Abstract
In this paper we investigate the problem of multiple expanding Newtonian stars that interact via their gravitational effect on each other. It is clear physically that if two stars at rest are separated initially, and start expanding as well as moving according to the laws of Newtonian gravity, they may eventually collide. Thus, one can ask whether each star can be given an initial position and velocity such that they can keep expanding without touching. We show that even with gravitational interaction between the bodies, a large class of initial positions and velocities give global-in-time solutions to the N Body Euler–Poisson system. To do this we use scaling mechanisms present in the compressible Euler system, shown in Parmeshwar (Quart Appl Math 79(2):273–334, 2021), gaining advantageous time weights and smallness in our estimates under the specific form of the Lagrangian flow maps, and assumption of small mass for each star, represented by a parameter delta . This is combined with a careful analysis of how the gravitational interaction between stars affects their dynamics.
Highlights
The compressible Euler–Poisson system for an inviscid, isentropic, ideal gas, acting under the influence of its own gravity, in its most basic form, is given by∂t ρ + ∇ · = 0, (1)ρ (∂t + u · ∇) u + ∇ p + ρ∇φ = 0, (2)Δφ = 4πρ, lim φ(t, x) = 0, (3) |x |→∞where ρ, u, p, and φ are the fluid density, velocity, pressure, and gravitational potential respectively
In the case of one body of incompressible fluid moving under its own gravity, equilibrium states have been studied by the likes of Newton, Maclaurin, Jacobi, Poincaré, and others
We exhibit an open set of initial positions and velocities that lead to global-in-time solutions of EP(N, γ ), with each star asymptotically behaving like an expanding star moving with constant velocity
Summary
The compressible Euler–Poisson system for an inviscid, isentropic, ideal gas, acting under the influence of its own gravity, in its most basic form, is given by. Jang [36,37] showed that for γ ∈ [6/5, 4/3), the Lane-Emden stars are nonlinearly unstable, whilst stability results for the range (4/3, 2) conditional on the existence of solutions close to the stationary ones have been shown in [55,66]. Jang and Makino [42,43], and Strauss and Wu [76,77], who built on techniques first employed by Lichtenstein [47] and Heilig [34], have constructed axisymmetric rotating solutions, by making use of the Implicit Function Theorem. This method allows for solutions with γ 4/3, and in the works by Strauss and Wu, they construct a continuous family of solutions parameterised by the prescribed angular velocity. Jang, Strauss, and Wu [44] used these techniques to construct a rotating magnetic star
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