Global existence of solutions for a fluid model of a neutron star

We consider an initial-boundary value problem for a fluid model of neutron star. In the case of a rapid cooling of the core of the star, the model used to describe the evolution of temperature in the star follows Lattimer et al. [2]. If a mechanical equilibrium is reached and the specific heat is a linear function of temperature, then the problem reduces to the study of a fast diffusion equation satisfied by the temperature in [3]. In a more general setting, suppose that the temperature is coupled to density and velocity fluctuations through a thermo-mechanical system; the simplest description of such a model is achieved through the compressible Navier-Stokes system in [4].

In this paper, we are interested in the 3D spherical symmetric solutions to the complete system, which has the general formulation as follows (see [5]):

in the domain \(\omega\times \mathbb{R}^{+}\) with \(\omega:=(R_{0},R_{1})\), where \(R_{0}\) is the radius of the internal rigid core of the star and \(R_{1}\) is the exterior boundary, and \(\rho(r,t)\) and \(v(r,t)\) denote the density and the velocity, respectively. Let \(\eta:=\frac{1}{\rho}\) be the specific volume and \(\theta(r,t)\) be the temperature, then the pressure \(p(\eta ,\theta)=\frac{A}{2}\frac{\theta^{2}}{\eta^{2-\beta}}\) and the internal energy \(e(\eta,\theta)=c_{v}\theta+\frac{A}{2(\beta-1)}\frac{\theta ^{2}}{\eta^{1-\beta}}\), where constants \(c_{v}>0\), \(A>0\) and \(1<\beta<2\). The heat flux Q is given by the Fourier law \(Q(\eta,\theta):=\kappa (\eta,\theta)\theta_{r}\) with the following constraints on the thermal conductivity:

$$\begin{aligned}& \underline{\kappa}\bigl(1+\theta^{q}\bigr)\leq\kappa(\eta,\theta)\leq \overline {\kappa}\bigl(1+\theta^{q}\bigr), \end{aligned}$$

(1.4)

$$\begin{aligned}& \bigl\vert \kappa_{\eta}(\eta,\theta)\bigr\vert +\bigl\vert \kappa_{\eta\eta}(\eta,\theta)\bigr\vert \leq\overline {K}_{1} \bigl(1+\theta^{q}\bigr), \end{aligned}$$

(1.5)

$$\begin{aligned}& \bigl\vert \kappa_{\theta}(\eta,\theta)\bigr\vert \leq \overline{K}_{2}\bigl(1+\theta^{q-1}\bigr), \end{aligned}$$

(1.6)

for any \(\theta\geq0\), with positive constants \(\underline{\kappa}\), κ̅, \(\overline{K}_{1}\), \(\overline{K}_{2}\) and \(q\geq4\). \(F(r,t)\) is a given external field force (gravitation). Finally, we also assume the bulk viscous coefficient μ is a positive constant and the shear viscous coefficient \(\nu=0\).

As in [1], we transform the system in Eulerian coordinates \((r,t)\) into that in Lagrangian (mass) coordinates \((x,t)\) by

$$ r(x,t):=r_{0}(x)+ \int_{0}^{t}v(x,s)\,ds, $$

(1.7)

where \(r_{0}(x):= (R_{0}^{3}+3\int_{0}^{x}\eta(y,0)\,dy )^{\frac{1}{3}}\) for \(x\in(0,M)\), we have

in the fixed domain \(\Omega\times \mathbb{R}^{+}\) with \(\Omega:=(0,M)\), where the specific volume η, the velocity v, the temperature θ and the radius r depend on the Lagrangian mass coordinates. Now the heat flux is \(Q(\eta,\theta)=\kappa(\eta,\theta)\frac{r^{4}\theta_{x}}{\eta}\) and the external field force is given by the Newtonian law \(f(x)=-G\frac {M_{0}}{r^{2}}\), where G and \(M_{0}\) are positive constants. Denote the stress σ by

$$\sigma(\eta,\theta):=-p+\mu\frac{(r^{2}v)_{x}}{\eta}. $$

System (1.8)-(1.11) is subjected to the following boundary and initial conditions:

$$\begin{aligned}& (\eta,v,r,\theta)|_{t=0}=(\eta_{0},v_{0},r_{0}, \theta_{0}) (x),\quad x\in [0,M], \end{aligned}$$

(1.12)

$$\begin{aligned}& v|_{x=0,M}=0,\qquad Q|_{x=0}=0,\qquad \theta|_{x=M}= \theta_{\Gamma},\quad t\geq0, \end{aligned}$$

(1.13)

with constant \(\theta_{\Gamma}>0\).

