Global existence of solutions for a fluid model of a neutron star
type: Item
instance of: scholarly article
title: Global existence of solutions for a fluid model of a neutron star
zbMATH Open document ID: 1351.35108
published in: Boundary Value Problems
publication date: 2016-08-08T00:00:00Z
review text: The authors analyze an initial-boundary value problem describing a fluid spherical model of neutron star, as introduced by \textit{J. M. Lattimer} et al. [``Rapid cooling and the structure of neutron stars'', Astrophys. J. 425, No. 2, 802--813 (1994; \url{doi:10.1086/174025})], in the case of a rapid cooling of the core of the star. When written in Eulerian coordinates \((r,t)\), the problem becomes \(\eta _{t}=(r^{2}v)_{x}\), \(v_{t}=r^{2}(-p+\mu \frac{(r^{2}v)_{x}}{\eta })_{x}+f\), \( e_{t}=Q_{x}+(-p+\mu \frac{(r^{2}v)_{x}}{\eta })(r^{2}v)_{x}\), \(r_{t}=v\), posed in \((0,M)\times \mathbb{R}^{+}\).\ Here \(\eta \) (resp. \(v\),\(e\)) denotes the inverse of the density (resp. the velocity, the internal energy), \(Q\) is the heat flux and \(f\) is the external field force. Initial data are added for \((\eta ,v,r,\theta )\) at \(t=0\) and Dirichlet boundary conditions are imposed to \(v\) at \(x=0,M\), to \(Q\) at \(x=0\) and to \(\theta \) at \(x=M\). The authors first recall the expression of the solution \((\overline{\eta },0, \overline{\theta })\) to the associated stationary problem. They define the notion of \(H^{i}([0,M])\) solution and the first main result proves the existence of a unique global \(H^{1}([0,M])\) solution \((\eta ,v,\theta )\) to the evolution problem, assuming hypotheses on the initial data and on the coefficients of the problem. Moreover, \(\eta \), \(\theta \) and \(r\) remain bounded from above and from below by positive constants. The authors here use classical functional analysis arguments. The authors also prove uniform energy estimates for the difference between this solution and the solution of the associated steady problem. Finally, the authors prove that if the initial data belong to \(H^{4}([0,M])\) and satisfy further hypotheses, then the evolution problem has a unique global \(H^{4}([0,M])\) solution. The authors first prove a \(H^{2}\)-regularity result computing higher derivatives of the unknowns and using functional analysis…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 85A30
zbMATH DE Number: 6609966
zbMATH Keywords: neutron star
zbMATH Keywords: spherical model
zbMATH Keywords: global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
Wikidata QID: Q59467329
author: Jianlin Zhang
MaRDI profile type: Publication
full work available at URL: s13661-016-0628-1
OpenAlex ID: W2469962279
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DOI: 10.1186/S13661-016-0628-1
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description: scientific article; zbMATH DE number 6609966