arXiv subjects

King Ming Lam

Publications and source records attributed to King Ming Lam.

4 recordsLinked to original sources

arXiv2026

A turning point principle for liquid Lane-Emden stars

Upon fixing the adiabatic index, the spherically symmetric liquid Lane-Emden stars governed by the Euler-Poisson system with a "stiffened gas" equation of state $p=ρ^γ-1$ form a one-parameter family, naturally parametrised by the central density $κ=\barρ_κ(0)\in(1,\infty)$. In contrast to the gaseous case, the liquid free boundary breaks the self-similarity of the family and bends the mass-radius curve, creating extrema of the mass when $γ<2(d-1)/d$. We formulate a turning point principle in the spirit of Zel'dovich, Wheeler and Thorne, and of its rigorous relativistic counterpart from the works of Hadžić, Lin and Rein: the number of radial growing modes equals the negative Morse index of the linearised operator, is locally constant along the family, and can change only at the turning points of the mass-radius curve, the bending orientation there dictating whether a growing mode is gained or lost. We describe the large central density limit in which the gaseous tail (read off from a planar dynamical system) determines whether the mass-radius curve spirals, and hence whether the number of growing modes remains bounded or tends to infinity. As corollaries we recover and sharpen the known radial (in)stability results for liquid Lane-Emden stars and, in combination with existing non-radial analysis, obtain a turning point criterion for non-radial stability.

math-ph↗

arXiv2026

Nonradial linear stability of liquid Lane-Emden stars

The classical model of a star is the Lane-Emden star with dynamics governed by the Euler-Poisson equations. We consider the case of a liquid star with a "stiffened gas" equation of state $p=ρ^γ-1$. We derive the full 3D linearised Euler-Poisson system around liquid Lane-Emden stars with no symmetry assumptions on the perturbations and show that the associated linear operator $\mathbf L$ is non-negative whenever the radial mode is non-negative. We show that $\mathbf L$ has an infinite-dimensional kernel each element of which corresponds to a linearly growing solution to the linearised system. When restricted to irrotational perturbations and modding out the three kernel elements corresponding to momentum conservation, however, we prove that $\mathbf L$ is strictly positive with coercivity bound $\langle\mathbf L\boldsymbolθ,\boldsymbolθ\rangle_{\barρ}\gtrsim\|\boldsymbolθ\|_{L^2(B_R)}^2$. Hence we demonstrate that the liquid Lane-Emden stars are stable against non-radial irrotational perturbations whenever the purely radial mode is stable, improving upon previous results that dealt only with purely radial perturbations. However, the stability might not be as strong as one might hope, as we prove that $\|\nabla\boldsymbolθ\|_{L^2(B_R)}^2$ cannot be controlled even in this case.

math.AP↗

arXiv2024

Nonradial stability of expanding Goldreich-Weber stars

Goldreich-Weber solutions constitute a finite-parameter of expanding and collapsing solutions to the mass-critical Euler-Poisson system. Two subclasses of this family correspond to compactly supported density profiles suitably modulated by the dynamic radius of the star that expands at the self-similar rate $λ(t)_{t\to\infty}\sim t^{\frac23}$ and linear rate $λ(t)_{t\to\infty}\sim t$ respectively. We prove two results: any linearly expanding Goldreich-Weber star is nonlinearly stable, while any given self-similarly expanding Goldreich-Weber star is codimension-4 nonlinearly stable against irrotational perturbations. The codimension-4 condition in the latter result is optimal and reflects the presence of 4 unstable directions in the linearised dynamics in self-similar coordinates, which are induced by the conservation of the energy and the momentum. This result can be viewed as a codimension-1 nonlinear stability of the moduli space of self-similarly expanding Goldreich-Weber stars against irrotational perturbations.

math.AP↗

arXiv2023

Linear Stability of liquid Lane-Emden stars

We establish various qualitative properties of liquid Lane-Emden stars in $\mathbb{R}^d$, including bounds for its density profile $ρ$ and radius $R$. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when $γ\geq 2(d-1)/d$; linearly stable when $γ<2(d-1)/d$ for stars with small relative central density $ρ(0)-ρ(R)$; and linearly unstable when $γ<2(d-1)/d$ for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.

math.AP↗