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Research article

Existence and uniqueness of solution for a class of non-Newtonian fluids with non-Newtonian potential and damping

  • Received: 07 February 2023 Revised: 28 February 2023 Accepted: 08 March 2023 Published: 17 March 2023
  • This paper discusses the existence and uniqueness of local strong solution for a class of 1D non-Newtonian fluids with non-Newtonian potential and damping term. Here we allow the initial vacuum and viscosity term to be fully nonlinear.

    Citation: Qiu Meng, Yuanyuan Zhao, Wucai Yang, Huifang Xing. Existence and uniqueness of solution for a class of non-Newtonian fluids with non-Newtonian potential and damping[J]. Electronic Research Archive, 2023, 31(5): 2940-2958. doi: 10.3934/era.2023148

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  • This paper discusses the existence and uniqueness of local strong solution for a class of 1D non-Newtonian fluids with non-Newtonian potential and damping term. Here we allow the initial vacuum and viscosity term to be fully nonlinear.



    We consider the following class of 1D non-Newtonian fluids

    {ρt+(ρu)x=0,ρΦx+(ρu2)x+(ρu)t(|ux|p2ux)x+Px=αρu,(|Φq2x|Φx)x=4πg(ρ1|Ω|Ωρdx), (1.1)

    where u, ρ, M=ρu and P=Aργ(A>0,γ>1) denote velocity, the unknown density, momentum and pressure, respectively, g>0 is the acceleration of gravity and Φ is the gravitational potential. The constant α>0 models friction. Without losing generality, throughout the paper we take α=1. The initial and boundary value conditions of Eq (1.1) are as follows

    {(ρ,u,Φ)|t=0=(ρ0,u0,Φ0), for all x[0,1],u(0,t)=u(1,t)=0, for all t[0,T],Φ(0,t)=Φ(1,t)=0, for all t[0,T]. (1.2)

    Ω is considered as a one-dimensional bounded interval here. Furthermore, for simplicity, we only assume the Ω=I=(0,1), ΩT=I×(0,T). The initial density ρ00, p and q are given constants, and they are both studied in the case of less than 2, where since the method of study is similar for 1<p<43 and 43<p<2, we next study only the case of 43<p<2.

    According to classical Newtonian fluid mechanics, in parallel fluids, the shear force is proportional to shear velocity, and its proportion is the viscosity coefficient, i.e.,

    Γ=Γ(ρ,u)=μu,μ>0.

    Generally, we call a fluid with the above properties a Newtonian fluid. Accordingly, a fluid does not have this property is called a non-Newtonian fluid. For non-Newton fluids, Γ(ρ,u) has a reasonable choice (see Ladyzhenskaya[1])

    Γij=(μ0+μ1|E(u)|p2)Eij(u),

    and

    Eij(u)=12(uixj+ujxi).

    In chemistry, biomechanics, glaciology, geology, and blood rheology, there are many problems in non-Newtonian fluids, which lead to an interest in studying non-Newtonian fluids [2,3,4]. There are many theoretical and experimental studies in non-Newtonian fluid flow.

    In this paper, the vacuum condition i.e., the initial density, is zero. Strictly speaking, a vacuum, a gas state below atmospheric pressure in a given space, is a physical phenomenon. In real life, vacuum distillation, vacuum drying, and vacuum concentration are typical vacuum cases. The role of vacuum in vacuum distillation is mainly to reduce the boiling point temperature of substances and reduce the influence of temperature factors on substances. Vacuum drying and concentration both use a vacuum environment to accelerate the volatilization and evaporation of specific substances or enable the whole process to be completed at lower conditions. In particular, non-Newtonian fluids can expand in a vacuum. According to the experiment, as the air in the vacuum bottle decreases, tiny bubbles gradually appear on the surface of the non-Newtonian fluid, and the bubbles expand until they spill the container out. In any case, the theoretical knowledge of non-Newtonian fluids under vacuum must be continuously refined. To this end, this article allows for an initial vacuum.

    In 1996, J. Málek, J. Neˇcas, M. Rokyta, M. R˙uˇziˇcka divided non-Newnewton flows with regard to p in the monograph [5] : when 1<p<2, we call such a fluid a shear thinning fluid, when p>2, we call such a fluid a shear thickening fluid. For non-Newtonian fluids, Yuan Hongjun and Xu Xiaojing [6] studied existence of a solution and whether it is unique of a class of non-Newtonian fluid solution with singularity and vacuum. Takashi Suzuki and Takayuki Kobayashi [7] proved the existence of weak solution to the Navier-Stokes-Poisson equation. Meng Qiu and Yuan Hongjun [8] proved the existence and uniqueness of a class of local solution under conditions where compressible non-Newtonian fluids with a non-Newtonian bit potential in a one-dimensional bounded interval. Song Yukun, Yuan Hongjun, and Yang Chen [9] investigated the existence and uniqueness of a class of local solution in the presence of isentropic compressible non-Newtonian fluids in a one-dimensional bounded area. Liu Hongzhi, Yuan Hongjun, Qiao Jiezeng and Li Fanpei [10] constructed the global existence of robust solution of Navier-Stokes equations with non-Newtonian potential. Li Huapeng and Yuan Hongjun [11] demonstrated the local existence and uniqueness of 1D non-Newtonian fluid solution with damping.

