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Research article

Adaptation of species as response to climate change: Predator-prey mathematical model

  • Most of the species currently threatened with extinction seem to be under the pressure of unsuitable environmental conditions; e.g., climate change, scarce food resource, habitat fragmentation. One should expect species to have forms of resilience against such extinction. The point here is to examine the effect of spatial gradients on species survival against increasing temperature arising from climate change. Therefore, we start with the question of whether, when faced with extinction stemming from climate change, a spatial gradient and a beachhead have the power to prevent extinction. This problem is addressed theoretically using a coupled reaction diffusion equation for a predator-prey system in which the prey experiences an Allee effect. It is demonstrated that there exists a relationship between the slope of the gradient and the beachhead at which the predator-prey system can stably survive. The tendency of the system can be defined by a function where the system includes the threshold point for extinction, that separates the areas of extinction and survival. The findings reveal that spatial gradient can be used as a precaution, when the species faces to extinction, for species to create new habitat and sustain its persistence. Therefore, in this paper, it is shown that, in theory, the recovery of species from unsuitable environmental conditions can be achieved. This can be possible by taking into account the spatial gradient to slow down the forthcoming ecological extinction, and thus extend the system a while as an adaptation mechanism.

    Citation: Yadigar Sekerci. Adaptation of species as response to climate change: Predator-prey mathematical model[J]. AIMS Mathematics, 2020, 5(4): 3875-3898. doi: 10.3934/math.2020251

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  • Most of the species currently threatened with extinction seem to be under the pressure of unsuitable environmental conditions; e.g., climate change, scarce food resource, habitat fragmentation. One should expect species to have forms of resilience against such extinction. The point here is to examine the effect of spatial gradients on species survival against increasing temperature arising from climate change. Therefore, we start with the question of whether, when faced with extinction stemming from climate change, a spatial gradient and a beachhead have the power to prevent extinction. This problem is addressed theoretically using a coupled reaction diffusion equation for a predator-prey system in which the prey experiences an Allee effect. It is demonstrated that there exists a relationship between the slope of the gradient and the beachhead at which the predator-prey system can stably survive. The tendency of the system can be defined by a function where the system includes the threshold point for extinction, that separates the areas of extinction and survival. The findings reveal that spatial gradient can be used as a precaution, when the species faces to extinction, for species to create new habitat and sustain its persistence. Therefore, in this paper, it is shown that, in theory, the recovery of species from unsuitable environmental conditions can be achieved. This can be possible by taking into account the spatial gradient to slow down the forthcoming ecological extinction, and thus extend the system a while as an adaptation mechanism.


    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 ρ,u,η, P(ρ)=aργ, Φ(x) denote the fluid density, velocity, the density of particles in the mixture, pressure, and the external potential respectively, a>0,γ>1. μ>0 is the viscosity coefficient, and 3λ+2μ0 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 Γ denotes the viscous stress tensor, which depends on Eij(u), and

    Eij(u)=uixj+ujxi,

    is the rate of strain. If the relation between the stress and rate of strain is linear, namely, Γ=μEij(u), where μ 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

    Γ=μ(uixj+ujxi)q,

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

    Γij=(μ0+μ1|E(xu)|p2)Eij(xu).

    For μ0=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 ρ,u,η, P(ρ)=aργ, Φ(x) denote the fluid density, velocity, the density of particles in the mixture, pressure, and the external potential respectively, a>0,γ>1,43<p,q<2. λ>0 is the viscosity coefficient, Ω is a one-dimensional bounded interval, for simplicity we only consider Ω=(0,1), ΩT=Ω×[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=λ=1 for simplicity. In the following sections, we will use simplified notations for standard Sobolev spaces and Bochner spaces, such as Lp=Lp(Ω),H10=H10(Ω),C([0,T];H1)=C([0,T];H1(Ω)).

