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Research article

Shifted Legendre polynomials-based single and double integral inequalities with arbitrary approximation order: Application to stability of linear systems with time-varying delays

  • This paper proposes novel single and double integral inequalities with arbitrary approximation order by employing shifted Legendre polynomials and Cholesky decomposition, and these inequalities could significantly reduce the conservativeness in stability analysis of linear systems with interval time-varying delays. The coefficients of the proposed single and double integral inequalities are determined by using the weighted least-squares method. Also former well-known integral inequities, such as Jensen inequality, Wirtinger-based inequality, auxiliary function-based integral inequalities, are all included in the proposed integral inequalities as special cases with lower-order approximation. Stability criterions with less conservatism are then developed for both constant and time-varying delay systems. Several numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.

    Citation: Deren Gong, Xiaoliang Wang, Peng Dong, Shufan Wu, Xiaodan Zhu. Shifted Legendre polynomials-based single and double integral inequalities with arbitrary approximation order: Application to stability of linear systems with time-varying delays[J]. AIMS Mathematics, 2020, 5(5): 4371-4398. doi: 10.3934/math.2020279

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  • This paper proposes novel single and double integral inequalities with arbitrary approximation order by employing shifted Legendre polynomials and Cholesky decomposition, and these inequalities could significantly reduce the conservativeness in stability analysis of linear systems with interval time-varying delays. The coefficients of the proposed single and double integral inequalities are determined by using the weighted least-squares method. Also former well-known integral inequities, such as Jensen inequality, Wirtinger-based inequality, auxiliary function-based integral inequalities, are all included in the proposed integral inequalities as special cases with lower-order approximation. Stability criterions with less conservatism are then developed for both constant and time-varying delay systems. Several numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.


    We consider the compressible isentropic Navier-Stokes equations with degenerate viscosities in R3, which gives the conservation laws of mass and momentum of fluids. This model comes from the Boltzmann equations through the Chapman-Enskog expansion to the second order, and the viscosities depend on the density ρ0 by the laws of Boyle and Gay-Lussac for ideal gas. This system can be written as

    {ρt+div(ρu)=0,(ρu)t+div(ρuu)+P=divT, (1)

    where xR3 is the spatial coordinate; t0 is the time; ρ is the density of the fluid; u=(u(1),u(2),u(3))R3 is the velocity of the fluid; P is the pressure, and for the polytropic fluid

    P=Aργ,γ>1, (2)

    where A is a positive constant, γ is the adiabatic index; T is the stress tensor given by

    T=μ(ρ)(u+(u))+λ(ρ)divuI3, (3)

    where I3 is the 3×3 unit matrix, μ(ρ) is the shear viscosity, λ(ρ) is the second viscosity, and

    μ(ρ)=αρ,λ(ρ)=βρ, (4)

    where the constants α and β satisfy

    α>0,2α+3β0.

    Here, the initial data are given by

    (ρ,u)|t=0=(ρ0,u0)(x),xR3, (5)

    and the far field behavior is given by

    (ρ,u)(0,0)as |x|,t0. (6)

    The aim of this paper is to prove a blow-up criterion for the regular solution to the Cauchy problem (1) with (5)-(6).

    Throughout the paper, we adopt the following simplified notations for the standard homogeneous and inhomogeneous Sobolev space:

    Dk,r={fL1loc(R3):|f|Dk,r=|kf|Lr<+},Dk=Dk,2(k2),D1={fL6(R3):|f|D1=|f|L2<},fXY=fX+fY,fs=fHs(R3),|f|p=fLp(R3),|f|Dk=fDk(R3).

    A detailed study of homogeneous Sobolev space can be found in [5].

    The compressible isentropic Navier-Stokes system is a well-known mathematical model, which has attracted great attention from the researchers, and some significant processes have been made in the well-posedness for this system.

    When (μ,λ) are both constants, with the assumption that there is no vacuum, the local existence of the classical solutions to system (1) follows from a standard Banach fixed point argument. For the existence results with vacuum and general data, the main breakthrough is due to Lions [16]. He established the global existence of weak solutions in R3, periodic domains or bounded domains, under the homogenous Dirichlet boundary conditions and the restriction γ>9/5. Later, the restriction on γ was improved to γ>3/2 by Feireisl-Novotný-Petzeltová [4]. Recently, Cho-Choe-Kim [2] introduced the following initial layer compatibility condition

    divT0+P(ρ0)=ρ0g

    for some gL2 to deal with the vacuum. They proved the local existence of the strong solutions in R3 or bounded domains with homogenous Dirichlet boundary conditions. Moreover, Huang-Li-Xin proved the global existence of the classical solutions to the Cauchy problem of the isentropic system with small energy and vacuum in [8].