    The damping item comes from resistance to fluid motion. Model (1.1) describes more natural phenomena. For example, porous media flow. We can refer to [12,13,14,15] contents of the damp item. However, for non-Newtonian fluid, there is no degradation result with non-Newtonian potential and damp item. We construct a system (1.1)–(1.2) with local existence and uniqueness of the strong solution of non-Newtonian fluids with the non-Newtonian potentials and friction damping. The result is the following theorem:

    Theorem 1. Assume that

    43<p<2,1<q<2,0ρ0H1(Ω),u0H10(I)H2(I),Φ0H20(I)H3(I)

    and that there is a function gL2(I), makes the following equation true almost everywhere on I:

    (|u0x|p2u0x)x+Px(ρ0)=ρ120g, (1.3)

    then there exists a small time T(0,+) and a unique strong solution (ρ,u,Φ) to the initial boundary value problem (1.1)–(1.2) such that:

    {ρC([0,T];H1(I)),ρtC([0,T];L2(I)),ΦL(0,T;H2(I)),ΦtL(0,T;H1(I)),uC([0,T];H10(I))L(0,T;H2(I)),utL2(0,T;H10(I)),ρutL(0,T;L2(I)),(|ux|p2ux)xC([0,T];L2(I)). (1.4)

    For the above theorem, we will be divided it into four parts to prove. In the first part, we use the iterative method to get the approximate solution system of problems (1.1)–(1.2) and then make a consistent estimate of its approximate solution. In the second part, the convergence of the approximate solution is proved by the weak convergence method. In the third and fourth parts, we demonstrated that a locally strong solution to problems (1.1)–(1.2) exists uniquely.

    Lemma 1. (Embedding inequality). Assume that f=0 on Ω, here ΩR1 is bounded and open, fC2+α(ˉΩ). Then

    |f|L(Ω)d12(Ω)|f|L2(Ω)

    where d(Ω) denotes the length of Ω.

    See the literature [4] for proof.

    The system of Eqs (1.1) we studied contains more unknowns, and (1.1)2 and (1.1)3 are non-linear, so we cannot get a direct solution (from Abelian theorem, the fifth and higher order algebraic equations have no analytical solution). Therefore, we are inspired to seek approximate solutions to the system of equations.

    Therefore, we apply an iterative approach to problems (1.1)–(1.2), which yields its approximate solution system.

    ρkt+uk1ρkx+uk1xρk=0, (2.1)
    ρkukt+ρkuk1ukx+ρkΦkx+Łεpuk+Pkx=ρkuk, (2.2)
    (|Φkx|q2Φkx)x=4πg(ρkm0), (2.3)
    {(ρk,uk,Φk)|t=0=(ρδ0,uε0,Φε0),x[0,1],uk(0,t)=uk(1,t)=0,t[0,T], (2.4)

    we take the initial mass m0=10ρ0(x)dx>0, Pk=P(ρk)=A(ρk)γ, A>0, γ>1, ρδ0=δ+ρ0Jδ, δ>0,

    Łεpuk=[(ε(ukx)2+1(ukx)2+ε)2p2ukx]x,

    For problem (2.5), uε0H2(I)H10(I) is a smooth solution to it

    {[(ε(uε0x)2+1(uε0x)2+ε)2p2u0x]x+Px(ρδ0)=(ρδ0)12g,uε0(0)=uε0(1)=0. (2.5)

    Then, we will conduct a consistent estimation of the approximate solution and prove that the limit of the approximate solution is just the solution of the Eqs (1.1)–(1.2).

    In order to do this, we will first get the uniform estimate on uε0. The uε0 is known from the smooth solution of the boundary value problem

    uε0xx=(ε(uε0x)2+1(uε0x)2+ε)p2((uε0x)2+ε)2(Px(ρδ0)(ρδ0)12g)(ε(uε0x)2+1)((uε0x)2+ε)(2p)(1ε2)(uε0x)2, (2.6)

    then

    |uε0xx|L2(I)|((uε0x)2+εε(uε0x)2+1)1p2|L(I)|Px(ρδ0)(ρδ0)12g|L2(I)(|uε0x|2L(I)+1)1p2(|(ρδ0)12g|L2(I)+|Px(ρδ0)|L2(I)) (2.7)
    (|uε0xx|2L2(I)+1)1p2(|(ρδ0)12g|L2(I)+|Px(ρδ0)|L2(I)).

    Using Young's inequality, we have

    |uε0xx|L2(I)C, (2.8)

    with the help of the Lemma 1, we get

    |uε0|L(I)+|uε0x|L(I)+|uε0xx|L2(I)C, (2.9)

    where C>0 is a constant that depends only on M0, which may not necessarily be fixed. Next, we denote

    M0=1+|ρ0|H1(I)+u0H10(I)H20(I)+|g|L2(I).

    For any fixed integer K, define

    JK(t)=max1kKsup0st(1+|ρk(s)|H1(I)+|uk(s)|W1,p0(I)+|ρkukt(s)|L2(I)), (2.10)

    then we will prove that JK(t) is locally bounded for 43<p<2. We estimate each term in JK(t) in the following sections.

