A Class of Global Solutions to the Euler–Poisson System

Abstract

Using recent developments in the theory of globally defined expanding compressible gases, we construct a class of global-in-time solutions to the compressible 3-D Euler-Poisson system without any symmetry assumptions in both the gravitational and the plasma case. Our allowed range of adiabatic indices includes, but is not limited to all γ of the form γ = 1 + 1 n , n ∈ N \ {1}. The constructed solutions have initially small densities and a compact support. As t → ∞ the density scatters to zero and the support grows at a linear rate in t.The three dimensional compressible Euler-Poisson system couples the equation for a compressible gas to a self-consistent force field created by the gas particles: if the interaction is gravitational, we refer to the model as the gravitational Euler-Poisson system and if the interaction is electrostatic we talk about the electrostatic Euler-Poisson system. In the gravitational case we obtain a model of a Newtonian star [45,1,2], while in the case of repelling forces between the particles, we arrive at a model for plasmas [12,14]. We shall work with the free-boundary formulation of the problem, wherein a moving *

Continued Gravitational Collapse for Newtonian Stars

“…Requirement (1.64) is important in establishing the well-posedness of (1.63). The local well-posedness theory for the physical vacuum problem was first developed in the Euler case [17,37], while the well-posedness statements for the gravitational Euler-Poisson system can be found in [26,29,31,34,42]. Nevertheless, the wellposedness theory cannot be directly applied to our setting, as (1.63) differs from the above mentioned works in two important aspects: the problem has explicit singularities at τ = 0 and the space time domain (τ, r ) ∈ (0, 1] × [0, 1] is strictly larger than the domain (s, r ) ∈ [0, 1 g(0) )×[0, 1] which only covers the star dynamics up to the first stipulated collapse time t * = 1 g(0) , see Section 1.4.…”

“…In the supercritical range 6 5 ≤ γ < 4 3 it has been shown by Jang [33,34] that the Lane-Emden stars are dynamically nonlinearly unstable. Besides the stationary states and the homologous collapsing stars in the mass-critical case γ = 4 3 , the only other global solutions of EP γ were constructed by Hadžić and Jang [29,31].…”

High-Order Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations With Nonlocal Free Energy

“…For this example we select an ideal-gas free energy with pressure P (ρ) = ρ (or, equivalently, Π(ρ) = ρ ln(ρ) − 1) together with an interaction potential with a kernel of the form W (x) = x 2 2 . In this case the steady state aimed to be preserved satisfies Free energies of this type are common in Euler-Poisson systems, in which the Euler equations for a compressible gas are coupled to a self-consistent force field created by the gas particles [55]. This interaction could be gravitational, leading to the modeling of Newtonian stars [7], or electrostatic with repelling forces between the Downloaded 03/25/21 to 92.0.10.177.…”

“…In our previous work [20] we extended the applicability of well-balanced schemes to the broad class of hydrodynamic models with attractive-repulsive interaction forces. In particular, we considered interactions associated with nonlocal convolutions or functions of convolutions, which is commonplace in applications such as the Keller-Segel model [9], more general Euler-Poisson systems [55], or in dynamic-density functional theory (DDFT) [47,48]. This class of balance laws may contain linear or nonlinear damping effects, such as the Cucker-Smale alignment term in collective behavior [36].…”

Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour

“…Moreover, (1) is also related to the models of self-gravitational gases [1,14,23]. However, the closest, in the authors' opinion, is a result regarding the pressureless Euler-Poisson system [20], see also [6]. The latter result is merely mono-dimensional and, instead of dissipation, a friction term is taken into account.…”