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CRITICAL THRESHOLDS IN 2D RESTRICTED EULER–POISSON EQUATIONS∗

We provide a complete description of the critical threshold phenomenon for the twodimensional 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.