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Global existence of solutions for a fluid model of a neutron star
Boundary Value Problems volume 2016, Article number: 121 (2016)
Abstract
In this paper, we consider an initial-boundary value problem for the equations of a fluid spherical model of neutron star considered by Lattimer et al. We establish the global existence and regularity of the spherically symmetric solutions in \(H^{i}\) (\(i=1,2,4\)) of this fluid model. These results improve and generalize the results of Ducomet and Necasova (Ann. Univ. Ferrara 55(1):153-193, 2009).
1 Introduction
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:
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
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
System (1.8)-(1.11) is subjected to the following boundary and initial conditions:
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.
2 Main results
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).
Definition 2.1
Function \((\eta(x,t),v(x,t),\theta(x,t))\) is called a global \(H^{i}([0,M])\)-solution to problem (1.8)-(1.13) if it satisfies the following conditions:
and
where \(i=1,2,4\).
For convenience, we first state a proposition from [1].
Proposition 2.1
The corresponding static problem to problem (1.8)-(1.13) has a unique solution \((\bar{\eta},\bar{v},\bar{\theta})\) given by
where the constant \(r_{0}\) only depends on the initial data.
We are now in a position to state our main result.
Theorem 2.1
Let the initial data \(0< C_{0}^{-1}<\eta_{0}(x)<C_{0}\), \((\eta_{0},v_{0},\theta _{0})\in(H^{1}[0,M])^{3}\). Assume that the heat conductivity κ satisfies (1.4)-(1.6) and the initial data are compatible with boundary conditions. Then problem (1.8)-(1.13) admits a unique global \(H^{1}([0,M])\)-solution \((\eta(x,t),v(x,t),\theta(x,t))\) verifying, for all \((x,t)\in[0,M]\times[0,T]\),
and
Theorem 2.2
Let the initial data \(0< C_{0}^{-1}<\eta_{0}(x)<C_{0}\), \((\eta_{0},v_{0},\theta _{0})\in(H^{2}[0,M])^{3}\). Assume that the heat conductivity κ satisfies (1.4)-(1.6) and the initial data are compatible with boundary conditions. Then problem (1.8)-(1.13) admits a unique global \(H^{2}([0,M])\)-solution \((\eta(x,t),v(x,t),\theta(x,t))\) verifying, for all \((x,t)\in[0,M]\times[0,T]\),
Theorem 2.3
Let the initial data \(0< C_{0}^{-1}<\eta_{0}(x)<C_{0}\), \((\eta_{0},v_{0},\theta _{0})\in(H^{4}[0,M])^{3}\). Assume that the heat conductivity κ satisfies (1.4)-(1.6) and the initial data are compatible with boundary conditions. Then problem (1.8)-(1.13) admits a unique global \(H^{4}([0,M])\)-solution \((\eta(x,t),v(x,t),\theta(x,t))\) verifying, for all \((x,t)\in[0,M]\times[0,T]\),
Corollary 2.1
Under the assumptions of Theorem 2.3 and some suitable compatibility conditions, the global solution \((\eta,v,\theta)\) to problem (1.8)-(1.13) is the classical solution verifying
Remark 2.1
The uniqueness of the global solutions has been obtained in [1].
Remark 2.2
Theorem 2.1 implies that problem (1.8)-(1.13) admits a unique global weak solution. Theorem 2.2 implies that problem (1.8)-(1.13) admits a unique global strong solution.
Remark 2.3
Our results generalize the previous work in [1].
3 Proofs of theorems
In this section, we will give some useful a priori estimates of the solutions to complete the proofs of the theorems.
3.1 Global existence of \(H^{1}\)-solution
In this subsection, we shall complete the proof of Theorem 2.1. As in [1], we have the following mass conservation and energy-entropy inequality.
