Exact solutions to 2D isothermal Euler-Poisson equations with qualitative analysis

The family of exact solutions with parameter λ of the 2D isothermal Euler-Poisson equations, which can be used to model the evolution of self-gravitating galaxies or gaseous stars, is investigated. By solving the Liouville equation in Yuen’s analytical solutions, a family of exact solutions is obtained for λ = 0. We show that although such solutions remain local with finite mass and finite potential energy for all time, they have unbounded kinetic energy for any given time. In physics terms, these (2 + 1)-dimensional solutions may correspond to a (3 + 1)-dimensional model of an infinite universe with finite mass and infinite kinetic energy. We also show that the mass and kinetic energy of the solutions are finite for λ < 0, thereby demonstrating in the (2 + 1)-dimensional case a universe model with finite total energy, provided that the finiteness of the potential energy is given. Copyright © 2020 The Author(s). Published by Elsevier B.V. on behalf of The Physical Society of the Republic of China (Taiwan).