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Experimental methods for screening parameters influencing the growth to product yield (Y(x/CH4)) of a biological methane production (BMP) process performed with Methanothermobacter marburgensis

  • New generation biofuels are a suitable approach to produce energy carriers in an almost CO2 neutral way. A promising reaction is the conversion of carbon dioxide (CO2) and molecular hydrogen (H2) to methane (CH4) and water (H2O). In this contribution, this so-called Sabatier reaction was performed biologically by using hydrogenotrophic and autotrophic methanogenic microorganisms from the archaea life domain. For the development of a biological methane production (BMP) process, one key parameter is the ratio of biomass production rate (rx) to methane evolution rate (MER) reflected in the growth to product yield (Y(x/CH4)) because it represents both a physiological and a scalable entity for the bioprocesses development as it quantify the selectivity of reaction with respect to the carbon. Y(x/CH4) needs also to be held constant in order to establish an adaptable media composition for developing a scalable feeding strategy. Identification of parameters and quantification of their impact on Y(x/CH4) is a necessary prerequisite for obtaining a growth kinetic model and developing advanced process control strategies especially for dynamic operation modes. In this work, process conditions and parameters impacting Y(x/CH4) were investigated by using a combination of multivariate and univariate chemostat cultures, as well as dynamic experiments. The proposed combination of methods is a novel modular approach for the development of BMP processes. It allowed determining the effects of multiple process factors on physiology and methane productivity of Methanothermobacter marburgensis. In fact, quantitative analysis of basal medium, sulphide and ammonium dilution rates, as well as the ammonium concentration revealed that all these variables vary rx without affecting MER. Hence Y(x/CH4) can be used to identify limiting or inhibiting conditions during media development tasks as well as for tuning the carbon flux of the bioprocess in an industrial application by reducing Y(x/CH4) to improve the carbon balance of the reaction.

    Citation: Sébastien Bernacchi, Simon Rittmann, Arne H. Seifert, Alexander Krajete, Christoph Herwig. Experimental methods for screening parameters influencing the growth to product yield (Y(x/CH4)) of a biological methane production (BMP) process performed with Methanothermobacter marburgensis[J]. AIMS Bioengineering, 2014, 1(2): 72-87. doi: 10.3934/bioeng.2014.2.72

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  • New generation biofuels are a suitable approach to produce energy carriers in an almost CO2 neutral way. A promising reaction is the conversion of carbon dioxide (CO2) and molecular hydrogen (H2) to methane (CH4) and water (H2O). In this contribution, this so-called Sabatier reaction was performed biologically by using hydrogenotrophic and autotrophic methanogenic microorganisms from the archaea life domain. For the development of a biological methane production (BMP) process, one key parameter is the ratio of biomass production rate (rx) to methane evolution rate (MER) reflected in the growth to product yield (Y(x/CH4)) because it represents both a physiological and a scalable entity for the bioprocesses development as it quantify the selectivity of reaction with respect to the carbon. Y(x/CH4) needs also to be held constant in order to establish an adaptable media composition for developing a scalable feeding strategy. Identification of parameters and quantification of their impact on Y(x/CH4) is a necessary prerequisite for obtaining a growth kinetic model and developing advanced process control strategies especially for dynamic operation modes. In this work, process conditions and parameters impacting Y(x/CH4) were investigated by using a combination of multivariate and univariate chemostat cultures, as well as dynamic experiments. The proposed combination of methods is a novel modular approach for the development of BMP processes. It allowed determining the effects of multiple process factors on physiology and methane productivity of Methanothermobacter marburgensis. In fact, quantitative analysis of basal medium, sulphide and ammonium dilution rates, as well as the ammonium concentration revealed that all these variables vary rx without affecting MER. Hence Y(x/CH4) can be used to identify limiting or inhibiting conditions during media development tasks as well as for tuning the carbon flux of the bioprocess in an industrial application by reducing Y(x/CH4) to improve the carbon balance of the reaction.


    Fluid-particle interaction model arises in many practical applications, and is of primary importance in the sedimentation analysis of disperse suspensions of particles in fluids. This model is one of the commonly used models nowadays in biotechnology, medicine, mineral processing and chemical engineering [27]-[25]. Usually, the fluid flow is governed by the Navier-Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation [4], the system has the form:

    $ {ρt+div(ρu)=0,(ρu)t+div(ρuu)+(P(ρ)+η)μΔuλdivu=(η+βρ)Φ,ηt+div(η(uΦ))Δη=0, $ (1)

    where $ \rho,u,\eta $, $ P(\rho) = a\rho^\gamma $, $ \Phi(x) $ denote the fluid density, velocity, the density of particles in the mixture, pressure, and the external potential respectively, $ a>0,\gamma>1 $. $ \mu>0 $ is the viscosity coefficient, and $ 3\lambda+2\mu\geq 0 $ are non-negative constants satisfied the physical requirements.

    There are many kinds of literatures on the study of the existence and behavior of solutions to Navier-Stokes equations (See [1]-[17]). Taking system (1) as an example, Carrillo $ et\ al $ [4] discussed the the global existence and asymptotic behavior of the weak solutions providing a rigorous mathematical theory based on the principle of balance laws, following the framework of Lions [18] and Feireisl $ et\ al $ [11,12]. Motivated by the stability arguments in [5], the authors also investigated the numerical analysis in [6]. Ballew and Trivisa [1] constructed suitable weak solutions and low stratification singular limit for a fluid particle interaction model. In addition, Mellet and Vasseur [20] proved the global existence of weak solutions of equations by using the entropy method on the asymptotic regime corresponding to a strong drag force and strong brownian motion. Zhang $ et\ al $ [31] establish the existence and uniqueness of classical solution to the system (1).

    Despite the important progress, there are few results of non-Newtonian fluid-particle interaction model. As we know, the Navier Stokes equations are generally accepted as a right governing equations for the compressible or incompressible motion of viscous fluids, which is usually described as

    $ {ρt+div(ρu)=0,(ρu)t+div(ρuu)div(Γ)+P=ρf, $

    where $ \Gamma $ denotes the viscous stress tensor, which depends on $ E_{ij}(\nabla u) $, and

    $ E_{ij}(\nabla u) = \frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}, $

    is the rate of strain. If the relation between the stress and rate of strain is linear, namely, $ \Gamma = \mu E_{ij}(\nabla u) $, where $ \mu $ is the viscosity coefficient, then the fluid is called Newtonian. If the relation is not linear, the fluid is called non-Newtonian. The simplest model of the stress-strain relation for such fluids given by the power laws, which states that

    $ \Gamma = \mu(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})^q, $

    for $ 0<q<1 $ (see[3]). In [16], Ladyzhenskaya proposed a special form for $ \Gamma $ on the incompressible model:

    $ \Gamma_{ij} = (\mu_0+\mu_1|\mathbb{E}(\nabla_x u)|^{p-2})\mathbb{E}_{ij}(\nabla_x u). $

    For $ \mu_0 = 0 $, if $ p<2 $ it is a pseudo-plastic fluid. In the view of physics, the model captures the shear thinning fluid for the case of $ 1<p<2 $ (see[19]).

