Critical Thresholds in 2D Restricted Euler-Poisson Equations

We provide a complete description of the critical threshold phenomenon for the two-dimensional localized Euler-Poisson equations, introduced by the authors in [Comm. Math. Phys., 228 (2002), pp. 435-466]. Here, the questions of global regularity vs. finite-time breakdown for the two-dimensional (2D) restricted Euler-Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.