Abstract
We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler–Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier–Stokes–Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in \(L^p\), and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum (\(\rho =0\)) and far-field (\(\rho =\infty \)) are carefully analyzed. Owing to the generality of the pressure law, only the \(W^{-1,p}_{\textrm{loc}}\)-compactness of weak entropy dissipation measures with \(p\in [1,2)\) can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the \(L^p\) compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.
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1 Introduction
We are concerned with global finite-energy solutions of the three-dimensional (3-D) compressible Euler–Poisson equations (CEPEs) that take the form:
for \((t,\textbf{x}):=(t,x_1,x_2,x_3)\in \mathbb {R}_+^4:=\mathbb {R}_{+}\times \mathbb {R}^{3}=(0,\infty )\times \mathbb {R}^3\). System (1.1) is used to model the motion of compressible gaseous stars under a self-consistent gravitational field (cf. [7]), where \(\rho \) is the density, \(P=P(\rho )\) is the pressure, \(\mathcal {M}\in \mathbb {R}^{3}\) is the momentum, \(\Phi \) represents the gravitational potential of gaseous stars as \({k_{g}}>0\), \(\nabla =(\partial _{x_1}, \partial _{x_2}, \partial _{x_3})\), and \(\Delta =\partial _{x_1x_1}+\partial _{x_2x_2}+\partial _{x_3x_3}\). Without loss of generality, by scaling, we take \({k_{g}}=1\) throughout this paper.
The constitutive pressure-density relation \(P(\rho )\) depends on the types of gaseous stars. The class of polytropic gases, i.e.,
has been widely investigated in mathematics. From the point view of astronomy, the constitutive pressure \(P(\rho )\) for certain gaseous stars is not of the polytropic form. For example, the pressure law of a white dwarf star takes the following form (cf. [7, 66]):
where \(\mathcal {C}_{1}, \mathcal {C}_{2}\), and \(\mathcal {C}_{3}\) are positive constants. It can be checked that \(P(\rho )\cong \kappa _1\rho ^\frac{5}{3}\) as \(\rho \rightarrow 0\) and \(P(\rho )\cong \kappa _2\rho ^\frac{4}{3}\) as \(\rho \rightarrow \infty \) for some positive constants \(\kappa _1\) and \(\kappa _2\).
In this paper, we consider a general pressure law in which any pressure function \(P(\rho )\) satisfies the following conditions:
-
(i)
The pressure function \(P(\rho )\) is in \(C^1([0,\infty ))\cap C^4(\mathbb {R}_+)\) and satisfies the hyperbolic and genuinely nonlinear conditions:
$$\begin{aligned} P'(\rho )>0,\quad 2P'(\rho )+\rho P''(\rho )>0\qquad \,\, \text {for }\rho >0. \end{aligned}$$(1.4) -
(ii)
There exists a constant \(\rho _{*}>0\) such that
$$\begin{aligned} P(\rho )=\kappa _{1} \rho ^{\gamma _1}\big (1+\mathcal {P}_1(\rho )\big ) \qquad \text { for } \rho \in [0, \rho _{*}), \end{aligned}$$(1.5)with some constants \(\gamma _1\in (1,3)\) and \(\kappa _1>0\), and a function \(\mathcal {P}_1(\rho )\in C^4(\mathbb {R}_+)\) satisfying that \(|\mathcal {P}_1^{(j)}(\rho )|\le C_{*}\rho ^{\gamma _1-1-j}\) for \(\rho \in (0,\rho _{*})\) and \(j=0,\cdots ,4\), where \(C_{*}>0\) is a constant depending only on \(\rho _{*}\).
-
(iii)
There exists a constant \(\rho ^{*}> \rho _{*}>0\) such that
$$\begin{aligned} P(\rho )=\kappa _2\rho ^{\gamma _2}\big (1+\mathcal {P}_2(\rho )\big ) \qquad \text { for } \rho \in [\rho ^{*},\infty ), \end{aligned}$$(1.6)with some constants \(\gamma _2\in (\frac{6}{5},\gamma _{1}]\) and \(\kappa _{2}>0\), and a function \(\mathcal {P}_2(\rho )\in C^{4}(\mathbb {R}_{+})\) satisfying that \(|\mathcal {P}_{2}^{(j)}(\rho )|\le C^{*}\rho ^{-\epsilon -j}\) for \(\rho \in [\rho ^{*},\infty )\) and \(j=0,\cdots ,4\), where \(\epsilon >0\), and \(C^{*}>0\) is a constant depending only on \(\rho ^{*}\).
It is direct to see that the polytropic gases in (1.2) satisfy assumptions (1.4)–(1.6). Moreover, the white dwarf star (1.3) is also included with
The restriction: \(\gamma _2>\frac{6}{5}\) is necessary to ensure the global existence of finite-energy solutions with finite total mass. Such a condition is also needed for the existence of the Lane–Emden solutions; see [7, 47].
We consider the Cauchy problem of (1.1) with the initial data:
subject to the far field condition:
The global existence of solutions of the Cauchy problem (1.1) and (1.8)–(1.9) is a longstanding open problem. Many efforts have been made for the polytropic gas case (1.2). Considerable progress has been made on the smooth or special solutions under some restrictions on the initial data. Among the most famous solutions of CEPEs (1.1) are the Lane–Emden steady solutions (cf. [47]), which describe spherically symmetric gaseous stars in equilibrium and minimize the energy among all possible configurations (cf. [46]). There exist expanding solutions for the non-steady CEPEs (1.1). Hadzić–Jang [34] proved the nonlinear stability of the affine solution (which is linearly expanding) under small spherically symmetric perturbations for \(\gamma =\frac{4}{3}\), while the stability problem for \(\gamma \ne \frac{4}{3}\) is still widely open. A class of linearly expanding solutions for \(\gamma =1+\frac{1}{k}\) with \(k\in \mathbb {N}\backslash \{1\}\), or \(\gamma \in (1,\frac{14}{13})\), was further constructed in [35]. For \(1< \gamma <\frac{4}{3}\), the concentration (collapse) phenomena may happen. Indeed, as \(\gamma =\frac{4}{3}\), there exists an homologous concentration solution; see [28, 30, 55]. More recently, Guo–Hadzić–Jang [31] observed a continued concentration solution for \(1< \gamma <\frac{4}{3}\); see also [37]. A kind of smooth radially symmetric self-similar solutions exhibiting gravitational collapse for \(1\le \gamma <\frac{4}{3}\) can be found in [32, 33]. We refer to [51, 54] for the local well-posedness of smooth solutions.
Owing to the strong nonlinearity and hyperbolicity, the smooth solutions of (1.1) with (1.2) may break down in a finite time, especially when the initial data are large (cf. [16, 55]). Therefore, weak solutions have to be considered for large initial data. For gaseous stars surrounding a solid ball, Makino [56] obtained the local existence of weak solutions for \(\gamma \in (1,\frac{5}{3}]\) with spherical symmetry; also see Xiao [69] for global weak solutions with a class of initial data. For this case, the possible singularity at the origin is prevented since the domain was considered outside a ball. Luo–Smoller [50] proved the conditional stability of rotating and non-rotating white dwarfs and rotating supermassive stars; see also Rein [61] for the conditional nonlinear stability of the Lane–Emden steady solutions.
Another fundamental question is whether global solutions can be constructed via the vanishing viscosity limit of the solutions of the compressible Navier–Stokes–Poisson equations (CNSPEs):
where \(D(\frac{\mathcal {M}}{\rho })=\frac{1}{2}\big (\nabla (\frac{\mathcal {M}}{\rho })+(\nabla (\frac{\mathcal {M}}{\rho }))^{\bot }\big )\) is the stress tensor, the Lamé (shear and bulk) viscosity coefficients \(\mu (\rho )\) and \(\lambda (\rho )\) depend on the density (that may vanish on the vacuum) and satisfy
and parameter \(\varepsilon >0\) is the inverse of the Reynolds number. Formally, as \(\varepsilon \rightarrow 0\), the sequence of the solutions of CNSPEs (1.10) converges to a corresponding solution of CEPEs (1.1). However, the rigorous proof has been one of the most challenging problems in mathematical fluid dynamics; see Chen-Feldman [9] and Dafermos [21].
The limit problem with vanishing physical viscosity dates back to the pioneering paper by Stokes [65]. Most of the known results were around the inviscid limit from the compressible Navier–Stokes to the Euler equations for the polytropic gas case (1.2). The first rigorous proof of the vanishing viscosity limit from the Navier–Stokes to the Euler equations was provided by Gilbarg [29], in which he established the existence and inviscid limit of the Navier–Stokes shock layers. For the case of large data, due to the lack of \(L^{\infty }\) uniform estimate, the \(L^{\infty }\) compensated compactness framework [22,23,24, 36, 48, 49] fails to work directly in the inviscid limit of the compressible Navier–Stokes equations. An \(L^{p}\) compensated compactness framework was first studied by LeFloch–Westdickenberg [43] for the isentropic Euler equations for the case \(\gamma \in (1,\frac{5}{3})\) in (1.2), and was further developed by Chen–Perepelitsa [13] to all \(\gamma >1\) for (1.2) with a simplified proof; see also [17] for spherically symmetric solutions of the M-D isentropic Euler equations. We also refer to [63, 64] for the 1-D case of asymptotically isothermal gas, i.e., \(\gamma _2=1\) in (1.6). More recently, Chen–He–Wang–Yuan [10] established both the strong inviscid limit of CNSPEs (1.10) and the global existence of spherically symmetric solutions of CEPEs (1.1) with large data for polytropic gases (1.2).
The main purpose of this paper is to establish the global existence of spherically symmetric finite-energy solutions of (1.1) with general pressure law (1.4)–(1.6):
subject to the initial condition:
and the asymptotic boundary condition:
Systems (1.1) and (1.10) for spherically symmetric solutions take the following respective forms:
and
The study of spherically symmetric solutions is motivated by many important physical problems such as stellar dynamics including gaseous stars and supernovae formation [7, 60, 68]. An important question is how the waves behave as they move radially inward near the origin, especially under the self-gravitational force for gaseous stars. The spherically symmetric solutions of the compressible Euler equations may blow up near the origin [20, 44, 57, 68] at certain time in some situations. Considering the effect of gravitation, a fundamental problem for CEPEs (1.1) is whether a concentration (delta-measure) is formed at the origin. This problem was answered in [10] for polytropic gases in (1.2) when the initial total-energy is finite that no delta-measure is formed for the density at the origin for the two cases: (i) \(\gamma >\frac{6}{5}\); (ii) \(\gamma \in (\frac{6}{5}, \frac{4}{3}]\) and the initial total-energy is finite and the total mass is less than a critical mass.
In this paper, we establish the global existence of finite-energy solutions of the Cauchy problem (1.1) and (1.13)–(1.14) with spherical symmetry as the inviscid limits of global weak solutions of CNSPEs (1.10) with general pressure law (1.4)–(1.6), especially including the white dwarf star (1.3). The \(L^p\) compensated compactness framework for the general pressure is also established. Moreover, it is proved that no delta-measure is formed for the density at the origin in the limit, and the critical mass for the white dwarf star is the same as the Chandrasekhar limit for the polytropic gas (1.2) with \(\gamma =\frac{4}{3}\). The precise statements of the main results are given in Sect. 2.
To achieve these, the main strategy is to develop entropy analysis, uniform estimates in \(L^p\), and a more general compensated compactness framework to prove that there exists a strongly convergent subsequence of solutions of CNSPEs (1.10) and show that the limit is the finite-energy weak solution of CEPEs (1.1) with general pressure law. This consists of the following three steps:
-
Establish the uniform \(L^p\) estimates of the solutions of CNSPEs (1.10) independent of \(\varepsilon \) for some \(p>1\);
-
Show the compactness for weak entropy dissipation measures;
-
Prove that the associated Young measure \(\nu _{(t,r)}\) is the delta measure almost everywhere which leads to a subsequence of solutions of CNSPEs (1.10) strongly converging to the global finite-energy solution of CEPEs (1.1).
The generality of pressure \(P(\rho )\) causes essential difficulties in the analysis for all of the above steps. We now describe these difficulties and show how they can be overcome:
(i) The crucial step in the \(L^p\) estimates is to show that \(\rho |u|^3\) (\(u:=\frac{m}{\rho }\) is the velocity) is uniformly bounded in \(L^1_\textrm{loc}\). This estimate might be obtained through constructing appropriate entropy \(\hat{\eta }\), which is a solution of \((\rho ,u)\) to the entropy equation:
with corresponding entropy flux \(\hat{q}\). If \((\rho ,u)\) is the solution of (1.16), any entropy-entropy flux pair (entropy pair, for short) \((\hat{\eta },\hat{q})\) satisfies
see (5.68) below. For the polytropic gas case (1.2), there is an explicit formula of the entropy kernel \(\chi (\rho ,u)\) so that \(\chi * \psi \) is the entropy, where \(*\) denotes the convolution and \(\psi (s)\) is any smooth function. By choosing \(\psi (s)=\frac{1}{2}s|s|\) as in [10], the corresponding entropy flux \(\hat{q}\) satisfies that \(\hat{q}\ge c_0\rho |u|^3\) and \(-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m\le 0\). Then the uniform bound of \(\rho |u|^3r^2\) in \(L^1_\textrm{loc}\) follows (cf. [10]).
However, there is no explicit formula of the entropy kernel \(\chi \) for the general pressure satisfying (1.4)–(1.6). Even for the special entropy pair generated by \(\psi (s)=\frac{1}{2}s|s|\), it is difficult to prove that \(\hat{q}\ge c_0\rho |u|^3\) and \(-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m\le 0\), due to the lack of explicit formula of the entropy kernel \(\chi \). Hence, the above approach does not apply directly, so we have to seek a new method to establish the uniform local integrability of \(\rho |u|^3\). One of the novelties of this paper is that a special entropy \(\hat{\eta }\) is constructed by solving a Goursat problem of the entropy equation (1.17) in the domain: \(|u|\le k(\rho ):=\int _{0}^{\rho }\sqrt{P'(y)}/{y}\,\textrm{d} y\), so that \(\hat{\eta }\) is chosen as the mechanical energy \(\eta ^*\) (see (2.13)) when \(u\ge k(\rho )\), \(-\eta ^*\) when \(u\le -k(\rho )\), and the boundary condition for the Goursat problem is given on the characteristics curves: \(u\pm k(\rho )=0\). One advantage of such a special entropy pair \((\hat{\eta },\hat{q})\) is that \(\hat{q}\ge c_0\rho |u|^3\) as \(|u|\ge k(\rho )\), and \(|\hat{q}|\le C\rho ^{\gamma _2+1}\) for large \(\rho \) as \(|u|\le k(\rho )\) via careful analysis for the Goursat problem; see Lemma 5.8 for details. Moreover, \(-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m\) vanishes as \(|u|\ge k(\rho )\). Similarly, \(|-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m|\le C\rho ^{\gamma _2+1}\) for large \(\rho \) as \(|u|\le k(\rho )\).
To show \(\rho |u|^3\) is uniformly bounded in \(L^1_\textrm{loc}\), it remains to prove that
is uniformly bounded for any \(T>0\) and \(d>0\). It should be noted that the local integrability \(\int _{0}^{T}\int _{d}^{D}\rho ^{\gamma _2+1}\,\textrm{d}r\textrm{d}t\le C\) was obtained in [10], but it is not enough yet to obtain the uniform \(L^1_\textrm{loc}\) estimate for \(\rho |u|^3\). Fortunately, we can obtain even stronger estimate than (1.18), i.e.,
by an elaborate analysis; see Lemma 5.6 and Corollary 5.7 for details.
(ii) For the polytropic gas case in (1.2), Chen–Perepelitsa [13, 14] and Chen–He–Wang–Yuan [10] proved the \(H_\textrm{loc}^{-1}\)–compactness for weak entropy dissipation measures via the explicit formula of the weak entropy kernel \(\chi \) by convolution with any test function of compact support, which also implies that the entropy pair \((\eta , q)\) is in \(L^r_\textrm{loc}, r>2\). However, it is not clear how the \(H_\textrm{loc}^{-1}\)–compactness for the general pressure satisfying (1.4)–(1.6) can be shown by using the expansions of the weak entropy kernel established in [11, 12]. Motivated by [64], we instead show the \(W_{\textrm{loc}}^{-1,p}\)–compactness for \(1\le p<2\), so that an improved div-curl lemma (cf. [19]) applies, which leads to the commutation identity for the entropy pairs. In fact, we can show that the entropy flux function q is bounded by \(\rho ^{\frac{\gamma _2+1}{2}}\) (see (4.81)) as \(\rho \) is large by careful analysis on the expansion of the entropy pair so that \(q\in L^2_\textrm{loc}\). Then the interpolation compactness yields the \(W^{-1,p}\) compactness for \(1\le p<2\); see Lemma 7.1 for details.
(iii) The argument for the reduction of the associated Young measure \(\nu _{(t,r)}(\rho , u)\), introduced in [10, 13, 14], for the polytropic gas case in (1.2), can be roughly stated as follows: Show first that every connected subset of the support of the Young measure is a bounded interval; then use the \(L^\infty \) reduction technique introduced in [8, 22, 24, 48] for a bounded supported Young measure to show that the Young measure is either a delta measure or supported on the vacuum line. This method essentially relies on the explicit formula of the weak entropy kernel \(\chi \). For the general pressure law satisfying (1.4)–(1.6), the above method does not apply directly, since it is difficult to show that every connected subset of the support of the Young measure is a bounded interval. Motivated by [11, 12, 48, 63, 64], we carefully analyze the singularities of \(\partial ^{\lambda _1+1} \chi \) with \(\lambda _1=\frac{3-\gamma _1}{2(\gamma _1-1)}\) for large \(\rho \) and fully exploit the property: \((\rho ^{\gamma _2+1},\rho |u|^3)\in L^1(\textrm{d}\nu _{(t,r)})\) so that the \(\partial ^{\lambda _1+1}-\)derivatives can be operated in the commutation relation; see Lemmas 4.11–4.14 for details. Then we prove that the Young measure is either a delta measure or supported on the vacuum line by similar arguments as in [11, 12, 48, 64]. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.
Finally, we remark that there are some related results on CNSPEs (1.10) and the compressible Euler equations. For weak solutions of CNSPEs (1.10), we refer to [26, 38, 40, 41] with constant viscosity, and [25, 27, 70] with density-dependent viscosity. Recently, Luo-Xin-Zeng [51,52,53] proved the large-time stability of the Lane–Emden solution for \(\gamma \in (\frac{4}{3},2)\). We also refer to the BD entropy developed in [2,3,4,5], which provides a new estimate for the gradient of the density. For the compressible Euler equations, we refer to [8, 15, 39, 44, 62] and the references cited therein.
The rest of this paper is organized as follows: In Sect. 2, the finite-energy solutions of the Cauchy problem (1.1) and (1.8)–(1.9) for CEPEs are introduced, and the main theorems of this paper are given. In Sect. 3, some elementary quantities and basic properties about the pressure and related internal energy are provided, and then some remarks on \(M_\textrm{c}\) are also given. The entropy analysis for weak entropy pairs for the general pressure satisfying (1.4)–(1.6) is presented in Sect. 4, especially a special entropy pair is constructed by solving a Goursat problem for the entropy equation (2.14). In Sect. 5, a free boundary problem (5.1)–(5.6) for (1.16) is analyzed, and some uniform estimates of solutions are derived, including the basic energy estimate, the BD-type entropy estimate, and the higher integrabilities of the density and the velocity. In Sect. 6, the global existence of weak solutions of CNSPEs (1.10) is established, and some uniform \(L^p\) estimates in Theorem 2.1 are also obtained. In Sect. 7, we prove the \(W_{\textrm{loc}}^{-1,p}\)–compactness of the entropy dissipation measures for the weak solutions of (1.16) and complete the proof of Theorem 2.1. In Sect. 8, the \(L^p\)–compensated compactness framework for the general pressure law (1.4)–(1.6) (Theorem 2.2) is established, which leads to the proof of Theorem 2.3 by taking the inviscid limit of weak solutions of CNSPEs (1.10) in Sect. 9. Appendix A is devoted to the presentation of both the sharp Sobolev inequality that is used in Sect. 5 and some variants of Grönwall’s inequality which are used in the proof of several estimates in Sect. 4.
Notations: Throughout this paper, we denote \(C^{\alpha } (\Omega ), L^{p}(\Omega ), W^{k, p}(\Omega )\), and \(H^{k}(\Omega )\) as the standard Hölder space, and the corresponding Sobolev spaces, respectively, on domain \(\Omega \) for \(\alpha \in (0,1)\) and \(p\in [1, \infty ]\). \(C_{0}^{k}(\Omega )\) represents the space of continuously differentiable functions up to the kth order with compact support over \(\Omega \), and \(\mathcal {D}(\Omega ):=C_{0}^{\infty }(\Omega )\). We also use \(L^{p}(I; r^{2}\textrm{d}r)\) or \(L^{p}([0, T) \times I; r^{2}\textrm{d} r\textrm{d} t)\) for an open interval \(I \subset \mathbb {R}_{+}\) with measure \(r^{2}\textrm{d}r\) or \(r^{2}\textrm{d}r \textrm{d}t\) correspondingly, and \(L_{\textrm{loc}}^{p}([0, \infty ); r^{2}\textrm{d}r)\) to represent \(L^{p}([0, R]; r^{2}\textrm{d} r)\) for any fixed \(R>0\).