    Non-Newtonian fluid flows are frequently encountered in many physical and industrial processes [8,9], such as porous flows of oils and gases [7], biological fluid flows of blood [30], saliva and mucus, penetration grouting of cement mortar and mixing of massive particles and fluids in drug production [13]. The possible appearance of the vacuum is one of the major difficulties when trying to prove the existence and strong regularity results. On the other hand, the constitutive behavior of non-Newtonian fluid flow is usually more complex and highly non-linear, which may bring more difficulties to study such flows.

    In recent years, there has been many research in the field of non-Newtonian flows, both theoretically and experimentally (see [14]-[26]). For example, in [14], Guo and Zhu studied the partial regularity of the generalized solutions to an incompressible monopolar non-Newtonian fluids. In [32], the trajectory attractor and global attractor for an autonomous non-Newtonian fluid in dimension two was studied. The existence and uniqueness of solutions for non-Newtonian fluids were established in [29] by applying Ladyzhenskaya's viscous stress tensor model.

    In this paper, followed by Ladyzhenskaya's model of non-Newtonian fluid, we consider the following system

    $ {ρt+(ρu)x=0,(ρu)t+(ρu2)x+ρΨxλ(|ux|p2ux)x+(P+η)x=ηΦx,(x,t)ΩT(|Ψx|q2Ψx)x=4πg(ρ1|Ω|Ωρdx),ηt+(η(uΦx))x=ηxx, $ (2)

    with the initial and boundary conditions

    $ {(ρ,u,η)|t=0=(ρ0,u0,η0),xΩ,u|Ω=Ψ|Ω=0,t[0,T], $ (3)

    and the no-flux condition for the density of particles

    $ (ηx+ηΦx)|Ω=0,t[0,T], $ (4)

    where $ \rho,u,\eta $, $ P(\rho) = a\rho^\gamma $, $ \Phi(x) $ denote the fluid density, velocity, the density of particles in the mixture, pressure, and the external potential respectively, $ a>0,\gamma>1,\frac{4}{3}<p,q<2 $. $ \lambda>0 $ is the viscosity coefficient, $ \Omega $ is a one-dimensional bounded interval, for simplicity we only consider $ \Omega = (0,1) $, $ \Omega_T = \Omega\times[0,T] $.

    The system describes a compressible shear thinning fluid-particle interaction system for the evolution of particles dispersed in a viscous non-Newtonian fluid and the particle is driven by non-Newtonian gravitational potential. To our knowledge, there still no existence results for (2)-(4) when $ 1<p,q<2 $. The aim of this paper is to study the existence and uniqueness of strong solutions to this system. Throughout the paper we assume that $ a = \lambda = 1 $ for simplicity. In the following sections, we will use simplified notations for standard Sobolev spaces and Bochner spaces, such as $ L^p = L^p(\Omega), H_0^1 = H_0^1(\Omega), C([0,T];H^1) = C([0,T];H^1(\Omega)) $.

    We state the definition of strong solution as follows:

    Definition 1.1. The $ (\rho,u,\Psi,\eta) $ is called a strong solution to the initial boundary value problem(2)-(4), if the following conditions are satisfied:

    (ⅰ)

    $ ρL(0,T;H1(Ω)),uL(0,T;W1,p0(Ω)H2(Ω)),ΨL(0,T;H2(Ω)),ηL(0,T;H2(Ω)),ρtL(0,T;L2(Ω)),utL2(0,T;H10(Ω)),ΨtL(0,T;H1(Ω)),ηtL(0,T;L2(Ω)),ρutL(0,T;L2(Ω)),(|ux|p2ux)xC(0,T;L2(Ω)). $

    (ⅱ) For all $ \phi \in L^\infty(0,T_*;H^1(\Omega)) $, $ \phi_t \in L^\infty(0,T_*;L^2(\Omega)) $, for a.e. $ t \in (0,T) $, we have

    $ Ωρϕ(x,t)dxt0Ω(ρϕt+ρuϕx)(x,s)dxds=Ωρ0ϕ(x,0)dx, $ (5)

    (ⅲ) For all $ \varphi \in L^\infty(0,T_*;W_0^{1,p}(\Omega)\cap H^2(\Omega)) $, $ \varphi_t \in L^2(0,T_*;H_0^1(\Omega)) $, for a.e. $ t \in (0,T) $, we have

    $ Ωρuφ(x,t)dxt0Ω{ρuφt+ρu2φxρΨxφλ|ux|p2uxφx+(P+η)φxηΦxφ}(x,s)dxds=Ωρ0u0φ(x,0)dx, $ (6)

    (ⅳ) For all $ \psi \in L^\infty(0,T_*;H^2(\Omega)) $, $ \psi_t \in L^\infty(0,T_*; H^1(\Omega)) $, for a.e. $ t\in (0,T) $, we have

    $ t0Ω|Ψx|q2Ψxψx(x,s)dxds=t0Ω4πg(ρ1|Ω|Ωρdx)ψ(x,0)dxds, $ (7)

    (ⅴ) For all $ \vartheta \in L^\infty(0,T_*;H^2(\Omega)) $, $ \vartheta_t \in L^\infty(0,T_*;L^2(\Omega)) $, for a.e. $ t\in (0,T) $, we have

    $ Ωηϑ(x,t)dxt0Ω[η(uΦx)ηx]ϑx(x,s)dxds=Ωη0ϑ(x,0)dx. $ (8)

    The main result of this paper is stated in the following theorem.

    Theorem 1.2. Let $ \Phi\in C^2(\Omega) $, $ \frac{4}{3}<p,q<2 $ and assume that the initial data $ (\rho_0,u_0,\eta_0) $ satisfy the following conditions

    $ 0ρ0H1(Ω),u0H10(Ω)H2(Ω),η0H2(Ω), $

    and the compatibility condition

    $ (|u0x|p2u0x)x+(P(ρ0)+η0)x+η0Φx=ρ120(g+Φx), $ (9)

    for some $ g\in L^2(\Omega) $. Then there exist a $ T_*\in(0,+\infty) $ and a unique strong solution $ (\rho,u,\Psi,\eta) $ to (2)-(4) such that

    $ ρL(0,T;H1(Ω)),uL(0,T;W1,p0(Ω)H2(Ω)),ΨL(0,T;H2(Ω)),ηL(0,T;H2(Ω)),ρtL(0,T;L2(Ω)),utL2(0,T;H10(Ω)),ΨtL(0,T;H1(Ω)),ηtL(0,T;L2(Ω)),ρutL(0,T;L2(Ω)),(|ux|p2ux)xC(0,T;L2(Ω)). $

    Remark 1. By using exactly the similar argument, we can prove the result also hold for the case $ 1 <p,q\leq\frac{4}{3} $. We omit the details here.

