Critical thresholds in 1D Euler equations with nonlocal forces
  We study the critical thresholds for the compressible pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity in one dimension. We provide a complete description for the critical threshold to the system without interaction forces leading to a sharp dichotomy condition between global in time existence or finite-time blow-up of strong solutions. When the interaction forces are considered, we also give a classification of the critical thresholds according to the different type of interaction forces. We also analyze conditions for global in time existence when the repulsion is modeled by the isothermal pressure law. 
Jos\'e A. Carrillo, Young-Pil Choi, Eitan Tadmor and Changhui Tan
math.AP