Lemma 3.1
Under the assumptions in Theorem 2.1, the following estimates hold, for any \(t\in[0,T]\),
Proof
See, e.g., Lemma 1 in [1]. □
Lemma 3.2
Under the assumptions in Theorem 2.1, the following estimates hold for all \((x,t)\in\Omega\times[0,T]\):
Proof
See, e.g., Propositions 2 and 5 in [1]. □
Lemma 3.3
Under the assumptions in Theorem 2.1, the following estimate holds for any \(t\in[0,T]\):
Proof
See, e.g., Propositions 3-5 and Lemma 5 in [1]. □
Lemma 3.4
Under the assumptions in Theorem 2.1, the following estimate holds for any \(t\in[0,T]\):
Proof
Multiplying (1.9) by \(v_{t}\) over \((0,M)\times (0,T)\), employing an integration by parts and using Lemmas 3.1-3.3 and the Young inequality, we have
which implies
Equation (1.10) can be rewritten as
Multiplying (3.7) by \(e_{\theta}^{-1}\theta_{xx}\), then integrating the result with respect to x over \((0,M)\), using Hölder’s inequality, the Sobolev embedding theorem, and Lemmas 3.1-3.3, we have, for any \(\varepsilon>0\),
Integrating (3.8) with respect to t over \((0,t)\), taking \(\varepsilon >0\) small enough, and using Lemmas 3.1 and 3.3, we can obtain
which, along with (3.6), leads to the estimate (3.5). □
Now combining Lemmas 3.1-3.4 and noting equation (1.8), we complete the proof of Theorem 2.1.
3.2 Global existence of \(H^{2}\)-solution
In this subsection, we shall deal with the \(H^{2}\)-regularity of the global solutions to problem (1.8)-(1.13).
Lemma 3.5
Under the assumptions in Theorem 2.2, the following estimate holds for any \(t\in[0,T]\):
Proof
See, e.g., Proposition 6 in [1]. □
Lemma 3.6
Under the assumptions in Theorem 2.2, the following estimate holds for any \(t\in[0,T]\):
Proof
Differentiating (1.9) with respect to x, we have
where
By Theorem 2.1 and Lemma 3.5, using Young’s inequality, we get, for any \(\varepsilon>0\),
Multiplying (3.12) by \(\frac{\eta_{xx}}{\eta}\), then integrating the result over \([0,M]\times[0,t]\) and using Young’s inequality and (3.13), taking \(\varepsilon>0\) sufficiently small, we can obtain (3.11). Thus we complete the proof. □
Lemma 3.7
Under the assumptions in Theorem 2.2, the following estimate holds for any \(t\in[0,T]\):
Proof
Differentiating (1.9) and (1.10) with respect to x, respectively, and using the Cauchy inequality, we easily obtain
and
By virtue of Theorem 2.1 and Lemmas 3.5-3.6, we complete the proof. □
Now combining Lemmas 3.5-3.7, we have completed the proof of Theorem 2.2.
3.3 Global existence of \(H^{4}\)-solution
In this subsection, we shall complete the proof of Theorem 2.3, which can be divided into the following lemmas.
Lemma 3.8
Under the assumptions of Theorem 2.3, we see that for any \(t\in[0,T]\) and for \(\varepsilon>0\) small enough,
Proof
Differentiating (1.9) and (1.10) with respect to x, respectively, using Theorems 2.1 and 2.2, we can get
Similarly, differentiating (1.9) and (1.10) with respect to x twice, respectively, we can infer from Theorems 2.1 and 2.2 that
or
It follows from (1.8) and (1.10) that
Differentiating (1.9) and (1.10) with respect to t, respectively, using Theorems 2.1-2.2 and (3.20)-(3.27), we have
Thus the estimate (3.17) follows from (3.20)-(3.23), (3.29), and (3.31).
Differentiating (1.9) with respect to t twice, multiplying the resultant by \(v_{tt}\) and performing an integration by parts in \(L^{2}(0,M)\), and using Theorem 2.2, the embedding theorem, and the Young inequality, we can derive
Thus, by Theorem 2.2,
which, together with (3.28) and (3.30), gives estimate (3.18).