    We state the definition of strong solution as follows:

    Definition 1.1. The (ρ,u,Ψ,η) 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 ϕL(0,T;H1(Ω)), ϕtL(0,T;L2(Ω)), for a.e. t(0,T), we have

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

    (ⅲ) For all φL(0,T;W1,p0(Ω)H2(Ω)), φtL2(0,T;H10(Ω)), for a.e. t(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 ψL(0,T;H2(Ω)), ψtL(0,T;H1(Ω)), for a.e. t(0,T), we have

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

    (ⅴ) For all ϑL(0,T;H2(Ω)), ϑtL(0,T;L2(Ω)), for a.e. t(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 ΦC2(Ω), 43<p,q<2 and assume that the initial data (ρ0,u0,η0) 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 gL2(Ω). Then there exist a T(0,+) and a unique strong solution (ρ,u,Ψ,η) 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,q43. 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

    Ωρ(t)dx=Ωρ0dx:=m0,(t>0,m0>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 u0H10(Ω)H2(Ω) 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 (ρ,u,η) is a smooth solution of (10)-(15) and ρ0δ, where 0<δ1 is a positive number. We denote by M0=1+μ0+μ10+|ρ0|H1+|g|L2.

    We first get the estimate of |u0xx|L2. 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 M0.

    Next, we introduce an auxiliary function

    Z(t)=sup0st(1+|ρ(s)|H1+|u(s)|W1,p0+|ρut(s)|L2+|ηt(s)|L2+|η(s)|H1).

    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 u0H10(Ω)H2(Ω), ρ0H1(Ω), η0H2(Ω), ΦC2(Ω), gL2(Ω), uε0 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 {uεj0}, j=1,2,3,..., of {uε0} and u0H10(Ω)H2(Ω) such that as εj0,

    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 L2 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 εj0,

    uεj0u0inC32(Ω),uεj0xxu0xxinL2(Ω)weakly.

    Thus, we can obtain {uεj0x} is a Cauchy subsequence of C32(Ω), for all α1>0, we find N, as i,j>N, and

    |uεi0xuεj0x|L(Ω)<α1.

    Now, we prove that {uε0xx} has a Cauchy sequence in L2 norm.

    Let

    |uεi0xuεj0x|L(Ω)<α1.

    For all α>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 |ρ0|H1(Ω), |g|L2(Ω) and |η0|H2(Ω). Using the following inequality,

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

    where 0<θ<1.

    By the simple calculation, we can get

    ϕ(s)2p1(1+sp2),

    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 {uεj0xx} and {uε0xx}, such that

    {uεj0xx}χinL2(Ω).

    By the uniqueness of the weak convergence, we have

    χ={uε0xx}.

    Since (P(ρ0)+η)x+η0Φxρ120(g+Φx) are independent of ε, the same that we obtain, as εj0,

    [(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 M0.

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

    [(εu2x+1u2x+ε)2p2ux]x=ρut+ρuux+ρΨx+(P+η)x+ηΦx.

    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 L2 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 L2-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 L2 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 ρ, integrating over Ω, 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 ρx, integrating over Ω, 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 M0.

    Proof. Multiplying (13) by η, integrating the resulting equation over ΩT, 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 ηt, integrating (by parts) over ΩT, 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 ηt, integrating (by parts) over ΩT, 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 M0.

    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 ut, integrating (by parts) over ΩT, 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 M0.

    Proof. Differentiating equation (11) with respect to t, multiplying the result equation by ut, and integrating it over Ω, 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

    ω=(u2x+1)p24,

    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 I11, we need to deal with the estimate of |Ψxt|L2. Differentiating (12) with respect to t, multiplying it by Ψt and integrating over Ω, 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

    βq=(Ψ2x+1)q24

    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 Ij(j=1,2,,11) into (42), and integrating over (τ,t)(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 |ρut(t)|2L2, we need to estimate limτ0|ρut(τ)|2L2. Multiplying (33) by ut and integrating over Ω, we get

    Ωρ|ut|2dx2Ω(ρ|u|2|ux|2+ρ|Ψx|2+ρ1|[(εu2x+1u2x+ε)2p2ux]x+(P+η)x+ηΦx|2)dx.