    When (μ,λ) depend on density in the following form

    μ(ρ)=αρδ1,λ(ρ)=βρδ2, (7)

    where δ1>0, δ20, α>0 and β are all real constants, system (1) has received a lot of attention. However, except for the 1D problems, there are few results on the strong solutions for the multi-dimensional problems, since the possible degeneracy of the Lamˊe operator caused by initial vacuum. This degeneracy gives rise to some difficulties in the regularity estimates because of the less regularizing effect of the viscosity on the solutions. Recently, Li-Pan-Zhu [11] have obtained the local existence of the classical solutions to system (1) in 2D space under

    δ1=1,δ2=0 or 1,α>0,α+β0, (8)

    and (6), where the vacuum cannot appear in any local point. They [12] also prove the same existence result in 3D space under

    (ρ,u)(ˉρ,0)as|x|, (9)

    with initial vacuum appearing in some open set or the far field, the constant ˉρ0 and

    1<δ1=δ2min(3,γ+12),α>0,α+β0. (10)

    We also refer readers to [3], [6], [10], [13], [18], [26] and references therein for other interesting progress for this compressible degenerate system, corresponding radiation hydrodynamic equations and magnetohydrodynamic equations.

    It should be noted that one should not always expect the global existence of solutions with better regularities or general initial data because of Xin's results [23] and Rozanova's results [20]. It was proved that there is no global smooth solutions to (1), if the initial density has nontrivial compact support (1D) or the solutions are highly decreasing at infinity(dD, d1). These motivate us to find the blow-up mechanisms and singularity structures of the solutions.

    For constant viscosity, Beale-Kato-Majda [1] first proved that the maximum norm of the vorticity controls the blow-up of the smooth solutions to 3D incompressible Euler equations

    limTTT0|curlu|dt=, (11)

    where T is the maximum existence time. Later, for the same problem, Ponce [19] proved that the maximum norm of the deformation tensor controls the blow-up of the smooth solutions

    limTTT0|D(u)|dt=, (12)

    where the deformation tensor D(u)=12(u+(u)). Huang-Li-Xin [7] proved that the criterion (12) holds for the strong solutions to the system (1). Sun-Wang-Zhang [22] proved that the upper bound of the density controls the blowup of the strong solution to the system (1). There are some other interesting results about infinite time blowup and finite time blowup results on the nonlinear wave equation with different initial energy levels, refer to [14], [15], [24] and references therein for detailed study.

    When the viscosities depend on density in the form of (4), S. Zhu [25] introduced the regular solutions, which can be defined as

    Definition 1.1. [25] Let T>0 be a finite constant, (ρ,u) is called a regular solution to the Cauchy problem (1) with (5)-(6) on [0,T]×R3 if (ρ,u) satisfies

    (A)(ρ,u)  in [0,T]×R3  satisfies the Cauchy problem (1)with (5)(6)in the sense of distributions;(B)ρ0,ργ12C([0,T];H2), (ργ12)tC([0,T];H1);(C)logρC([0,T];D1), (logρ)tC([0,T];L2);(D)uC([0,T];H2)L2([0,T];D3), utC([0,T];L2)L2([0,T];D1).

    The local existence of the regular solutions has been obtained by Zhu [25].

    Theorem 1.2. [25] Let 1<γ2 or γ=3. If the initial data (ρ0,u0) satisfies the regularity conditions

    ργ1200,(ργ120,u0)H2,logρ0D1, (13)

    then there exist a small time T and a unique regular solution (ρ,u) to the Cauchy problem (1) with (5)-(6). Moreover, we also have ρ(t,x)C([0,T]×R3) and

    ρC([0,T];H2),ρtC([0,T];H1).

    Based on Theorem 1.2, we establish the blow-up criterion for the regular solution in terms of logρ and the deformation tensor D(u), which is similar to the Beale-Kato-Majda criterion for the ideal incompressible Euler equations and the compressible Navier-Stokes equations.

