Glassey, Robert T.; Strauss, Walter A. Absence of shocks in an initially dilute collisionless plasma. (English) Zbl 0646.35072 Commun. Math. Phys. 113, 191-208 (1987). 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 Cited in 2 ReviewsCited in 64 Documents 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 Keywords:Cauchy problem; relativistic Vlasov-Maxwell equations; initial data; unique global \(C^ 1\)-solution; iteration method Citations:Zbl 0593.35076 PDFBibTeX XMLCite \textit{R. T. Glassey} and \textit{W. A. Strauss}, Commun. Math. Phys. 113, 191--208 (1987; Zbl 0646.35072) Full Text: DOI References: [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 · doi:10.1007/BF00250732 [4] Glassey, R., Strauss, W.: High velocity particles in a collisionless plasma. Math. Methods Appl. Sci.9, 46-52 (1987) · Zbl 0649.35079 · doi:10.1002/mma.1670090105 [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 · doi:10.1007/BF01647974 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.