    According to the smoothness of (ρ,u,η), 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 τ in (45), as τ0, 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,˜C are positive constants, depending only on M0. This means that there exist a time T1>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 M0.

    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 u0=0 and assuming that uk1 was defined for k1, let ρk,uk,ηk 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

    Lpθk=[(ε(θkx)2+1(θkx)2+ε)2p2θkx]x.

    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 M0, but independent of k.

    In addition, we first find ρk from the initial problem

    ρkt+uk1ρkx+uk1xρk=0,
    ρk|t=0=ρ0,

    with smooth function uk1, obviously, there is a unique solution ρk on the above problem and also we could obtain that

    ρk(x,t)δexp[T10|uk1x(.,s)|Lds]>0,for all  t(0,T1).

    Next, we will prove the approximate solution (ρk,uk,ηk) converges to a limit (ρε,uε,ηε) in a strong sense. To this end, let us define

    ˉρk+1=ρk+1ρk,ˉuk+1=uk+1uk,ˉηk+1=ηk+1ηk,ˉΨk+1=Ψk+1Ψk.

    By a direct calculation, we can verify that the functions ˉρk+1,ˉuk+1,ˉηk+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 ˉρk+1, integrating over Ω 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 Cζ is a positive constant, depending on M0 and ζ for all t<T1 and k1.

    Multiplying (57) by ˉuk+1, integrating over Ω 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

    σ(s)=(εs2+1s2+ε)2p2s,

    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 ˉΨk+1, integrating over Ω, 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 Bξ(t)=C(1+|ukxt(t)|2L2, for all tT1 and k1. Using (55) we derive

    t0Bξ(s)dsC+Ct.

    Multiplying (59) by ˉηk+1, integrating over Ω, 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 Eζ(t) is depending only on Bζ(t) and Cξ, for all tT1 and k1. Using (55), we obtain

    t0Eξ(s)dsC+Cξt.

    Integrating (67) over (0,t)(0,T1) 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 ξ>0 and 0<T<T1 such that Cexp(CξT)<12, 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 M0.

    Therefore, as k+, the sequence (ρk,uk,ηk) converges to a limit (ρε,uε,ηε) 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 (ρε,uε,ηε) 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 ε, there exists a subsequence (ρεj,uεj,ηεj) of (ρε,uε,ηε), without loss of generality, we denote to (ρε,uε,ηε). Let ε0, 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 δ>0, let ρδ0=Jδρ0+δ, where Jδ is a mollifier on Ω, and uδ0H10(Ω)H2(Ω) is a smooth solution of the boundary value problem

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

    where gδC0 and satisfies |gδ|L2|g|L2, limδ0+|gδg|L2=0.

    We deduce that (ρδ,uδ,ηδ) is a solution of the following initial boundary value problem

    {ρt+(ρu)x=0,(ρu)t+(ρu2)x+ρΨxλ(|ux|p2ux)x+(P+η)x=ηΦx,(|Ψx|q2Ψx)x=4πg(ρ1|Ω|Ωρdx),ηt+(η(uΦx))x=ηxx,(ρ,u,η)|t=0=(ρδ0,uδ0,ηδ0),u|Ω=(ηx+ηΦx)|Ω=0,

    where ρδ0δ,43<p,q<2.

    By the proof of Lemma 2.1, there exists a subsequence {uδj0} of {uδ0}, as δj0+, uδ0u0 in H10(Ω)H2(Ω), (|uδj0x|p2uδj0x)x(|u0x|p2u0x)x in L2(Ω), Hence, u0 satisfies the compatibility condition (9) of Theorem 1.2. By virtue of the lower semi-continuity of various norms, we deduce that (ρ,u,η) satisfies the following uniform estimate

    esssup0tT1(|ρ|H1+|u|W1,p0H2+|η|H2+|ηt|L2+|ρut|L2+|ρt|L2)+T0(|ρut|2L2+|uxt|2L2+|ηx|2L2+|ηt|2L2+|ηxt|2L2)dsC, (79)

    where C is a positive constant, depending only on M0. 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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