This work splits into three parts. In Chapter 1 we prove existence of weak solutions of (VM) for given (and suitable) 𝑓˚,𝐸˚,𝐻˚,𝒦𝛼,𝑔𝛼, and𝑢. To this end, we firstly define in Section 1.1 what we call weak solutions of (VM). The strategy to construct a weak solution follows the strategy of Guo [Guo93], who considered (VM) with𝜀=𝜇=Id, 𝑢=0, and (VM.4) and (VM.5) only imposed onΩand equipped with perfect conductor boundary conditions on𝜕Ω. Firstly, we consider the Vlasov part in Section 1.2 and state some important results of Beals and Protopopescu [BP87], who dealt with transport equations with Lipschitz continuous vector field subject to boundary conditions; here, we also refer to the book of Greenberg, Mee, and Protopopescu [GMP87]. Going to the
0.2 Outline 5 level of characteristics and exploiting that the characteristic flow is measure preserving (which follows from the fact that the Lorentz force of electrodynamics has no sources and sinks with respect to momentum), 𝐿𝑝-bounds on 𝑓𝛼 and 𝑓𝛼
+ are derived. After shortly discussing the Maxwell part in Section 1.3, we proceed with the construction of a weak solution in Section 1.4. Additionally to𝐿𝑝-bounds on 𝑓𝛼 and 𝑓𝛼
+, we make use of an energy consideration. For classical solutions of (VM) one can easily derive the energy balance
𝑑 𝑑𝑡
𝑁
Õ
𝛼=1
∫
Ω
∫
R3
q
𝑚2𝛼+ |𝑣|2𝑓𝛼𝑑𝑣𝑑𝑥+ 1 8𝜋
∫
R3
𝜀𝐸·𝐸+𝜇𝐻·𝐻𝑑𝑥
!
≤𝐶−
∫
R3
𝐸·𝑢 𝑑𝑥,
if𝒦𝛼 takes the form𝒦𝛼 = 𝑎𝛼𝐾with 0≤ 𝑎𝛼 ≤ 1, and where𝐶is some expression in the𝑔𝛼; if𝑎𝛼 =1 for all𝛼, equality holds above. In order to apply a quadratic Gronwall argument and to conclude that the left bracket is bounded for each time, the map
(𝐸, 𝐻) ↦→
∫
R3
𝜀𝐸·𝐸+𝜇𝐻·𝐻𝑑𝑥 12
should be a norm on𝐿2 R3;R6
which is equivalent to the standard𝐿2-norm. Thus, assumptions about uniform positive definiteness of 𝜀and𝜇will be made. Then, it is natural to search for a weak solution in those spaces for whose norms the above a priori bounds have been established. It turns out that firstly a cut-off system has to be investigated in Section 1.4.2. Afterwards, the cut-off is removed in Section 1.4.3 and the main result is proved in Theorem 1.4.4.
As already mentioned, in Section 1.5 we turn to the redundancy of the divergence part of Maxwell’s equations. Guo [Guo93] proved that the divergence equations are redundant if one imposes them onΩ. However, in our set-up the Maxwell equations are imposed on whole space. Thus, things are more complicated since we have to
“cross over”𝜕Ω. Whereas (0.5b) is easy to handle, the consideration of (0.5a) is much more difficult and requires the property of local conservation of charge and the correct definition of the charge density𝜌. The idea is to show that the weak form—(1.1.2), in particular—also holds for test functions that do not depend on𝑣and thus to have a weak form of conservation of internal charge at hand. Therefore, we have to perform some technical approximations under a smoothness assumption about𝜕Ω. It turns out that a part of𝜌is a distribution which is supported on𝜕Ωand arises due to the boundary conditions. The main result is stated in Theorem 1.5.6.
In Chapter 2 we analyze an optimal control problem. A typical aim in fusion plasma physics is to keep the amount of particles hitting𝜕Ωas small as possible (since they damage the reactor wall), while the control costs should not be too exhaustive (to ensure efficiency). This leads to a minimization problem where a certain objective function shall be driven to a minimum over a certain set of functions satisfying (VM) in a weak sense. More precisely, the objective function is
1 𝑞
𝑁
Õ
𝛼=1
𝑤𝛼 𝑓+𝛼
𝑞 𝐿𝑞
𝛾+𝑇•,𝑑𝛾𝛼
+1 𝑟k𝑢k𝑟
𝒰.
Here, 1 < 𝑞 < ∞,𝑤𝛼 >0, and𝒰 =𝑊1,𝑟 ]0, 𝑇•[ ×Γ;R3
with 43 < 𝑟 < ∞. Thus, the objective function penalizes hits of the particles on𝜕Ωand exhaustive control costs. In addition to (VM), it is necessary to impose two inequality constraints, namely, (2.1.1) and (2.1.2), which are natural in the sense that they come from formal a priori bounds.
After discussing the minimization problem in detail in Section 2.1, we firstly prove existence of a minimizer in Section 2.2; see Theorem 2.2.1. Secondly, we establish an approach to derive first order optimality conditions for a minimizer under the assumption𝑞 >2 in Sections 2.3 and 2.4. To this end, the one main idea is to write the weak form of (VM) equivalently as an identity
𝒢 𝑓𝛼, 𝑓+𝛼
𝛼, 𝐸, 𝐻, 𝑢
=0 inΛ∗,
where𝒢is differentiable,Λis a uniformly convex, reflexive test function space, andΛ∗ is its topological dual space; see Section 2.3. The other main idea, which is motivated by approaches of Lions [Lio85], is to introduce an approximate minimization problem with a penalization parameter𝑠 >0 which is driven to infinity later; see Section 2.4.
In particular, we add the differentiable term 𝑠
2
𝒢 𝑓𝛼, 𝑓+𝛼
𝛼, 𝐸, 𝐻, 𝑢
2 Λ∗
to the original objective function and abolish the constraint that (VM) be solved. For this approximate problem, we prove existence of a minimizer and establish first order optimality conditions; see Theorems 2.4.3 and 2.4.11. After that, we let𝑠 → ∞and prove that, along a suitable sequence, a minimizer of the original problem is obtained in the limit, and the convergence of the controls𝑢is even strong; see Theorem 2.4.13.
Lastly, we briefly discuss in Section 2.5 how these results can also be verified in case of similar set-ups or different objective functions. We should point out that the main problem we have to deal with is that existence of global-in-time solutions to (VM) is only known in a weak solution concept. In fact, one cannot expect 𝐶1-solutions in general as a result of the boundary conditions for the plasma particles; this was observed by Guo [Guo95] even in a one-dimensional setting. It is an open problem whether or not such weak solutions are unique for given𝑢. Thus, standard approaches to derive first order optimality conditions via introducing a (preferably differentiable) control-to-state operator, as is, for example, done in the books of Hinze et al. [Hin+09]
and Tröltzsch [Trö10], cannot be applied.
In Chapter 3 we consider the case that only an external magnetic field influences the internal system. The aim then is to answer the following two questions: Firstly, for a given time-independent external magnetic field, is there a corresponding stationary solution? Secondly, are there stationary solutions that are confined in Ω, i.e., the particles stay away from the boundary of their container, if the external magnetic field is adjusted suitably? Results are obtained in the case thatΩis an infinitely long cylinder (hence no longer bounded) and that the electromagnetic fields are subject to perfect conductor boundary conditions on𝜕Ω. In particular, proceeding similarly to Degond [Deg90], Batt and Fabian [BF93], Knopf [Kno19], and Skubachevskii [Sku14], we state some basic assumptions on the symmetry of the appearing functions and
0.3 Further literature 7