2 Mathematical Problem and Main Theorems
The spherically symmetric initial data function \((\rho _{0},\mathcal {M}_{0})(\textbf{x})\) given in (1.13) is assumed to be of both finite initial total-energy:
and initial total-mass:
where the internal energy \(e(\rho )\) is related to the pressure by
and \(\omega _n:=\frac{2\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2})}\) denotes the surface area of the unit sphere in \(\mathbb {R}^n\). The initial potential \(\Phi _0(\textbf{x})\) is determined by
For \(\gamma _{2}\in (\frac{6}{5},\frac{4}{3}]\), we define the critical mass \(M_\textrm{c}\) as follows:
(i) When \(\gamma _2=\frac{4}{3}\),
where \(M_\textrm{ch}\) is the Chandrasekhar limit that is the total mass of the Lane–Emden steady solution \((\rho _{s}(|\textbf{x}|),0)\) for \(P(\rho )=\kappa _2\rho ^{\frac{4}{3}}\): \(\rho _{s}(|\textbf{x}|)\) has compact support and is determined by the equations:
with the center density \(\rho _{s}(0)=\varrho \). It is well-known that \(M_\textrm{ch}\) is a uniform constant with respect to the center density \(\varrho \) (cf. [7]).
(ii) When \(\gamma _2\in (\frac{6}{5},\frac{4}{3})\),
with
and
It is clear in (2.6)–(2.8) that \(M_\textrm{c}(\beta )\) is well determined for \(\beta >0\) and \(\gamma _2\in (\frac{6}{5},\frac{4}{3})\). Some useful properties of \(M_\textrm{c}:=\sup _{\beta >0}M_\textrm{c}(\beta )\) will be presented in Proposition 3.3 below. We also point out that \(M_\textrm{c}\) in (2.5) is strictly larger than the one obtained in [10, (2.8)] for \(\gamma _2=\frac{4}{3}\) (cf. [18]).
For the spherically symmetric initial data \((\rho _{0}, m_{0},\Phi _{0})(r)\) imposed in (1.12)–(1.14) satisfying (2.1)–(2.2), using similar arguments as in [10, Appendix A], we can construct a sequence of approximate initial data functions \((\rho _{0}^{\varepsilon },m_{0}^{\varepsilon }, \Phi _{0}^{\varepsilon })(r)\) satisfying
Moreover, as \(\varepsilon \rightarrow 0\), \((E_{0}^{\varepsilon }, E_{1}^{\varepsilon }) \rightarrow (E_{0}, 0)\) and
where \(\tilde{q}\in \{1,\gamma _{2}\}\). Furthermore, there exists \(\varepsilon _0\in (0,1]\) such that, for any \(\varepsilon \in (0,\varepsilon _0]\),
where \(M_\textrm{c}^{\varepsilon }\) is defined in (2.5)–(2.8) by replacing \(E_0\) with \(E_0^{\varepsilon }\).
Now we introduce the weak entropy pairs of the 1-D isentropic Euler system (cf. [11, 42]):
A pair of functions \((\eta (\rho ,m),q(\rho ,m))\) is called an entropy pair of (2.11) if
Moreover, \(\eta (\rho ,m)\) is called a weak entropy if \(\eta (\rho ,m)\vert _{\rho =0}=0\), and a convex entropy if \(\nabla ^2\eta (\rho , m)\ge 0\). The mechanical energy and energy flux pair is defined as
which is a convex weak entropy pair. From (2.12), any entropy satisfies
with \(u=\frac{m}{\rho }\). It is known in [11, 12, 48, 49] that any regular weak entropy can be generated by the convolution of a smooth function \(\psi (x)\) with the fundamental solution \(\chi (\rho ,u,s)\) of the entropy equation (2.14), i.e.,
The corresponding entropy flux is generated from the flux kernel \(\sigma (\rho ,u,s)\) (see (4.56)), i.e.,
We first consider the Cauchy problem of CNSPEs (1.10) with approximate initial data:
subject to the far field condition:
For concreteness, we take \(\varepsilon \in (0,1]\) and the viscosity coefficients \((\mu (\rho ),\lambda (\rho ))=(\rho ,0)\) in (1.10).
Definition 2.1
A triple \((\rho ^{\varepsilon }, \mathcal {M}^{\varepsilon },\Phi ^{\varepsilon })(t,\textbf{x})\) is said to be a weak solution of the Cauchy problem (1.10) and (2.17) if
-
(i)
\(\rho ^{\varepsilon }(t, \textbf{x}) \ge 0\), and \((\mathcal {M}^{\varepsilon }, \frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}})(t, \textbf{x})=\textbf{0}\,\) a.e. on \(\{(t, \textbf{x})\,:\,\rho ^{\varepsilon }(t, \textbf{x})=0\}\,\)(vacuum),
$$\begin{aligned} \begin{aligned}&\rho ^{\varepsilon } \in L^{\infty }(0, T ; L^{\gamma _2}(\mathbb {R}^{3})), \quad \nabla \sqrt{\rho ^{\varepsilon }} \in L^{\infty }(0, T ; L^{2}(\mathbb {R}^{3})), \\ {}&\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \in L^{\infty }(0, T ; L^{2}(\mathbb {R}^{3})),\quad \Phi ^{\varepsilon }\in L^{\infty }(0,T;L^{6}(\mathbb {R}^3)),\quad \nabla \Phi ^{\varepsilon }\in L^{\infty }(0,T;L^2(\mathbb {R}^3)). \end{aligned} \end{aligned}$$ -
(ii)
For any \(t_{2} \ge t_{1} \ge 0\) and any \(\zeta (t, \textbf{x}) \in C_{0}^{1}([0, \infty ) \times \mathbb {R}^{3})\), the mass equation (1.10)\(_{1}\) holds in the sense:
$$\begin{aligned} \int _{\mathbb {R}^{3}}(\rho ^{\varepsilon } \zeta )(t_{2}, \textbf{x})\,\textrm{d} \textbf{x} -\int _{\mathbb {R}^{3}}(\rho ^{\varepsilon } \zeta )(t_{1}, \textbf{x})\,\textrm{d} \textbf{x} =\int _{t_{1}}^{t_{2}} \int _{\mathbb {R}^{3}}(\rho ^{\varepsilon } \zeta _{t}+\mathcal {M}^{\varepsilon } \cdot \nabla \zeta )(t, \textbf{x}) \,\textrm{d} \textbf{x} \textrm{d} t. \end{aligned}$$ -
(iii)
For any \(\Psi =(\Psi _{1}, \Psi _{2}, \Psi _{3})(t,\textbf{x}) \in (C_{0}^{2}([0, \infty ) \times \mathbb {R}^{3}))^3\), the momentum equations (1.10)\(_{2}\) hold in the sense:
$$\begin{aligned}&\int _{\mathbb {R}_{+}^4}\Big (\mathcal {M}^{\varepsilon } \cdot \Psi _{t} +\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \big (\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \nabla \big ) \Psi +P(\rho ^{\varepsilon }) \nabla \cdot \Psi \Big )\,\textrm{d} \textbf{x} \textrm{d} t +\int _{\mathbb {R}^{3}} \mathcal {M}_{0}^{\varepsilon }(\textbf{x}) \cdot \Psi (0, \textbf{x})\,\textrm{d} \textbf{x} \\&\quad =-\varepsilon \int _{\mathbb {R}_{+}^{4}}\Big (\frac{1}{2} \mathcal {M}^{\varepsilon } \cdot \big (\Delta \Psi +\nabla (\nabla \cdot \Psi )\big ) +\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}}\cdot \big (\nabla \sqrt{\rho ^{\varepsilon }} \cdot \nabla \big )\Psi \Big )\,\textrm{d} \textbf{x} \textrm{d} t\\&\qquad -\varepsilon \int _{\mathbb {R}_{+}^{4}} \nabla \sqrt{\rho ^{\varepsilon }} \cdot \big (\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \nabla \big ) \Psi \,\textrm{d}\textbf{x}\textrm{d}t +\int _{\mathbb {R}_{+}^{4}}\big ( \rho ^{\varepsilon } \nabla \Phi ^{\varepsilon } \cdot \Psi \big )(t, \textbf{x})\,\textrm{d}\textbf{x}. \end{aligned}$$ -
(iv)
For any \(t\ge 0\) and \(\xi (\textbf{x})\in C_{0}^1(\mathbb {R}^3)\),
$$\begin{aligned} \int _{\mathbb {R}^3}\nabla \Phi ^{\varepsilon }(t,\textbf{x})\cdot \nabla \xi (\textbf{x})\,\textrm{d}\textbf{x}=-\int _{\mathbb {R}^3}\rho ^{\varepsilon }(t,\textbf{x})\xi (\textbf{x})\,\textrm{d}\textbf{x}. \end{aligned}$$
Then we have
Theorem 2.1
(Global existence of spherically symmetric solutions for CNSPEs). Assume that the initial data function \((\rho _{0}^{\varepsilon },\mathcal {M}_{0}^{\varepsilon },\Phi _{0}^{\varepsilon })(\textbf{x})\) is given in (2.17)–(2.18) with \((\rho _{0}^{\varepsilon },m_{0}^{\varepsilon },\Phi _{0}^{\varepsilon })(r)\) satisfying (2.9)–(2.10). Then, for each fixed \(\varepsilon \in (0, 1]\), there exists a global weak solution \((\rho ^{\varepsilon },\mathcal {M}^{\varepsilon }, \Phi ^{\varepsilon })(t,\textbf{x})\) of the Cauchy problem (1.10) and (2.17)–(2.18) in the sense of Definition 2.1 with following spherically symmetric form:
such that, for \(t\ge 0\),
Furthermore, for \((\rho ^{\varepsilon }, m^{\varepsilon }, \Phi ^{\varepsilon })(t, r)\), there exists a measurable function \(u^{\varepsilon }(t,r)\) with
and \(u^{\varepsilon }(t,r):=0~{ a.e.}\text { on }\big \{(t,r)\,:\,\rho ^{\varepsilon }(t,r)=0\text { or }r=0\big \}\) such that \(m^{\varepsilon }(t,r)=(\rho ^{\varepsilon }u^{\varepsilon })(t,r)\) a.e. on \(\mathbb {R}_{+}^2:=\mathbb {R}_+\times \mathbb {R}_+\). Moreover, the following properties hold:
for any \(T \in \mathbb {R}_{+}\) and interval \([d, D]\Subset (0, \infty )\), where \(C(M, E_{0})\), \(C(M, E_{0}, T)\), and \(C(d, D, M, E_{0}, T)\) are positive constants independent of \(\varepsilon \). In addition, for \(\varepsilon \in (0,1]\),
for any \(p\in [1,2)\), where \(\psi (s)\) is any smooth function with compact support on \(\mathbb {R}\).
Remark 2.1
In this paper, we require the density-dependent viscosity coefficients \(\mu (\rho )\) and \(\lambda (\rho )\) to satisfy the BD entropy relation (cf. [2,3,4,5]):
which is important for us to derive the estimate for the derivative of the density. Under the physical restriction (1.11) and the BD entropy relation (2.27), \(\lambda (\rho )\) cannot be a non-zero constant. Since we focus mainly on the global existence of weak solutions for CEPEs by the vanishing viscosity limit of weak solutions of CNSPEs which means the viscous terms will vanish eventually, we consider only the special case \((\mu (\rho ),\lambda (\rho ))=(\rho ,0)\) in the present paper, which corresponds to the well-known Saint-Venant model of shallow water.
Recently, in [6, 27], the global existence of weak solutions was established for the compressible Navier–Stokes equations and CNSPEs for a class of general density-dependent viscous coefficients satisfying the BD entropy relation, respectively. Motivated by [6, 27], it should be able to extend our results to a class of more general viscous coefficients. However, for such general viscous coefficients \(\mu (\rho )\) and \(\lambda (\rho )\) satisfying the BD relation, we have to check the uniform estimates of the solutions and the validity of vanishing viscosity limit \(\varepsilon \rightarrow 0\) so that major modifications to our present paper are required, which is out of scope of this paper.
Now we introduce the notion of finite-energy solutions of CEPEs (1.1).
Definition 2.2
A measurable vector function \((\rho ,\mathcal {M},\Phi )\) is said to be a finite-energy solution of the Cauchy problem (1.1) and (1.8)–(1.9) provided that
-
(i)
\(\rho (t,\textbf{x})\ge 0\) a.e., and \((\mathcal {M},\frac{\mathcal {M}}{\sqrt{\rho }})(t, \textbf{x})=\textbf{0}\) a.e. on \(\{(t,\textbf{x})\in \mathbb {R}_{+}^{4}:\,\rho (t,\textbf{x})=0\}\) (vacuum).
-
(ii)
For a.e. \(t>0\), the total energy is finite:
$$\begin{aligned} \left\{ \begin{aligned}&\int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )+\frac{1}{2}|\nabla \Phi |^{2}\Big )(t, {\textbf {x}}) \,\text {d} {\textbf {x}} \le C(E_{0}, M), \\&\int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )-\frac{1}{2}|\nabla \Phi |^{2}\Big )(t, {\textbf {x}})\,\text {d} {\textbf {x}}\\ {}&\quad \,\, \le \int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}_{0}}{\sqrt{\rho _{0}}}\Big |^{2} +\rho _{0} e(\rho _{0})-\frac{1}{2}|\nabla \Phi _{0}|^{2}\Big )({\textbf {x}})\,\text {d} {\textbf {x}}. \end{aligned}\right. \end{aligned}$$(2.28) -
(iii)
For any \(\zeta (t, \textbf{x})\in C_{0}^{1}([0,\infty )\times \mathbb {R}^{3})\),
$$\begin{aligned} \int _{\mathbb {R}_{+}^{4}}(\rho \zeta _{t}+\mathcal {M}\cdot \nabla \zeta )\,\textrm{d}\textbf{x}\textrm{d}t+\int _{\mathbb {R}^3}(\rho _{0}\zeta )(0,\textbf{x})\,\textrm{d}\textbf{x}=0. \end{aligned}$$(2.29) -
(iv)
For any \(\Psi (t, \textbf{x})=(\Psi _{1},\Psi _{2},\Psi _{3})(t,\textbf{x})\in (C_{0}^{1}([0,\infty )\times \mathbb {R}^3))^{3}\),
$$\begin{aligned}&\int _{\mathbb {R}_{+}^{4}}\Big (\mathcal {M}\cdot \partial _{t}\Psi +\frac{\mathcal {M}}{\sqrt{\rho }}\cdot (\frac{\mathcal {M}}{\sqrt{\rho }}\cdot \nabla )\Psi +P(\rho )\, \nabla \cdot \Psi \Big )\,\textrm{d}\textbf{x}\textrm{d}t+\int _{\mathbb {R}^3}\mathcal {M}_{0}(\textbf{x})\cdot \Psi (0,\textbf{x})\,\textrm{d}\textbf{x}\nonumber \\&\quad =\int _{\mathbb {R}_{+}^{4}}(\rho \nabla \Phi \cdot \Psi )(t, \textbf{x})\,\textrm{d} \textbf{x}. \end{aligned}$$(2.30) -
(v)
For any \(\xi (\textbf{x})\in C_{0}^{1}(\mathbb {R}^3)\),
$$\begin{aligned} \int _{\mathbb {R}^{3}} \nabla \Phi (t, {\textbf {x}}) \cdot \nabla \xi ({\textbf {x}}) \,\text {d}{} {\textbf {x}} =- \int _{\mathbb {R}^{3}} \rho (t, {\textbf {x}}) \xi ({\textbf {x}})\,\text {d}{} {\textbf {x}}\qquad \,\, \text{ for }\,\,{ a.e.}\,\, t \ge 0. \end{aligned}$$(2.31)
Remark 2.2
In the spherically symmetric form, Definition 2.2 becomes the following: A measurable vector function \((\rho ,\mathcal {M},\Phi )(t,\mathbf{{x}})=(\rho (t,r),m(t,r)\frac{\mathbf{{x}}}{r},\Phi (t,r))\) is said to be a spherically symmetric finite-energy solution of the Cauchy problem (1.1) and (1.13)–(1.14) provided that
-
(i)
\(\rho (t,r)\ge 0\) a.e., and \((m,\frac{m}{\sqrt{\rho }})(t,r)=\textbf{0}\) a.e. on \(\{(t,r)\in \mathbb {R}_{+}^2:\,\rho (t,r)=0\}\) (vacuum).
-
(ii)
For a.e. \(t>0\), the total energy is finite:
$$\begin{aligned} \left\{ \begin{aligned}&\int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )+\frac{1}{2}|\Phi _{r}|^{2}\Big )(t, r) \,r^2\text {d} r \le C(E_{0}, M), \\ {}&\int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )-\frac{1}{2}|\Phi _{r}|^{2}\Big )(t, r)\,r^2\text {d}r\\ {}&\quad \,\, \le \int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m_{0}}{\sqrt{\rho _{0}}}\Big |^{2}+\rho _{0} e(\rho _{0})-\frac{1}{2}| \Phi _{0r}|^{2}\Big )(r)\,r^2\text {d}r. \end{aligned}\right. \end{aligned}$$(2.32) -
(iii)
For any \(\zeta (t, r)\in C_{0}^{1}([0,\infty )\times \mathbb {R})\),
$$\begin{aligned}&\int _{\mathbb {R}_{+}^2}(\rho \zeta _{t}+m\zeta _{r})(t,r)\,r^2\textrm{d}r\textrm{d}t +\int _{0}^{\infty }\rho _{0}(r)\zeta (0,r)\,r^2\textrm{d}r=0. \end{aligned}$$(2.33) -
(iv)
For any \(\psi (t,r)\in C_{0}^{1}([0,\infty )\times \mathbb {R})\) with \(\psi (t,0)=0\) for all \(t\ge 0\),
$$\begin{aligned}&\int _{\mathbb {R}_{+}^2}m(t,r)\psi _{t}(t,r)\,r^2\textrm{d}r\textrm{d}t +\int _{\mathbb {R}_{+}^2}\big (\frac{m^2}{\rho }\big )(t,r)\,\psi _{r}(t,r)\,r^2\textrm{d}r\textrm{d}t\nonumber \\&\qquad +\int _{\mathbb {R}_{+}^2}P(\rho (t,r))\,(\psi _{r}+\frac{2}{r}\psi )(t,r)\,r^2\textrm{d}r\textrm{d}t+\int _{0}^{\infty }m_{0}(r)\,\psi (0,r)\,r^2\textrm{d}r\nonumber \\&\quad =\int _{\mathbb {R}_{+}^2}(\rho \Phi _{r})(t,r)\,\psi (t,r)\,r^2\textrm{d} r\textrm{d}t. \end{aligned}$$(2.34) -
(v)
For any \(\xi (r)\in C_{0}^{1}(\mathbb {R})\) and a.e. \(t\ge 0\),
$$\begin{aligned}&\int _{0}^{\infty }\Phi _{r}(t,r)\,\xi _{r}(r)\,r^2\textrm{d}r=-\int _{0}^{\infty } \rho (t,r)\,\xi ( r)\,r^2\textrm{d}r. \end{aligned}$$(2.35)
To establish the strong convergence of the inviscid limit of solutions \((\rho ^{\varepsilon }, \mathcal {M}^{\varepsilon }, \Phi ^{\varepsilon })(t,\textbf{x})\) of CNSPEs (1.10) obtained in Theorem 2.1 as \(\varepsilon \rightarrow 0\), we establish the following \(L^p\) compensated compactness framework for the 1-D Euler equations (2.11) with general pressure law (1.4)–(1.6), in which restriction \(\gamma _{2}\in (\frac{6}{5}, \gamma _{1}]\) in (1.6) can be relaxed to \(\gamma _{2}\in (1, \gamma _1]\).
Theorem 2.2
(\(L^p\) compensated compactness framework). Let
be a sequence of measurable functions with \(\rho ^{\varepsilon }\ge 0\) a.e. on \(\mathbb {R}_{+}^2\) satisfying the following two conditions:
-
(i)
For any \(T>0\) and \(K\Subset \mathbb {R}_+\), there exists \(C(K,T)>0\) independent of \(\varepsilon \) such that
$$\begin{aligned} \int _0^T\int _K\big ((\rho ^\varepsilon )^{\gamma _2+1}+\rho ^{\varepsilon }|u^{\varepsilon }|^3\big )\,\textrm{d}r\textrm{d}t\le C(K,T). \end{aligned}$$ -
(ii)
For any entropy pair \((\eta ^\psi ,q^\psi )\) defined in (2.15)–(2.16) with any smooth function \(\psi (s)\) of compact support on \(\mathbb {R}\),
$$\begin{aligned} \partial _{t} \eta ^{\psi }(\rho ^{\varepsilon }, m^{\varepsilon }) +\partial _{r} q^{\psi }(\rho ^{\varepsilon }, m^{\varepsilon }) \qquad \text{ is } \text{ compact } \text{ in }\, W_{\textrm{loc}}^{-1,1}(\mathbb {R}_{+}^{2}). \end{aligned}$$
Then there exists a subsequence (still denoted) \((\rho ^{\varepsilon },m^{\varepsilon })(t,r)\) and a vector function \((\rho ,m)(t,r)\) such that, as \(\varepsilon \rightarrow 0\),
where \(L_{\textrm{loc}}^{p}(\mathbb {R}_{+}^{2})\) represents \(L^{p}([0, T] \times K)\) for any \(T>0\) and compact set \(K \Subset \mathbb {R}_+\).
