Upper-thresholds for shock formation in two-dimensional weakly restricted Euler-Poisson equations

The multi-dimensional Euler-Poisson system describes the dynamic behavior of many important physical flows, yet as a hyperbolic system its solution can blow up for some initial configurations. This paper strives to advance our understanding on the critical threshold phenomena through the study of a two-dimensional weakly restricted Euler-Poisson (WREP) system. This system can be viewed as an improved model of the restricted Euler-Poisson (REP) system introduced in [H. Liu and E. Tadmor, Comm. Math. Phys., 228:435-466, 2002]. We identify upper-thresholds for finite time blow up of solutions for WREP equations with attractive/repulsive forcing. It is shown that the thresholds depend on the size of the initial density relative to the initial velocity gradient through both trace and a nonlinear quantity.