Absence of shocks in an initially dilute collisionless plasma
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); \textit{C. Bardos} and \textit{P. Degond}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 101-118 (1985; Zbl 0593.35076)].
scientific article; zbMATH DE number 4054200