Now, we are ready to state our main theorem.
Theorem 2.3
(Global existence of finite-energy solutions). Let the pressure function \(P(\rho )\) satisfy (1.4)–(1.6), and let the spherically symmetric initial data \((\rho _0,\mathcal {M}_0, \Phi _{0})(\textbf{x})\) be given in (1.13)–(1.14) with \((\rho _{0},m_{0},\Phi _{0})(r)\) satisfying (2.1)–(2.2) and (2.4). Assume that \(\gamma _2>\frac{4}{3}\), or \(M<M_\textrm{c}\) as \(\gamma _2\in (\frac{6}{5},\frac{4}{3}]\). Then there exists a global finite-energy solution \((\rho ,\mathcal {M},\Phi )(t,\textbf{x})\) of (1.1) and (1.13)–(1.14) with spherical symmetry form (1.12) in the sense of Definition 2.2.
Remark 2.3
For the steady gaseous star problem, there is no white dwarf star if the total mass is larger than the so-called Chandrasekhar limit when \(\gamma \in (\frac{6}{5},\frac{4}{3}]\); see [7]. Theorem 2.3 requires similar restriction on the total mass when \(\gamma _2\in (\frac{6}{5},\frac{4}{3}]\) for non-steady gaseous stars. Moreover, in view of (2.5), for the non-steady white dwarf star, the critical mass is exactly the Chandrasekhar limit in the case that \(P(\rho )=\kappa _2\rho ^{\frac{4}{3}}\). It would be interesting to analyze whether the critical mass defined in (2.6)–(2.8) for \(\gamma _2\in (\frac{6}{5},\frac{4}{3})\) is optimal.
Remark 2.4
Theorem 2.3 can be extended to the 3-D compressible Euler equations, i.e., (1.1) with \(\Phi =0\). Moreover, the inviscid limit from the compressible Navier–Stokes equations to Euler equations with far-field vacuum can also be justified.
Remark 2.5
Theorem 2.3 also holds for the plasmas case, i.e., \(k_{g}=-1\) in (1.1), by a similar proof. In this case, the restriction: \(M<M_\textrm{c}\) can be removed, and condition \(\gamma _{2}>\frac{6}{5}\) can be relaxed to \(\gamma _2>1\) if the additional assumption: \(\rho _{0}\in L^{\frac{6}{5}}(\mathbb {R}^3)\) is imposed. We omit the proof in this paper for brevity and, instead, refer the reader to [10] for details.
3 Properties of the General Pressure Law and Related Internal Energy
In this section, we present some useful estimates involving the general pressure \(P(\rho )\) with (1.4)–(1.6) and the corresponding internal energy \(e(\rho )\), which are used in the subsequent development.
Denote \(c(\rho ):=\sqrt{P'(\rho )}\) as the speed of sound, and
By direct calculation, we can obtain the following asymptotic behaviors of \(P(\rho )\), \(e(\rho )\), and \(k(\rho )\).
Lemma 3.1
Assume that \(\rho _{*}\) given in (1.5) is small enough and \(\rho ^{*}\) given in (1.6) is large enough such that the following estimates hold:
-
(i)
When \(\rho \in (0,\rho _{*}]\),
$$\begin{aligned} \left\{ \begin{aligned}&\underline{\kappa }_{1}\rho ^{\gamma _1}\le P(\rho )\le \bar{\kappa }_{1}\rho ^{\gamma _1},\\&\underline{\kappa }_{1}\gamma _1\rho ^{\gamma _1-1}\le P'(\rho )\le \bar{\kappa }_{1}\gamma _1\rho ^{\gamma _1-1},\\&\underline{\kappa }_{1}\gamma _1(\gamma _1-1)\rho ^{\gamma _1-2}\le P''(\rho )\le \bar{\kappa }_{1}\gamma _1(\gamma _1-1)\rho ^{\gamma _1-2}, \end{aligned}\right. \end{aligned}$$(3.2)and when \(\rho \in [\rho ^{*},\infty )\),
$$\begin{aligned} \left\{ \begin{aligned}&\underline{\kappa }_{2}\rho ^{\gamma _2}\le P(\rho )\le \bar{\kappa }_{2}\rho ^{\gamma _2},\\&\underline{\kappa }_{2}\gamma _2\rho ^{\gamma _2-1}\le P'(\rho )\le \bar{\kappa }_{2}\gamma _2\rho ^{\gamma _2-1},\\&\underline{\kappa }_{2}\gamma _2(\gamma _2-1)\rho ^{\gamma _2-2}\le P''(\rho )\le \bar{\kappa }_{2}\gamma _2(\gamma _2-1)\rho ^{\gamma _2-2}, \end{aligned}\right. \end{aligned}$$(3.3)where we have denoted \(\,\underline{\kappa }_{i}:=(1-\mathfrak {a}_0)\kappa _{i}\) and \(\bar{\kappa }_{i}:=(1+\mathfrak {a}_0) \kappa _{i}\) with \(\mathfrak {a}_0=\frac{3-\gamma _1}{2(\gamma _1+1)}\) and \(i=1,2\).
-
(ii)
For \(e(\rho )\) and \(k(\rho )\), there exists \(C>0\) depending on \((\gamma _1, \gamma _2, \kappa _1,\kappa _2, \rho _{*}, \rho ^{*})\) such that
$$\begin{aligned}&C^{-1}\rho ^{\gamma _1-1}\le e(\rho )\le C\rho ^{\gamma _1-1},\,\, C^{-1}\rho ^{\gamma _1-2}\le e'(\rho )\le C\rho ^{\gamma _1-2} \,\,\,\,\, \text { for }\rho \in (0,\rho _{*}], \end{aligned}$$(3.4)$$\begin{aligned}&C^{-1}\rho ^{\gamma _2-1}\le e(\rho )\le C\rho ^{\gamma _2-1},\,\, C^{-1}\rho ^{\gamma _2-2}\le e'(\rho )\le C\rho ^{\gamma _2-2} \,\,\,\,\, \text { for }\rho \in [\rho ^{*},\infty ), \end{aligned}$$(3.5)and, for \(i=0,1\),
$$\begin{aligned}&\frac{\rho ^{\theta _{1}-i}}{C}\le k^{(i)}(\rho )\le C\rho ^{\theta _{1}-i},\,\, \frac{\rho ^{\theta _{1}-2}}{C}\le |k''(\rho )|\le C\rho ^{\theta _1-2}\,\,\,\,\, \text {for } \rho \in (0,\rho _{*}],\nonumber \\ \end{aligned}$$(3.6)$$\begin{aligned}&\frac{\rho ^{\theta _{2}-i}}{C}\le k^{(i)}(\rho )\le C\rho ^{\theta _{2}-i}, \,\, \frac{\rho ^{\theta _{2}-2}}{C}\le |k''(\rho )|\le C\rho ^{\theta _2-2} \,\,\,\,\,\text {for } \rho \in [\rho ^{*},\infty ), \nonumber \\ \end{aligned}$$(3.7)where \(\theta _{1}=\frac{\gamma _1-1}{2}\) and \(\theta _2=\frac{\gamma _2-1}{2}\).
It follows from (3.2)–(3.3) that
when \(\rho \in [0,\rho _{*}]\cup [\rho ^{*},\infty )\). For later use, we denote
Then it follows from (3.8) that
Motivated by [64], we have
Lemma 3.2
\(0<d(\rho )\le C\) for all \(\rho >0\), and
Proof
It follows from (1.4) that \( d(\rho )=1+\frac{\rho P''(\rho )}{2P'(\rho )}>0. \) Moreover, by (3.10), it is direct to see that \(d(\rho )\) is bounded. Using (1.6), we see that, for \(\rho \ge \rho ^{*}\),
Then, for \(\rho \ge \max \{\rho ^{*},(8C^{*})^{1/\epsilon }\}\),
where we have used that \(|\mathcal {P}_2^{(j)}(\rho )|\le C^{*}\rho ^{-\epsilon -j}\) for \(j=0,1,2,\) in the last inequality. \(\square \)
Hereafter, for simplicity of notation, we assume that (3.11) holds for \(\rho \ge \rho ^{*}\). Furthermore, using (1.6) and \(e'(\rho )=\frac{P(\rho )}{\rho ^2}\), we obtain that, for \(\rho \ge \rho ^{*}\),
which, with \(e(0)=0\) and \(|\mathcal {P}_{2}(\rho )|\le C^{*}\rho ^{-\epsilon }\), yields that, for any parameter \(\beta >0\),
Then we see that
With a careful analysis of \(C_{\max }(\beta )\), we obtain some estimates of \(M_\textrm{c}\) defined in (2.6).
Proposition 3.3
Let \(h(\rho )=P(\rho )\rho ^{-1}-(\gamma _2-1)e(\rho )\), and let \(\tilde{M}_\textrm{c}\) be the critical mass obtained in [10, (2.8)] for the polytropic gases in (1.2) with \(\gamma \in (\frac{6}{5},\frac{4}{3})\). Then \(M_\textrm{c}\) defined in (2.6)–(2.8) satisfies that \(M_\textrm{c}\le \tilde{M}_\textrm{c}\); in particular, \(M_\textrm{c}<\tilde{M}_\textrm{c}\) when \(h'(\rho )> 0\ \mathrm{for~all}\ \rho > 0\). For example,
satisfies conditions (1.4)–(1.6). If \(M_\textrm{c}(\delta )\) is the critical mass defined in (2.6)–(2.8) for pressure \(P_{\delta }(\rho )\), then \(M_\textrm{c}(\delta )<\tilde{M}_\textrm{c}\) for any \(\delta >0\).
Proof
For \(\gamma _2\in (\frac{6}{5},\frac{4}{3})\), it follows from (2.6)–(2.8) and (3.13) that, for any fixed \(\beta >0\),
which yields that \(M_\textrm{c}\le \tilde{M}_\textrm{c}\).
Let \(g(\rho ):=\rho ^{\frac{\gamma _2-1}{5\gamma _2-6}}\big (\beta +e(\rho )\big )^{-\frac{1}{5\gamma _2-6}}\). Then \(C_{\max }(\beta )=\max _{\rho \ge 0}g(\rho )\). Since \(e'(\rho )=\frac{P(\rho )}{\rho ^2}\), a direct calculation shows that
If \(h'(\rho )> 0\) for all \(\rho > 0\), then \(h(\rho )\ge h(0)=0\). Let \(K_{0}:=\max _{\rho> 0}h(\rho )>0\). For \(\beta \) small enough such that \(0<\beta <\frac{K_{0}}{\gamma _2-1}\), there exists a unique point \(\rho _{\beta }>0\) such that \(g'(\rho _{\beta })=0\), i.e.,
and \( C_{\max }(\beta )=g(\rho _{\beta }) =(\gamma _2-1)^{\frac{1}{5\gamma _2-6}} \big (P(\rho _{\beta })\rho _{\beta }^{-\gamma _2}\big )^{-\frac{1}{5\gamma _2-6}}. \) Moreover, it follows from (3.18) that \(\lim _{\beta \rightarrow 0+}\rho _{\beta }=0\). Thus, we see from (1.5) that
which, with (3.16\(_{1}\)), implies that \(\lim _{\beta \rightarrow 0+}M_\textrm{c}(\beta )=0\).
On the other hand, it follows directly from (3.13) and (3.16)\(_{1}\) that \(\lim _{\beta \rightarrow \infty }M_\textrm{c}(\beta )=0\). Therefore, the maximum value \(M_\textrm{c}\) of \(M_\textrm{c}(\beta )\) must be attained at some point \(\beta _{0}\in (0,\infty )\) with \(M_\textrm{c}=M_\textrm{c}(\beta _0)< \tilde{M}_\textrm{c}\) due to (3.16)\(_{1}\).
For the pressure function \(P_{\delta }(\rho )\) in (3.15), it is direct to check that conditions (1.4)–(1.6) are satisfied and \(\gamma _2=\frac{4}{3}-\frac{\epsilon _{0}}{6}\in (\frac{6}{5},\frac{4}{3})\). Let \(e_{\delta }(\rho )\) be the corresponding internal energy with \(e_{\delta }'(\rho )=\frac{P_{\delta }(\rho )}{\rho ^2}\) and \(e_{\delta }(0)=0\), and \(h_{\delta }(\rho ):=P_{\delta }(\rho )\rho ^{-1}-(\gamma _2-1)e_{\delta }(\rho )\). It follows from a direct calculation that
\(T_{\delta }(0)=0\), and \( T_{\delta }'(\rho )=-(\gamma _2-1)P_{\delta }'(\rho )+\rho P_{\delta }''(\rho )=\frac{2+\epsilon _{0}}{18}\delta \rho ^{\frac{2}{3}}(\delta +\rho ^{\frac{2+\epsilon _{0}}{3}})^{-\frac{3}{2}}>0 \) for \(\rho >0\). Thus, \(T_{\delta }(\rho )>0\) for \(\rho >0\), which implies that \(h_{\delta }'(\rho )> 0\) for \(\rho >0\), so that \(M_\textrm{c}(\delta )<\tilde{M}_\textrm{c}\) for any \(\delta >0\). \(\square \)
4 Entropy Analysis: Weak Entropy Pairs
Compared with the polytropic gas case in [10], there is no explicit formula of the entropy kernel for the general pressure law (1.4)–(1.6) so that we have to analyze the entropy equation (2.14) carefully to obtain several desired estimates.
4.1 A special entropy pair
In order to obtain the higher integrability of the velocity, we are going to construct a special entropy pair such that \(\rho |u|^3\) can be controlled by the entropy flux. Indeed, such a special entropy \(\hat{\eta }(\rho ,u)\) is constructed as
for \(k(\rho )=\int _{0}^{\rho }\frac{\sqrt{P'(y)}}{y}\,\textrm{d}y\) and, in the intermediate region \(-k(\rho )\le u\le k(\rho )\), \(\hat{\eta }(\rho ,u)\) is the unique solution of the Goursat problem of the entropy equation (2.14):
see Fig. 1. Set
Then (4.2) can be rewritten as
The corresponding characteristic boundary conditions become
Since \(\frac{k''(\rho )}{k'(\rho )}\) has the singularity at vacuum \(\rho =0\), the Goursat problem (4.4)–(4.5) is singular, which requires a careful analysis.
It follows from (4.4) that there exist two characteristic curves originating from origin O(0, 0) in the \((\rho ,u)\)–plane:
For any given point \(O_1(\rho _0,u_0)\) with \(u_0=0\), we can draw two backward characteristic curves \(\ell _{0}^{\pm }\) through \(O_1(\rho _0,u_0)\); see Fig. 1. Let \(O_2(\rho _{0}^{+},u_{0}^{+})\) be the intersection point of \(\ell _{0}^{+}\) and \(\ell _{+}\), and let \(O_3(\rho _{0}^{-},u_{0}^{-})\) be the intersection point of \(\ell _{0}^{-}\) and \(\ell _{-}\). Let \(\Sigma \) be the region surrounded by arc \(\widehat{OO_2O_{1}O_{3}}\), and let \(\overline{\Sigma }\) be the closure of \(\Sigma \).
Lemma 4.1
The Goursat problem (4.2) admits a unique solution \(\hat{\eta }\in C^2(\mathbb {R}_{+} \times \mathbb {R})\) such that
-
(i)
\(|\hat{\eta }(\rho ,u)|\le C\big (\rho |u|^2+\rho ^{\gamma (\rho )}\big )\) for \((\rho ,u)\in \mathbb {R}_{+}\times \mathbb {R}\), where \(\gamma (\rho )=\gamma _1\) if \(\rho \in [0,\rho _{*}]\) and \(\gamma (\rho )=\gamma _2\) if \(\rho \in (\rho _{*},\infty )\).
-
(ii)
If \(\hat{\eta }\) is regarded as a function of \((\rho , u)\),
$$\begin{aligned} |\hat{\eta }_{\rho }(\rho ,u)|\le C\big (|u|^2+\rho ^{2\theta (\rho )}\big ),\,\,\, |\hat{\eta }_{u}(\rho ,u)|\le C\big (\rho |u|+\rho ^{\theta (\rho )+1}\big ) \quad \,\,\,\,\text {for }(\rho ,u)\in \mathbb {R}_{+}\times \mathbb {R}, \end{aligned}$$and, if \(\hat{\eta }\) is regarded as a function of \((\rho , m)\),
$$\begin{aligned} |\hat{\eta }_{\rho }(\rho ,m)|\le C\big (|u|^2+\rho ^{2\theta (\rho )}\big ),\,\,\,|\hat{\eta }_{m}(\rho ,m)|\le C(|u|+\rho ^{\theta (\rho )}) \quad \,\,\,\,\text {for}\, (\rho ,m)\in \mathbb {R}_{+}\times \mathbb {R}, \end{aligned}$$where \(\theta (\rho ):=\frac{\gamma (\rho )-1}{2}\).
-
(iii)
If \(\hat{\eta }_{m}\) is regarded as a function of \((\rho ,u)\),
$$\begin{aligned} \qquad |\hat{\eta }_{m\rho }(\rho ,u)|\le C\rho ^{\theta (\rho )-1},\quad |\hat{\eta }_{mu}(\rho ,u)|\le C, \end{aligned}$$and, if \(\hat{\eta }_{m}\) is regarded as a function of \((\rho ,m)\),
$$\begin{aligned} |\hat{\eta }_{m\rho }(\rho ,m)|\le C\rho ^{\theta (\rho )-1},\quad |\hat{\eta }_{mm}(\rho ,m)|\le C\rho ^{-1}. \end{aligned}$$ -
(iv)
If \(\hat{q}\) is the corresponding entropy flux determined by (2.12), then \(\hat{q}\in C^2(\mathbb {R}_{+}\times \mathbb {R})\) and
$$\begin{aligned}&\hat{q}(\rho ,u)=\frac{1}{2}\rho |u|^3\pm \rho u\big (e(\rho )+\rho e'(\rho )\big )\qquad{} & {} \text{ for } \pm u\ge k(\rho ),\\ {}&|\hat{q}(\rho ,u)|\le C\rho ^{\gamma (\rho )+\theta (\rho )}\qquad \qquad \qquad \,\,\,\,\,\,{} & {} \text{ for } |u|<k(\rho ),\\ {}&\hat{q}(\rho ,u)\ge \frac{1}{2}\rho |u|^3\qquad \qquad \qquad \qquad \quad \,\,\,{} & {} \text{ for } |u|\ge k(\rho ),\\ {}&|\hat{q}-u\hat{\eta }|\le C\big (\rho ^{\gamma (\rho )}|u|+\rho ^{\gamma (\rho )+\theta (\rho )}\big )\qquad{} & {} \text{ for } (\rho ,u)\in \mathbb {R}_{+}\times \mathbb {R}. \end{aligned}$$
Proof
To prove that (4.2) has a unique \(C^2\)–solution \(\hat{\eta }\) in \(\mathbb {R}_{+}\times \mathbb {R}\), it suffices to prove that (4.4)–(4.5) admits a unique \(C^1\)–solution \((V_1, V_2)\) in \(\Sigma \) for any given point \(O(\rho _0, u_0)\). We use the Picard iteration and divide the proof into six steps.
1. For any point \(A(\rho ,u)\in \Sigma \), there are two backward characteristic curves through \(A(\rho , u)\):
Let \(B(\rho _1,u_1)\) be the intersection point of \(\ell _{+}\) and \(\ell _1\), and let \(C(\rho _2,u_2)\) be the intersection point of \(\ell _{-}\) and \(\ell _2\). It follows from (4.6)–(4.7) that
Using (4.4) and integrating \(V_1\) and \(V_2\) along the characteristic curves \(\ell _{1}\) and \(\ell _2\) respectively, we have
Denote \(V_{i}^{(0)}(\rho ,u):=V_{i}(\rho _{i},u_{i})\). It follows from (4.5) and (4.8) that
We define the iterated scheme:
Then we obtain two sequences \(\{V_{i}^{(n)}\}_{n=0}^{\infty }\) for \(i=1,2\). We now prove that \(\{V_{i}^{(n)}\}_{n=0}^{\infty }\) are uniformly convergent in \(\overline{\Sigma }\), which is equivalent to proving that
are uniformly convergent in \(\overline{\Sigma }\).
