 1.2.7.(ii). Note that𝒦_𝛼 =𝑎^𝛼𝐾in (0.1b) yields (0.3). Since the time variable in the sets above is somewhat unnecessary, we abbreviate

𝛾⁺𝑇 B (𝑡, 𝑥, 𝑣) ∈ [0, 𝑇[ ×𝜕Ω×R^{3 |𝑣·𝑛(𝑥)}>0 , 𝛾⁻𝑇 B (𝑡, 𝑥, 𝑣) ∈ [0, 𝑇[ ×𝜕Ω×R^{3 |𝑣·𝑛(𝑥)}<0 , 𝛾𝑇⁰ B (𝑡, 𝑥, 𝑣) ∈ [0, 𝑇[ ×𝜕Ω×R^{3 |𝑣·𝑛(𝑥)=}0 , 𝛾˜⁺ B _{(𝑥, 𝑣) ∈}

𝜕Ω×R^{3 |𝑣·𝑛(𝑥)}>0 , 𝛾˜⁻ B _{(𝑥, 𝑣) ∈}

𝜕Ω×R^{3 |𝑣·𝑛(𝑥)}<0 , 𝛾˜⁰ B _{(𝑥, 𝑣) ∈}

𝜕Ω×R^{3 |𝑣·𝑛(𝑥)=}0

for 0 < 𝑇 ≤ ∞. For ease of notation it will be convenient to introduce a surface measure on[0,∞[ ×𝜕Ω×R³, namely,

𝑑𝛾𝛼=|

b^𝑣𝛼·𝑛(𝑥)|𝑑𝑣𝑑𝑆_𝑥𝑑𝑡.

Furthermore, the Vlasov part is coupled with Maxwell’s equations, which describe the time evolution of the electromagnetic fields:

𝜀𝜕^𝑡𝐸−curl^𝑥𝐻=−4𝜋𝑗, (0.4a)

𝜇𝜕^𝑡𝐻+curl^𝑥𝐸=0, (0.4b)

(𝐸, 𝐻)(0)= 𝐸,˚ 𝐻˚

. (0.4c)

Here, the current𝑗=𝑗^int+𝑢is typically the sum of the internal currents 𝑗^intB

𝑁

Õ

𝛼=1

𝑞_𝛼

∫

R³

b^𝑣𝛼𝑓^𝛼𝑑𝑣

and some external current𝑢, that is supported in some open setΓ⊂R³. We will always extend 𝑗^int(𝑢) by zero outsideΩ(Γ). Concerning set-ups with boundary conditions on the plasma, the papers we are aware of deal with perfect conductor boundary conditions for the electromagnetic fields; see, for example, [Guo93]. Such a set-up can model no interaction between the interior and the exterior. However, considering fusion reactors, there are external currents in the exterior, for example, in field coils.

These external currents induce electromagnetic fields and thus influence the behavior

of the internal plasma. Even more important, the main aim of fusion plasma research is to adjust these external currents “suitably”. Thus, we impose Maxwell’s equations globally in space.

Actually, Maxwell’s equations additionally include conditions on the divergence of 𝐷=𝜀𝐸and𝐵=𝜇𝐻, namely,

div^𝑥(𝜀𝐸)=4𝜋𝜌, (0.5a)

div^𝑥 𝜇𝐻

=0, (0.5b)

where 𝜌 denotes the charge density. Usually, these equations are known to be re-dundant if all functions are smooth enough, local conservation of charge is satisfied, i.e.,

𝜕^𝑡𝜌+div^𝑥𝑗 =0,

and (0.5) holds initially, which we then view as a constraint on the initial data. There-fore, in Chapters 1 and 2 we largely ignore (0.5) and discuss in Section 1.5 in what sense (0.5) is satisfied in the context of a weak solution concept.

We thus arrive at the following Vlasov–Maxwell system, which is (0.1) and (0.4) combined, on a time interval with given final time 0<𝑇_•≤ ∞:

𝜕^𝑡𝑓^𝛼+

b^𝑣𝛼·𝜕^𝑥𝑓^𝛼+𝑞_𝛼(𝐸+

b^𝑣𝛼×𝐻) ·𝜕^𝑣𝑓^𝛼 =0 on𝐼_𝑇

• ×Ω×R^3, (VM.1) 𝑓₋^𝛼 =𝒦_𝛼𝑓₊^𝛼+𝑔^𝛼 on𝛾𝑇⁻_•, (VM.2) 𝑓^𝛼(0)= 𝑓˚𝛼 onΩ×R^3, (VM.3) 𝜀𝜕^𝑡𝐸−curl^𝑥𝐻=−4𝜋𝑗 on𝐼_𝑇

•×R^3, (VM.4) 𝜇𝜕^𝑡𝐻+curl^𝑥𝐸=0 on𝐼_𝑇

•×R^3, (VM.5) (𝐸, 𝐻)(0)=

𝐸,˚ 𝐻˚

onR^3, (VM.6) where (VM.1) to (VM.3) have to hold for all 𝛼 =1, . . . , 𝑁 and 𝐼_𝑇

• denotes the given time interval. Here and in the following,𝐼_𝑇B^[0, 𝑇]for 0≤𝑇<∞and𝐼_∞ B^[0,∞[.