    In this section, we will prove the local existence of strong solutions. From the continuity equation $ (2)_1 $, we can deduce the conservation of mass

    $ \int_\Omega\rho(t){\rm d} x = \int_\Omega\rho_0{\rm d} x: = m_0,\quad(t > 0,m_0 > 0) $

    Because equation $ (2)_2 $ possesses always with singularity, we overcome this difficulty by introduce a regularized process, then by taking the limiting process back to the original problem. Namely, we consider the following system

    $ ρt+(ρu)x=0, $ (10)
    $ (ρu)t+(ρu2)x+ρΨx[(εu2x+1u2x+ε)2p2ux]x+(P+η)x=ηΦx, $ (11)
    $ [(ϵΨ2x+1Ψ2x+ϵ)2q2Ψx]x=4πg(ρm0), $ (12)
    $ ηt+(η(uΦx))x=ηxx, $ (13)

    with the initial and boundary conditions.

    $ (ρ,u,η)|t=0=(ρ0,u0,η0),xΩ, $ (14)
    $ u|Ω=Ψ|Ω=(ηx+ηΦx)|Ω=0,t[0,T], $ (15)

    and $ u_0\in H_0^1(\Omega)\cap H^2(\Omega) $ is the smooth solution of the boundary value problem

    $ {[(εu20x+1u20x+ε)2p2u0x]x+(P(ρ0)+η0)x+η0Φx=ρ120(g+Φx),u0|Ω=0. $ (16)

    Provided that $ (\rho,u,\eta) $ is a smooth solution of (10)-(15) and $ \rho_0\geq \delta $, where $ 0<\delta\ll 1 $ is a positive number. We denote by $ M_0 = 1+ \mu_0+ \mu^{-1}_0+|\rho_0|_{H^1}+|g|_{L^2} $.

    We first get the estimate of $ |u_{0xx}|_{L^2} $. From (16), we have

    $ {[(εu20x+1u20x+ε)2p2u0x]x+(P(ρ0)+η0)x+η0Φx=ρ120(g+Φx),u0|Ω=0. $ (16)

    Then

    $ |u0xx|L21p1|(u20x+εεu20x+1)1p2|L|(P(ρ0)+η0)x+η0Φxρ120(g+Φx)|L21p1(|u0x|2L+1)1p2(|(P(ρ0)+η0)x+η0Φxρ120(g+Φx)|L2)1p1(|u0xx|2L2+1)1p2(|Px(ρ0)|L2+|η0x|L2+|η0|L|Φx|L2+|ρ0|12L|g|L2+|ρ0|12L|Φx|L2). $

    Applying Young's inequality, we have

    $ |u0xx|L2C(|Px(ρ0)|L2+|η0x|L2+|η0|L|Φx|L2+|ρ0|12L|g|L2+|ρ0|12L|Φx|L2)1p1C, $

    thus

    $ |u0|L+|u0x|L+|u0xx|L2C, $ (17)

    where $ C $ is a positive constant, depending only on $ M_0 $.

    Next, we introduce an auxiliary function

    $ Z(t) = \sup\limits_{0\leq s\leq t}(1+|\rho(s)|_{H^1}+|u(s)|_{W_0^{1,p}}+|\sqrt\rho u_t(s)|_{L^2}+|\eta_t(s)|_{L^2}+|\eta(s)|_{H^1}). $

    We will derive some useful estimate to each term of $ Z(t) $ in terms of some integrals of $ Z(t) $, then apply arguments of Gronwall's inequality to prove $ Z(t) $ is locally bounded.

    In order to prove the main Theorem, we first give some useful lemmas for later use.

    Lemma 2.1. Let $ u_{0}\in H^{1}_{0}(\Omega)\cap H^{2}(\Omega) $, $ \rho_0\in H^{1}(\Omega) $, $ \eta_0\in H^{2}(\Omega) $, $ \Phi\in C^2(\Omega) $, $ g\in L^2(\Omega) $, $ u_0^\varepsilon $ is a solution of the boundary value problem

    $ {[(ε(uε0x)2+1(uε0x)2+ε)2p2uε0x]x+(P(ρ0)+η0)x+η0Φx=ρ120(g+Φx),uε0(0)=uε0(1)=0. $ (18)

    Then there are a subsequence $ \{u^{\varepsilon_j}_0\} $, $ j = 1,2,3,..., $ of $ \{u^{\varepsilon }_0\} $ and $ u_{0}\in H^{1}_{0}(\Omega)\cap H^{2}(\Omega) $ such that as $ \varepsilon_j\rightarrow 0 $,

    $ uεj0u0inH10(Ω)H2(Ω),[(εj(uεj0x)2+1(uεj0x)2+εj)2p2uεj0x]x(|u0x|p2u0x)xinL2(Ω). $

    Proof. According to (18), we have

    $ uεj0u0inH10(Ω)H2(Ω),[(εj(uεj0x)2+1(uεj0x)2+εj)2p2uεj0x]x(|u0x|p2u0x)xinL2(Ω). $

    Taking it by the $ L^2 $ norm, we have

    $ |uε0xx|L2|(ε(uε0x)2+1(uε0x)2+ε)1p2|L|(P(ρ0)+η0)x+η0Φx+ρ120(g+Φx)|L2(|uε0x|2L+1)1p2|(P(ρ0)+η0)x+η0Φx+ρ120(g+Φx)|L2, $

    then

    $ |uε0xx|L2C(1+|(P(ρ0)+η0)x+η0Φx+ρ120(g+Φx)|L2)1p1C. $ (19)

    Therefore, by the above inequality, as $ \varepsilon_j\rightarrow 0 $,

    $ uεj0u0inC32(Ω),uεj0xxu0xxinL2(Ω)weakly. $

    Thus, we can obtain $ \{u^{\varepsilon_j}_{0x}\} $ is a Cauchy subsequence of $ C^{\frac{3}{2}}(\Omega) $, for all $ \alpha_1>0 $, we find $ N $, as $ i, j > N $, and

    $ |uεi0xuεj0x|L(Ω)<α1. $

    Now, we prove that $ \{u^{\varepsilon}_{0xx}\} $ has a Cauchy sequence in $ L_2 $ norm.

    Let

    $ |uεi0xuεj0x|L(Ω)<α1. $

    For all $ \alpha>0 $, there exists $ N $, as $ i, j > N $, we can deduce that

    $ |uεi0xxuεj0xx|L2(Ω)|ϕiϕj|L(Ω)|(P(ρ0)+η0)x+η0Φxρ120(g+Φx)|L2(Ω). $

    With the assumption, we can obtain

    $ |(P(ρ0)+η0)x+η0Φxρ120(g+Φx)|L2(Ω)C, $

    where $ C $ is a positive constant, depending only on $ |\rho_0|_{H^{1}(\Omega)} $, $ |g|_{L^{2}(\Omega)} $ and $ |\eta_0|_{H^{2}(\Omega)} $. Using the following inequality,

    $ |ϕiϕj|L(Ω)|10ϕ(θ(uεi0x)2+(1θ)(uεj0x)2)dθ((uεi0x)2(uεj0x)2)|L(Ω), $ (20)

    where $ 0<\theta<1 $.

