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.

Keywords

critical thresholds, restricted Euler–Poisson equations

2010 Mathematics Subject Classification

Primary 35Q35. Secondary 35B30.

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Published 24 February 2017