Now let us recall some known results for the related system. For the full 3D compressible Navier-Stokes system with heat conductivity, we can refer to the basic references on the global existence of a weak solution, such as Lions [6], Feireisl [7], Feireisl and Novotný [8] and Bresch and Desjardins [9] and references therein. For the large-time behavior of the global solutions, we would also like to mention the work of Feireisl and Petzeltová [10, 11] and Feireisl and Novotný [12]. On the subject of the global existence and large-time behavior of smooth/strong solutions for the one-dimensional motions of viscous polytropic ideal gas under various conditions, we refer the reader to Kazhikhov and Shelukhin [13], Kawohl [14], Chen [15], Jiang [16–18], Zheng and Qin [19], Qin [20–22], and so on. For the free boundary problems, we can also refer to the work of Nagasawa [23, 24], Tani [25, 26] and Hsiao and Luo [27]. For the free and pure Neumann boundary value problem, we refer the reader to Umehara and Tani [28, 29], Qin and Huang [30], Qin et al. [31], and the references therein.

However, in the major part of astrophysical literature, for example, at least when rotation and magnetic aspects are neglected, a quite reliable approximation is spherical symmetry; see also [4, 32, 33]. In this quasi-monodimensional situation, the global existence and large-time behavior of a classical solution have been established in some spherically symmetric cases, and we refer to [5, 18, 20, 34–38] and the references therein. In addition, for the cylindrically symmetric Navier-Stokes equations with various boundary conditions, the global well-posedness of the solutions has been studied by many researchers, and we can refer to [14, 15, 25, 31, 39–46] and the references therein. For problem (1.8)-(1.13), Ducomet and Nečasová [1] have proved the global well-posedness and large-time asymptotics for the initial data \((\eta_{0},v_{0},\theta_{0})\in H^{1}\times H^{1}\times H^{1}\). Ducomet and Nečasová [3] considered a fast diffusion equation satisfied by the temperature and proved well-posedness and large-time asymptotics of global solutions with the initial data \(\theta _{0}\in L^{2}\).

In this paper, we shall establish the global existence and regularity of solutions for the spherical symmetric model of neutron star with the initial data \((\eta_{0},v_{0},\theta_{0})\in H^{i}\times H^{i}\times H^{i}\) (\(i=1,2,4\)). The main novelty is to establish the \(H^{i}\) (\(i=1,2,4\)) regularity of the global solutions to problem (1.8)-(1.13). It is worth pointing out that the boundary condition on the temperature θ is different from general Dirichlet or Neumann boundary condition. Our results improve and generalize the results in [1].

In the following, the notations \(L^{p} \) (\(1\leq p\leq+\infty\)) and \(W^{k,p}\) (in particular, \(W^{k,2}\) is also denoted by \(H^{k}\) and \(H_{0}^{1}=W_{0}^{1,2}\)) stand for the usual Lebesgue spaces and the usual Sobolev spaces on \((0,M)\), respectively. \(\Vert \cdot \Vert _{B}\) denotes the norm in the space B, \(\Vert \cdot \Vert :=\Vert \cdot \Vert _{L^{2}}\). \(C^{\alpha,\beta }=C^{\alpha,\beta}([0,M]\times[0,T])\) stands for uniformly Hölder continuous space with exponents α in x and β in t. We use \(C_{0}\) and \(C_{1}\) to denote a generic positive constant depending only on the parameters of the system and the bounds of the initial data \((\eta_{0},v_{0},\theta_{0})\in(H^{1}([0,M]))^{3}\), but being independent of t. Furthermore, \(C_{i}(T) \) (\(i=1,2,4\)) is a universal constant only dependent on the given time T, the physical constants and the initial data \((\eta_{0},v_{0},\theta_{0})\in(H^{i}([0,M]))^{3}\).

The rest of the paper is arranged as follows. In Section 2, we will state our main theorems about the global existence of the solutions to problem (1.8)-(1.13). Subsequently, by a series of lemmas, we shall prove our main theorems in Section 3.

Let T be an arbitrary positive number. Now we give the definition of \(H^{i}([0,M])\)-solution to the initial-boundary problem (1.8)-(1.13).

For convenience, we first state a proposition from [1].

We are now in a position to state our main result.

In this section, we will give some useful a priori estimates of the solutions to complete the proofs of the theorems.