    By the simple calculation, we can get

    $ \phi^{'}(s)\leq \frac{2}{p-1}(1+s^{-\frac{p}{2}}), $

    where $ C $ depending only on $ p $, then

    $ |ϕiϕj|L(Ω)2p1|(1+10(θ(uεi0x)2+(1θ)(uεj0x)2)dθ)((uεi0x)2(uεj0x)2)|L(Ω)2p1|uεi0xuεj0x|L(Ω)|uεi0x+uεj0x|L(Ω)+4(2p)(p1)|uεi0xuεj0x|2p2L(Ω)|uεi0x+uεj0x|2p2L(Ω)α. $

    Substituting this into (18), we have

    $ |uεi0xxuεj0xx|L(Ω)<α, $

    then there is a subsequence $ \{u^{\varepsilon_j}_{0xx}\} $ and $ \{u^{\varepsilon}_{0xx}\} $, such that

    $ {uεj0xx}χinL2(Ω). $

    By the uniqueness of the weak convergence, we have

    $ χ={uε0xx}. $

    Since $ (P(\rho_0)+\eta)_x+\eta_0 \Phi_x-\rho_0^{1\over 2}(g+\Phi_x) $ are independent of $ \varepsilon $, the same that we obtain, as $ \varepsilon_j\rightarrow 0 $,

    $ [(εj(uεj0x)2+1(uεj0x)2+εj)2p2uεj0x]x(|u0x|p2u0x)xinL2(Ω). $

    This completes the proof of Lemma 2.1.

    Lemma 2.2.

    $ sup0tT|ρ(t)|2H1Cexp(Ct0Z6γ(3p4)(q1)(s)ds), $ (21)

    where $ C $ is a positive constant, depending only on $ M_0 $.

    Proof. We estimates for $ u $ and $ \eta $ for later use. It follows from (11) that

    $ \Big[\Big(\frac{\varepsilon u_x^2+1}{u_x^2+\varepsilon}\Big)^{\frac{2-p}{2}}u_x\Big]_x = \rho u_t+\rho u u_x+\rho\Psi_x+(P+\eta)_x+\eta\Phi_x. $

    We note that

    $ |uxx|1p1(u2x+ε)1p2|ρut+ρuux+ρΨx+(P+η)x+ηΦx|1p1(|ux|2p+1)|ρut+ρuux+ρΨx+(P+η)x+ηΦx|. $

    Taking it by the $ L^2 $ norm and using Young's inequality, we have

    $ |uxx|p1L2C(1+|ρut|L2+|ρuux|L2+|ρΨx|L2+|(P+η)x|L2+|ηΦx|L2)C(1+|ρ|12L|ρut|L2+|ρ|L|u|L|ux|p2Lp|ux|1p2L+|ρ|γ1L|ρx|L2+|ηx|L2+|η|L|Φx|L2+|ρ|L2|Ψxx|L2)C[1+|ρ|12L|ρut|L2+(|ρ|L|u|L|ux|p2Lp)2(p1)3p4+|ρ|γ1L|ρx|L2+|ηx|L2+|η|L|Φx|L2+|ρ|L2|Ψxx|L2]+12|uxx|p1L2. $ (22)

    On the other hand, by $ (12) $, we have

    $ |Ψxx|1q1(|Ψx|2q+1)|4πg(ρm0)|. $

    Taking it by $ L^2 $-norm, using Young's inequality, which gives

    $ |Ψxx|L2CZ1q1(t). $ (23)

    This implies that

    $ |uxx|L2CZmax{qq1,(p1)(4+p)3p4γ}(t)CZ6γ(3p4)(q1)(t). $ (24)

    By (13), taking it by the $ L^2 $ norm, we have

    $ |ηxx|L2|ηt+(η(uΦx))x|L2|ηt|L2+|ηx|L2|u|L+|ηx|L2|Φx|L+|η|L2|uxx|L2+|η|L|Φxx|L2CZ6γ+2(3p4)(q1)(t). $ (25)

    Multiplying (10) by $ \rho $, integrating over $ \Omega $, we deduce that

    $ 12ddtΩ|ρ|2ds+Ω(ρu)xρdx=0. $

    Integrating it by parts, using Sobolev inequality, we obtain

    $ ddt|ρ(t)|2L2Ω|ux||ρ|2dx|uxx|L2|ρ|2L2. $ (26)

    Differentiating $ (10) $ with respect to $ x $, and multiplying it by $ \rho_x $, integrating over $ \Omega $, and using Sobolev inequality, we have

    $ ddtΩ|ρx|2dx=Ω[32ux(ρx)2+ρρxuxx](t)dxC[|ux|L|ρx|2L2+|ρ|L|ρx|L2|uxx|L2]C|ρ|2H1|uxx|L2. $ (27)

    From (26) and (27) and the Gronwall's inequality, then lemma 2.2 holds.

    Lemma 2.3.

    $ |η|2H1+|ηt|2L2+t0(|ηx|2L2+|ηt|2L2+|ηxt|2L2)(s)dsC(1+t0Z4(s)ds), $ (28)

    where $ C $ is a positive constant, depending only on $ M_0 $.

    Proof. Multiplying $ (13) $ by $ \eta $, integrating the resulting equation over $ \Omega_T $, using the boundary conditions (4) and Young's inequality, we have

    $ t0|ηx(s)|2L2ds+12|η(t)|2L2ΩT(|ηuηx|+|ηΦxηx|)dxds14t0|ηx(s)|2L2ds+Ct0|ux|2Lp|η|2H1ds+Ct0|η|2H1ds+C14t0|ηx(s)|2L2ds+C(1+t0Z4(t)ds). $ (29)

    Multiplying $ (13) $ by $ \eta_t $, integrating (by parts) over $ \Omega_T $, using the boundary conditions (4) and Young's inequality, we have

    $ t0|ηt(s)|2L2ds+12|ηx(t)|2L2ΩT|η(uΦx)ηxt|dxds14t0|ηxt(s)|2L2ds+Ct0|η|2H1|ux|2Lpds+Ct0|η|2H1ds+C14t0|ηxt(s)|2L2ds+C(1+t0Z4(t)ds). $ (30)

    Differentiating $ (13) $ with respect to $ t $, multiplying the resulting equation by $ \eta_t $, integrating (by parts) over $ \Omega_T $, we get

    $ t0|ηxt(s)|2L2ds+12|ηt(t)|2L2=ΩT(η(uΦx))tηxtdxdsC+ΩT(|ηtuηxt|+|ηtΦxηxt|+|ηxutηt|+|ηuxtηt|)dxdsC(1+t0(|ηt|2L2||ux|2Lp+|ηt|2L2+|ηx|2L2|ηt|2L2+|η|2H1|ηt|2L2)dx)+12t0|ηxt|2L2+12t0|uxt|2L2C(1+t0Z4(s)ds). $ (31)

    Combining (29)-(31), we obtain the desired estimate of Lemma 2.3.

