The two-dimensional Euler-Poisson system with spherical symmetry

This article concerns the global-in-time existence of smooth solutions with small amplitude to two space dimensional Euler-Poisson system. The main difficulty lies in the slow time decay (1 + t)−1 of the linear system. Inspired by the work of Ozawa et al., [“Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions,” Math. Z. 222, 341–362 (1996)10.1007/BF02621870; “Remarks on the Klein-Gordon equation with quadratic nonlinearity in two space dimensions,” in Nonlinear Waves, Gakuto International Series: Mathematical Sciences and Applications Vol. 10 (Gakkotosho, Tokyo, 1997), pp. 383–392,] we show that such smooth solutions exist for radially symmetric flows.