    Multiplying (2.2) by ukt, Integrating over (0,1) concerning x and integrating over (0,t) to s gives, we can get

    t010ρk|ukt|2dxds+t010[(ε(ukx)2+1(ukx)2+ε)2p2ukx]ukxtdxds=10Pkukx(0)dx+10Pkukx(t)dxt010(Pktukx+ρkΦkxukt+ρkukukt+ρkuk1ukxukt)dxds. (2.11)

    We firstly compute the second term of (2.11), we obtain

    10[(ε(ukx)2+1(ukx)2+ε)2p2ukx]ukxtdx=12ddt10((ukx)20(εs+1s+ε)2p2ds)dx, (2.12)

    and

    (ukx)20(εs+1s+ε)2p2ds(ukx)20(s+1)p22ds=2p(((ukx)2+1)p21). (2.13)

    Substituting (2.12), (2.13) into (2.11), by (2.9) and Young's inequality, we have

    t0|ρkukt(s)|2L2(I)ds+1p10|ukx(t)|pdxC+10|Pkukx(t)|dxt010|Pktukx|dxdst010ρk|uk||ukt|dxdst010ρk|uk1||ukx||ukt|dxdst010ρk|Φkx||ukt|dxds. (2.14)

    By (2.1), we get

    Pkt=γPkuk1Pkxuk1.

    Then the above inequality can be expressed as

    t0|ρkukt(s)|2L2(I)ds+|ukx(t)|pLp(I)t010(|ρkukukt|+|ρkuk1ukxukt|+|ρkΦkxukt|dxds+10|Pkukx|dx+t010|Pkxuk1ukx|+γ|Puk1xukx|)dxds+CCηt0|ρkuk|2L2(I)ds+Cηt0|ρk(s)|L(I)|uk1x(s)|2Lp(I)|ukxx(s)|2L2(I)ds+C|Pk(t)|pp1Lpp1(I)+t0(Aγ|ρk|γ1L(I)|ρkx|L2(I)|uk1|L(I)|ukx(s)|L(I)+γ|P(s)|L(I)|uk1x(s)|Lp(I)|ukx(s)|L(I))ds+Cηt0|ρk|H1(I)|Φkxx|2L2(I)ds+12t0|ρkukt|2L2(I)(s)ds+12|ukx(t)|pLp(I)+C, (2.15)

    where 0<η1. To estimate the right part of the (2.14), we have the following estimates

    |ρk(t)|L(I)+|Pk(t)|H1(I)CJγK(t). (2.16)

    Using (2.1), we have

    10|Pk(t)|pp1dx=10|Pk(0)|pp1dx+t0s(10(Pk(s))pp1dx)dsC(1+t0J2γ+1p1K(s)ds). (2.17)

    By virtue of (2.2), we have

    [(ε(ukx)2+1(ukx)2+ε)2p2ukx]x=ρkukt+ρkΦkx+ρkuk1ukx+Pkx+ρkuk,

    then we have

    |ukxx|=(ε(ukx)2+1(ukx)2+ε)p2[(ukx)2+ε]2(ε(ukx)2+1)((ukx)2+ε)(2p)(1ε2)(ukx)2|ρkukt+ρkuk1ukx+ρkΦkx+Pkx+ρkuk|1p1(|ukx|2p+1)|ρkukt+ρkuk1ukx+ρkΦkx+Pkx+ρkuk|, (2.18)

    taking the above inequality by L2 norm, using Young's inequality, we obtain

    |ukxx|p1L2(I)C[1+|ρkukt|L2(I)+|ρkuk1ukx|L2(I)+|ρkΦkx|L2(I)+|Pkx|L2(I)+|ρkuk|L2(I)]C[1+|ρk|12L(I)|ρkukt|L2(I)+(|ρk|L(I)|uk1x|Lp(I)|ukx|p2Lp(I))2(p1)3p4+|ρk|H1(I)|Φkxx|L2(I)+|Pkx|L2(I)+|ρkuk|L2(I)]+12|ukxx|p1L2(I).

    We deal with |Φkxx|L2(I), by (2.3) we have

    |Φkxx|1q1|Φkx|2q|4πg(ρkm0)|,

    taking it by L2-norm, using Young's inequality and Lemma 1, we get

    |Φkxx|L2(I)CJ1q1K(t), (2.19)

    then

    |ukxx(t)|L2(I)CJ(4+p)γ3p4K(t)CJ6γ3p4K(t). (2.20)

    Using (2.14) and the above inequality, we get

    t0|ρkukt(s)|2L2(I)ds+|ukx(t)|pLp(I)C(1+t0J24γ3p4K(s)ds), (2.21)

    for all k,1kK.

    We differentiate (2.2) with respect to t, and multiply it by ukt, and integrating it over (0,1) with respect to x, we obtain

    12ddt10ρk|ukt|2(t)dx+10[(ε(ukx)2+1(ukx)2+ε)2p2ukx]tukxt(t)dx=10[(ukuktuk1ukxΦkx)ρktρkuk1tukxρkΦkxtρkukt]uktdx+10Pktukxtdx. (2.22)

    Since

    [(ε(ukx)2+1(ukx)2+ε)2p2ukx]tukxt=(ε(ukx)2+1(ukx)2+ε)p2(ε(ukx)2+1)((ukx)2+ε)(2p)(1ε2)(ukx)2((ukx)2+ε)2(ukxt)2(p1)((ukx)2+1)p22(ukxt)2, (2.23)

    let

    βk=((ukx)2+1)p24.

    by (2.20), we have

    |β1k|L(I)=|((ukx)2+1)2p4|L(I)(|ukx|2L(I)+1)2p4|ukx|2p2L(I)+1CJ3γ3p4K(t).