From Lemma 3.1 and (4.10), we know that \(V_{i}^{(0)}, i=1,2\), are continuous in \(\overline{\Sigma }\) and there exists a constant \(C_1>0\) depending only on \(\rho _{*}\) and \(\rho ^{*}\) such that
which, with the fact that \(\rho _{i}\le \rho \), yields that
where \(\tilde{C}_{1}\ge C_{1}(\rho _{*})^{\theta _{2}-M_{1}}\), \(\hat{C}_{1}\ge C_{1}\), and \(M_{1}\) are positive constants to be chosen later.
It follows from (3.9) and (3.10) that there exist a constant \(\nu <1\) and a constant \(C_{0}\gg 1\) depending on \(\rho _{*}\) and \(\rho ^{*}\) such that
For the estimate of \(|V_{i}^{(1)}-V_{i}^{(0)}|\), we divide it into six cases:
Case 1. \(\rho _{i}\le \rho \le \rho _{*}\): It follows from (4.11) and (4.13)–(4.14) that
where \(\varpi _1:=\frac{\nu }{1+\theta _1}\in (0,1)\).
Case 2. \(\rho _{i}\le \rho _{*}\le \rho \le \rho ^{*}\): Then
where \(\varpi _{M_1}:=\frac{C_0}{1+M_{1}}\) and, in the last inequality of (4.16), we have chosen
Case 3. \(\rho _{*}\le \rho _{i}\le \rho \le \rho ^{*}\): It is direct to see that
Case 4. \(\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho \): Then
where \(\varpi _2:=\frac{\nu }{1+\theta _2}\in (0,1)\) and, in the last inequality of (4.19), we have used (4.17) and chosen
Case 5. \(\rho _{*}\le \rho _{i}\le \rho ^{*}\le \rho \): It follows similarly that
where we have used (4.20) in the last inequality of (4.21).
Case 6. \(\rho ^{*}\le \rho _{i}\le \rho \): We see that
Combining (4.15)–(4.22), we obtain
To utilize the induction arguments, we make the induction assumption for \(n=k\):
We now make the estimate for \(n=k+1\). To estimate \(\big |V_{i}^{(k+1)}(\rho ,u)-V_{i}^{(k)}(\rho ,u)\big |\), it suffices to consider the case: \(\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho \), since the other cases can be done by similar arguments as in (4.15)–(4.22). Noting (4.24) and \(\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho \), and using similar arguments in (4.19), we have
where we have chosen \(\tilde{C}_1\) and \(\hat{C}_1\) such that
Therefore, under assumption (4.25), we obtain
Recalling that \(\theta _{1}\ge \theta _2\), we can take \(C_{0}\) and \(M_{1}\) large enough such that
Combining (4.13) with (4.23)–(4.26), and taking
by induction, we conclude that, for any \(n\ge 1\),
Noting that (4.26) and \(\rho \le \rho _0\) for \((\rho ,u)\in \overline{\Sigma }\), we have proved that the two sequences in (4.12), \(i=1,2\), are uniformly convergent in \(\overline{\Sigma }\) so that sequence \(\{(V_1^{(n)}, V_2^{(n)})\}\) is uniformly convergent in \(\overline{\Sigma }\). Let \((V_1, V_2)\) be the limit function of sequence \((V_{1}^{(n)},V_{2}^{(n)})\). Noting the continuity and the uniform convergence of \((V_{1}^{(n)},V_{2}^{(n)})\), \((V_{1},V_2)\) is continuous in \(\overline{\Sigma }\). Taking the limit: \(n\rightarrow \infty \) in (4.11), we conclude that \((V_{1},V_{2})\) is the continuous solution of (4.9).
2. It follows from (4.13) and (4.28) that, for \((\rho , u)\in \{\rho \ge 0,\,|u|\le k(\rho )\}\) and \(i=1,2\),
On the other hand, we see from (4.3) that, for \(|u|\le k(\rho )\),
Hence, for \(|u|\le k(\rho )\), it holds that
where \((\bar{\rho },u)\) is the point satisfying \(k(\bar{\rho })=|u|\), and we have used the boundary data in (4.2).
3. We now show that \(V_1\) and \(V_2\) have continuous first-order derivatives with respect to \((\rho , u)\). Using (4.7)–(4.8) and Lemma 3.1, we have
Applying \(\partial _u\) to (4.11) and using (4.32) yield that, for \(i=1,2\),
It follows from (4.10), (4.32), Lemma 3.1, and a direct calculation that, for \(i=1,2\),
where \(C_{2}\) is chosen to be a common, fixed, and large enough constant in (4.32) and (4.34) depending only on \(\rho _{*}\) and \(\rho ^{*}\), and \(\bar{C}_2 \ge C_2,\tilde{C}_2\ge C_2(\rho _{*})^{-M_2}, \hat{C}_2\ge C_2\), and \(M_2\) are some large positive constants to be chosen later.
To estimate \(\big |(\frac{\partial V_{i}^{(1)}}{\partial u})(\rho ,u)-(\frac{\partial V_{i}^{(0)}}{\partial u})(\rho ,u)\big |\), we divide it into six cases:
Case 1. \(\rho _{i}\le \rho \le \rho _{*}\): It follows from (4.13)–(4.14) and (4.33)–(4.34) that
where, in the last inequality of (4.35), we have chosen
Case 2. \(\rho _{i}\le \rho _{*}\le \rho \le \rho ^{*}\): Then, similarly, we have
where \(\varpi _{M_2}:=\frac{C_{0}}{1+M_{2}}\) and, in the last inequality of (4.37), we have used (4.36) and chosen
Case 3. \(\rho _{*}\le \rho _{i}\le \rho \le \rho ^{*}\): It follows that
where, in the last inequality of (4.39), we have chosen
Case 4. \(\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho \): For this case, similarly, we have
where, in the last inequality of (4.41), we have used (4.36) and (4.38) and chosen
Case 5. \(\rho _{*}\le \rho _{i}\le \rho ^{*}\le \rho \): Then
where we have used (4.40) and (4.42) in the last inequality of (4.43).
Case 6. \(\rho ^{*}\le \rho _{i}\le \rho \): It follows similarly that
where, in the last inequality of (4.44), we have chosen
Combining (4.35)–(4.45), we conclude that, for \(i=1,2\),
provided (4.36), (4.38), (4.40), (4.42), and (4.45) hold.
To use the induction arguments, we make the induction assumption for \(n=k\): For \(i=1,2\),
To estimate \(\vert \frac{\partial (V_{i}^{(k+1)}- V_{i}^{(k)})}{\partial u}(\rho ,u)\vert \), it suffices to consider the case: \(\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho \) for simplicity of presentation, since the other cases can be estimated by similar arguments in (4.35)–(4.45). In fact, for the case: \(\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho \), it follows from (4.28) and (4.47) that
where we have chosen
Thus, under assumption (4.48), we conclude that, for \(i=1,2\),
Combining (4.27) with (4.46)–(4.48) and taking
we have proved that, for any \(n\ge 1\) and \(i=1,2\),
Noting that \(\nu <1\) and \(\rho \le \rho _0\) for \((\rho ,u)\in \overline{\Sigma }\), we know that \(\big \{\frac{\partial V_{i}^{(n)}}{\partial u}\big \}\) is uniformly convergent in \(\overline{\Sigma }\). It is direct to check that the limit function is \(\frac{\partial V_{i}}{\partial u}\). Due to the continuity and uniform convergence of \(\{\frac{\partial V_{i}^{(n)}}{\partial u}\}\), it is clear that \(\frac{\partial V_{i}}{\partial u}\) is continuous in \(\overline{\Sigma }\).
On the other hand, it follows from (4.9) that
which, with (4.14), (4.28), and (4.49), yields that, for \(k\ge 0\) and \(i=1,2\),
for some large constant \(C>0\). Thus, \(\frac{\partial V_{i}^{(n)}}{\partial \rho }\) converges uniformly to \(\frac{\partial V_{i}}{\partial \rho }\) in \(\overline{\Sigma }\). It is clear that \(\frac{\partial V_{i}}{\partial \rho }\) is continuous. Therefore, \((V_{1}(\rho ,u), V_{2}(\rho ,u))\) is a \(C^1\)–solution of the Goursat problem (4.4)–(4.5), which implies that \(\hat{\eta }\) is a \(C^2\)–solution of (4.2).
4. From (4.34) and (4.49), we obtain that, for \(i=1,2\),
for \(\rho \ge 0\) and \(|u|\le k(\rho )\). Similarly, using (4.50), we see that, for \(i=1,2\),
Therefore, for \(|u|\le k(\rho )\), it follows from (4.3) and (4.51) that
If \(\hat{\eta }_{m}\) is regarded as a function of \((\rho ,u)\), we have
If \(\hat{\eta }_{m}\) is regarded as a function of \((\rho ,m)\), we see that, for \(|u|\le k(\rho )\),
5. We now prove the uniqueness of \(\hat{\eta }\), which is equivalent to the uniqueness of solutions of (4.4)–(4.5) in the class of \(C^1\)–solutions satisfying (4.29). Suppose that there exist two \(C^{1}\) solutions \((V_{1},V_{2})\) and \((\tilde{V}_{1},\tilde{V}_{2})\) of (4.4)–(4.5) satisfying the uniform estimate (4.29). Then it follows from (4.9) that, for \(i=1,2\),
Applying the uniform estimates (4.29) and similar arguments as in (4.28)–(4.52) yields
for any \(n\ge 0\), where \(C \gg 1\) is independent of n. Taking \(n\rightarrow \infty \), we obtain that \(V_{i}(\rho ,u)\equiv \tilde{V}_{i}(\rho ,u)\) for \(|u|\le k(\rho )\) which, with (4.3) and \(\hat{\eta }(0,u)\equiv 0\), yields the uniqueness of \(\hat{\eta }\).
6. We now estimate the entropy flux \(\hat{q}\). It follows from (2.12) that, for all entropy pairs,
Then there exists an entropy flux \(\hat{q}(\rho ,u)\in C^2(\mathbb {R}_{+}\times \mathbb {R})\) corresponding to the special entropy \(\hat{\eta }\):
It follows from (4.30) and (4.53) that \(|\hat{q}_{\rho }(\rho ,u)|=|u\hat{\eta }_{\rho }+\rho k'(\rho )^2\hat{\eta }_{u}|\le C\rho ^{\gamma (\rho )+\theta (\rho )-1}\) for \(|u|\le k(\rho )\), which implies
where \((\bar{\rho },u)\) is the point satisfying \(k(\bar{\rho })=|u|\).
For \(|u|\le k(\rho )\), it follows from (4.31) and (4.54) that \(|\hat{q}-u\hat{\eta }|\le |\hat{q}|+|u||\hat{\eta }|\le C\rho ^{\gamma (\rho )+\theta (\rho )}\). In region \(\{(\rho ,u):\,|u|\ge k(\rho )\}\), it is direct to check that all the estimates in Lemma 4.1 hold by using (4.1). Therefore, the proof of Lemma 4.1 is now complete. \(\square \)
4.2 Estimates of the weak entropy pairs
In order to show the compactness of the weak entropy dissipation measures below, we now derive some estimates of the weak entropy pairs. To achieve this, from (2.15)–(2.16), it requires to analyze the entropy kernel and entropy flux kernel, respectively.
The entropy kernel \(\chi =\chi (\rho ,u,s)\) is a fundamental solution of the entropy equation (2.14):
As pointed out in [11] that equation (4.55) is invariant under the Galilean transformation, which implies that \(\chi (\rho ,u,s)=\chi (\rho ,u-s,0)=\chi (\rho ,0,s-u)\). For simplicity, we write it as \(\chi (\rho ,u,s)=\chi (\rho ,u-s)\) below when no confusion arises.
The corresponding entropy flux kernel \(\sigma (\rho ,u,s)\) satisfies the Cauchy problem for \(\sigma -u\chi \):
We recall from [11] that \(\sigma -u\chi \) is also Galilean invariant. From (1.4)–(1.6), \(P(\rho )\) satisfies all the conditions in [11, 12].
For later use, we introduce the definition of fractional derivatives (cf. [8, 11, 48]). For any real \(\alpha >0\), the fractional derivative \(\partial _{s}^{\alpha }f\) of a function \(f=f(s)\) is
where \(\Gamma (x)\) is the Gamma function and the convolution should be understood in the sense of distributions. The following formula:
holds for fractional derivatives. We now present two useful lemmas for the entropy kernel \(\chi (\rho ,u)\) and the entropy flux kernel \(\sigma (\rho , u)\) when \(\rho \) is bounded.
Lemma 4.2
([11, Theorems 2.1–2.2]). The entropy kernel \(\chi (\rho ,u)\) admits the expansion:
where \(k(\rho )=\int _{0}^{\rho }\frac{\sqrt{P'(y)}}{y}\,\textrm{d}y\) and
Moreover, \({\text {supp}}\chi (\rho ,u)\subset \{(\rho ,u)\,:\, |u|\le k(\rho )\}\), and \(\chi (\rho ,u)>0\) in \(\{(\rho ,u)\,: \, |u|<k(\rho )\}\). The remainder term \(g_1(\rho ,\cdot )\) and its fractional derivative \(\partial _{u}^{\lambda _1+1}g_1(\rho ,\cdot )\) are Hölder continuous. Furthermore, for any fixed \(\rho _{\max }>0\), there exists \(C(\rho _{\max })>0\) depending only on \(\rho _\textrm{max}\) such that
for any \(0\le \rho \le \rho _{\max }\) and some \(\alpha _0\in (0,1)\). In addition, for any \(0\le \rho \le \rho _{\max }\),
Proof
Since (4.57)–(4.59) have been derived in [11, Theorem 2.2], it suffices to prove (4.60). From (3.6), we find that \(|a_{1}(\rho )|\le C(\rho _{\max })\) for \(0\le \rho \le \rho _{\max }\). For \(|a_1'(\rho )|\), a direct calculation shows that
It follows from (1.5) that \(k(\rho )=C_{1}\rho ^{\theta _1}\big (1+O(\rho ^{2\theta _1})\big )\) as \(\rho \in [0,\rho _{\max }]\) for some constant \(C_1>0\) that may depend on \(\kappa _1\) and \(\gamma _1\). Then, by direct calculation, we observe that the term involving \(\rho ^{-1}\) in \(a_1'(\rho )\) vanishes so that \(|a_1'(\rho )|\le C(\rho _{\max })\rho ^{2\theta _1-1}\). Similarly, we obtain that \(|a_1''(\rho )|\le C(\rho _{\max })\rho ^{2\theta _1-2}\). Finally, using (4.58)\(_3\), we can obtain the estimates of \(a_2(\rho )\) in (4.60) by a direct calculation. This completes the proof. \(\square \)
Lemma 4.3
([11, Theorem 2.3]). The entropy flux kernel \(\sigma (\rho ,u)\) admits the expansion
where
The remainder term \(g_2(\rho ,\cdot )\) and its fractional derivative \(\partial _{u}^{\lambda _1+1}g_2(\rho ,\cdot )\) are Hölder continuous. Moreover, for any fixed \(\rho _{\max }>0\), there exists \(C(\rho _{\max })>0\) depending only on \(\rho _{max}\) such that
for any \(0\le \rho \le \rho _{\max }\) and some \(\alpha _0\in (0,1)\). Furthermore, similar to the proof of (4.60), for any \(0\le \rho \le \rho _{\max }\),
Remark 4.1
In [11, Theorem 2.2], it is proved that \(a_2(\rho )\) and \(b_{2}(\rho )\) satisfy \(|a_2(\rho )|+|b_2(\rho )|\le C\rho k(\rho )^{-2}\) for the pressure law given in [11, (2.1)]. In this paper, we have improved them to be (4.60) and (4.62) under conditions (1.4)–(1.6).
For later use, we recall a useful representation formula for \(\chi (\rho ,u)\).
Lemma 4.4
(First representation formula, [64, Lemma 3.4]). Given any \((\rho ,u)\) with \(|u|\le k(\rho )\) and \(0\le \rho _0<\rho \),
where \(\tilde{d}(\rho ):=2+(\rho -\rho _0)\frac{k''(\rho )}{k'(\rho )}. \)
Remark 4.2
In the statement of [64, Lemma 3.4], \(\rho _{0}\) is positive. However, the proof of [64, Lemma 3.4] is also valid for \(\rho _{0}=0\) without modification; see also [11, (3.38)].
Given any \(\psi \in C_{0}^2(\mathbb {R})\), a regular weak entropy pair \((\eta ^{\psi },\,q^{\psi })\) can be given by
It follows from (4.55) that
Using Lemmas 4.2–4.4, we obtain the following lemma for the weak entropy pair \((\eta ^{\psi },q^{\psi })\).
Lemma 4.5
For any weak entropy \((\eta ^{\psi },q^{\psi })\) defined in (4.63), there exists a constant \(C_{\psi }>0\) depending only on \(\rho ^{*}\) and \(\psi \) such that, for all \(\rho \in [0,2\rho ^{*}]\),
If \(\eta ^{\psi }\) is regarded as a function of \((\rho ,m)\), then
Moreover, if \(\eta _m^{\psi }\) is regarded as a function of \((\rho ,u)\), then
Proof
All the estimates can be found in [63, Lemma 3.8] or [64, Lemma 4.13] except the estimate of \(\eta _{\rho }^{\psi }(\rho ,m)\). In fact, applying Lemma 4.4 to (4.64) and using \(\displaystyle d(\rho ):=2+\frac{\rho k''(\rho )}{k'(\rho )}\), we have
We regard \(\eta ^{\psi }\) as a function of \((\rho ,m)\). Then we have
Without loss of generality, we assume that \({\text {supp}}\psi \subset [-L,L]\) for some \(L>0\). Then a direct calculation shows that \(\eta ^{\psi }(\rho ,u)=0\) if \(|u|\ge k(\rho )+L\). Noticing \(\eta _{u}^{\psi }(\rho ,u)=\rho \eta _{m}^{\psi }(\rho ,m)\), we have
Thus, it suffices to calculate \(\partial _{\rho }\eta ^{\psi }(\rho ,u)\). It follows from (4.65) that \(\partial _{\rho }\eta ^{\psi }(\rho ,u)=\partial _{\rho }I_1+\partial _{\rho }I_2\). A direct calculation shows that
Using (3.6) and Lemma 3.2, we obtain that
Similarly, we obtain that \(|\partial _{\rho }I_2|\le C_{\psi }(1+\rho ^{\theta _1})\). Thus, we conclude that \( |\partial _{\rho }\eta ^{\psi }(\rho ,u)|\le |\partial _{\rho }I_1|+|\partial _{\rho }I_2|\le C_{\psi }(1+\rho ^{\theta _1}), \) which, with (4.66)–(4.67), implies that \( |\partial _{\rho }\eta ^{\psi }(\rho ,m)|\le C_\psi (1+\rho ^{\theta _1}).\) \(\square \)
We notice that all the above estimates for the weak entropy pairs in Lemmas 4.2–4.5 hold when the density is bounded. To establish the \(L^p\)-compensated compactness framework, we need the entropy pair estimates when the density is large, namely \(\rho \ge \rho ^{*}\). From now on in this subsection, we use the representation formula of Lemma 4.4 to estimate \((\eta ^{\psi },q^{\psi })\) in the large density region \(\rho \ge \rho ^{*}\).
Lemma 4.6
There exists a positive constant \(C>0\) depending only on \(\rho ^{*}\) such that
Proof
For \(\rho \ge \rho ^{*}\), \(\chi (\rho ,u)\) satisfies
where \(\chi (\rho ^{*},u)\) and \(\chi _{\rho }(\rho ^{*},u)\) are given in Lemma 4.2. Then, applying Lemma 4.4, we obtain that, for \(\rho > \rho ^{*}\),
where \(d_{*}(\rho ):=2+(\rho -\rho ^{*})\frac{k''(\rho )}{k'(\rho )}\). By a similar proof to that for Lemma A.3, we have
which, with (3.7), yields that \(\Vert \chi (\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\rho \) for \(\rho \ge 2\rho ^{*}\). For \(\rho ^{*}\le \rho \le 2\rho ^{*}\), it follows from Lemma 4.2 that \(\Vert \chi (\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\le C\rho \). \(\square \)
Lemma 4.7
Let \(\rho \ge \rho ^{*}\) and \(\psi \in C_{0}^2(\mathbb {R})\). Then, in the \((\rho ,u)\)–coordinates,
In the \((\rho ,m)\)–coordinates, \(\,|\eta ^{\psi }_{\rho }(\rho ,m)|+ \rho ^{\theta _2}|\eta _{m}^{\psi }(\rho ,m)|+\rho ^{1+\theta _2}|\eta _{mm}^{\psi }(\rho ,m)|\le C_{\psi }\rho ^{\theta _2}\).
If \(\eta ^{\psi }_{m}(\rho ,m)\) is regarded as a function of \((\rho , u)\), then \(\,|\eta _{mu}^{\psi }|+\rho ^{1-\theta _2}|\eta _{m\rho }^{\psi }|\le C_{\psi }\).
All the above constants \(C_{\psi }>0\) depend only on \(\Vert \psi \Vert _{C^2}\) and \({\text {supp}}\psi \).
