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Research article

A recent survey of permutation trinomials over finite fields

  • Constructing permutation polynomials is a hot topic in the area of finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials were constructed. In 2015, Hou surveyed the achievements of permutation polynomials and novel methods. But, very few were known at that time. Recently, many permutation binomials and trinomials have been constructed. Here we survey the significant contribution made to the construction of permutation trinomials over finite fields in recent years. Emphasis is placed on significant results and novel methods. The covered material is split into three aspects: the existence of permutation trinomials of the respective forms xrh(xs), λ1xa+λ2xb+λ3xc and x+xs(qm1)+1+xt(qm1)+1, with Niho-type exponents s,t.

    Citation: Varsha Jarali, Prasanna Poojary, G. R. Vadiraja Bhatta. A recent survey of permutation trinomials over finite fields[J]. AIMS Mathematics, 2023, 8(12): 29182-29220. doi: 10.3934/math.20231495

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  • Constructing permutation polynomials is a hot topic in the area of finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials were constructed. In 2015, Hou surveyed the achievements of permutation polynomials and novel methods. But, very few were known at that time. Recently, many permutation binomials and trinomials have been constructed. Here we survey the significant contribution made to the construction of permutation trinomials over finite fields in recent years. Emphasis is placed on significant results and novel methods. The covered material is split into three aspects: the existence of permutation trinomials of the respective forms xrh(xs), λ1xa+λ2xb+λ3xc and x+xs(qm1)+1+xt(qm1)+1, with Niho-type exponents s,t.



    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 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

    \begin{align} &\frac{1}{2}\frac{d}{dt}\int_\Omega|\bar\eta^{k+1}|^2{\rm d} x+\int_\Omega|\bar\eta_x^{k+1}|^2{\rm d} x\\ &\leq\int_\Omega|\bar\eta^{k+1}||u^k-\Phi_x||\bar\eta_x^{k+1}|{\rm d} x+\int_\Omega(|\eta^k||\bar u^k|)_x|\bar\eta^{k+1}|{\rm d} x\\ &\leq|\bar\eta^{k+1}|_{L^2}|u^k-\Phi_x|_{L^\infty}|\bar\eta_x^{k+1}|_{L^2}+|\eta_x^k|_{L^2}|\bar u^k|_{L^\infty}|\bar\eta^{k+1}|_{L^2}+|\eta^k|_{L^\infty}|\bar u_x^k|_{L^2}|\bar\eta^{k+1}|_{L^2}\\ &\leq C_\xi|\bar\eta^{k+1}|_{L^2}^2+\xi|\bar\eta_x^{k+1}|_{L^2}^2+\xi|\bar u_x^k|_{L^2}^2. \end{align} (66)

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

    \begin{align} &\frac{d}{dt}\Big(|\bar\rho^{k+1}(t)|_{L^2}^2+|\sqrt{\rho^{k+1}}\bar u^{k+1}(t)|_{L^2}^2+|\bar\eta^{k+1}(t)|_{L^2}^2\Big)+|\bar u_x^{k+1}(t)|_{L^2}^2+|\bar\eta_x^{k+1}|_{L^2}^2\\ &\leq E_\xi(t)|\bar\rho^{k+1}(t)|_{L^2}^2+C|\sqrt{\rho^k} \bar u^k|_{L^2}^2+C_\xi|\bar\eta^{k+1}|_{L^2}^2+\xi|\bar u_x^k|_{L^2}^2, \end{align} (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

    \begin{align} |\bar\rho^{k+1}(t)|_{L^2}^2&+|\sqrt{\rho^{k+1}}\bar u^{k+1}(t)|_{L^2}^2+|\bar\eta^{k+1}(t)|_{L^2}^2+\int_0^t|\bar u_x^{k+1}(t)|_{L^2}^2{\rm d} s+\int_0^t|\bar\eta_x^{k+1}|_{L^2}^2{\rm d} s\\ &\leq C\exp(C_\xi t)\int_0^t(|\sqrt{\rho^k} \bar u^k(s)|_{L^2}^2+|\bar u_x^k(s)|_{L^2}^2){\rm d} s. \end{align} (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

    \begin{align} \sum\limits_{k = 1}^K[\sup\limits_{0\leq t\leq T_*}(|\bar\rho^{k+1}(t)|_{L^2}^2&+|\sqrt{\rho^{k+1}}\bar u^{k+1}(t)|_{L^2}^2+|\bar\eta^{k+1}(t)|_{L^2}^2{\rm d} t\\ &+\int_0^{T_*}|\bar u_x^{k+1}(t)|_{L^2}^2+\int_0^{T_*}|\bar\eta_x^{k+1}(t)|_{L^2}^2{\rm d} t] < C, \end{align} (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

    \begin{align} &\rho^k\rightarrow\rho^\varepsilon\quad\mbox{in}\ \ L^\infty(0,T_*;L^2(\Omega)), \end{align} (70)
    \begin{align} &u^k\rightarrow u^\varepsilon\quad\mbox{in}\ \ L^\infty(0,T_*;L^2(\Omega))\cap L^2(0,T_*;H_0^1(\Omega)), \end{align} (71)
    \begin{align} &\eta^k\rightarrow \eta^\varepsilon\quad\mbox{in}\ \ L^\infty(0,T_*;L^2(\Omega))\cap L^2(0,T_*;H^1(\Omega)). \end{align} (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

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

    \begin{align} &\rho^\varepsilon\rightarrow\rho^\delta\quad\mbox{in}\ \ L^\infty(0,T_*;L^2(\Omega)), \end{align} (74)
    \begin{align} &u^\varepsilon\rightarrow u^\delta\quad\mbox{in}\ \ L^\infty(0,T_*;L^2(\Omega))\cap L^2(0,T_*;H_0^1(\Omega)), \end{align} (75)
    \begin{align} &\eta^\varepsilon\rightarrow \eta^\delta\quad\mbox{in}\ \ L^\infty(0,T_*;L^2(\Omega))\cap L^2(0,T_*;H^1(\Omega)), \end{align} (76)

    and there also holds

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

    \begin{align} \begin{cases} \begin{aligned} &L_p u_0^\delta+\big(P(\rho_0^\delta)+\eta_0^\delta\big)_x+\eta_0^\delta\Phi_x = (\rho_0^\delta)^{1\over2}(g^\delta+\Phi_x),&\\ &u_0^\delta|_{\partial\Omega} = 0,& \end{aligned} \end{cases} \end{align} (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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