    Then (2.22) can be rewritten as

    12ddt10ρk|ukt|2dx+(p1)10((ukx)2+1)p22(ukxt)2dx102ρk|uk1||ukt||ukxt|dx+10|ρkx||uk1|2|ukx||ukt|dx+10|ρkx|uk1|ukukt|dx+10ρk|uk1||uk1x||ukx||ukt|dx+10ρk|uk1x||uk||ukt|dx+10|Pkx||uk1||ukxt|dx+10γPk|uk1x||ukxt(t)|dx+10|ρkx||uk1||Φkx||ukt|dx+10ρk|uk1x||Φkx||ukt|dx+10ρk|uk1t||ukx||ukt|dx+10ρk|ukt||ukt|dx+10ρk|Φkxtukt|dx=12j=1Ij. (2.24)

    Using Sobolev embedding theorem and Young's inequality, we obtain

    I1=102ρk|uk1||ukt||ukxt|dxCJ16γ3p4K(t)+p18|βkukxt(t)|2L2(I),I2=10ρkx|uk1|2|ukx||ukt|dxCJ30γ3p4K(t)+p18|βkukxt(t)|2L2(I),I3=10|ρkx||uk1||ukt||uk|dxCJ24γ3p4K(t)+p18|βkukxt(t)|2L2(I),I4=10ρk|uk1||uk1x||ukt||ukx|dxCJ34γ3p4K(t),I5=10ρk|uk1x||uk||ukt|dx|ρk|12L(I)|uk1x|L(I)|uk|L2(I)|ρkukt|L2(I)CJ24γ3p4K(t),I6=10|Pkx||uk1||ukxt|dxCJ14γ3p4K(t)+p18|βkukxt(t)|2L2(I),I7=10γPk|uk1x||ukxt|dxCJ22γ3p4K(t)+p18|βkukxt|2L2(I),I8=10|ρkx||uk1Φkx||ukt|dxCJ34γ3p4K(t)+p18|βkukxt|2L2(I),I9=10ρk|uk1x||Φkx||ukt|dx|ρk|Lpp1(I)|uk1x|Lp(I)|Φkx|L(I)|ukt|L(I)CJ34γ3p4K(t)+p18|βkukxt|2L2(I),I10=10ρk|uk1t||ukx||ukt|dx|ρk|12L(I)|uk1t|L(I)|ukx|L(I)|ρkukt|L2(I)CJ24γ3p4K(t)+p12|βk1uk1xt|2L2(I)+p18|βkukxt|2L2(I),I11=10ρk|ukt||ukt|dx|ρk|12L(I)|ρkukt|L2(I)|ukt|L2(I)CJ32K(t).

    In order to estimate I12, we need to deal with Φkxt. Differentiating (2.3) with respect to t, multiplying it by Φkt and integrating over (0,1), we have

    10[(ε(Φkx)2+1(Φkx)2+ε)2q2Φkx]tΦkxtdx=4πg10(ρkuk1)xΦktdx.

    By (2.23), we have

    10[(ε(Φkx)2+1(Φkx)2+ε)2q2Φkx]tΦkxtdx(q1)10[(Φkx)2+1]q22|Φkxt|2dx.

    Let

    βqk=[(Φkx)2+1]q24,

    then

    |(βqk)1|L(I)=|[(Φkx)2+1]2q4|L(I)C(|Φkxx|2q2L2(I)+1)CJ2q2(q1)K(t),

    we have

    10|βqkΦkxt|2dx=C10(ρkuk1)ΦkxtdxC|ρk|L2(I)|uk1|L(I)|βqkΦkxt|L2(I)|(βqk)1|L(I).

    Using Young's inequality, combining the above estimate we obtain

    I12=10ρk|Φkxt||ukt|dx|ρk|L(I)|ρkukt|L2(I)|βqkΦkxt|L2(I)|(βqk)1|L(I)CJ6K(t).

    Substituting Ij(j=1,2,,12) into (2.24), integrating over (τ,t) on time variable, we have

    |ρkukt(t)|2L2(I)+(p1)tτ|βkukxt|2L2(I)(s)dsCtτJ48γ3p4K(s)ds+sup0kK(1+|ρkukt(τ)|2L2(I))+p12tτ|βk1uk1xt|2L2(I)(s)ds, (2.25)

    then, from the above recursive relation, for 1kK, we obtain

    (p1)tτ|βkukxt|2L2(I)(s)ds(1+12+14++12K)C[tτJ34γ3p4K(s)ds+sup0kK(1+|ρkukt(τ)|2L2(I))]2C[tτJ34γ3p4K(s)ds+sup0kK(1+|ρkukt(τ)|2L2(I))].

    Thus, we deduce from (2.25) that

    |ρkukt(s)|2L2(I)+tτ|βkukxt|2L2(I)(s)dsCtτJ48γ3p4K(s)ds+sup0kK(1+|ρkukt(τ)|2L2(I)), (2.26)

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

    To obtain the estimate of |ρkukt(t)|2L2(I), we need to estimate

    limτ0sup0kK(1+|ρkukt(τ)|2L2(I)).

    Using (2.2), we get

    10ρk|ukt|2(t)dx210(ρk|uk1|2|ukx|2+ρkΦkx+ρk|uk|2+(ρk)1|Łεpuk+Pkx|2)dx,
    Łεpuk=[(ε(ukx)2+1(ukx)2+ε)2p2ukx]x.