Similarly, differentiating (1.10) with respect to t twice, multiplying the result by \(\theta_{tt}\) and performing an integration by parts over \(L^{2}(0,M)\), and using the embedding theorem and the Young inequality, we have
By virtue of Theorems 2.1-2.2 and the embedding theorem, we deduce that, for any \(\varepsilon\in(0,1)\),
which, by Hölder’s inequality, implies
Thus it follows from (3.33)-(3.40) that, for any \(\varepsilon\in(0,1)\) small enough,
Therefore taking the supremum in t on the left-hand side of (3.41) and choosing \(\varepsilon\in(0,1)\) small enough, we can derive estimate (3.19) from (3.30). The proof is complete. □
Lemma 3.9
Under the assumptions of Theorem 2.3, the following estimates hold for any \(t\in[0,T]\) and for \(\varepsilon>0\) small enough:
Proof
Differentiating (1.9) with respect to x and t, multiplying the result by \(v_{xt}\) and integrating by parts in \(L^{2}(0,M)\), we have
with
Using Theorem 2.2 and Lemma 3.8, the interpolation inequality, and Poincaré’s inequality, we can get
where
and
Applying Young’s inequality several times, we have, for any \(\varepsilon \in(0,1)\),
and
Thus it follows from (3.45)-(3.47) and Theorem 2.1 and Lemma 3.8 that
which, along with Theorem 2.2, further yields
Analogously, from Lemma 3.8, Theorem 2.1, and the embedding theorem, we can also derive that, for any \(\varepsilon\in(0,1)\),
which, combined with (3.44), (3.49), and Theorem 2.2, shows that, for any \(\varepsilon\in(0,1)\) small enough,
On the other hand, differentiating (1.9) with respect to x and t, we can derive from Theorem 2.2 and Lemma 3.8 that
Thus inserting (3.52) into (3.51) leads to (3.42).
Similarly, by (1.10), we have
where
By virtue of the embedding theorem and the Young inequality, we derive from Lemmas 3.1, 3.8, and (3.42) that, for any \(\varepsilon\in(0,1)\),
Differentiating (1.10) with respect to x and t, we can derive from Theorems 2.1-2.2 and Lemma 3.8 that
Inserting (3.54)-(3.58) into (3.53) yields (3.43). □
Lemma 3.10
Under the assumptions of Theorem 2.3, we have, for any \(t\in[0,T]\),
Proof
Adding (3.42)-(3.43) and choosing \(\varepsilon>0\) small enough, we get
Now multiplying (3.18) and (3.19) by ε and \(\varepsilon ^{\frac{3}{2}}\), respectively, then adding the results to (3.62) and taking ε sufficiently small, we obtain (3.59).
Differentiating (1.9) with respect to x and noting that \(\eta _{xxt}=(r^{2}v)_{xxx}\), we get
where
Differentiating (3.63) with respect to x, we have
where
Obviously, it follows from Theorem 2.1 and Lemmas 3.8-3.9 that
and
Multiplying (3.64) by \(\frac{\eta_{xxx}}{\eta}\) over \(L^{2}(0,M)\), we can obtain
which, along with (3.66), gives
It follows from (1.8)-(1.10) that
Using the embedding theorem, Theorems 2.1-2.2 and Lemmas 3.8-3.9, we can derive from (3.24)-(3.25), (3.59), and (3.68)-(3.70) that, for any \(t\in[0,T]\),
Differentiating (1.9)-(1.10) with respect to t and using Theorems 2.1-2.2 and Lemmas 3.8-3.9, we can deduce from (3.59), (3.68)-(3.71) that
which, combined with (3.24)-(3.25) and (3.72), implies
Therefore it follows from (3.71), (3.74), and the embedding theorem that
Now differentiating (3.64) with respect to x, we find
where
From the embedding theorem and Lemmas 3.8-3.9 and (3.68)-(3.75), we can derive
whence
It follows from (3.28)-(3.31) that
which, along with (3.52) and (3.59), gives
Thus from (3.68), (3.74), (3.77), and (3.79), it follows that
Multiplying (3.76) by \(\frac{\eta_{xxxx}}{\eta}\) in \(L^{2}(0,M)\), we can get
whence, by (3.80),
Differentiating (1.10) with respect to x and t, we can derive from Theorems 2.1-2.2 and Lemmas 3.8-3.9 that
Thus,
Differentiating (1.9) with respect to x three times, applying Lemmas 3.8-3.9, Theorems 2.1-2.2, and Poincaré’s inequality, we have
Thus it follows from (3.74), (3.79), and (3.82) that
Similarly, we can differentiate (1.10) with respect to x three times and use Lemmas 3.8-3.9, Theorems 2.1-2.2, Poincaré’s inequality, (3.74), (3.82), and (3.84) to find
Hence, (3.60)-(3.61) follow from (3.74), (3.82), (3.86), and (3.87). □
Finally, combining Lemmas 3.8-3.10, we complete the proof of Theorem 2.3.
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Zhang, J. Global existence of solutions for a fluid model of a neutron star. Bound Value Probl 2016, 121 (2016). https://doi.org/10.1186/s13661-016-0628-1
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DOI: https://doi.org/10.1186/s13661-016-0628-1