Proof
We divide the proof into five steps.
1. Using (4.63) and Lemma 4.6, we obtain that, for \(\rho \ge \rho ^{*}\),
2. For the estimate of \(\eta _{\rho }^{\psi }(\rho ,u)\), the proof is similar to Lemma 4.5. Indeed, \(\eta ^{\psi }\) satisfies
It follows from (4.70) and Lemma 4.4 that
where \(d_{*}(\rho )=2+(\rho -\rho ^{*})\frac{k''(\rho )}{k'(\rho )}\) and \(0<d_{*}(\rho )\le 3\) for \(\rho \ge \rho ^{*}\) from (3.10). Then, following the similar arguments as in the proof of Lemma 4.5, we can obtain that \(|\eta _{\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2}\) for \(\rho \ge 2\rho ^{*}\). Moreover, from Lemma 4.5, \(|\eta _{\rho }^{\psi }(\rho ,u)|\le C_{\psi }\le C_{\psi }\rho ^{\theta _2}\) for \(\rho \in [\rho ^{*},2\rho ^{*}]\) so that
3. For \(\eta _{\rho \rho }^{\psi }(\rho ,u)\), it follows from (4.69)–(4.70) that
4. In the \((\rho ,m)\)–coordinates, it is clear that
On the other hand, if \(\eta _{m}^{\psi }\) is regarded as a function \((\rho ,u)\), it is direct to obtain
Thus, using (4.69) and (4.72),
where we have used that \({\text {supp}}\psi \subset [-L,L]\) and \(\eta _{u}^{\psi }(\rho ,u)=\rho \eta _{m}^{\psi }(\rho ,m)\).
5. For the estimates of \(\eta _{m\rho }^{\psi }(\rho ,u)=\partial _{\rho }\eta _{m}^{\psi }(\rho ,u)\), it follows from (4.71) that
and \(\partial _{\rho }\eta _{m}^{\psi }(\rho ,u)=\partial _{\rho }J_1+\partial _{\rho }J_2+\partial _{\rho }J_3\).
A direct calculation shows that
Clearly, we have
which, with (4.69) and \(0<d_{*}(\rho )\le 3\) for \(\rho \ge \rho ^{*}\), yields
It follows from (4.69) and \(0<d_{*}(\rho )\le 3\) for \(\rho \ge \rho ^{*}\) that
For \(J_{1,3}+J_{2,3}\), it is direct to see that
For \(\partial _{\rho }J_3\), we notice that
which, with \(0<\theta _2\le 1\), yields
Combining (4.74)–(4.79) with (4.73) yields that \(|\eta _{m\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2-1}\) for \(\rho \ge 2\rho ^{*}\). For \(\rho ^{*}\le \rho \le 2\rho ^{*}\), it follows from Lemma 4.5 that \( |\eta _{m\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _1-1}\le C_{\psi } \) for \(\rho ^{*}\le \rho \le 2\rho ^{*}\). Thus, we conclude that \(|\eta _{m\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2-1}\) for \(\rho \ge \rho ^{*}\). \(\square \)
We now estimate \(q^{\psi }\) for \(\rho \ge \rho ^{*}\). It follows from (4.56) that \(h:=\sigma -u\chi \) satisfies
where \((\sigma -u\chi )(\rho ^{*},u)\) and \((\sigma -u\chi )_{\rho }(\rho ^{*},u)\) are given by Lemma 4.3. Similar to Lemma 4.4, we have the following representation formula for h.
Lemma 4.8
(Second representation formula [64, Lemmas 3.4 and 3.9]). For any \((\rho ,u)\) with \(|u|\le k(\rho )\) and \(\rho >\rho ^{*}\),
where \(d_{*}(\rho )=2+(\rho -\rho ^{*})\frac{k''(\rho )}{k'(\rho )}\).
Lemma 4.9
There exists a constant \(C>0\) depending only on \(\rho ^{*}\) such that
Proof
It follows from (3.3), (4.80), and Lemma 4.6 that
which, with (3.7) and a similar proof to that for Lemma A.3, yields
For \(\rho ^{*}\le \rho \le 2\rho ^{*}\), it follow from Lemma 4.3 that \(\Vert h(\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\le C\rho ^{1+\theta _2}\). \(\square \)
Lemma 4.10
For \(\rho \ge \rho ^{*}\) and \(\psi \in C_{0}^2(\mathbb {R})\),
Proof
Recall that
It follows from Lemma 4.9 that
Since there exists \(L>0\) such that \({\text {supp}}\psi \subset [-L,L]\), then it follows from Lemma 4.7 that \(|u\eta ^{\psi }(\rho ,u)|\le (k(\rho )+L)|\eta ^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{1+\theta _2}\) for \(\rho \ge \rho ^{*}\), which, with (4.82)–(4.83), yields (4.81). \(\square \)
4.3 Singularities of the entropy kernel and the entropy flux kernel
As indicated in [11, 63, 64], understanding the singularities of the entropy kernel and the entropy flux kernel is essential for the reduction of the Young measure. Thus, it requires some detailed estimates of the singularities of the entropy kernel and the entropy flux kernel. The arguments in this subsection are similar to [64, Sect. 6], the main difference is that a more subtle Grönwall inequality (see Lemma A.3) is needed to obtain the desired estimates of the singularities.
Lemma 4.11
For \(\rho \ge \rho ^{*}\), the coefficient functions \(a_1(\rho )\) and \(a_2(\rho )\) and the remainder term \(g_1(\rho ,u)\) in Lemma 4.2 satisfy
where \(\alpha _0\in (0,1)\) is the Hölder exponent.
Proof
We divide the proof into five steps.
1. It follows from (4.60) that, for \(\rho \in [0,\rho _{*}]\),
For \(\rho \ge \rho ^{*}\), it follows from (3.7), (4.58), and a direct calculation that
Moreover, calculating the derivatives explicitly, we conclude
2. For the remainder term \(g_1(\rho ,u)\), it follows from [11, Proof of Theorem 2.1] that
where \(f_{\lambda _1}(y)=[1-y^2]_{+}^{\lambda _1}\) and \(A(\rho )=-a_2''(\rho )k(\rho )^{2\lambda _1+3}\). By (4.84), \(A(\rho )\sim O(\rho ^{-1+2\theta _1})\) as \(\rho \rightarrow 0\) and \(|A(\rho )|\le \rho ^{-\frac{3}{2}+\theta _2(1+\frac{1}{2\theta _1})}\) for \(\rho \ge \rho ^{*}\). Similar to Lemma 4.4, we have the following representation formula for \(g_1(\rho ,u)\):
where \(d(\rho )=2+\rho \frac{k''(\rho )}{k'(\rho )}\) satisfying (3.11). Since \(\rho A(\rho )k(\rho )^{-1}\sim O(\rho ^{\theta _1})\) as \(\rho \rightarrow 0\), the second integral is well-defined. Then it follows directly from (3.6)–(3.7) and (4.87) that
Since \(g_1(\rho ,u)\) is Hölder continuous and \({\text {supp}}g_1(\rho ,\cdot )\subset [-k(\rho ),k(\rho )]\), it follows from (3.6)–(3.7) that \(\Vert k'(\rho )g_1(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\) is locally integrable with respect to \(\rho \in [0,\infty )\). Applying Lemma A.3 to (4.88), we obtain that, for \(\rho \ge \rho ^{*}\),
which, with (3.7), yields that, for \(\rho \ge \rho ^{*}\),
3. Applying \(\partial _{u}\) to (4.87), we have
Since \(|f_{\lambda _1+1}(s)|\le 1\), by similar arguments as in Step 2, we can obtain
4. Applying the fractional derivative \(\partial _{u}^{\lambda _1}\) to (4.89), we have
where we have taken into account the homogeneity of the factional derivative in the last term. Using the Fourier transform relation as in [48, (I.26)–(I.27)], we can obtain
for some positive constants \(C_{\lambda _1+1}\) and \(\tilde{C}_{\lambda _1+1}\) depending only on \(\lambda _1+1\), where we have used the asymptotic relations for the first kind of Bessel functions \(J_{\lambda _1+\frac{3}{2}}(|\xi |)\) to obtain the final inequality. Since \((1+|\xi |^2)^{-1}\) is integrable, applying the Fourier inversion theorem, we see that \((\partial _{u}^{\lambda _1}f_{\lambda _1+1})(u)\) is uniformly bounded. Hence, by similar arguments as in Step 2, we have
5. By Lemma 4.2, we assume that \(\alpha _0\in (0,1)\) is the Hölder exponent of \((\partial _{u}^{\lambda _1+1})g_1(\rho ,u)\). Then, applying the fractional derivative \(\partial _{u}^{\lambda _1}\) to (4.89), we have
Noting
and using the Fourier inversion theorem, we find that \((\partial _{u}^{\lambda _1+\alpha _0}f_{\lambda _1+1})(u)\) is uniformly bounded. By similar arguments as in Step 2, we obtain that \(\Vert \big (\partial _u^{\lambda _1+1+\alpha _0}g_1\big )(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho \) for \(\rho \ge \rho ^{*}\). This completes the proof. \(\square \)
From Lemmas 4.2 and 4.11, we conclude
Corollary 4.12
\(\chi (\rho , \cdot )\) is Hölder continuous and
Lemma 4.13
For \(\rho \ge \rho ^{*}\), the coefficient functions \(b_{1}(\rho )\) and \(b_2(\rho )\) and the remainder term \(g_2(\rho ,u)\) in Lemma 4.3 satisfy
where \(\alpha _0\in (0,1)\) is the Hölder exponent.
Proof
We divide the proof into five steps.
1. It follows from (4.62) that, for \(\rho \in [0,\rho _{*}]\),
From (4.61) and (3.7), we have
Using (4.84)–(4.85) and (4.90)–(4.91), we obtain that, for \(\rho \ge \rho ^{*}\),
which, with (4.61), yields that \(|b_2(\rho )|\le C\rho ^{\frac{1}{2}-\theta _2(1+\frac{1}{2\theta _1})}\) for \(\rho \ge \rho ^{*}\). Moreover, by calculating the derivatives explicitly, we obtain
2. For the remainder term \(g_2(\rho ,u)\), recalling from [11, Proof of Theorem 2.2], \(g_2\) satisfies
where \(f_{\lambda _1}(y)=[1-y^2]_{+}^{\lambda _1}\). Similar to the arguments for Lemma 4.8, we obtain
which yields
It follows from (3.2)–(3.3), (3.6)–(3.7), (4.90), (4.92), and Lemma 4.11 that, for \(\rho \ge \rho ^{*}\)
Substituting (4.95)–(4.96) into (4.94), applying Lemma A.3 and Corollary A.4, and using (3.7), we obtain that, for \(\rho \ge \rho ^{*}\),
3. Denoting \(\widetilde{f}(s)=sf_{\lambda _1+1}(s)\) and applying \(\partial _{u}\) to (4.93), we have
Since \(|\widetilde{f}(s)|\le 1\), by similar arguments as in Step 2, we obtain
4. Applying \(\partial _{u}^{\lambda _1}\) to (4.97), we have
where \(\widetilde{f}^{(\lambda _1)}:=\partial _{s}^{\lambda _1}\widetilde{f}(s)\). Since \(\widetilde{f}^{(\lambda _1)}\) is uniformly bounded, similar arguments as in Step 2 yield
5. Applying \(\partial _{u}^{\lambda _1+\alpha _0}\) to (4.97), we have
Noting that \(\widetilde{f}^{(\lambda _1+\alpha _0)}(s)\) is uniformly bounded, by similar arguments as in Step 2, we have
This completes the proof. \(\square \)
The following lemma provides the explicit singularities of \(\chi (\rho ,u-s)\) and \((\sigma -u\chi )(\rho ,u-s)\).
Lemma 4.14
The fractional derivatives \(\partial _{u}^{\lambda _1+1}\chi \) and \(\partial _{u}^{\lambda _1+1}(\sigma -u\chi )\) admit the expansions:
where \(\delta \) is the Dirac measure, H is the Heaviside function, PV is the principle value distribution, and Ci is the Cosine integral:
for some constant \(C_0>0\). The remainder terms \(r_{\chi }\) and \(r_{\sigma }\) are Hölder continuous functions. Moreover, there exists a positive constant \(C=C(\gamma _1,\gamma _2,\rho _{*},\rho ^{*})\) such that, for \(\rho \ge \rho ^{*}\),
where \(\alpha _1\in (0,\alpha _0]\) is the common Hölder exponent of \(r_{\chi }\) and \(r_{\sigma }\).
Proof
From [64, Lemma 6.4], we obtain (4.98)–(4.99), where the coefficients are given by
where \(A_{i}^{\lambda _1}, i=1,\ldots ,4\), are constants depending only on \(\lambda _1\), and \(\tilde{r}\) and \(\tilde{q}\) are uniformly bounded Hölder continuous functions. Thus, using Lemma 4.11, we see that, for \(\rho \ge \rho ^{*}\),
Similarly, we have
where \(\tilde{\ell }\) is also a uniformly bounded Hölder continuous function. Using Lemma 4.13, we conclude that, for \(\rho \ge \rho ^{*}\),
This completes the proof. \(\square \)
5 Uniform Estimates of Approximate Solutions
As in [10], we construct the approximate solutions via the following approximate free boundary problem for CNSPEs:
for \((t,r)\in \Omega _{T}\) with
where \(\{r=b(t):\,0\le t\le T\}\) is a free boundary determined by
and \(a=b^{-1}\) with \(b\gg 1\). On the free boundary \(r=b(t)\), we impose the stress-free boundary condition:
On the fixed boundary \(r=a=b^{-1}\), we impose the Dirichlet boundary condition:
The initial condition is
5.1 Basic estimates
Denote
For given total energy \(E_{0}^{\varepsilon ,b}>0\), the critical mass \(M_\textrm{c}^{\varepsilon ,b}\) is defined in (2.5)–(2.8) by replacing \(E_{0}\) with \(E_{0}^{\varepsilon ,b}\).
For the approximate initial data \((\rho _{0}^{\varepsilon },m_{0}^{\varepsilon })\) imposed in (2.17) satisfying (2.9)–(2.10), using similar arguments in [10, Appendix A], we can construct a sequence of smooth functions \((\rho _{0}^{\varepsilon ,b},u_{0}^{\varepsilon ,b})\) defined on [a, b], which is compatible with the boundary conditions (5.4)–(5.5), such that
-
(i)
There exists a constant \(C_{\varepsilon ,b}>0\) depending on \((\varepsilon ,b)\) so that, for all \(\varepsilon \in (0,1]\) and \(b>1\),
$$\begin{aligned} 0<C_{\varepsilon ,b}^{-1}\le \rho _{0}^{\varepsilon ,b}(r)\le C_{\varepsilon ,b}<\infty . \end{aligned}$$(5.7) -
(ii)
For all \(\varepsilon \in (0,1]\) and \(b>1\),
$$\begin{aligned}&\int _{a}^{b}\rho _{0}^{\varepsilon ,b}(r)r^{2}dr=\frac{M}{\omega _3}, \qquad E_{0}^{\varepsilon ,b}\le C(1+E_{0}),\qquad E_{1}^{\varepsilon ,b}\le C(1+M)\varepsilon , \end{aligned}$$(5.8)$$\begin{aligned}&\rho _{0}^{\varepsilon ,b}(b)\cong b^{-(3-\alpha )}\qquad \,\, \text {with }\alpha :=\min \{\frac{1}{2},\frac{3(\gamma _1-1)}{\gamma _1}\}. \end{aligned}$$(5.9) -
(iii)
For each fixed \(\varepsilon \in (0,1]\), as \(b\rightarrow \infty \), \((E_{0}^{\varepsilon ,b},E_{1}^{\varepsilon ,b})\rightarrow (E_0^{\varepsilon },E_{1}^{\varepsilon })\) and
$$\begin{aligned}{} & {} (\rho _{0}^{\varepsilon ,b},\rho _{0}^{\varepsilon ,b}u_{0}^{\varepsilon ,b})\longrightarrow (\rho _0^{\varepsilon },m_{0}^{\varepsilon }) \qquad \text {in}\, L^{\tilde{q}}([a,b];r^{2}dr)\times L^1([a,b];r^{2}dr)\nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \, \text {for }\tilde{q}\in \{1,\gamma _{2}\}. \end{aligned}$$(5.10) -
(iv)
For each fixed \(\varepsilon \in (0,\varepsilon _0]\), there exists a large constant \(\mathcal {B}(\varepsilon )>0\) such that
$$\begin{aligned} M<M_\textrm{c}^{\varepsilon ,b}\qquad \text {for }b\ge \mathcal {B}(\varepsilon )\text { and }\gamma _2\in (\frac{6}{5},\frac{4}{3}], \end{aligned}$$(5.11)where \(M_\textrm{c}^{\varepsilon ,b}\) is defined in (2.5)–(2.8) by replacing \(E_{0}\) with \(E_{0}^{\varepsilon ,b}\).
We point out that (5.9) is important for us to close the BD-type entropy estimate in Lemma 5.4 and to obtain the higher integrability of the density in Lemma 5.6 below.
Once the free boundary problem (5.1)–(5.6) is solved, we define the potential function \(\Phi \) to be the solution of the Poisson equation:
with \(\Omega _{t}:=\{\textbf{x}\in \mathbb {R}^3\,:\,a\le |\textbf{x}|\le b(t)\}\), for which \(\rho \) has been extended to be zero outside \(\Omega _{t}\). In fact, we can show that \(\Phi (t, \textbf{x})=\Phi (t, r)\) with
so that \(\Phi (t,r)\) can be recovered by integrating (5.12).
In this section, parameters \((\varepsilon ,b)\) are fixed with \(\varepsilon \in (0,\varepsilon _0]\) and \(b\ge \max \{\rho _{*}^{-\frac{\gamma _1}{3}},\mathcal {B}(\varepsilon )\}\) such that (5.11) holds and \(\rho _{0}^{\varepsilon ,b}(b)\le \rho _{*}\). The global existence of smooth solutions of our approximate problem (5.1)–(5.6) whose initial data satisfy (5.7)–(5.11) and pressure satisfies (1.4)–(1.6) can be obtained by using similar arguments in [25, Sect. 3] with \(\gamma _2\in (\frac{4}{3},\infty )\), or with \( \gamma _2\in (\frac{6}{5},\frac{4}{3}]\) and \(M<M_\textrm{c}^{\varepsilon ,b}(\gamma _2)\), so the details are omitted here for simplicity.
Noting that the upper and lower bounds of \(\rho ^{\varepsilon ,b}\) in [25] depend on parameters \((\varepsilon , b)\), we now establish some uniform estimates, independent of b, such that the limit: \(b\rightarrow \infty \) can be taken to obtain the global weak solutions of problem (1.10) and (2.17)–(2.18) in Sect. 6 below as approximate solutions of problem (1.1) and (1.13)–(1.14). Throughout this section, we drop the superscript in both the approximate solutions \((\rho ^{\varepsilon ,b},u^{\varepsilon ,b})(r)\) and the approximate initial data \((\rho _{0}^{\varepsilon ,b},u_{0}^{\varepsilon ,b})\) for simplicity.
For smooth solutions, it is convenient to analyze (5.1)–(5.6) in the Lagrangian coordinates. It follows from (5.3) that
which implies
For \(r\in [a,b(t)]\) and \(t\in [0,T]\), the Lagrangian coordinates \((\tau , x)\) are defined by
which translate \([0,T]\times [a,b(t)]\) into a fixed domain \([0,T]\times [0,\frac{M}{\omega _3}]\). By direct calculation, we see that \(\nabla _{(t,r)}x=(-\rho u r^{2},\rho r^{2})\), \(\nabla _{(t,r)}\tau =(1,0)\), \(\nabla _{(\tau ,x)}r=(u,\rho ^{-1}r^{-2})\), and \(\nabla _{(\tau ,x)}t=(1,0)\). In the Lagrangian coordinates, the initial-boundary value problem (5.1)–(5.6) becomes
for \((\tau ,x)\in [0,T]\times [0,\frac{M}{\omega _3}]\), and
where \(r=r(\tau ,x)\) is defined by \(\,\frac{\textrm{d}}{\textrm{d}\tau }r(\tau ,x)=u(\tau ,x)\) for \((\tau ,x)\in [0,T]\times [0,\frac{M}{\omega _3}]\), and the fixed boundary \(x=\frac{M}{\omega _3}\) corresponds to the free boundary: \(b(\tau )=r(\tau ,\frac{M}{\omega _3})\) in the Eulerian coordinates.
Lemma 5.1
(Basic energy estimate). The smooth solution \((\rho , u)(t,r)\) of problem (5.1)–(5.6) satisfies
where \(\rho (t, r)\) has been understood to be 0 for \(r\in [0, a]\cup (b(t),\infty )\) in the second term of the left-hand side (LHS) and the second term of the right-hand side (RHS). In particular, there exists a positive constant \(C(E_{0},M)\) depending only on the total initial energy \(E_{0}\) and initial-mass M such that the following estimates hold for the two separate cases:
Case 1. \(\displaystyle \gamma _2\in (\frac{6}{5},\frac{4}{3}]\) and \(M<M_\textrm{c}^{\varepsilon ,b}\):
Case 2. \(\displaystyle \gamma _2>\frac{4}{3}\):
Proof
We divide the proof into three steps.