    Since (ρk,uk,Φk) is a smooth solution, we obtain

    limt010(ρk|uk1|2|ukx|2+ρk|uk|2+ρkΦkx+(ρk)1|Łεpuk+Pkx|2)(x,t)dx|ρ0|L(I)|uε0|L(I)|uε0x|2L2(I)+|ρ0|L(I)|uk|L2(I)+|ρ0|L(I)|Φkx|L2(I)+|g|2L2(I).

    Thus, using (2.9), we deduce

    limτ0sup10ρk|ukt|2(τ)dxC.

    Taking a limit on τ for inequality (2.26), we obtain, as τ0,

    |ρkukt(t)|2L2(I)+t0|βkukxt|2L2(I)(s)dsC(1+t0J48γ3p4K(s)ds). (2.27)

    We differentiate (2.1) concerning x, multiply it by ρkx, integrating it over (0, 1) for x, and using Sobolev embedding theorem, we obtain

    ddt|ρkx|2L2(I)dx=10(32uk1x(ρkx)2+ρkρkxuk1xx)(t)dx32(|uk1x|L(I)|ρkx|2L2(I)+|ρk|L(I)|uk1xx|L2(I))3|ρkx|2L2(I)|uk1xx|L2(I),

    applying Gronwall's inequality, it follows that

    sup0tT|ρk(t)|2H1(I)|ρk0|2H1(I)exp(Ct0|uk1xx(,s)|L2(I)ds). (2.28)

    Substituting (2.20) into the above inequality, we get

    |ρk(t)|2H1(I)C|ρk0(t)|2H1(I)exp(t0J6γ3p4K(s)ds). (2.29)

    Using (2.29) and (2.1), we have

    |ρkt(t)|L2(I)|ρkx(t)|L2(I)|uk1(t)|L(I)+|ρk(t)|L(I)|uk1xx(t)|L2(I)J8γ3p4K(t). (2.30)

    By virtue of (2.20), (2.27), (2.29) and (2.30), we conclude that

    |ukx(t)|pLp(I)+|uxx(t)|L2(I)+|ρk(t)|H1(I)+|ρkukt(t)|L2(I)+t0(|ρkukt(s)|2L2(I)+|ukxt(s)|2L2(I))dsC1exp(C2t0J48γ3p4K(s)ds), (2.31)

    where C1,C2 are two positive constants, depending only on M0. By the definition of JK(t), we obtain

    JK(t)C1exp(C2t0J48γ3p4K(s)ds). (2.32)

    If

    T0J48γ3p4K(s)ds1,

    then we take T1=T. On the other hand, if

    t0J48γ3p4K(s)ds>1,

    we can find t0(0,T), such that

    t00J48γ3p4K(s)ds=1.

    So we have

    sup0tt0JK(t)C1ec2,

    and

    t00J48γ3p4K(s)ds=1t00C48γ3p4e48γc3p4dsC48γ3p4e48γc3p4t0,

    so

    T1=c48γ3p4e48γC3p4,

    then we have

    sup0tT1JK(s)CecC. (2.33)

    Given this inequality, we can acquire a short-time T1>0 such that:

     ess sup0tT1(|ρk(s)|H1(I)+|uk|w1,p0(I)H2(I)+|ρkukt(s)|L2(I)+|ρkt|L2(I))+T10|uxt|2L2(I)dtC. (2.34)

    It is demonstrated that the approximate solution (ρk,uk,Φk) strongly converge to the solution of the Eqs (1.1)–(1.2) with positive density. We give the following definition

    ˉρk+1=ρk+1ρk,ˉuk+1=uk+1uk,ˉΦk+1=Φk+1Φk,

    then we verify that (ˉρk+1,ˉuk+1,ˉΦk+1) satisfy the system of equations

    ˉρk+1t+(ˉρk+1uk)x+(ρkˉuk)x=0, (3.1)
    ρk+1ˉuk+1t+ρk+1ukˉuk+1x+(Łεpuk+1Łεpuk)+(Pk+1xPkx)=ˉρk+1(ukˉuk+1uktukukxΦkx)ρkˉuk+1ρk+1(ˉukukx+ˉΦk+1x), (3.2)
    ŁεqΦk+1ŁεqΦk=4πgˉρk+1, (3.3)

    the initial boundary value conditions are given as follows

    ˉuk+1=0,ˉΦk+1=0onΩ×(0,T),ρk+1(x,0)=0,ˉuk+1(x,0)=0,xΩ.

    Multiplying (3.1) by ˉρk+1, integrating over I with respect to x, we deduce that

    ddt|ˉρk+1(t)|2L2(I)C|ukx(t)|L(I)|ˉρk+1(t)|2L2(I)+|ρk(t)|H1(I)|ˉukx(t)|L2(I)|ˉρk+1(t)|L2(I)Bkη|ˉρk+1(t)|2L2(I)+η|ˉukx(t)|2L2(I), (3.4)

    where Bkη=C|ukxx|L2(I)+Cη|ρk|2H1(I), for all tT1 and k1.

