Critical Thresholds in 2D Restricted Euler-Poisson Equations

We provide a complete description of the critical threshold phenomena for the two-dimensional localized Euler-Poisson equations, introduced by the authors in [Liu & Tadmor, Comm. Math Phys., To appear]. Here, the questions of global regularity vs. finite-time breakdown for the 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 size of three quantities: the initial density, the initial divergence as well as the initial spectral gap, that is, the difference between the two eigenvalues of the $2 \times 2$ initial velocity gradient.