    Theorem 1.3. Let (ρ,u) be a regular solution obtained in Theorem 1.2. Then if ¯T<+ is the maximal existence time, one has both

    limT¯T(sup0tT|logρ|6+T0|D(u)| dt)=+, (14)

    and

    limsupT¯TT0D(u)LD1,6 dt=+. (15)

    The rest of the paper can be organized as follows. In Section 2, we will give the proof for the criterion (14). Section 3 is an appendix which will present some important lemmas which are frequently used in our proof, and also give the detail derivation for the desired system used in our following proof.

    In this section, we give the proof for Theorem 1.3. We use a contradiction argument to prove (14), let (ρ,u) be the unique regular solution to the Cauchy problem (1) with (5)-(6) and the maximal existence time ¯T. We assume that ¯T<+ and

    limT¯T(sup0tT|logρ|6+T0|D(u)|dt)=C0<+ (16)

    for some constant 0<C0<. If we prove that under assumption (16), ¯T is actually not the maximal existence time for the regular solution, there will be a contradiction, thus (14) holds.

    Notice that, one can also prove (15) by contradiction argument. Assume that

    limsupT¯TT0D(u)LD1,6 dt=C0<+ (17)

    for some constant 0<C0<. Combing (17) with the mass equation, we know that

    limT¯Tsup0tT|logρ|6CC0,

    which implies that under assumption (17), we have (16). Thus, if we prove that (14) holds, then (15) holds immediately.

    In the rest part of this section, based on the assumption (16), we will prove that ˉT is not the maximal existence time for the regular solution.

    From the definition of the regular solution, we know for

    ϕ=ργ12,ψ=2γ1logϕ, (18)

    (ϕ,ψ,u) satisfies

    {ϕt+uϕ+γ12ϕdivu=0,ψt+(uψ)+divu=0,ut+uu+2θϕϕ+Lu=ψQ(u), (19)

    where L is the so-called Lamˊe operator given by

    Lu=div(α(u+(u))+βdivuI3), (20)

    and terms (Q(u),θ) are given by

    Q(u)=α(u+(u))+βdivuI3,θ=Aγγ1. (21)

    See our appendix for the detailed process of the reformulation.

    For (19)2, we have the equivalent form

    ψt+3l=1Allψ+Bψ+divu=0. (22)

    Here Al=(a(l)ij)3×3 (i,j,l=1,2,3) are symmetric with a(l)ij=u(l) when i=j; and a(l)ij=0, otherwise. B=(u), so (22) is a positive symmetric hyperbolic system. By direct computation, one knows

    ψ=2γ1ϕϕ=2γ1ργ12ργ12=ρρ=logρ, (23)

    combing this with (16), one has ψL([0,T];L6).

    Under (16) and (19), we first show that the density ρ is uniformly bounded.

    Lemma 2.1. Let (ρ,u) be the unique regular solution to the Cauchy problem (1) with (5)-(6) on [0,¯T)×R3 satisfying (16). Then

    ρL([0,T]×R3)+ϕL([0,T];Lq)C,0T<¯T,

    where C>0 depends on C0, constant q[2,+] and ¯T.

    Proof. First, it is obvious that ϕ can be represented by

    ϕ(t,x)=ϕ0(W(0,t,x))exp(γ12t0divu(s,W(s,t,x))ds), (24)

    where WC1([0,T]×[0,T]×R3) is the solution to the initial value problem

    {ddtW(t,s,x)=u(t,W(t,s,x)),0tT,W(s,s,x)=x, 0sT, xR3.

    Then it is clear that

    ϕL([0,T]×R3)|ϕ0|exp(CC0)C. (25)

    Similarly,

    ρL([0,T]×R3)C. (26)

    Next, multiplying (19)1 by 2ϕ and integrating over R3, we get

    ddt|ϕ|22C|divu||ϕ|22, (27)

    from (16), (27) and the Gronwall's inequality, we immediately obtain

    ϕL([0,T];L2)C. (28)

    Combing (25)-(28) together, one has

    ϕL([0,T];Lq)C,q[2,+].

    We complete the proof of this lemma.

    Before go further, notice that

    |ϕ|6=|ϕlogϕ|6=2γ1|ϕlogρ|6C|ϕ||logρ|6C, (29)

    where we have used (16) and Lemma 2.1. Next, we give the basic energy estimates on u.

    Lemma 2.2. Let (ρ,u) be the unique regular solution to the Cauchy problem (1) with (5)-(6) on [0,¯T)×R3 satisfying (16). Then

    sup0tT|u(t)|22+T0|u(t)|22dtC,0T<¯T,

    where C only depends on C0 and ¯T.