1. Using (2.3) and similar calculations as in the proof [10, Lemma 3.1], we have
2. We now control the second term on the LHS of (5.18) and the second term on the RHS of (5.18) to close the estimates. By similar calculations as in [10, Lemma 3.1], one can obtain
where we have understood \(\rho \) to be zero for \(r\in [0,a)\cup (b(t),\infty )\) in (5.19).
3. Now we use the internal energy to control the gravitational potential term. First, we obtain from (3.12) that there exist two constants \(C_{1},C_{2}>0\) depending only on \(\rho ^{*}\) such that
Thus, we have
where \(K>\rho ^{*}\) is some large constant to be chosen later.
Multiplying (5.11) by \(\Phi \) and integrating by parts yield
where we have used the positive constant \(A_3:=\frac{4}{3}\omega _{4}^{-\frac{2}{3}}>0\) that is the sharp constant for the Sobolev inequality in \(\mathbb {R}^3\) (see Lemma A.1). Then it follows from (5.19) and (5.21) that
where \(B_{\beta }\) is the constant defined in (2.8).
When \(\gamma _2>\frac{4}{3}\), i.e., \(\frac{1}{3(\gamma _2-1)}<1\), it follows from (5.22) by taking \(\beta =1\) that
which, with (5.18), yields (5.17).
When \(\gamma _{2}=\frac{4}{3}\), i.e., \(\frac{1}{3(\gamma _2-1)}=1\). It has been proved in [18, Theorem 3.1] that there exists an optimal constant \(C_{\min }=6\kappa _2M_\textrm{ch}^{-\frac{3}{2}}\) such that
which, with (5.20), yields
Since \(M<M_\textrm{ch}\), we can always choose \(K>\rho ^{*}\) large enough such that
Then one can deduce (5.16) for \(\gamma _2=\frac{4}{3}\) from (5.18), (5.25), and the fact that \(\rho ^{\gamma _2}\ge C\rho e(\rho )\).
When \(\gamma _{2}\in (\frac{6}{5},\frac{4}{3})\), we define
A direct calculation shows that
which yields that \(\frac{\textrm{d}^2F(s;\beta )}{\textrm{d}s^2}<0\) for \(s>0\) since \(\gamma _2<\frac{4}{3}\). Thus, \(F(s;\beta )\) is concave with respect to \(s>0\). We denote
which is the critical point of F(s) satisfying \(\frac{\textrm{d}F(s;\beta )}{\textrm{d}s}(s_{*}(\beta ))=0\). The maximum of \(F(s;\beta )\) with respect to \(s>0\) is
It follows from the definition of \(M_\textrm{c}^{\varepsilon ,b}\) that, if \(M<M_\textrm{c}^{\varepsilon ,b}\), there exists \(\beta _{0}>0\) such that \(M<M_\textrm{c}^{\varepsilon ,b}(\beta _0)\). Then, from (5.26)–(5.27), we have
where we have used that \(\frac{1}{4-3\gamma _2}>\frac{5}{2}>1\) for \(\gamma _2\in (\frac{6}{5},\frac{4}{3})\). Then, combining (5.18) and (5.22) with (5.28)–(5.29), we obtain
Hence, due to the continuity of \( \int _{a}^{b(t)}\big (\rho e(\rho )\big )(t,r)\,r^{2}\textrm{d}r\) with respect to t, the strict inequality:
must hold. Otherwise, there exists \(t_0>0\) such that \(\int _{a}^{b(t_0)}\big (\rho e(\rho )\big )(t_{0},r)\,r^{2}\textrm{d}r=s_{*}(\beta _0)\), which yields
This contradicts (5.30). Thus, we prove (5.31) under condition (5.11).
Therefore, under condition (5.11), it follows from (5.26) and (5.31) that
Combining (5.18) and (5.25) with (5.32), we conclude (5.16). \(\square \)
Corollary 5.2
Under the assumptions of Lemma 5.1 and noting (3.5),
Corollary 5.3
Under the assumptions of Lemma 5.1, it follows from (5.12), (5.16)–(5.17), and (5.19) that, for \(t\ge 0\) and \(r\ge 0\),
For later use, we analyze the boundary value of density \(\rho \). Using (5.14)\(_1\) and (5.15), we have
which yields that \(\rho (\tau , \frac{M}{\omega _3})\le \rho _{0}(\frac{M}{\omega _3})\). In the Eulerian coordinates, it is equivalent to
Moreover, noting (5.8) and \(b\ge (\rho _{*})^{-\gamma _1/3}\), we see that \(\rho (t,b(t))\le \rho _0(b)\le \rho _{*}\) for all \(t\ge 0\). From (3.2)\(_1\) and (5.33), there exists a positive constant \(\tilde{C}\) depending only on \((\gamma _1, \kappa _1)\) such that \(\rho _{\tau }(\tau ,\frac{M}{\omega _3})=-\frac{1}{\varepsilon }\, P(\tau ,\frac{M}{\omega _3}) \ge -\frac{\tilde{C}}{\varepsilon }\big (\rho (\tau ,\frac{M}{\omega _3})\big )^{\gamma _1}\), which implies
Therefore, in the Eulerian coordinates,
Lemma 5.4
(BD-type entropy estimate). Under the conditions of Lemma 5.1, for any given \(T>0\),
Proof
We divide the proof into three steps.
1. Using (2.3) and similar calculations as in the proof [10, Lemma 3.3], we have
which, with Lemma 5.1, yields
2. For the second term on the RHS of (5.37), it follows from (5.8) and (3.2)\(_1\) that
For the last term on the RHS of (5.37), using (5.34), we have
3. To close the estimates, we need to control the third term on the RHS of (5.37), that is,
We divide the estimate of the above term into the following two cases:
Case 1. For \(\gamma _2\ge 2\), it follows from Corollary 5.2 that
Case 2. For \(\displaystyle \gamma _2\in (\frac{6}{5},2)\), then \(3\gamma _2>2\). A direct calculation shows that
Denote \(\sqrt{F(\rho )}:=\int _{0}^{\rho }\sqrt{\frac{P'(s)}{s}}\,\textrm{d}s\). Then it follows from (3.3)\(_2\) that
which, with Corollary 5.2, implies that, for \(\overline{\vartheta }=\frac{3(2-\gamma _2)}{4}\),
For \(B_{R}({\textbf {0}})\subset \mathbb {R}^3\), the following Sobolev’s inequality holds:
It follows from (5.13) and Corollary 5.2 that
which yields
Using (3.2)\(_2\)–(3.3)\(_2\) leads to \(F(\rho )\le C(\rho +\rho ^{\gamma _2})\), which, with (5.43)–(5.44) and Corollary 5.2, implies
Substituting (5.45) into (5.42), we obtain
where we have used \(\frac{2\bar{\vartheta }}{\gamma _2}\in (0,1)\) for \(\gamma _2>\frac{6}{5}\). Finally, substituting (5.38)–(5.41) and (5.46) into (5.37), we conclude (5.36). \(\square \)
In order to take the limit: \(b\rightarrow \infty \), we need to make sure that domain \(\Omega _{T}\) can be expanded to \([0,T]\times \mathbb {R}_{+}\) for fixed \(\varepsilon >0\): \(\lim \limits _{b\rightarrow \infty }b(t)=\infty \).
Lemma 5.5
(Expanding of domain \(\Omega _{T}\)). Given \(T>0\) and \(\varepsilon \in (0,\varepsilon _0]\), there exists \(C_1(M,E_0,T,\varepsilon )>0\) such that, if \(b\ge C_1(M,E_0,T,\varepsilon )\),
Proof
Noting \(b(0)=b\) and the continuity of b(t), we first make the a priori assumption:
Integrating (5.3) over [0, t] yields
It follows from (5.35), (5.48), and Lemma 5.1 that
We take \( C_{1}(M,E_0,T,\varepsilon ):= \max \big \{\rho _{*}^{-\frac{\gamma _1}{3}}, (4C(M,E_0,T,\rho _{*},\gamma _1,\gamma _2,\varepsilon ))^{\frac{2}{\alpha }}, \mathcal {B}(\varepsilon )\big \}, \) which, with (5.8) and (5.50), implies that
provided that \(b\ge C_1(M,E_0,T,\varepsilon )\). Combining (5.51) with (5.49), we have
Thus, we have closed the a priori assumption (5.48). Finally, using (5.52) and the continuity argument, we can conclude (5.47). \(\square \)
5.2 Higher integrability of the density and the velocity
As implied in [13], the higher integrabilities of the density and the velocity are important for the \(L^p\) compensated compactness framework. However, for the general pressure law, due to the lack of an explicit formula for the entropy kernel, for the special entropy pair \((\eta ^{\psi },q^{\psi })\) by taking the test function \(\psi =\frac{1}{2} s|s|\) in (2.15)–(2.16), we can not obtain that \(q^{\psi } \gtrsim \rho |u|^3 + \rho ^{\gamma +\theta }\) in general. To derive the higher integrability of the velocity, we use the special entropy pair constructed in Lemma 4.1, at the cost of the higher integrability of the density over domain \([0,T]\times [d, b(t)]\) for some \(d>0\). Since \(b(t)\rightarrow \infty \) as \(b\rightarrow \infty \), we indeed need the higher integrability of the density on the unbounded domain. We point out that this is different from the case of [10] in which only the higher integrability on the bounded domain \([0,T]\times [d,D]\) for any given \(0<d<D<\infty \) is needed.
Lemma 5.6
(Higher integrability on the density). Let \((\rho ,u)\) be a smooth solution of (5.1)–(5.6). Then, under the assumption of Lemma 5.1, for any given \(d>2b^{-1}>0\),
Proof
Let \(\omega (r)\) be a smooth function with \({\text {supp}}\omega \subset (\frac{d}{2},\infty )\) and \(\omega (r)=1\) for \(r\in [d,\infty )\). Multiplying (5.1)\(_2\) by \(w(y)y^{2}\), we have
Integrating (5.54) with respect to y from \(\frac{d}{2}\) to r and then multiplying the equation by \(\rho (t,r)\) yield
Using (5.1)\(_1\), we have
which, with (5.55), yields that
Multiplying (5.56) by \(\omega (r)\) leads to
Using Lemma 5.1 and (5.13), we have
which yields
For \(I_2\), using (5.8), (5.34), (5.51), and \(b\gg 1\), we have
For \(I_{11}\), we obtain
We divide the estimate of \(\int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\varepsilon \rho ^3\omega ^2\,r^{2}\textrm{d}r\textrm{d}t\) into two cases:
Case 1. \(\gamma _2\in (\frac{6}{5},2)\): For \(t\in [0,T]\), denoting \( A(t):=\{r\in [\frac{d}{2},b(t)]\,:\, \rho (t,r)\ge \rho ^{*}\}, \) then it follows from (5.13) that \(|A(t)|\le C(d,\rho ^{*})M\). For any \(r\in A(t)\), let \(r_{0}\) be the closest point to r so that \(\rho (t,r_{0})=\rho ^{*}\) with \(|r-r_{0}|\le |A(t)|\le C(d,\rho ^{*})M\). Then, for any smooth function \(f(\rho )\),
Recalling (3.3) and (3.5), we notice that \(P(\rho )\cong \rho ^{\gamma _2}\) and \(e(\rho )\cong \rho ^{\gamma _2-1}\) for any \(r\in A(t)\). Then
A direct calculation shows that
Combining (5.62)–(5.64), we obtain that, for \(\gamma _2\in (\frac{6}{5},2)\),
Case 2. \(\gamma _2\in [2,3)\): Using (5.13) and the same argument as for (5.64), we have
Finally, integrating (5.57) over \([0,T]\times [\frac{d}{2},b(t)]\) and using (5.58)–(5.61) and (5.65)–(5.66), we conclude (5.53). \(\square \)
Corollary 5.7
It follows from (3.3) and Lemma 5.6 that
In order to use the \(L^p\) compensated compactness framework, we still need to obtain the higher integrability of the velocity (see [13]). With the help of Lemma 5.6, we use the special entropy pair constructed in Lemma 4.1 to achieve this.
Lemma 5.8
(Higher integrability of the velocity). Let \((\rho ,u)\) be the smooth solution of (5.1)–(5.6). Then, under the assumption of Lemma 5.1,
Proof
Considering \((5.1)_1\times \hat{\eta }_{\rho }r^{2}+(5.1)_2\times \hat{\eta }_{m}r^{2}\), we can obtain
Using (5.3), a direct calculation yields
Integrating (5.68) over [r, b(t)) and using (5.69), we have
We now control the terms on the RHS of (5.70). For the third term on the RHS of (5.70), it follows from (5.9), (5.34), (5.44), and Lemmas 4.1, 5.1, and 5.4–5.5 that
For the first term on the RHS of (5.70), integrating by parts yields
It follows from (5.4) and Lemma 4.1 that
which, with similar arguments as in (5.71), yields
Hence, using (5.36), (5.72)–(5.73), and Lemma 4.1, we have
For the second term, third term, and sixth term on the RHS of (5.70), using (3.4)–(3.5) and Lemmas 4.1 and 5.1, we obtain
For the fifth term on the RHS of (5.70), we note from (3.6)–(3.7) and Lemma 4.1 that
Then it follows from (5.78)–(5.79) and Corollary 5.7 that
where we have used \(\theta _2\in (0,1)\) since \(\gamma _2\in (\frac{6}{5},3)\). Combining (5.70)–(5.71), (5.74)–(5.77), and (5.80), we obtain that \(\int _{0}^{T}\int _{d}^{D}\hat{q}\,r^{2}\textrm{d}r\textrm{d}t\le C(d, D, M, E_0,T)\), which, along with (5.67) and Lemma 4.1, gives
On the other hand, we have
Combining (5.81) with (5.82), we obtain that \(\int _{0}^{T}\int _{d}^{D}\rho |u|^3\,r^{2}\textrm{d}r\textrm{d}t\le C(d, D, M, E_0, T)\). This completes the proof of Lemma 5.8. \(\square \)
6 Existence of Global Weak Solutions of CNSPEs
In this section, for fixed \(\varepsilon >0\), we take the limit: \(b \rightarrow \infty \) to obtain the global existence of solutions of the Cauchy problem for (1.10). Meanwhile, some uniform estimates in Theorem 2.1 are obtained. To take the limit, some careful attention is required, since the weak solutions may involve the vacuum. We use similar compactness arguments as in [10, 17] to handle the limit: \(b \rightarrow \infty \). Throughout this section, we denote the smooth solutions of (5.1)–(5.6) as \((\rho ^{\varepsilon , b}, u^{\varepsilon , b})\) for simplicity.
First of all, we extend our solutions \((\rho ^{\varepsilon , b}, u^{\varepsilon , b})\) to be zero on \(([0, T] \times [0, \infty )) \backslash \Omega _{T}\). It follows from Lemma 5.5 that
which implies that domain \([0, T] \times [a, b(t)]\) expands to \([0, T] \times (0, \infty )\) as \(b \rightarrow \infty \). That is, for any set \(K \Subset (0, \infty )\), when \(b \gg 1, K \Subset (a, b(t))\) for all \(t \in [0, T]\). Now we define
where \(m^{\varepsilon ,b}:=\rho ^{\varepsilon ,b}u^{\varepsilon ,b}\). Then it is direct to check that the corresponding functions \((\rho ^{\varepsilon ,b},\mathcal {M}^{\varepsilon ,b}, \Phi ^{\varepsilon ,b})(t,\textbf{x})\) are classical solutions of
for \((t,\textbf{x})\in [0,\infty )\times \Omega _{t}\) with \(\mathcal {M}^{\varepsilon ,b}\vert _{\partial B_{a}(\textbf{0})}=0\).
Based on the estimates obtained in Sect. 5, by the same arguments as in [10, Sect. 4], we have
Lemma 6.1
For fixed \(\varepsilon >0\), as \(b\rightarrow \infty \) (up to a subsequence), there exists a vector function \((\rho ^{\varepsilon }, m^\varepsilon )(t,r))\) such that
-
(i)
\((\sqrt{\rho ^{\varepsilon ,b}},\rho ^{\varepsilon ,b})\rightarrow (\sqrt{\rho ^{\varepsilon }}, \rho ^{\varepsilon })\) a.e. and strongly in \(C(0,T;L_{\textrm{loc}}^p)\) for any \(p\in [1,\infty )\), where \(L_{\textrm{loc}}^p\) denotes \(L^p(K)\) for any compact set \(K\Subset (0,\infty )\). In particular, \(\rho ^{\varepsilon }\ge 0\) a.e. on \(\mathbb {R}_{+}^2\).
-
(ii)
The pressure function sequence \(P(\rho ^{\varepsilon , b})\) is uniformly bounded in \(L^{\infty }(0, T; L_{\textrm{loc}}^{p}(\mathbb {R}))\) for all \(p \in [1, \infty ]\), and
$$\begin{aligned} P(\rho ^{\varepsilon , b}) \longrightarrow P(\rho ^{\varepsilon }) \quad \text { strongly in } L^{p}(0, T; L_{\textrm{loc }}^{p}(\mathbb {R}))\qquad \text {for } p\in [1, \infty ). \end{aligned}$$ -
(iii)
The momentum function sequence \(m^{\varepsilon , b}\) converges strongly in \(L^{2}(0, T; L_{\textrm{loc}}^{p}(\mathbb {R}))\) to \(m^{\varepsilon }\) for all \(p \in [1, \infty )\). In particular,
$$\begin{aligned} m^{\varepsilon , b}(t,r)=(\rho ^{\varepsilon , b} u^{\varepsilon , b})(t,r) \longrightarrow m^{\varepsilon }(t, r) \qquad { a.e.}\,\text { in }[0, T] \times (0, \infty ). \end{aligned}$$ -
(iv)
\(m^{\varepsilon }(t, r)=0\) a.e. on \(\{(t, r)\,:\,\rho ^{\varepsilon }(t, r)=0\}\). Furthermore, there exists a function \(u^{\varepsilon }(t, r)\) such that \(m^{\varepsilon }(t, r)=\rho ^{\varepsilon }(t, r) u^{\varepsilon }(t, r)\) a.e., \(u^{\varepsilon }(t, r)=0\) a.e. on \(\{(t, r)\,:\,\rho ^{\varepsilon }(t, r)=0\}\), and
$$\begin{aligned} \begin{aligned} m^{\varepsilon , b}&\longrightarrow m^{\varepsilon }=\rho ^{\varepsilon } u^{\varepsilon } \qquad \text { strongly in}\, L^{2}\left( 0, T ; L_{\textrm{loc}}^{p}(\mathbb {R})\right) \,\text { for }\, p \in [1, \infty ), \\ \frac{m^{\varepsilon , b}}{\sqrt{\rho ^{\varepsilon , b}}}&\longrightarrow \frac{m^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} =\sqrt{\rho ^{\varepsilon }} u^{\varepsilon } \qquad \text { strongly in}\, L^{2}(0, T ; L_{\textrm{loc}}^{2}(\mathbb {R})). \end{aligned} \end{aligned}$$
Let \((\rho ^{\varepsilon },m^{\varepsilon })(t,r)\) be the limit function obtained above. Using (5.13), (6.1), Lemmas 5.1–5.6, 5.8, and 6.1, Corollaries 5.2–5.3 and 5.7, Fatou’s lemma, and the lower semicontinuity, we conclude the proof of (2.22)–(2.25).
Now we show the convergence of the potential functions \(\Phi ^{\varepsilon ,b}\). Using the similar arguments as in [10, Lemma 4.6], we have
Lemma 6.2
For fixed \(\varepsilon >0\), there exists a function \(\Phi ^{\varepsilon }(t,\textbf{x})=\Phi ^{\varepsilon }(t,r)\) such that, as \(b\rightarrow \infty \) (up to a subsequence),
Moreover, since \(\gamma _2>\frac{6}{5}\),
Using (6.5), Fatou’s lemma, and Lemmas 5.1 and 6.1, we obtain the following energy inequality:
We denote
Then (2.20) follows directly from (6.6). Moreover, we can prove that \((\rho ^{\varepsilon },\mathcal {M}^{\varepsilon }, \Phi ^{\varepsilon })\) is a global weak solution of the Cauchy problem (1.10) and (2.17)–(2.18) in the sense of Definition 2.1. In fact, by the same arguments in [10, Remark 4.7 and Lemmas 4.9–4.11], we have
Lemma 6.3
Let \(0 \le t_{1}<t_{2} \le T\), and let \(\zeta (t, \textbf{x}) \in C_{0}^{1}([0, T] \times \mathbb {R}^{3})\) be any smooth function with compact support. Then
Moreover, (2.21) holds, and the total mass is conserved:
Lemma 6.4
Let \(\Psi (t, \textbf{x}) \in (C_{0}^{2}([0, T] \times \mathbb {R}^{3}))^{3}\) be any smooth function with compact support so that \(\Psi (T, \textbf{x})=0\). Then
with \(V^{\varepsilon }(t, \textbf{x}) \in L^{2}(0, T; L^{2}(\mathbb {R}^{3}))\) as a function satisfying
where \(C\left( E_{0}, M\right) >0\) is a constant independent of \(T>0\).
