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Absence of shocks in an initially dilute collisionless plasma. (English) Zbl 0646.35072
The Cauchy problem for the relativistic Vlasov-Maxwell equations \[ \partial_ tf_{\alpha}+\hat v_{\alpha}\cdot \nabla_ xf_{\alpha}+e_{\alpha}+e_{\alpha}[E+c^{-1}\hat v_{\alpha}\wedge B]\cdot \nabla_ vf_{\alpha}=0 \]
\[ E_ t=c Cur B-j,\quad \nabla \cdot E=\rho;\quad B_ t=-c Curl E,\quad \nabla \cdot B=0 \] is studied in three dimensions. The authors prove: If the initial data satisfy the constraints \((\nabla \cdot E_ 0=\rho_ 0\equiv 4\pi \int_{k^ 3}\sum_{\alpha}e_{\alpha}f_{\alpha_ 0}dv\), \(\nabla \cdot B_ 0=0)\) and have compact support and sufficiently small \(C^ 2\) norm, then there exists a unique global \(C^ 1\)-solution. This is proved using the iteration method. This class of problems have been studied by the authors, Bardos and Degond [the authors, Math. Methods Appl. Sci. 9, 46- 52 (1987); C. Bardos and P. Degond, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 101-118 (1985; Zbl 0593.35076)].
Reviewer: B.Guo

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
82C70 Transport processes in time-dependent statistical mechanics
35B40 Asymptotic behavior of solutions to PDEs
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[1] Bardos, C., Degond, P.: Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. Ann. Inst. Henri Poincaré, Analyse non linéaire2, 101-118 (1985) · Zbl 0593.35076
[2] Bardos, C., Degond, P., Ha, T.-N.: Existence globale des solutions des équations de Vlasov-Poisson relativistes en dimension 3. C. R. Acad. Sci. Paris301, 265-268 (1985) · Zbl 0598.35109
[3] Glassey, R., Strauss, W.: Singularity formation in a collisionless plasma could occur only at high velocities. Arch. Ration. Mech. Anal.92, 59-90 (1986) · Zbl 0595.35072
[4] Glassey, R., Strauss, W.: High velocity particles in a collisionless plasma. Math. Methods Appl. Sci.9, 46-52 (1987) · Zbl 0649.35079
[5] Horst, E.: Global solutions of the relativistic Vlasov-Maxwell system of plasma physics. Habilitationsschrift, Universität München 1986
[6] John, F.: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscr. Math.28, 235-268 (1979) · Zbl 0406.35042
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