    Lemma 2.4.

    $ t0|ρut(s)|2L2(s)ds+|ux(t)|pLpC(1+t0Z10+4γ(3p4)(q1)(s)ds), $ (32)

    where $ C $ is a positive constant, depending only on $ M_0 $.

    Proof. Using (10), we rewritten the (11) as

    $ ρut+(ρu)ux+ρΨx[(εu2x+1u2x+ε)2p2ux]x+(P+η)x=ηΦx. $ (33)

    Multiplying (33) by $ u_t $, integrating (by parts) over $ \Omega_T $, we have

    $ ΩTρ|ut|2dxds+ΩT(εu2x+1u2x+ε)2p2uxuxtdxds=ΩT(ρuux+ρΨx+Px+ηx+ηΦx)utdxds. $ (34)

    We deal with each term as follows:

    $ Ω(εu2x+1u2x+ε)2p2uxuxtdx=12Ω(εu2x+1u2x+ε)2p2(u2x)tdx=12ddtΩ(u2x0(εs+1s+ε)2p2ds)dx, $
    $ u2x0(εs+1s+ε)2p2dsu2x0(s+1)2p2ds=2p[(u2x+1)p21], $
    $ ΩTPxutdxds=ΩTPuxtdxds=ddtΩTPuxdxdsΩTPtuxdxds. $

    By virtue of $ (10) $, we have

    $ Pt=γPuxPxu,ΩTηxutdxds=ΩTηuxtdxds=ddtΩTηuxdxdsΩTηtuxdxds.ΩTηΦxutdxds=ddtΩTηΦxudxds+ΩTηtΦxudxds. $ (35)

    Substituting the above into (34), using Sobolev inequality and Young's inequality, we have

    $ t0|ρut(s)|2L2ds+|ux(t)|pLpΩT(|ρuuxut|+|ρΨxut|+|γPu2x|+|Pxuux|+|ηtux|+|ηtΦxu|)dxds+Ω(|Pux|+|ηux|+|ηΦxu|)dx+CC+t0(|ρ|12L|u|L|ux|p2Lp|ux|1p2L|ρut|L2+|ρ|12L|Ψx|L|ρut|L2)ds+t0(γ|P|L2|ux|p2Lp|ux|1p2L|uxx|L2+aγ|ρ|γ1L|ρx|L2|u|L|ux|L+|ηt|L2|ux|p2Lp|ux|1p2L+|ηt|L2|Φx|L2|u|L)ds+|P|Lpp1|ux|Lp+|η|Lpp1|ux|Lp+|η|Lpp1|Φx|Lp|u|LC(1+t0(|ρ|L|ux|2+pLp|uxx|2pL2+|ρ|H1|Ψxx|2L2+|P|L|ux|p2Lp|uxx|2p2L2+|ρ|γ1L|ρx|L2|ux|Lp|uxx|L2+|ηt|L2|ux|p2Lp|uxx|1p2L2+|ηt|L2|ux|Lp)ds)+|P|pp1Lpp1+|η|pp1Lpp1+12t0|ρut(s)|2L2ds+12|ux(t)|pLp. $ (36)

    To estimate (36), combining (35) we have the following estimates

    $ Ω|P(t)|pp1dx=Ω|P(0)|pp1dx+t0s(ΩP(s)pp1dx)dsΩ|P(0)|pp1dx+pp1t0Ωaγργ1P(s)1p1(ρxuρux)dxdsC+Ct0|ρ|γ1L|P|1p1L|ρ|H1|ux|LpdsC(1+t0Zγp1+γ+1(s)ds), $ (37)

    In exactly the same way, we also have

    $ Ω|η(t)|pp1dxC(1+t0Z1p1+1(s)ds), $ (38)

    which, together with (36) and (37), implies (32) holds.

    Lemma 2.5.

    $ |ρut(t)|2L2+t0|uxt|2L2(s)dsC(1+t0Z26γ(3p4)(q1)(s)ds), $ (39)

    where $ C $ is a positive constant, depending only on $ M_0 $.

    Proof. Differentiating equation $ (11) $ with respect to $ t $, multiplying the result equation by $ u_t $, and integrating it over $ \Omega $, we have

    $ 12ddtΩρ|ut|2dx+Ω[(εu2x+1u2x+ε)2p2ux]tuxtdx=Ω[(ρu)x(u2t+uuxut+Ψxut)ρuxu2t+(P+η)tuxtηtΦxutρΨxtut]dx. $ (40)

    Note that

    $ Ω[(εu2x+1u2x+ε)2p2ux]tuxtdx=Ω[(εu2x+1u2x+ε)p2ux](εu2x+1)(u2x+ε)(2p)(1ε2)u2x(u2x+ε)2u2xtdx(p1)Ω(u2x+1)p22|uxt|2dx, $ (41)

    Let

    $ \omega = (u_x^2+1)^{\frac{p-2}{4}}, $

    from (24), it follows that

    $ |ω1|L=|(u2x+1)2p4|LC(|uxx|2p2L2+1)CZ2γ(3p4)(q1)(t). $

    Combining (35), (40) can be rewritten into

    $ ddtΩ|ρ|ut|2dx+Ω|ωuxt|2dx2Ωρ|u||ut||uxt|dx+Ωρ|u||ux|2|ut|dx+Ω|ρx||u|2|ux||ut|dx+Ω|ρx||u||Ψx||ut|dx+Ωρ|ux||Ψx||ut|dx+Ωρ|ux||ut|2dx+ΩγP|ux||uxt|dx+Ω|Px||u||uxt|dx+Ω|ηt||uxt|dx+Ω|ηt||Φx||ut|dx+Ωρ|Ψxt||ut|dx=11j=1Ij. $ (42)

    Using Sobolev inequality, Young's inequality, $ (11) $, (24) and (25), we obtain

    $ ddtΩ|ρ|ut|2dx+Ω|ωuxt|2dx2Ωρ|u||ut||uxt|dx+Ωρ|u||ux|2|ut|dx+Ω|ρx||u|2|ux||ut|dx+Ω|ρx||u||Ψx||ut|dx+Ωρ|ux||Ψx||ut|dx+Ωρ|ux||ut|2dx+ΩγP|ux||uxt|dx+Ω|Px||u||uxt|dx+Ω|ηt||uxt|dx+Ω|ηt||Φx||ut|dx+Ωρ|Ψxt||ut|dx=11j=1Ij. $ (42)
    $ ddtΩ|ρ|ut|2dx+Ω|ωuxt|2dx2Ωρ|u||ut||uxt|dx+Ωρ|u||ux|2|ut|dx+Ω|ρx||u|2|ux||ut|dx+Ω|ρx||u||Ψx||ut|dx+Ωρ|ux||Ψx||ut|dx+Ωρ|ux||ut|2dx+ΩγP|ux||uxt|dx+Ω|Px||u||uxt|dx+Ω|ηt||uxt|dx+Ω|ηt||Φx||ut|dx+Ωρ|Ψxt||ut|dx=11j=1Ij. $ (42)