Lemma 6.5
It follows from (6.3) that \(\Phi ^{\varepsilon }\) satisfies Poisson’s equation in the classical sense except for the origin: \((t,{\textbf {x}})\in [0,\infty )\times (\mathbb {R}^{3}\backslash \{\varvec{0}\})\). Moreover, for any smooth function \(\xi (\textbf{x})\in C_{0}^{1}(\mathbb {R}^3)\) with compact support,
7 \(W_{\textrm{loc}}^{-1,p}\)–Compactness of Weak Entropy Dissipation Measures
In this section, using the estimates of the weak entropy pairs obtained in Lemmas 4.5, 4.7, and 4.10, we establish the compactness of weak entropy dissipation measures: \(\partial _{t}\eta ^{\psi }(\rho ^\varepsilon ,m^\varepsilon )+\partial _{r}q^{\psi }(\rho ^\varepsilon ,m^\varepsilon )\) for each weak entropy pair \((\eta ^{\psi },q^{\psi })\). Unfortunately, we fail to obtain the same \(H_{\textrm{loc}}^{-1}\)-compactness as in [10, 17], since we only obtain that \(q^{\varepsilon }\) is uniformly bounded in \(L_{\textrm{loc}}^{2}\) from Lemma 4.10 and Corollary 5.7. Instead, using similar arguments as in [10, Lemma 4.2], we can obtain the compactness in \(W_{\textrm{loc}}^{-1,p}\) for any \(p\in [1,2)\).
Lemma 7.1
(Compactness of the entropy dissipation measures). Let \((\eta ^{\psi },q^{\psi })\) be a weak entropy pair defined in (4.63) for any smooth and compactly supported function \(\psi (s)\) on \(\mathbb {R}\). Then, for \(\varepsilon \in (0,\varepsilon _0]\),
Proof
To establish (7.1), we first need to study the equation: \(\partial _{t}\eta ^{\psi }(\rho ^{\varepsilon },m^{\varepsilon })+\partial _{r}q(\rho ^{\varepsilon },m^{\varepsilon })\) in the distributional sense, which is more complicated than that in [13, 14]. For simplicity, we denote \((\eta ^{\varepsilon ,b},q^{\varepsilon ,b})=(\eta ^{\psi }(\rho ^{\varepsilon ,b},m^{\varepsilon ,b}),q^{\psi }(\rho ^{\varepsilon ,b},m^{\varepsilon ,b}))\) and \((\eta ^{\varepsilon },q^{\varepsilon })=(\eta ^{\psi }(\rho ^{\varepsilon },m^{\varepsilon }),q^{\psi }(\rho ^{\varepsilon },m^{\varepsilon }))\). We divide it into four steps.
1. Considering \((5.1)_1\times \eta _{\rho }^{\varepsilon ,b}+(5.1)_2\times \eta _{m}^{\varepsilon ,b}\), we obtain
where \(\rho ^{\varepsilon ,b}\) is understood to be zero in domain \([0,T]\times [0,a)\) so that \(\int _{a}^{r}\rho ^{\varepsilon ,b}(t,z)\,z^{2}\textrm{d}z\) can be written as \(\int _{0}^{r}\rho ^{\varepsilon ,b}(t,z)\,z^{2}\textrm{d}z\) in the potential term. Let \(\phi (t, r) \in C_{0}^{\infty }\left( \mathbb {R}_{+}^{2}\right) \) and \(b \gg 1\) so that \({\text {supp}}\phi (t, \cdot ) \in (a, b(t))\). Multiplying (7.2) by \(\phi \) and integrating by parts yield
2. From Lemmas 4.2 and 6.1, it is clear to see that
In \(\{(t,r):\,\rho ^{\varepsilon }(t,r)=0\}\), it follows from Lemmas 4.5 and 4.7 that
Combining (7.4)–(7.5), we obtain
Similarly, it follows from Lemmas 4.3, 4.10, and 6.1 that
For \(\gamma _2 \in (1,3)\) and any subset \(K\Subset (0,\infty )\), it follows from Lemmas 4.5, 4.7, 4.10, and Corollary 5.7 that
which implies that \((\eta ^{\varepsilon , b}, q^{\varepsilon , b})\) is uniformly bounded in \(L_{\textrm{loc}}^{2}(\mathbb {R}_{+}^{2})\). This, with (7.6)–(7.7), yields that, up to a subsequence,
Thus, for any \(\phi \in C_{0}^{1}(\mathbb {R}_{+}^{2})\), as \(b \rightarrow \infty \) (up to a subsequence),
Furthermore, \((\eta ^{\varepsilon }, q^{\varepsilon })\) is uniformly bounded in \(L_{\textrm{loc }}^{2}(\mathbb {R}_{+}^{2})\), which implies that
Since \(q^{\varepsilon ,b}\) is only uniformly bounded in \(L_{\textrm{loc}}^{2}(\mathbb {R}_{+}^2)\) in view of Lemma 4.10 and Corollary 5.7, we cannot conclude that \(\partial _{t}\eta ^{\varepsilon }+\partial _{r}q^{\varepsilon }\) is uniformly bounded in \(\varepsilon >0\) in \(W_{\textrm{loc}}^{-1,p}(\mathbb {R}_{+}^2)\) with \(p>2\), which is different from [10].
3. For the terms on the RHS of (7.3), using Lemmas 4.5, 4.7, and 4.10, and similar calculations as in [10, Lemma 5.11], we obtain that
and there exist local bounded Radon measures \((\mu _{1}^{\varepsilon }, \mu _{2}^{\varepsilon }, \mu _{3}^{\varepsilon })\) on \(\mathbb {R}_{+}^{2}\) such that, as \(b \rightarrow \infty \) (up to a subsequence),
In addition, for \(i=1,2,3\),
Then, up to a subsequence, we have
Moreover, there exists a function \(f^{\varepsilon }\) such that, as \(b \rightarrow \infty \) (up to a subsequence),
Then it follows from (7.13) that
4. Taking \(b \rightarrow \infty \) (up to a subsequence) on both sides of (7.3), it follows from (7.8), (7.10), (7.12), and (7.14) that
in the distributional sense. It follows from (7.10)–(7.11) that
is a locally uniformly bounded Radon measure sequence. From (7.13), we know that
Then it follows from (7.15)–(7.16) that the sequence:
which also implies that
On the other hand, the interpolation compactness theorem (cf. [8, 22]) indicates that, for \(p_{2}>1, p_{1} \in \left( p_{2}, \infty \right] \), and \(p_{0} \in \left[ p_{2}, p_{1}\right) \),
which is a generalization of Murat’s lemma in [58, 67]. Combining this theorem for \(1<p_{2}<2\) and \(p_{1}=2\) with the facts in (7.9) and (7.17), we conclude that
Combining (7.19) with (7.18), we conclude (7.1). \(\square \)
8 \(L^p\) Compensated Compactness Framework
In this section, with the help of our understanding of the singularities of the entropy kernel and entropy flux kernel obtained in Lemma 4.14, we now establish the \(L^p\) compensated compactness framework and complete the proof of Theorem 2.2. The key ingredient is to prove the reduction of the Young measure. The arguments are similar to [63, Sect. 4] and [64, Sect. 7], based on [11, 13], so we only sketch the proof for self-containedness.
We denote the upper half-plane by
and consider the following subset of continuous functions:
where \(\mathbb {S}^{1} \subset \mathbb {R}^{2}\) is the unit circle. Since \(\overline{C}(\mathbb {H})\) is a complete sub-ring of the continuous functions on \(\mathbb {H}\) containing the constant functions, there exists a compactification \(\overline{\mathcal {H}}\) of \(\mathbb {H}\) such that \(C(\overline{\mathcal {H}})\) is isometrically isomorphic to \(\overline{C}(\mathbb {H})\) (cf. [43]), written \(C(\overline{\mathcal {H}})\cong \overline{C}(\mathbb {H})\). The topology of \(\overline{\mathcal {H}}\) is the weak-star topology induced by \(C(\overline{\mathcal {H}})\), i.e., a sequence \(\{v_{n}\}_{n \in \mathbb {N}}\) in \(\overline{\mathcal {H}}\) converges to \(v \in \overline{\mathcal {H}}\) if \(|\varphi (v_{n})-\varphi (v)| \rightarrow 0\) for all \(\varphi \in C(\overline{\mathcal {H}})\), which is separable and metrizable (cf. [43]). Denote by V the weak-star closure of the vacuum states \(\{(\rho ,u) \in \mathbb {R}^2:\, \rho =0\}\) and define \(\mathcal {H}:=\mathbb {H}\cup V\). In view of the functions that lie in \(\overline{C}(\mathbb {H})\), the topology of \(\overline{\mathcal {H}}\) does not distinguish points in V. Since \(\overline{\mathcal {H}}\) is homeomorphic to a compact metric space, we may apply the fundamental theorem of Young measures in Alberti-Müller [1, Theorem 2.4].
Lemma 8.1
([1, Theorem 2.4]). Given any sequence of measurable functions \(\left( \rho ^{\varepsilon }, \rho ^{\varepsilon }u^{\varepsilon }\right) : \mathbb {R}_{+}^{2} \rightarrow \overline{\mathbb {H}}\), there exists a subsequence (still denoted) \((\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\) generating a Young measure \(\nu _{(t, r)} \in {\text {Prob}}(\overline{\mathcal {H}})\) in the sense that, for any \(\phi \in \overline{C}(\mathbb {H})\),
where \(\iota :\overline{C}(\mathbb {H})\rightarrow C(\overline{\mathcal {H}})\) is an isometrically isomorphism. Moreover, sequence \((\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\) converges to \((\rho , \rho u):\,\mathbb {R}_{+}^2\rightarrow \overline{\mathcal {H}}\) a.e. if and only if
in the phase coordinates \((\rho ,m)\) with \(m=\rho u\).
From now on, we often use the same letter \(\nu _{(t,r)}\) for an element of \(\big (\overline{C}(\mathbb {H})\big )^{*}\) or \(\big (C(\overline{\mathcal {H}})\big )^{*}\), and use the same letter for \(\iota (\phi )\) and \(\phi \) for simplicity, when no confusion arises.
The following lemma shows that the Young measure \(\nu _{(t,r)}\), generated by the sequence of measurable approximate solutions \((\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\) satisfying the assumptions of Theorem 2.2, is only supported on the interior of \(\mathcal {H}\). Moreover, the Young measure \(\nu _{(t,r)}\) can be extended to a larger class of test functions than just \(\overline{C}(\mathbb {H})\). This is proved in [13, Proposition 5.1]; also see [43, Proposition 2.3].
Lemma 8.2
([13, Proposition 5.1]). The following statements hold:
-
(i)
For the Young measure \(\nu _{(t,r)}\) generated by the sequence of measurable approximate solutions \((\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\) satisfying the assumptions in Theorem 2.2,
$$\begin{aligned} (t,r)\mapsto \int _{\mathbb {H}}(\rho ^{\gamma _2+1}+\rho |u|^3)\,\textrm{d}\nu _{(t,r)}(\rho ,u)\in L_{\textrm{loc}}^1(\mathbb {R}_{+}^2) . \end{aligned}$$ -
(ii)
Let \(\phi (\rho ,u)\) be a function such that
-
(a)
\(\phi \) is continuous on \(\overline{\mathbb {H}}\) and \(\phi =0\) on \(\partial \mathbb {H}\),
-
(b)
there exists a constant \(\mathfrak {a}>0\) such that \({\text {supp}}\phi \subset \{u+k(\rho )\ge -\mathfrak {a},u-k(\rho )\le \mathfrak {a}\}\),
-
(c)
\(|\phi (\rho ,u)|\le \rho ^{\beta (\gamma _2+1)}\) for all \((\rho ,u)\) with large \(\rho \) and some \(\beta \in (0,1)\).
Then \(\phi \) is \(\nu _{(t,r)}\)–integrable for \((t,r)\in \mathbb {R}_{+}^2\) a.e. and
$$\begin{aligned} \phi (\rho ^{\varepsilon }(t,r), u^{\varepsilon }(t,r))\rightharpoonup \int _{\mathbb {H}}\phi (\rho ,u) \,\textrm{d}\nu _{(t,r)}(\rho ,u)\qquad \text {in }\;L_{\textrm{loc}}^1(\mathbb {R}_{+}^2). \end{aligned}$$ -
(a)
-
(iii)
\(\nu _{(t,r)}\in {\text {Prob}}(\mathcal {H})\) for \((t,r)\in \mathbb {R}_{+}^2\) a.e., that is, \(\, \nu _{(t,r)}\big (\overline{\mathcal {H}}\backslash (\mathbb {H}\cup V)\big )=0. \)
We now prove the commutation relation. Since we only have the \(W_{\textrm{loc}}^{-1,p}\)–compactness of the entropy dissipation measures for \(p\in [1,2)\), the classical div-curl lemma in [58] fails to obtain the commutation relation. Thus, we adopt an improved version of the div-curl lemma.
Lemma 8.3
([19, Theorem]). Let \(\Omega \subset \mathbb {R}^n\) be an open bounded set, and \(p,q\in (1,\infty )\) with \(\frac{1}{p}+\frac{1}{q}=1\). Let \(\textbf{v}^{\varepsilon }\) and \(\textbf{w}^{\varepsilon }\) are sequences of vector fields such that
Suppose that \(\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\) is equi-integrable uniformly in \(\varepsilon \), and
Then \(\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\,\rightharpoonup \,\textbf{v}\cdot \textbf{w}\) in \(\mathcal {D}^{\prime }(\Omega )\).
Lemma 8.4
(Commutation relation). Let \(\{(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\}_{\varepsilon >0}\) be the measurable approximate solutions satisfying the assumptions of Theorem 2.2, and let \(\nu _{(t,r)}\) be a Young measure generated by the family \(\{(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\}_{\varepsilon >0}\) in Lemma 8.2. Then
for all \(s_1,s_2\in \mathbb {R}\), where \(\displaystyle \overline{f}:=\int f\, \textrm{d}\nu _{(t,r)}\), \(\chi (s_i)=\chi (\cdot , \cdot -s_i)\), and \(\sigma (s_i)=\sigma (\cdot ,\cdot -s_i)\).
Proof
For any \(\psi \in C_0^2(\mathbb {R})\), it follows from Lemmas 4.5, 4.7, and 4.10 that
It is clear that the support of \((\eta ^{\psi }, q^{\psi })\) is contained in \(\left\{ k(\rho )+u \ge -L, u-k(\rho ) \le L\right\} \) for some \(L>0\) depending only on \({\text {supp}}\,\psi \). For any \(\psi _{1}, \psi _{2} \in C_{0}^{2}(\mathbb {R})\), we consider the sequences of vector fields:
Noting \(\rho ^{\varepsilon }\in L_{\textrm{loc}}^{1+\gamma _2}(\mathbb {R}_{+}^2)\) and (8.2), we see that both \(\textbf{v}^{\varepsilon }\) and \(\textbf{w}^{\varepsilon }\) are uniformly bounded sequences in \(L_{\textrm{loc}}^2(\mathbb {R}_{+}^2)\). Moreover, by Lemma 8.2 and the uniqueness of weak limits, we obtain
By direct calculation, we see that
Using (8.2), we obtain that \( \vert \textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\vert \le C\big ((\rho ^{\varepsilon })^2+(\rho ^{\varepsilon })^{2+\theta _2}\big ) \) for \(\rho >0\) which, with (2.25), yields that \(\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\in L_{\textrm{loc}}^{\frac{1+\gamma _2}{2+\theta _2}}(\mathbb {R}_{+}^2)\) uniformly in \(\varepsilon \). Thus, \(\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\) is equi-integrable uniformly in \(\varepsilon \) since \(\frac{1+\gamma _2}{2+\theta _2}>1\). It follows from Lemma 8.3 that
On the other hand, using (8.2) and Lemma 8.2, we find that
which, with (8.3), yields that
It follows from (8.4) and Fubini’s theorem that
Since \(\psi _{1}, \psi _{2} \in C_{0}^{2}(\mathbb {R})\) are arbitrary, we conclude
This completes the proof. \(\square \)
Theorem 8.5
(Reduction of the Young measure). Let \(\nu _{(t,r)}\in \textrm{Prob}(\mathcal {H})\) be the Young measure generated by sequence \(\{(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\}_{\varepsilon >0}\) in Lemma 8.2. Then either \(\nu _{(t,r)}\) is contained in V or the support of \(\nu _{(t,r)}\) is a single point in \(\mathbb {H}\).
Proof
The proof is similar to [11, 48, 63, 64]. Since the estimates of the entropy kernel and entropy flux kernel are different, we sketch the proof for self-containedness.
Taking \(s_1,s_2,s_3\in \mathbb {R}\) and multiplying (8.1) by \(\overline{\chi (s_3)}\), one obtains
Cyclically permuting index \(s_j\) and adding the resultant equations together, we have
Applying the fractional derivative operators \(P_{2}:=\partial _{s_{2}}^{\lambda _1+1}\) and \(P_{3}:=\partial _{s_{3}}^{\lambda _1+1}\) in the sense of distributions to obtain
where, for example, distribution \(\overline{P_2\chi (s_2)}\) is defined by
We take two standard but different functions \(\phi _{2}, \phi _{3} \in C_{0}^{\infty }(-1,1)\) such that \(\displaystyle \int _{\mathbb {R}} \phi _{j}(s_{j})\,\textrm{d} s_{j}=1\) with \(\phi _{j} \ge 0\) for \(j=2,3\). For \(\tau >0\), denote \(\phi _{j}^{\tau }(s_{j}):=\frac{1}{\tau }\phi _{j}(\frac{s_{j}}{\tau })\). As indicated in [43], we can always choose \(\phi _2\) and \(\phi _3\) such that
Multiplying (8.5) by \(\phi _{2}^{\tau }(s_1-s_2)\phi _3^{\tau }(s_1-s_3)\) and integrating the resultant equation with respect to \((s_2, s_3)\) yield
where we have used the notion: \( \overline{P_{j} \chi _{j}^{\tau }}=\overline{P_{j} \chi _{j}} * \phi _{j}^{\tau }(s_{1}) =\int _{\mathbb {R}} \overline{\partial _{s_{j}}^{\lambda } \chi (s_{j})} \frac{1}{\tau ^{2}} \phi _{j}^{\prime }(\frac{s_{1}-s_{j}}{\tau }) \,\textrm{d}s_{j}\) for \(j=2,3\).
Multiplying (8.7) by \(\psi (s_1)\in \mathcal {D}(\mathbb {R})\), integrating the resultant equation with respect to \(s_1\), then taking limit \(\tau \rightarrow 0\) and applying Lemmas 8.8–8.9 below, we obtain
Noting that \(Z(\rho )>0\) for \(\rho >0\) from Lemma 8.7 below, \(Y(\phi _2,\phi _3)> 0\) from (8.6), and \(\psi (s)\) is an arbitrary test function, we deduce from (8.8) that
We define \(\mathbb {S}=\{s\in \mathbb {R}:\,\overline{\chi (s)}>0\}\). It follows from [63] that \(\mathbb {S}\) admits the representation:
For the case: \(\mathbb {S}=\emptyset \), it is clear that \(\overline{\chi (s)}=0\) for all \(s\in \mathbb {R}\) so that \({\text {supp}}\nu _{(t,r)}\subset V\), since \(\chi (s)>0\) for all \(\rho >0\) and \(s\in (u-k(\rho ), u+k(\rho ))\).
For the case: \(\mathbb {S}\ne \emptyset \), it follows from (8.15) below that \(s\mapsto \overline{\chi (s)}\) is a continuous map. Then \(\mathbb {S}\) is an open set so that \(\mathbb {S}\) is at most a countable union of open intervals. Thus, we may write
for at most countably many numbers \(\zeta _k:=u_{k}-k(\rho _{k})\) and \(\xi _k:=u_{k}+k(\rho _{k})\) with \((\rho _{k},u_{k})\in \textrm{supp}\nu _{(t,r)}\) in the extended real line \(\mathbb {R}\cup \{\pm \infty \}\) such that \(\zeta _k<\xi _k\le \zeta _{k+1}\) for all k. For later use, we denote the Riemann invariants \(z(\rho ,u):=u-k(\rho ) \) and \(w(\rho ,u):=u+k(\rho )\). Thus, noting that \(\textrm{supp}\,\chi (s)=\{(\rho ,u)\,:\,z(\rho ,u)\le s\le w(\rho ,u)\}\), we obtain
If \(\zeta _{k}\) and \(\xi _{k}\) are both finite, due to the fact that \(k(\rho )\) is a strictly monotone increasing and unbounded function of \(\rho \), it is clear that \(\left\{ (\rho , u) \,:\, \zeta _{k} \le z(\rho , u)\le w(\rho , u) \le \xi _{k}\right\} \) is bounded. Now we deduce from (8.9) that
Thus, the support of measure \(\nu _{(t,r)}\) must be contained in the vacuum set V and at most a countable union of points \(P_k(\rho _{k},\,u_{k})\):
Therefore, we may write
with measure \(\nu _{V}\) supported on the vacuum set V. For later use, we denote
We claim that, if \(\chi (P_{k},s)>0\), then \(\chi (P_{k'},s)=0\) for all \(k\ne k'\). Indeed, recall that \({\text {supp}}\chi (s)=\{(\rho ,u)\,:\,z(\rho ,u)\le s\le w(\rho ,u)\}\) and that \(\chi (\rho , u, s)>0\) if and only if \(z(\rho ,u)<s<w(\rho ,u)\). If \(\chi (P_{k},s)>0\), then \(\zeta _{k}<s<\xi _{k}\). If, in addition, \(\chi (P_{k'},s)>0\) for some \(k\ne k'\), it must hold that \(\zeta _{k'}< s < \xi _{k'}\). However, since \(\xi _{k-1}\le \zeta _{k}< \xi _{k}\le \zeta _{k+1}\), this is impossible for any \(P_{k'}\) with \(k' \ne k\).
