Thresholds in three-dimensional restricted Euler–Poisson equations

This work provides a description of the critical threshold phenomenon in multi-dimensional restricted Euler–Poisson (REP) equations, introduced in [H. Liu, E. Tadmor. Spectral dynamics of the velocity gradient field in restricted fluid flows, Comm. Math. Phys. 228 (2002) 435–466]. For three-dimensional REP equations, we identified both upper thresholds for the finite-time blow up of solutions and subthresholds for the global existence of solutions, with the thresholds depending on the relative size of the eigenvalues of the initial velocity gradient matrix and the initial density. For the attractive forcing case, these one-sided threshold conditions of the initial configurations are optimal, and the corresponding results also hold for arbitrary n dimensions ( n ≥ 3 ).