    In order to estimate $ I_{11} $, we need to deal with the estimate of $ |\Psi_{xt}|_{L^2} $. Differentiating (12) with respect to $ t $, multiplying it by $ \Psi_t $ and integrating over $ \Omega $, we have

    $ Ω[(ϵΨ2x+1Ψ2x+ϵ)2q2Ψx]tΨxtdx=4πgΩ(ρu)xΨtdx, $ (43)

    and

    $ Ω[(ϵΨ2x+1Ψ2x+ϵ)2q2Ψx]tΨxtdx(q1)Ω(Ψ2x+1)q22|Ψxt|2dx. $ (44)

    Let

    $ \beta^q = (\Psi_x^2+1)^{\frac{q-2}{4}} $

    then

    $ |(βq)1|L=|(Ψ2x+1)2q4|LC(|Ψxx|2q2L2+1)CZ2q2(q1)(t). $

    Then (43) can be rewritten into

    $ Ω|βqΨxt|2dxCΩ(ρu)ΨxtdxC|ρ|L2|u|L|βqΨxt|L2|(βq)1|L. $

    Using Young's inequality, combining the above estimates we deduce that

    $ I11|ρ|12L|ρut|L2|βqΨxt|L2|(βq)1|LCZ5q32(q1)(t). $

    Substituting $ I_j(j = 1,2,\ldots,11) $ into (42), and integrating over $ (\tau,t)\subset (0,T) $ on the time variable, we have

    $ |ρut(t)|2L2+t0|ωuxt|2L2(s)ds|ρut(τ)|2L2+t0Z26γ(3p4)(q1)(s)ds. $ (45)

    To obtain the estimate of $ |\sqrt{\rho}u_t(t)|_{L^2}^2 $, we need to estimate $ \lim\limits_{\tau\rightarrow 0}|\sqrt{\rho}u_t(\tau)|_{L^2}^2 $. Multiplying (33) by $ u_t $ and integrating over $ \Omega $, we get

    $ \int_\Omega\rho|u_t|^2{\rm d} x\leq 2\int_\Omega(\rho|u|^2|u_x|^2+\rho|\Psi_x|^2+\rho^{-1}\big|-[(\frac{\varepsilon u_x^2+1}{u_x^2+\varepsilon})^{\frac{2-p}{2}}u_x]_x+(P+\eta)_x+\eta\Phi_x\big|^2){\rm d}x. $

    According to the smoothness of $ (\rho,u,\eta) $, we have

    $ limτ0Ω(ρ|u|2|ux|2+ρ|Ψx|2+ρ1|[(εu2x+1u2x+ε)2p2ux]x+(P+η)x+ηΦx|2)dx=Ω(ρ0|u0|2|u0x|2+ρ0|Ψx|2+ρ10|[(εu20x+1u20x+ε)2p2u0x]x+(P0+η0)x+η0Φx|2)dx|ρ0|L|u0|2L|u0x|2L2+|ρ0|L|Ψx|2+|g|2L2+|Φx|2L2C. $

    Then, taking a limit on $ \tau $ in (45), as $ \tau\rightarrow 0 $, we can easily obtain

    $ |ρut(t)|2L2+t0|uxt|2L2(s)dsC(1+t0Z26γ(3p4)(q1)(s)ds), $ (46)

    This complete the proof of Lemma 2.5.

    With the help of Lemma 2.2 to Lemma 2.5, and the definition of $ Z(t) $, we conclude that

    $ Z(t)Cexp(˜Ct0Z26γ(3p4)(q1)(s)ds), $ (47)

    where $ C,\tilde{C} $ are positive constants, depending only on $ M_0 $. This means that there exist a time $ T_1>0 $ and a constant $ C $, such that

    $ esssup0tT1(|ρ|H1+|u|W1,p0H2+|η|H2+|ηt|L2+|ρut|L2+|ρt|L2)+T10(|ρut|2L2+|uxt|2L2+|ηx|2L2+|ηt|2L2+|ηxt|2L2)dsC, $ (48)

    where $ C $ is a positive constant, depending only on $ M_0 $.

    In this section, the existence of strong solutions can be established by a standard argument. We construct the approximate solutions by using the iterative scheme, derive uniform bounds and thus obtain solutions of the original problem by passing to the limit. Our proof will be based on the usual iteration argument and some ideas developed in [10]. Precisely, we first define $ u^0 = 0 $ and assuming that $ u^{k-1} $ was defined for $ k\geq 1 $, let $ \rho^k,u^k,\eta^k $ be the unique smooth solution to the following system

    $ ρkt+ρkxuk1+ρkuk1x=0, $ (49)
    $ ρkukt+ρkuk1ukx+ρkΨkx+Lpuk+Pkx+ηkx=ηkΦx, $ (50)
    $ LqΨk=4πg(ρkm0), $ (51)
    $ ηkt+(ηk(uk1Φx))x=ηkxx, $ (52)

    with the initial and boundary conditions

    $ (ρk,uk,ηk)|t=0=(ρ0,u0,η0), $ (53)
    $ uk|Ω=(ηkx+ηkΦx)|Ω=0, $ (54)

    where

    $ L_p \theta^k = -\Big[\Big(\frac{\varepsilon(\theta_x^k)^2+1}{(\theta_x^k)^2+\varepsilon}\Big)^{\frac{2-p}{2}}\theta_x^k\Big]_x.\qquad $

    With the process, the nonlinear coupled system has been deduced into a sequence of decoupled problems and each problem admits a smooth solution. And the following estimates hold

    $ esssup0tT1(|ρk|H1+|uk|W1,p0H2+|ηk|H2+|ηkt|L2+|ρkukt|L2+|ρkt|L2)+T10(|ρkukt|2L2+|ukxt|2L2+|ηkx|2L2+|ηkt|2L2+|ηkxt|2L2)dsC, $ (55)

    where $ C $ is a generic constant depending only on $ M_0 $, but independent of $ k $.