Thus, taking \(s_{1}, s_{2} \in \mathbb {R}\) such that \(\chi (P_{k},s_{1})\chi (P_{k},s_{2})>0\), we deduce from the commutation relation (8.1) and (8.10) that
Now, choosing \(s_{1}\) and \(s_{2}\) such that the second factor in this expression is non-zero, we obtain that \(\alpha _k=0\) or 1 for all k. This completes the proof. \(\square \)
Combining Theorem 8.5 with Lemma 8.1, we conclude that \((\rho ^{\varepsilon },m^{\varepsilon })\) converges to \((\rho ,m)\) almost everywhere. Moreover, noting that \(|m^{\varepsilon }|^{\frac{3(\gamma _{2}+1)}{\gamma _{2}+3}}\le C\big ((\rho ^{\varepsilon })^{\gamma _{2}+1}+\rho ^{\varepsilon }|u^{\varepsilon }|^3\big )\) for any \(T,d,D>0\), we have
which implies that \(m^{\varepsilon }\) is uniformly bounded in \(L_{\textrm{loc}}^{\frac{3(\gamma _{2}+1)}{\gamma _{2}+3}}(\mathbb {R}_{+}^2)\) with respect to \(\varepsilon \). This implies that (2.36) holds. Therefore, the proof of Theorem 2.2 is complete.
Now, we are going to prove the auxiliary lemmas, Lemmas 8.8–8.9, which are used in the proof of Theorem 8.5. We first recall two useful lemmas in [11, 43].
Lemma 8.6
([43, Lemmas 3.8–3.9]). Let \(\mathfrak {R} \in C_{\textrm{loc }}^{\alpha }(\mathbb {R})\) be a Hölder continuous function for some \(\alpha \in (0,1)\), and let \(g \in C_{0}^{\alpha }(\mathbb {R})\) be a Hölder continuous function with compact support. Assume \(L_{0}>2\) such that \({\text {supp}} g \subset B_{L_0-2}(0)\).
-
(i)
For any pair of distributions \(T_{2}, T_{3} \in \mathcal {D}^{\prime }(\mathbb {R})\) from the following collection:
$$\begin{aligned} (T_{2}, T_{3})=(\delta , Q_{3}), \,\, (PV, Q_{3}), \,\, (Q_{2}, Q_{3}) \end{aligned}$$with \(Q_{2}, Q_{3} \in \{H, Ci, {\mathfrak {R}}\}\), there exists a constant \(C>0\) independent of \((\rho , u)\) such that
$$\begin{aligned}&\sup \limits _{\tau \in (0,1)}\Big \vert \int _{-\infty }^{\infty } g(s_{1})\Big \{\big (T_{2}(s_{2}-u \pm k(\rho )) T_{3}(s_{3}-u \pm k(\rho ))\\&\qquad \qquad \qquad \qquad \qquad \quad -T_{2}(s_{3}-u\pm k(\rho ))T_{3}(s_{2}-u\pm k(\rho ))\big ) * \phi _{2}^{\tau } * \phi _{3}^{\tau }\Big \} \left( s_{1}\right) \,\textrm{d} s_{1}\Big \vert \\&\quad \le C\Vert g\Vert _{C^{ \alpha }(\mathbb {R})} \big (1+\Vert \mathfrak {R}\Vert _{C^{ \alpha }(\overline{B_{L_0}(0)})}\big )^{2}. \end{aligned}$$ -
(ii)
For any pair of distributions from
$$\begin{aligned} (T_{2}, T_{3})=(\delta , \delta ), \,\,(PV, PV),\,\, (Q_{2}, Q_{3}),\,\, (\delta , PV), \,\, (PV, Q_{3}), \end{aligned}$$with \(Q_{2}, Q_{3} \in \{H, Ci, \mathfrak {R}\}\), there exists \(C>0\) independent of \((\rho ,u)\) such that
$$\begin{aligned}&\sup \limits _{\tau \in (0,1)} \Big \vert \int _{-\infty }^{\infty }\Big \{\big ((s_{2}-s_{3}) T_{2}(s_{2} -u \pm k(\rho ))T_{3}(s_{3}-u \pm k(\rho ))\big )* \phi _{2}^{\tau } * \phi _{3}^{\tau }\Big \}(s_{1}) \,\textrm{d} s_{1} \Big \vert \\&\quad \le C\Vert g\Vert _{C^{ \alpha }(\mathbb {R})}\big (1+\Vert \mathfrak {R}\Vert _{C^{ \alpha }(\overline{B_{L_0}(0)})}\big )^{2}. \end{aligned}$$
Motivated by [11], it follows from Lemmas 4.2–4.3, (1.4), and a direct calculation that
Lemma 8.7
([11, Lemmas 4.2–4.3]). The mollified fractional derivatives of the entropy kernel and the entropy flux kernel satisfy the following convergence properties:
-
(i)
When \(0\le \rho <\infty \),
$$\begin{aligned} P_{2} \chi _{2}^{\tau } P_{3} \sigma _{3}^{\tau }-P_{3} \chi _{3}^{\tau } P_{2} \sigma _{2}^{\tau } \longrightarrow Y(\phi _{2}, \phi _{3}) Z(\rho ) \sum _{\pm }(K^{\pm })^{2} \delta _{s_{1}=u \pm k(\rho )} \end{aligned}$$as \(\tau \rightarrow 0\) weakly-star in measures in \(s_{1}\) and locally uniformly in \((\rho , u)\), where \(Y(\phi _2,\phi _3)\) satisfies (8.6), \(Z(\rho ):=(\lambda _1+1) M_{\lambda }^{-2} k(\rho )^{2 \lambda } D(\rho )>0\) with \(D(\rho )\) defined in (8.11), and \(K^{\pm }\ne 0\) are some constants.
-
(ii)
For \(j=2,3\), \(\chi _1\,P_{j}\sigma _{j}^{\tau }-\sigma _1\,P_{j}\chi _{j}^{\tau }\) are Hölder continuous in \((\rho ,u,s_1)\), uniformly in \(\tau \), and there exists a Hölder continuous function \(X=X(\rho ,u,s_1)\), independent of the mollifying sequence \(\phi _j\), such that
$$\begin{aligned} \chi (s_1) P_{j} \sigma _{j}^{\tau }-P_{j} \chi _{j}^{\tau } \sigma (s_1) \longrightarrow X(\rho , u, s_{1}) \qquad \text {as}\, \tau \rightarrow 0 \end{aligned}$$uniformly in \(\left( \rho , u, s_{1}\right) \) on the sets on which \(\rho \) is bounded.
Lemma 8.8
For any test function \(\psi \in \mathcal {D}(\mathbb {R})\),
where \(Y(\phi _2,\phi _3)\) is defined by (8.6) and \(Z(\rho )\) is given in Lemma 8.7.
Proof
It follows from Lemma 8.7 that, when \(\rho \) is bounded,
locally uniform in \((\rho ,u)\) and hence pointwise for all \((\rho ,u)\). Therefore, we have
For \(\rho \ge \rho ^{*}\), we notice that
Using Lemma 4.14, we see that (8.14) consists of a sum of terms of the form:
with \(T_{2}, T_{3} \in \{\delta , \textrm{PV},H, \textrm{Ci}\}\), the terms of the form:
and the terms of the form:
with \(T_{2} \in \{\delta , H, \textrm{PV}, \textrm{Ci}, r_{\chi }\}\) and \(T_{3} \in \{H, \textrm{Ci}, r_{\sigma }\}\). We emphasize that, in the last two cases, when \(T_2,T_3\in \{r_{\chi },r_{\sigma }\}\), \(A_{i,\pm }(\rho )B_{j,\pm }(\rho )\) or \(A_{i,\pm }(\rho )B_{j,\pm }(\rho )(s_{k}-u)\) should be replaced by 1.
Before applying Lemma 8.6, we now show that \(\overline{\chi (s)}\) is Hölder continuous. In fact, it follows from Corollary 4.12 and Lemma 8.2 that, for any \(s,s'\in \mathbb {R}\) and \(\alpha \in (0,\min \{\lambda _1,1\}]\),
which implies that \(\overline{\chi (s)}\) is Hölder continuous. Hence, using Lemma 8.6 and the fact that \(|s_{j}-u|\le k(\rho )\) for \(j=2,3\), we obtain
with \(L_{0}:=|{\text {supp}}\psi |+2\) and \(\beta =\frac{\theta _2+2}{\gamma _2+1}\in (0,1)\). Thus, using Lemmas 8.2 and 8.7, and Lebesgue’s dominated convergence theorem, we obtain
which, with (8.13), yields (8.12). This completes the proof. \(\square \)
Lemma 8.9
For any test function \(\psi \in \mathcal {D}(\mathbb {R})\),
Proof
Fix \((\rho ,u)\in \mathbb {H}\). It follows from Lemma 8.7 that
It is clear that
It follows from Lemma 4.14 that \(P_{j} \chi _{j}^{\tau }, j=2,3\), are measures in \(s_{1}\) such that
where \(\Vert \mu \Vert _{\mathfrak {M}, \alpha } =\sup \left\{ |\langle \mu , f\rangle |\,:\, f\in C_{0}^{ \alpha }(\mathbb {R})\right. \) and \(\left. \Vert f\Vert _{C^{ \alpha }(\mathbb {R})} \le 1\right\} \) with \(\alpha \in (0,1)\). Then we use Lemma 8.2 and Lebesgue’s dominated convergence theorem to pass the limit inside the Young measure to obtain
pointwise in \((\rho ,u)\) as \(\tau \rightarrow 0\). Now we are going to prove
for some constants \(C>0\) and \(\beta \in (0,1]\), which are both independent of \(\tau \). Once (8.17) is proved, it follows from Lebesgue’s dominated convergence theorem that
Since \(X(\rho ,u,s_{1})\) is independent of the choice of the mollifying functions \(\phi _{2}^{\tau }\) and \(\phi _{3}^{\tau }\) from Lemma 8.7, we may interchange the roles of \(s_{2}\) and \(s_{3}\) to conclude the proof of (8.16).
To see the validity of (8.17), we begin by observing that, for \(j=2,3\), \(\overline{P_j\chi _{j}^{\tau }}(s_1)\) and \(\psi (s_1)\) are independent of \((\rho ,u)\). Then it suffices to estimate the function:
It follows from Lemmas 4.2–4.3 and 4.14 (also see [11, Proof of Lemma 4.2]) that
with
and \(E^{4,\tau }\) is the remainder term which consists of the mollification of continuous functions, where we have used the notation: \(G_{\lambda _1}(s_1)=[k(\rho )^2-(u-s_1)^2]^{\lambda _1}\), and \(g_{i}(s_1)=g_{i}(\rho ,u-s_1)\) for \(i=1,2\).
We first demonstrate the uniform bound on the term involving the delta measures. By direct calculation, we have
Noting that
using (8.19)–(8.20) and Lemmas 4.2–4.3 and 4.11–4.14, we obtain
For the term involving with the principal value distribution, a direct calculation shows that
If \(|x|\le 2\tau \), we have
On the other hand, if \(|x|\ge 2\tau \), we can obtain
Notice that
which, with (8.20), (8.22)–(8.24), and Lemmas 4.2–4.3 and 4.11–4.14, yields
Combining (8.21) with (8.25) yields that there exists \(\beta _1=\frac{3+\theta _2}{2(\gamma _2+1)}\in (0,1)\) such that
For \(E^{4,\tau }\) consisting of the mollification of continuous functions, direct calculations show that
with \(\beta _2=\frac{4+3\theta _2}{2(\gamma _2+1)}\in (0,1)\). Combining (8.27) with (8.26), we conclude the proof of (8.17). \(\square \)
9 Existence of Global Finite-Energy Solutions of CEPEs
In this section, we complete the proof of Theorem 2.3. Since the proof is similar to [10], we sketch the proof for the self-containedness of this paper. We divide the proof into four steps.
1. Since \((\rho ^{\varepsilon },m^{\varepsilon })(t,r)\) obtained in Theorem 2.1 satisfies all the assumptions of Theorem 2.2, then it follows from Theorem 2.2 that there exists a vector function \((\rho ,m)(t,r)\) such that, up to a subsequence as \(\varepsilon \rightarrow 0\),
for \(p_{1} \in [1, \gamma _2+1)\) and \(p_{2} \in [1, \frac{3(\gamma _2+1)}{\gamma _2+3})\), where \(L_{\textrm{loc}}^{p_{j}}(\mathbb {R}_{+}^{2})\) represents \(L^{p_{j}}([0, T] \times K)\) for any \(T>0\) and \(K \Subset (0, \infty ), j=1,2\).
Noting (9.1) and \(\rho ^{\varepsilon }\ge 0\) a.e. from Lemma 6.1, it is direct to show that \(\rho (t,r)\ge 0\) a.e. on \(\mathbb {R}_{+}^2\). Moreover, it follows from (2.22) that \(\sqrt{\rho ^{\varepsilon }}u^{\varepsilon }r=\frac{m^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}}r\) is uniformly bounded in \(L^{\infty }(0,T;L^2(\mathbb {R}))\). Then using Fatou’s lemma yields
Thus, \(m(t,r)=0\) a.e. on \(\{(t,r)\,:\,\rho (t,r)=0\}\), and we can define the limit velocity u(t, r) as
Therefore, \(m(t,r)=\rho (t,r)u(t,r)\) a.e. on \(\mathbb {R}_{+}^2\). Also, we can define \(\big (\frac{m}{\sqrt{\rho }}\big )(t,r):=\sqrt{\rho (t,r)}u(t,r)\), which is zero a.e. on \(\{(t,r):\,\rho (t,r)=0\}\). Moreover, using (2.24) and Fatou’s lemma, we obtain
for any \([d,D]\Subset (0,\infty )\).
By similar calculations as in [10, Sect. 5], we obtain that, as \(\varepsilon \rightarrow 0\),
for any \(T>0\) and \([d, D] \Subset (0, \infty )\).
From (9.2)–(9.3), we also obtain the convergence of the mechanical energy as \(\varepsilon \rightarrow 0\):
Using (9.2), (9.4), and Fatou’s lemma, and taking limit \(\varepsilon \rightarrow 0\) in (2.21)–(2.22), we have
which implies
This indicates that \(\rho (t,r)\in L^{\infty }([0,T];L^{\gamma _2}(\mathbb {R};r^{2}\textrm{d}r))\), which implies that \(\rho (t,\textbf{x})\) is a function in \(L^{\infty }([0,T];L^{\gamma _2}(\mathbb {R}^3))\) with \(\gamma _2>1\) (rather than a measure in \((t,\textbf{x})\)). Therefore, no delta measure (i.e., concentration) is formed in the density in the time interval [0, T], especially at the origin: \(r=0\).
2. For the convergence of the gravitational potential functions \(\Phi ^{\varepsilon }(t,r)\), by similar calculation in [10, Sect. 5], we obtain that, as \(\varepsilon \rightarrow 0\) (up to a subsequence),
Thus, using (6.3), (9.1), (9.7), Fatou’s lemma, and similar arguments as in (9.5)–(9.6), we have
On the other hand, it follows from (6.4) that there exists a function \(\Phi (t, \textbf{x})=\Phi (t, r)\) such that, as \(\varepsilon \rightarrow 0\) (up to a subsequence),
Thus, by (9.7) and the uniqueness of limit, we obtain that \(\Phi _{r}(t, r) r^{2}=\int _{0}^{r} \rho (t, z) z^{2} \,\textrm{d} z\) a.e. \((t, r) \in \mathbb {R}_{+}^{2}\). By similar arguments in [10, Sect. 5], we also have the strong convergence of the potential functions:
3. Now we define
Then it follows from (2.20), (9.8), and Fatou’s lemma that
which implies that, for a.e. \(t \ge 0\),
On the other hand, using (2.22), (9.6), and (9.8), we obtain
Combining (9.9) with (9.10), we complete the proof of (2.28).
4. Using (6.7), (6.9)–(6.10), and similar arguments as in [17, Sect. 5], we conclude the proof of (2.29)–(2.31) which, along with Steps 1–3, shows that \((\rho , \mathcal {M}, \Phi )(t, \textbf{x})\) is indeed a global weak solution of problem (1.1) and (1.13)–(1.14) in sense of Definition 2.2. This completes the proof. \(\square \)
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Acknowledgements
Gui-Qiang G. Chen’s research is partially supported by the UK Engineering and Physical Sciences Research Council Awards EP/L015811/1, EP/V008854/1, and EP/V051121/1. Feimin Huang’s research is partially supported by the National Natural Science Foundation of China No. 12288201 and the National Key R &D Program of China No. 2021YFA1000800. Tianhong Li’s research is partially supported by the National Natural Science Foundation of China No. 10931007. Yong Wang’s research is partially supported by the National Natural Science Foundation of China Nos. 12022114 and 12288201, and CAS Project for Young Scientists in Basic Research, Grant No. YSBR-031.
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Appendix A. Some Inequalities
Appendix A. Some Inequalities
1.1 A.1. A sharp Sobolev inequality
In this subsection, we recall a sharp Sobolev inequality, which is used in Sect. 5.1. The proof can be found in [45, Sect. 8.3].
Lemma A.1
(Sobolev inequality). Let \(n\ge 3\) and \(\nabla f\in L^2(\mathbb {R}^n)\) with \(\lim _{|\textbf{x}|\rightarrow \infty }f(\textbf{x})=0\). Then
where \(A_{n}=\frac{4}{n(n-2)}\omega _{n+1}^{-\frac{2}{n}}\) is the best constant and \(\omega _{k}=\frac{2\pi ^{\frac{k}{2}}}{\Gamma (\frac{k}{2})}\) is the surface area of the unit sphere in \(\mathbb {R}^k\).
1.2 A.2. Some variants of the Grönwall inequality
In this subsection, we introduce some variants of the Grönwall inequality, which plays an essential role in identifying the singularities of the entropy kernel and entropy flux kernel; see also [64].
Lemma A.2
(A variant of Grönwall inequality [59, Theorem 1.2.4]). Let x(t), y(t), z(t), and w(t) be non-negative continuous functions on \(J=[t_0,t_1]\) with \(t_0\ge 0\). If
then
Lemma A.3
Let \(\theta \ge 0\), and let d(s) be defined in (3.9). Assume that \(x(t)\ge 0\) is measurable and locally integrable, and satisfies
for some constant \(C>0\). Then there exists a possibly larger constant \(\tilde{C}>0\) independent of t such that, for \(t\ge \rho ^{*}\),
Proof
Since x(t) is positive and locally integrable, then, using Lemma 3.2, there exists a constant \(C>0\) that may depend on \(\rho ^{*}\), but independent of t, such that
This, with (A1), yields that \(x(t)\le Ct^{\theta }+\frac{1}{t}\int _{\rho ^{*}}^{t}d(s)x(s)\,\textrm{d} s\) for \(t\ge \rho ^{*}\). Applying Lemma A.2, we obtain
It is clear that
It follows from Lemma 3.2 that \( \frac{|d(r)-(1+\theta _2)|}{r}\le Cr^{-\epsilon -1} \) for \(r\ge \rho ^{*}\) which, with (A3), yields
Combining (A2) with \(|d(s)|\le 3\), we obtain that, for \(t\ge \rho ^{*}\),
Case 1. If \(0\le \theta <\theta _2\), it follows from (A4) that
Case 2. If \( \theta =\theta _2\), it follows from (A4) that
Case 3. If \( \theta >\theta _2\), then it follows from (A4) that
This completes the proof. \(\square \)
Corollary A.4
If x(t) satisfies
with \(\theta >\theta _2\), then \(\,x(t)\le Ct^{\theta } \ln t\,\) for \(t\ge \rho ^{*}\).
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Chen, GQ.G., Huang, F., Li, T. et al. Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry. Commun. Math. Phys. 405, 77 (2024). https://doi.org/10.1007/s00220-023-04916-1
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DOI: https://doi.org/10.1007/s00220-023-04916-1