    Multiplying (3.2) by ˉuk+1, integrating over I with respect to x, using (3.1), Hölder inequality and Lemma 1, we obtain

    12ddt10ρk+1|ˉuk+1|2dx+10(Łεpuk+1Łεpuk)ˉuk+1dx=10(ˉρk+1(ukuktuk1ukxΦkx)ˉuk+1ρk+1(ˉΦk+1x+ˉukukx)ˉuk+1(Pk+1xPkx)ˉuk+1)dx|ˉρk+1|L2(I)|uk|L2(I)|ˉuk+1x|L2(I)+|ˉρk+1|L2(I)|ukxt|L2(I)|ˉuk+1x|L2(I)+|ˉρk+1|L2(I)|ˉuk1x|L2(I)|ukxx|L2(I)|ˉuk+1x|L2(I)+|ˉρk+1|L2(I)|Φkxx|L2(I)|ˉuk+1x|L2(I)+|ρk+1|H1(I)|ˉΦk+1x|L2(I)|ˉuk+1x|L2(I)+|ρk+1|12H1(I)|ˉukx|L2(I)|ukxx|L2(I)|ρk+1ˉuk+1|L2(I)+|Pk+1Pk|L2(I)|ˉuk+1x|L2(I). (3.5)

    Let

    ω(s)=[(εs2+1s2+ε)2p2s]p1(s2+ε)2p2,

    so

    10[10ω(θuk+1x+(1θ)ukx)]dθ(¯uk+1x)2dxC10(¯uk+1x)2dx, (3.6)

    using (3.4) and (3.5), we have

    10(Łεpuk+1Łεpuk)ˉuk+1dxC10(¯uk+1x)2dx, (3.7)

    Using (2.34), (3.7) and Young's inequality, (3.5) could be rewritten as

    ddt10ρk+1|ˉuk+1|2dx+10|ˉuk+1x|2dxEkη(t)|ˉρk+1(t)|2L2(I)+η|ˉukx|2L2(I), (3.8)

    where Ekη(t)=C(1+|ukxt(t)|2L2(I)), for all t<T1 and k1. Using (2.31), we derive

    t0Ekη(s)dsC+Ct,t0Bkη(s)dsC+Ct.

    According to Eq (2.3)

    [|Φk+1x|q2Φk+1x]x[|Φkx|q2Φkx]x=4πgˉρk+1,

    let's multiply both sides of this equation by ˉΦk+1, about x in (0,1) integral, available

    10([|Φk+1x|q2Φk+1x]x[|Φkx|q2Φkx]x)ˉΦk+1dx=4πg10ˉΦk+1ˉρk+1dx,z(s)=(εs2+1s2+ε)2p2s

    then

    z(s)=[(εs2+1s2+ε)2p2s]p1(s2+ε)2p2,10([(Φk+1x)q2Φk+1x]x[(Φkx)q2Φkx]x)ˉΦk+1dx=10[10z(θΦk+1x+(1θ)Φkx)]dθ(ˉΦk+1x)2dxC10(ˉΦk+1x)2dx.

    By combining the above formula, H¨older inequality and Lemma 1 are obtained

    |ˉΦk+1x|2L2(I)C|ˉρk+1|2L2(I). (3.9)

    Collecting (3.4), (3.8) and (3.9), we deduce that

    ddt(|ˉρk+1(t)|2L2(I)+|ρk+1ˉuk+1(t)|2L2(I))+|ˉuk+1x(t)|2L2(I)+|ˉΦk+1x|2L2(I)C(|ˉρk+1(t)|2L2(I)+|ρk+1ˉuk+1(t)|2L2(I))+η|ˉukx(t)|2L2(I). (3.10)

    Using Gronwall's inequality, we have

    |ˉρk+1(t)|2L2(I)+|ρk+1ˉuk+1(t)|2L2(I)+t0(|ˉuk+1x(s)|2L2(I)+|ˉΦk+1x|2L2(I))dsCexp(Cηt)t0(|ρkˉuk(s)|2L2(I)+|ˉukx(s)|2L2(I))ds.

    Then, we choose η>0 and then T>0 so small that T<T1 and Cexp(CηT)<1/2, we get

    |ˉρ1(t)|2L2(I)+|ρ1ˉu1(t)|2L2(I)+t0(|ˉu1x(s)|2L2(I)+|ˉΦ1x|2L2(I))ds12t0(|ρ0ˉu0(s)|2L2(I)+|ˉu0x(s)|2L2(I))ds,|ˉρ2(t)|2L2(I)+|ρ2ˉu2(t)|2L2(I)+t0(|ˉu2x(s)|2L2(I)+|ˉΦ2x|2L2(I))ds12t0(|ρ1ˉu1(s)|2L2(I)+|ˉu1x(s)|2L2(I))ds,......|ˉρk+1(t)|2L2(I)+|ρk+1ˉuk+1(t)|2L2(I)+t0(|ˉuk+1x(s)|2L2(I)+|ˉΦk+1x|2L2(I))ds12t0(|ρkˉuk(s)|2L2(I)+|ˉukx(s)|2L2(I))ds.

    Hence, we combine the above inequalities, in view of Gronwall's inequality, we deduce that

    Kk=1[sup0tT(|ˉρk+1(t)|2L2(I)+|ρk+1ˉuk+1(t)|2L2(I))+T0(|ˉuk+1x(s)|2L2(I)+|ˉΦk+1x|2L2(I))dt]<C. (3.11)

    Therefore, we conclude that the full sequence (ρk,uk,Φk) converges to a limit (ρ,u,Φ) in the following strong sense:

    ρkρ in L(0,T;L2(I)), (3.12)
    uku in L(0,T;L2(I))L2(0,T;H10(I)). (3.13)

    Combining (3.3) and the convergence of (3.12), we can get

    ΦkΦ in L(0,T;H2(I)). (3.14)

    From the lower semi-continuity of the norm, we get:

     ess sup0tT1(|ρ(t)|H1(I)+|u(t)|W1,p0H2(I)+|ρut(t)|L2(I)+|ρt(t)|L2(I))+T0|uxt(t)|2L2(I)dtC. (3.15)

    The proof of existence should be completed in three steps, namely, taking limits on k, ε0+ and δ0+. Since the method is similar, we will only describe the process of taking limits on δ0 below. The first two steps can be found in the literature [4].