    In addition, we first find $ \rho^k $ from the initial problem

    $ \rho_t^k+u^{k-1}\rho_x^k+u_x^{k-1}\rho^k = 0, $
    $ \quad \rho^k|_{t = 0} = \rho_0, $

    with smooth function $ u^{k-1} $, obviously, there is a unique solution $ \rho^k $ on the above problem and also we could obtain that

    $ \rho^k(x,t)\geq\delta\exp\big[-\int_0^{T_1}|u_x^{k-1}(.,s)|_{L^\infty}{\rm d} s\big] > 0, \mbox{for all} \ \ t\in(0,T_1). $

    Next, we will prove the approximate solution $ (\rho^k,u^k,\eta^k) $ converges to a limit $ (\rho^\varepsilon,u^\varepsilon, \eta^\varepsilon) $ in a strong sense. To this end, let us define

    $ \bar\rho^{k+1} = \rho^{k+1}-\rho^k,\quad\bar u^{k+1} = u^{k+1}-u^k,\quad\bar\eta^{k+1} = \eta^{k+1}-\eta^k,\quad\bar\Psi^{k+1} = \Psi^{k+1}-\Psi^k. $

    By a direct calculation, we can verify that the functions $ \bar\rho^{k+1},\bar u^{k+1},\bar\eta^{k+1} $ satisfy the system of equations

    $ ˉρk+1t+(ˉρk+1uk)x+(ρkˉuk)x=0, $ (56)
    $ ρk+1ˉuk+1t+ρk+1ukˉuk+1x+(Lpuk+1Lpuk)=ˉρk+1(ukt+ukukx+Ψk+1x)(Pk+1Pk)xˉηk+1x+ρk(ˉukukxˉΨk+1x)ˉηk+1Φx, $ (57)
    $ LqΨk+1LqΨk=4πgˉρk+1, $ (58)
    $ ˉηk+1t+(ηkˉuk)x+(ˉηk+1(ukΦx))x=ˉηk+1xx. $ (59)

    Multiplying (56) by $ \bar\rho^{k+1} $, integrating over $ \Omega $ and using Young's inequality, we obtain

    $ ddt|ˉρk+1|2L2C|ˉρk+1|2L2|ukx|L+|ρk|H1|ˉukx|L2|ˉρk+1|L2C|ukxx|L2|ˉρk+1|2L2+Cξ|ρk|2H1|ˉρk+1|2L2+ξ|ˉukx|2L2Cξ|ˉρk+1|2L2+ξ|ˉukx|2L2, $ (60)

    where $ C_\zeta $ is a positive constant, depending on $ M_0 $ and $ \zeta $ for all $ t<T_1 $ and $ k\geq 1 $.

    Multiplying (57) by $ \bar u^{k+1} $, integrating over $ \Omega $ and using Young's inequality, we obtain

    $ 12ddtΩρk+1|ˉuk+1|2dx+Ω(Lpuk+1Lpuk)ˉuk+1dxCΩ[|ˉρk+1|(|ukt|+|ukukx|+|Ψk+1x|)+|Pk+1xPkx|+|ˉηk+1x|+|ρk|ˉuk||ukx|+|ρk||ˉΨk+1x|+|ˉηk+1Φx|]|ˉuk+1|dxC(|ˉρk+1|L2|ukxt|L2|ˉuk+1x|L2+|ˉρk+1|L2|ukx|Lp|ukxx|L2|ˉuk+1x|L2+|ˉρk+1|L2|Ψk+1x|L2|ˉuk+1x|L2+|Pk+1Pk|L2|ˉuk+1x|L2+|ˉηk+1|L2|ˉuk+1x|L2+|ρk|12L2|ρkˉuk|L2|ukxx|L2|ˉuk+1x|L2+|ρk|H1|ˉΨk+1x|L2|ˉuk+1x|L2+|ˉηk+1|L2|ˉuk+1x|L2). $ (61)

    Let

    $ \sigma(s) = \big(\frac{\varepsilon s^2+1}{s^2+\varepsilon}\big)^{\frac{2-p}{2}}s, $

    then

    $ σ(s)=(εs2+1s2+ε)p2(εs2+1)(s2+ε)(2p)(1ε2)s2(s2+ε)2p1(s2+ε)2p2. $

    To estimate the second term of (61), we have

    $ Ω(Lpuk+1Lpuk)ˉuk+1dx=Ω10σ(θuk+1x+(1θ)ukx)dθ|ˉuk+1x|2dxΩ[10dθ|θuk+1x+(1θ)ukx|2pL+1](ˉuk+1x)2C1Ω|ˉuk+1x|2dx. $ (62)

    On the other hand, multiplying (58) by $ \bar\Psi^{k+1} $, integrating over $ \Omega $, we obtain

    $ Ω(LqΨk+1LqΨk)ˉΨk+1dx=4πgΩˉρk+1ˉΨk+1dx. $ (63)

    Since

    $ Ω(LqΨk+1LqΨk)ˉΨk+1xdx=(q1)Ω(10|θΨk+1x+(1θ)Ψkx|q2dθ)(ˉΨk+1x)2dx, $

    and

    $ 10|θΨk+1x+(1θ)Ψkx|q2dθ=101|θΨk+1x+(1θ)Ψkx|2qdθ101(|Ψk+1x|+|Ψkx|2q)dθ=1(|Ψk+1x|+|Ψkx|)2q, $

    then

    $ Ω[|Ψk+1x|q2Ψk+1x|Ψkx|q2Ψkx]ˉΨk+1xdx1(|Ψk+1x(t)|L+|Ψkx(t)|L)2qΩ(ˉΨk+1x)2dx, $

    which implies

    $ Ω(ˉΨk+1x)2dxC|ˉρk+1|2L2. $ (64)

    From (55), (62) and (64), (61) can be re-written as

    $ ddtΩρk+1|ˉuk+1|2dx+C1Ω|ˉuk+1x|2dxBξ(t)|ˉρk+1|2L2+C(|ρkˉuk|2L2+|ˉηk+1|2L2)+ξ|ˉuk+1x|2L2, $ (65)

    where $ B_\xi(t) = C(1+|u_{xt}^k(t)|_{L^2}^2 $, for all $ t\leq T_1 $ and $ k\geq 1 $. Using (55) we derive

    $ \int_0^tB_\xi(s){\rm d} s\leq C+Ct. $

    Multiplying (59) by $ \bar \eta^{k+1} $, integrating over $ \Omega $, using (55) and Young's inequality, we have

    $ 12ddtΩ|ˉηk+1|2dx+Ω|ˉηk+1x|2dxΩ|ˉηk+1||ukΦx||ˉηk+1x|dx+Ω(|ηk||ˉuk|)x|ˉηk+1|dx|ˉηk+1|L2|ukΦx|L|ˉηk+1x|L2+|ηkx|L2|ˉuk|L|ˉηk+1|L2+|ηk|L|ˉukx|L2|ˉηk+1|L2Cξ|ˉηk+1|2L2+ξ|ˉηk+1x|2L2+ξ|ˉukx|2L2. $ (66)

    Combining (60), (65) and (66), we have

    $ ddt(|ˉρk+1(t)|2L2+|ρk+1ˉuk+1(t)|2L2+|ˉηk+1(t)|2L2)+|ˉuk+1x(t)|2L2+|ˉηk+1x|2L2Eξ(t)|ˉρk+1(t)|2L2+C|ρkˉuk|2L2+Cξ|ˉηk+1|2L2+ξ|ˉukx|2L2, $ (67)

    where $ E_\zeta(t) $ is depending only on $ B_\zeta(t) $ and $ C_\xi $, for all $ t\leq T_1 $ and $ k\geq 1 $. Using (55), we obtain

