Global existence of solutions for a fluid model of a neutron star
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 arguments.
scientific article; zbMATH DE number 6609966
scientific article