    We take δ to be a very small positive number, let ρδ0=Jδρ0+δ, Jδ is a mollifier on I, uδ0H10(I)H2(I) is the unique smooth solution of the boundary value problem:

    {(|uδ0x|p2uδ0x)x=Px(ρδ0)+(ρδ0)12gδ,uδ0(0)=uδ0(1)=0,

    there exists gδC0(I) satisfies

    |gδ|L2(I)|g|L2(I),limδ0+|gδg|L2(I)=0.

    For ρδ0=Jδρ0+δ, there is a subsequence {(ρδj0,uδj0)}of{(ρδ0,uδ0)}, as δj0+ satisfing

    Px(ρδj0)+(ρδj0)12gδjPx(ρ0)+ρ120g in L2(I),
    (|uδj0x|p2uδj0x)x(|u0x|p2u0x)x in L2(I).

    Therefore, (ρ0,u0) satisfies the following problem

    (|u0x|p2u0x)x=Px(ρ0)+ρ120g a.e xI.

    There exists a T(0,+), the initial-boundary value problem

    {ρt+(ρu)x=0,(x,t)ΩT(ρu)t+(ρu2)x+ρΦx(|u0x|p2u0x)x+Px=ρu,(x,t)ΩT[|Φkx|q2Φkx]x=4πg(ρk1|Ω|Ωρkdx),PP(ρ)=Aργ,A>0,γ>1,(ρ,u,Φ)|t=0=(ρδ0,uδ0,Φδ0),x[0,1]u|x=0=u|x=1=0,t[0,T]

    admits a unique solution (ρδ,uδ,Φδ). Moreover, (ρδ,uδ,Φδ) satisfies the uniform estimate

    esssup0tT1(|ρδ(t)|H1(I)+|uδ(t)|W1,p0(I)H2(I)+|ρδuδt(t)|L2(I)+|ρδt(t)|L2(I))+T0|uδxt(t)|2L2(I)dtC.

    According to the above uniform estimate, by the lower semi-continuity of norm, as δj0, we deduce the following uniform estimate:

    esssup0tT1(|ρ(t)|H1(I)+|u(t)|W1,p0(I)H2(I)+|ρut(t)|L2(I)+|ρt(t)|L2(I))+T0|uxt(t)|2L2(I)dtC.

    Suppose (ρ1,u1,Φ1) is a strong solution to the problem (1.1)–(1.2), (ρ2,u2,Φ2) is also a strong solution to the problem (1.1)–(1.2), then we have

    1210ρ1(u1u2)2dx+t010(Łpu1Łpu2)(u1u2)dxds=t010((ρ1ρ2)(hΦ2x)(u1u2)+ρ1(u1u2)2u2xρ1(u1u2)2+(P1P2)(u1u2)x)dxds+t010ρ1|(Φ1Φ2)x||u1u2|dxds=5j=1Ij, (5.1)

    where h=u2u2tu2u2xL2(0,T;L2(I)).

    Then using Hölder inequality together with Lemma 1, we have

    I1t0|ρ1ρ2|L2(I)|hΦ2x|L2(I)|u1u2|L(I)(I)dsCεt0|ρ1ρ2|2L2(I)|hΦ2x|2L2(I)ds+εt0|u1xu2v|2L2(I)ds,I2=t010ρ1|u1u2|2u2xdxdst0|ρ1(u1u2)|2L2(I)|u2x|L(I)ds,I3=t010ρ1|u1u2|2dxdst0ρ1(u1u2)|2L2(I)ds,I4=t010|P1P2||u1xu2x|dxdst0|P1P2|L2(I)|u1xu2x|L2(I)dsCεt0|P1P2|2L2(I)ds+εt0|u1xu2x|2L2(I)ds,I5=t010ρ1|(Φ1Φ2)x||u1u2|dxdst0|ρ1(ρ1ρ2)|L2(I)|u1u2|L2(I)ds.

    By (3.7), we have

    t010(Łεpu1Łεpu2)(u1u2)dxdsμp220t0|u1xu2x|2L2(I)ds,

    where μ0=1(|ux(t)|L(0,t;L(I))+|ˉux(t)|L(0,t;L(I)))2p.

    Then, following from (5.1), by choosing ε=(μ0)p22/8, we derive

    12|ρ1(u1u2)|2L2(I)+34(μ0)p22t0|u1xu2x|2L2(I)dst0(C|ρ1ρ2|2L2(I)|hΦ2x|2L2(I)+|ρ1(u1u2)|2L2(I)(1+|u2x|L(I))+C|P1P2|2L2(I))ds, (5.2)

    where 0<A(t)=C(1+|hΦ2x|2L2(I)+|u2x|2L(I))L1(0,T).

