CRITICAL THRESHOLDS IN MULTI-DIMENSIONAL EULER-POISSON EQUATIONS WITH RADIAL SYMMETRY

Abstract

We study the global regularity of multi-dimensional repulsive Euler-Poisson equations in the radial setup. We show that the question of global regularity vs. finite breakdown of smooth solutions depends on whether the initial configuration crosses an initial critical threshold in configuration space. Specifically, there exists a global-in-time smooth solution if and only if the initial configuration of density rho(0), radial velocity R(0), and electrical charge e(0) satisfies R'(0) >= F(rho(0), e(0), R(0)) for a certain threshold F. Similarly, we characterize the critical threshold for global smooth solutions subject to two-dimensional radially symmetric data with swirl. We also discuss a possible framework for global regularity analysis beyond the radial case, which indicates that the main difficulty lies with bounding the spectral gap, lambda(2)(del u) - lambda(1)(del u)