Existence of radial global smooth solutions to the pressureless Euler-Poisson equations with quadratic confinement

Abstract

We consider the pressureless Euler-Poisson equations with quadratic confinement. For spatial dimension $d\ge 2,d\ne 4$, we give a necessary and sufficient condition for the existence of radial global smooth solutions, which is formulated explicitly in terms of the initial data. This condition appears to be much more restrictive than the critical-threshold conditions commonly seen in the study of Euler-type equations. To obtain our results, the key observation is that every characteristic satisfies a periodic ODE system, and the existence of global smooth solution requires the period of every characteristic to be identical.

References in corpus (9)

• Critical thresholds in flocking hydrodynamics\\with nonlocal alignment
• Spectral Dynamics of the Velocity Gradient Field in Restricted Flows
• Rotation Prevents Finite-Time Breakdown
• Global regularity for the fractional Euler alignment system
• Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces
• Flocking hydrodynamics with external potentials
• On the Euler-Alignment system with weakly singular communication weights
• Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems
• The 2D Euler-Poisson System with Spherical Symmetry