    As is known from the definition of a strong solution, we take φ=ρ1ρ2, then

    1210|ρ1ρ2|2dx=t010(ρ1u1ρ2u2)(ρ1ρ2)xdxds=t010(ρ1(u1u2)+(ρ1ρ2)u2)(ρ1ρ2)dxds=t010(ρ1x(u1u2)(ρ1ρ2)+ρ1(u1u2)x(ρ1ρ2)t0(|ρ1x|L2(I)|u1u2|L(I)|ρ1ρ2|L2(I)+|ρ1|L(I)|u1xu2x|+12|u2x|L2(I)|ρ1ρ2|2L2(I))dst0(B(s)|ρ1ρ2|2L2(I)+18(μ0)p22|u1xu2x|2L2(I))ds, (5.3)

    where 0<B(t)=C(|ρ1|H1(I)+|u2x|L(I))L1(0,T). Similarly, we have

    t012ddt10|P1P2|2dxdst0(D(s)|P1P2|2L2(I)+18(μ0)p22|u1xu2x|2L2(I))ds, (5.4)

    where 0<B(t)=C(|ρ1|H1(I)+|u2x|L(I))L1(0,T). Similarly, we have

    Combining (5.2), (5.3) and (5.4), we obtain

    12(μ0)p22t0|u1xu2x|2L2(I)ds+12[|ρ1(u1u2)|2L2(I)+|ρ1ρ2|2L2(I)+|P1P2|2L2(I)]t0H(t)(|ρ1(u1u2)|2L2(I)+|ρ1ρ2|2L2(I)+|P1P2|2L2(I))ds,

    where H(t)=A(t)+B(t)+D(t)L1(0,T). Using Gronwall's inequality, we can get

    12(μ0)p22|(u1xu2x)(t)|2L2(I)+12esssup0tT1(|ρ1(u1u2)(t)|2L2(I)+|(ρ1ρ2)(t)|2L2(I)+|(P1P2)(t)|2L2(I))0,

    then

    ρ1=ρ2,ρ1(u1u2)=0,u1x=u2x,

    we can get

    10(Φ1xΦ2x)2dxC|ρ1ρ2|2L2(I).

    Therefore

    ρ1=ρ2,u1=u2,Φ1=Φ2.

    The authors thank Beihua University for funding and supporting this work through The Science and Technology Research Project of the Jilin Provincial Education Department (Grant No.JJKH20220040KJ).

    The authors declare there is no conflicts of interest.



    [1] O. A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Steklov Inst. Math., 102 (1967), 95–118.
    [2] L. Yang, K. Du, A comprehensive review on the natural, forced, and mixed convection of non-Newtonian fluids (nanofluids) inside different cavities, J. Therm. Anal. Calorim., 140 (2020), 2033–2054. https://doi.org/10.1007/s10973-019-08987-y doi: 10.1007/s10973-019-08987-y
    [3] Z. P. Xin, S. G. Zhu, Well-posedness of the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum, J. Math. Pures Appl., 152 (2020), 94–144. https://doi.org/10.1016/j.matpur.2021.05.004 doi: 10.1016/j.matpur.2021.05.004
    [4] W. R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, 1978.
    [5] S. Whitaker, Introduction to Fluid Mechanics, Krieger, Melbourne, FL, 1986.
    [6] H. J. Yuan, X. J. Xu, Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum, J. Differ. Equations, 245 (2008), 2871–2916. https://doi.org/10.1016/j.jde.2008.04.013 doi: 10.1016/j.jde.2008.04.013
    [7] T. Kobayashi, T. Suzuki, Weak solutions to the Navier-Stokes-Poisson equation, Adv. Math. Sci. Appl., 18 (2008), 141–168.
    [8] H. Yuan, M. Qiu, Local existence of strong solution for a class of compressible non-Newtonian fluids with non-Newtonian potential, Comput. Math. Appl., 65 (2013), 563–575. https://doi.org/10.1016/j.camwa.2012.10.010 doi: 10.1016/j.camwa.2012.10.010
    [9] Y. Song, H. Yuan, Y. Chen, On the strong solutions of one-dimensional Navier-Stokes-Poisson equations for compressible non-Newtonian fluids, J. Math. Phys., 54 (2013), 229–240. https://doi.org/10.1063/1.4803485 doi: 10.1063/1.4803485
    [10] H. Liu, H. Yuan, J. Qiao, F. Li, Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids, Acta Math. Sci., 32 (2012), 1467–1486. https://doi.org/10.1016/s0252-9602(12)60116-7 doi: 10.1016/s0252-9602(12)60116-7
    [11] H. Li, H. Yuan, Existence and uniqueness of solutions for a class of non-Newtonian fluids with vacuum and damping, J. Math. Anal. Appl., 391 (2012), 223–239. https://doi.org/10.1016/j.jmaa.2012.02.015 doi: 10.1016/j.jmaa.2012.02.015
    [12] C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, in Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, Birkhäuser Basel, (1995), 294–307. https://doi.org/10.1007/978-3-0348-9229-2_16
    [13] C. M. Dafermos, R. Pan, Global BV solutions for the P-System with frictional damping, SIAM J. Math. Anal., 41 (2009), 1190–1205. https://doi.org/10.1137/080735126 doi: 10.1137/080735126
    [14] F. Huang, R. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differ. Equations, 220 (2006), 207–233. https://doi.org/10.1016/J.JDE.2005.03.012 doi: 10.1016/J.JDE.2005.03.012
    [15] R. Pan, K. Zhao, Initial boundary value problem for compressible Euler equations with damping, Indiana Univ. Math. J., 57 (2008), 2257–2282. https://doi.org/10.1512/iumj.2008.57.3366 doi: 10.1512/iumj.2008.57.3366
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