    $ \int_0^t E_\xi(s){\rm d} s\leq C+C_\xi t. $

    Integrating (67) over $ (0,t)\subset (0,T_1) $ with respect to t, using Gronwall's inequality, we have

    $ |ˉρk+1(t)|2L2+|ρk+1ˉuk+1(t)|2L2+|ˉηk+1(t)|2L2+t0|ˉuk+1x(t)|2L2ds+t0|ˉηk+1x|2L2dsCexp(Cξt)t0(|ρkˉuk(s)|2L2+|ˉukx(s)|2L2)ds. $ (68)

    From the above recursive relation, choose $ \xi>0 $ and $ 0<T_*<T_1 $ such that $ C\exp(C_\xi T_*)<\frac{1}{2} $, using Gronwall's inequality, we deduce that

    $ Kk=1[sup0tT(|ˉρk+1(t)|2L2+|ρk+1ˉuk+1(t)|2L2+|ˉηk+1(t)|2L2dt+T0|ˉuk+1x(t)|2L2+T0|ˉηk+1x(t)|2L2dt]<C, $ (69)

    where $ C $ is a positive constant, depending only on $ M_0 $.

    Therefore, as $ k\rightarrow +\infty $, the sequence $ (\rho^k,u^k,\eta^k) $ converges to a limit $ (\rho^\varepsilon,u^\varepsilon,\eta^\varepsilon) $ in the following strong sense

    $ ρkρεin  L(0,T;L2(Ω)), $ (70)
    $ ukuεin  L(0,T;L2(Ω))L2(0,T;H10(Ω)), $ (71)
    $ ηkηεin  L(0,T;L2(Ω))L2(0,T;H1(Ω)). $ (72)

    By virtue of the lower semi-continuity of various norms, we deduce from the uniform estimate (55) that $ (\rho^\varepsilon,u^\varepsilon,\eta^\varepsilon) $ satisfies the following uniform estimate

    $ esssup0tT1(|ρε|H1+|uε|W1,p0H2+|ηε|H2+|ηεt|L2+|ρεuεt|L2+|ρεt|L2)+T0(|ρεuεt|2L2+|uεxt|2L2+|ηεx|2L2+|ηεt|2L2+|ηεxt|2L2)dsC. $ (73)

    Since all of the constants are independent of $ \varepsilon $, there exists a subsequence $ (\rho^{\varepsilon_j},u^{\varepsilon_j}, \eta^{\varepsilon_j}) $ of $ (\rho^{\varepsilon},u^{\varepsilon},\eta^{\varepsilon}) $, without loss of generality, we denote to $ (\rho^{\varepsilon},u^{\varepsilon},\eta^{\varepsilon}) $. Let $ \varepsilon \rightarrow 0 $, we can get the following convergence

    $ ρερδin  L(0,T;L2(Ω)), $ (74)
    $ uεuδin  L(0,T;L2(Ω))L2(0,T;H10(Ω)), $ (75)
    $ ηεηδin  L(0,T;L2(Ω))L2(0,T;H1(Ω)), $ (76)

    and there also holds

    $ esssup0tT1(|ρδ|H1+|uδ|W1,p0H2+|ηδ|H2+|ηδt|L2+|ρδuδt|L2+|ρδt|L2)+T0(|ρδuδt|2L2+|uδxt|2L2+|ηδx|2L2+|ηδt|2L2+|ηδxt|2L2)dsC. $ (77)

    For each small $ \delta>0 $, let $ \rho_0^\delta = J_\delta * \rho_0+\delta $, where $ J_\delta $ is a mollifier on $ \Omega $, and $ u_0^\delta\in H_0^1(\Omega)\cap H^2(\Omega) $ is a smooth solution of the boundary value problem

    $ {Lpuδ0+(P(ρδ0)+ηδ0)x+ηδ0Φx=(ρδ0)12(gδ+Φx),uδ0|Ω=0, $ (78)

    where $ g^\delta\in C_0^\infty $ and satisfies $ |g^\delta|_{L^2}\leq |g|_{L^2} $, $ \lim\limits_{\delta\rightarrow 0^+}|g^\delta-g|_{L^2} = 0 $.

    We deduce that $ (\rho^\delta,u^\delta,\eta^\delta) $ is a solution of the following initial boundary value problem

    $ \begin{align*} \begin{cases} \begin{aligned} &\rho_t+(\rho u)_x = 0,\\ &(\rho u )_t+(\rho u^2)_x+\rho\Psi_x-\lambda(|u_x|^{p-2}u_x)_x+(P+\eta)_x = -\eta\Phi_x,\\ &(|\Psi_x|^{q-2}\Psi_x)_x = 4\pi g(\rho-\frac{1}{|\Omega|} \int_\Omega \rho {\rm d} x),\\ &\eta_t+(\eta(u-\Phi_x))_x = \eta_{xx},\\ &(\rho,u,\eta)|_{t = 0} = (\rho_0^\delta,u_0^\delta,\eta_0^\delta),\\ &u|_{\partial\Omega} = (\eta_x+\eta\Phi_x)|_{\partial\Omega} = 0, \end{aligned} \end{cases} \end{align*} $

    where $ \rho_0^\delta\geq\delta, \frac{4}{3}<p,q<2 $.

    By the proof of Lemma 2.1, there exists a subsequence $ \{u_0^{\delta_j}\} $ of $ \{u_0^{\delta}\} $, as $ \delta_j\rightarrow 0^+ $, $ u_0^{\delta}\rightarrow u_0 $ in $ H_0^1(\Omega)\cap H^2(\Omega) $, $ -(|u_{0x}^{\delta_j}|^{p-2}u_{0x}^{\delta_j})_x\rightarrow -(|u_{0x}|^{p-2}u_{0x})_x $ in $ L^2(\Omega) $, Hence, $ u_0 $ satisfies the compatibility condition (9) of Theorem 1.2. By virtue of the lower semi-continuity of various norms, we deduce that $ (\rho,u,\eta) $ satisfies the following uniform estimate

    $ \begin{align} \mbox{ess}\sup\limits_{0\leq t\leq T_1}(&|\rho|_{H^1}+|u|_{W_0^{1,p} \cap H^2}+|\eta|_{H^2}+|\eta_t|_{L^2}+|\sqrt{\rho}u_t|_{L^2}+|\rho_t|_{L^2})\\ &+\int_0^{T_*}(|\sqrt\rho u_t|_{L^2}^2+|u_{xt}|_{L^2}^2+|\eta_x|_{L^2}^2+|\eta_t|_{L^2}^2+|\eta_{xt}|_{L^2}^2) {\rm d} s\leq C, \end{align} $ (79)

    where $ C $ is a positive constant, depending only on $ M_0 $. The uniqueness of solution can also be obtained by the same method as the above proof of convergence, we omit the details here. This completes the proof.

    The authors would like to thank the anonymous referees for their valuable suggestions.

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