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Global weak solutions of the Vlasov-Maxwell system with boundary conditions. (English) Zbl 0787.35072
The Vlasov-Maxwell system \[ \partial_ tf_ \beta+{v \over m_ \beta} \cdot \nabla_ xf_ \beta+{e_ \beta \over m_ \beta}(E+{1\over c}v \times B)\cdot \nabla_ vf_ \beta=0,\;1 \leq \beta \leq N \]
\[ \partial_ tE-c \text{curl} B=-j=-4 \pi \sum_ \beta e_ \beta \int_{\mathbb{R}^ 3} v f_ \beta dv,\;\partial_ tB+c \text{curl} E=0, \]
\[ \text{div} E=\rho=4 \pi \sum_ \beta e_ \beta \int_{\mathbb{R}^ 3}f_ \beta dv,\text{ div} B=0 \] models a collisionless plasma consisting of \(N\) different particle species which are described by time- dependent density functions \(f_ \beta=f_ \beta(t,x,v)\) on phase space \(\Omega \times \mathbb{R}^ 3\) with \(\Omega \subset \mathbb{R}^ 3\) open; the particles interact only by a selfconsistent electromagnetic field \((E,B)\).
Earlier investigations of the corresponding initial value problem were restricted to the case \(\Omega=\mathbb{R}^ 3\), i.e. boundary effects were not included. Global existence for general data is established only in a weak sense [cf. R. J. DiPerna and P. L. Lions, Commun. Pure Appl. Math. 42, No. 6, 729-757 (1989; Zbl 0698.35128)].
The author extends the latter result to the mathematically more difficult and physically more realistic case of \(\Omega \subset \mathbb{R}^ 3\) with \(C^{1,\mu}\) boundary, \(\mu>0\). The boundary is taken to be perfectly conducting, and the particles are specularly reflected or partially absorbed at the boundary. After making precise the notion of a weak solution, the author proceeds as follows:
The phase space is approximated by a sequence of bounded domains. On each of these a sequence of linear Vlasov and linear Maxwell systems is solved iteratively, based on the linear transport theory of P. Beals and V. Protopopescu [J. Math. Anal. Appl. 121, 370-405 (1987; Zbl 0657.45007)]. The compactness result of DiPerna and Lions (velocity averaging) yields a weak limit, first on each of the approximating domains and then on the original phase space \(\Omega \times \mathbb{R}^ 3\). This weak limit is the desired global weak solution.
The paper is carefully written and readable, a point which has to be stressed considering the technically difficult matter.
Reviewer: G.Rein (München)

35Q35 PDEs in connection with fluid mechanics
35L50 Initial-boundary value problems for first-order hyperbolic systems
35D05 Existence of generalized solutions of PDE (MSC2000)
35Q60 PDEs in connection with optics and electromagnetic theory
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
Full Text: DOI
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