Continued Gravitational Collapse for Newtonian Stars
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Continued Gravitational Collapse for Newtonian Stars Yan Guo, Mahir Hadži´c
& Juhi Jang
Communicated by N. Masmoudi
Abstract The classical model of an isolated selfgravitating gaseous star is given by the Euler–Poisson system with a polytropic pressure law P(ρ) = ρ γ , γ > 1. For any 1 < γ < 43 , we construct an infinite-dimensional family of collapsing solutions to the Euler–Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface, with continuous mass absorption at the origin. The leading order singular behavior is described by an explicit collapsing solution of the pressureless Euler–Poisson system.
1. Introduction The basic model of a Newtonian star is given by the 3-dimensional compressible Euler–Poisson system [1,11,62] ∂t ρ + div (ρu) = 0, ρ (∂t u + (u · ∇)u) + ∇ P(ρ) + ρ∇ = 0, = 4π ρ, lim (t, x) = 0. |x|→∞
(1.1a) (1.1b) (1.1c)
Here ρ, u, P(ρ), denote the gas density, the gas velocity vector, the gas pressure, and the gravitational potential respectively. To close the system we impose the socalled polytropic equation of state: P(ρ) = ρ γ , γ > 1.
(1.2)
The power γ is called the adiabatic exponent. Here a star is modelled as a compactly supported compressible gas surrounded by vacuum, which interacts with a self-induced gravitational field. To describe
Y. Guo et al.
the motion of the boundary of the star we must consider the corresponding freeboundary formulation of (1.1). In this case, a further unknown in the problem is the support of ρ(t, ·) denoted by (t). We prescribe the natural boundary conditions ρ = 0, V(∂(t)) = u · n
on ∂(t), on ∂(t),
(1.3a) (1.3b)
and the initial conditions (ρ(0, ·), u(0, ·)) = (ρ0 , u0 ) , (0) = .
(1.4)
Here V(∂(t)) is the normal velocity of the moving boundary ∂(t) and condition (1.3b) simply states that the movement of the boundary in normal direction is determined by the normal component of the velocity vector field. We refer to the system (1.1)–(1.3) as the EPγ -system. We point the reader to the classical text [11] where the existence of static solutions of EPγ is studied under the natural boundary condition (1.3a). We next impose the physical vacuum condition on the initial data: dP (ρ) · n∂ < 0. (1.5) −∞ < ∇ dρ Condition (1.5) implies that the normal derivative of the squared speed of sound cs2 (ρ) = ddρP (ρ) is discontinuous at the vacuum boundary. This condition is famously satisfied by the well-known class of steady states of the EPγ -system known as the Lane–Emden stars. At the same time, condition (1.5) is the key assumption that guarantees the well-posedness of the Euler–Poisson system with vacuum regions. For any ε¯ > 0 consider the mass preserving rescaling applied to the EPγ -system: ˜ ρ = ε¯ −3 ρ(s, y), ˜ y), u = ε¯ −1/2 u(s, ˜ y), = ε¯ −1 (s,
(1.6)
where s = ε¯ −3/2 t, y = ε¯ −1 x. It is easy to see that the above rescaling is mass-critical, that is M[ρ] = M[ρ]. ˜ A simple calculation reveals that if (ρ, u, ) solve the EPγ -system, then the
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