Continued Gravitational Collapse for Newtonian Stars
Yan Guo, Mahir Hadžić, Juhi Jang
Abstract
Abstract The classical model of an isolated selfgravitating gaseous star is given by the Euler–Poisson system with a polytropic pressure law $$P(\rho )=\rho ^\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mi>γ</mml:mi> </mml:msup> </mml:mrow> </mml:math> , $$\gamma >1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . For any $$1<\gamma <\frac{4}{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>γ</mml:mi> <mml:mo><</mml:mo> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> , 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.