    Proof. Multiplying (19)3 by 2u and integrating over R3, we have

    ddt|u|22+2R3(α|u|2+(α+β)(divu)2)dx=R32(uuuθϕ2u+ψQ(u)u)dx:L1+L2+L3. (30)

    The right-hand side terms can be estimated as follows.

    L1=R32uuudxC|divu||u|22,L2=2R3θϕ2divudxC|ϕ|22|divu|C|divu|,L3=R32ψQ(u)udxC|ψ|6|u|2|u|3C|u|2|u|122|u|122α2|u|22+C|u|2|u|2α|u|22+C|u|22, (31)

    where we have used (16), (23) and the facts

    |u|3C|u|122|u|122. (32)

    Thus (30) and (31) yield

    ddt|u|22+α|u|22C(|divu|+1)|u|22+C|divu|. (33)

    By the Gronwall's inequality, (16) and (33), we have

    |u(t)|22+t0|u(s)|22dsC,0tT. (34)

    This completes the proof of this lemma.

    The next lemma provides the key estimates on ϕ and u.

    Lemma 2.3. Let (ρ,u) be the unique regular solution to the Cauchy problem (1) with (5)-(6) on [0,¯T)×R3 satisfying (16). Then

    sup0tT(|u(t)|22+|ϕ(t)|22)+T0(|2u|22+|ut|22)dtC,0T<¯T,

    where C only depends on C0 and ¯T.

    Proof. Multiplying (19)3 by Luθϕ2 and integrating over R3, we have

    12ddtR3(α|u|2+(α+β)|divu|2)dx+R3(Luθϕ2)2 dx=αR3(uu)×curlu dx+R3(2α+β)(uu)divu dx+θR3(ψQ(u))ϕ2dxθR3(uu)ϕ2 dx+R3(ψQ(u))Lu dxθR3utϕ2 dx≡:9i=4Li, (35)

    where we have used the fact that

    u+divu=curl(curlu)=×curlu.

    First, from the standard elliptic estimate shown in Lemma 3.3, we have

    |2u|22C|θϕ2|22C|div(α(u+(u))+βdivuI3)|22C|θϕ2|22C|div(α(u+(u))+βdivuI3)θϕ2|22=CR3(Luθϕ2)2dx. (36)

    Second, we estimate the right-hand side of (35) term by term. According to

    {u×curlu=12(|u|2)uu,×(a×b)=(b)a(a)b+(divb)a(diva)b,

    Hölder's inequality, Young's inequality, (16), (23), (29), Lemma 2.2, Lemma 3.1 and (19)1, one can obtain that

    |L4|=α|R3(u)u×curlu dx|=α|R3(curlu×((u)u))dx|=α|R3(curlu×(u×curlu))dx|=α|R3(12|curlu|2divucurluD(u)curlu)dx|C|u||u|22,|L5|=|R3(2α+β)(u)udivu dx||R3(2α+β)(u:udivu+12(divu)3)dx|+C|ϕ|6|u|3|u|2|divu|C(|u|22|divu|+|u|122|u|122|u|2|divu|)C(|u|22+|u|2|u|2)|divu|C|divu|(|u|22+1),L6=θR3(ψQ(u))ϕ2 dxC|ψ|6|u|3|ϕ2|2C|u|122|2u|122|ϕ|2|ϕ|C|ϕ|22+C(ϵ)|u|22+ϵ|2u|22,|L7|=θ|R3(uu)ϕ2dx|=θ|R3u:(u)ϕ2dxR3ϕ2u(divu)dx| (37)
    =θ|R3u:(u)ϕ2dx+R3(divu)2ϕ2dx+R2uϕ2divu dx|C(|u|22|ϕ2|+|u|2|ϕ|2|ϕ||divu|)C(|u|22+|divu||ϕ|2)C(|u|22+|divu|+|divu||ϕ|22),L8=R3(ψQ(u))Lu dxC|ψ|6|u|3|2u|2C|2u|322|u|122C(ϵ)|u|22+ϵ|2u|22,L9=θR3utϕ2dx=θR3ϕ2divut dx=θddtR3ϕ2divu dxθR3(ϕ2)tdivu dx=θddtR3ϕ2divu dxθR32ϕϕtdivu dx=θddtR3ϕ2divu dx+θR3uϕ2divu dx+θ(γ1)R3ϕ2(divu)2dxθddtR3ϕ2divu dx+C(|u|2|ϕ|2|ϕ||divu|+|u|22|ϕ2|)θddtR3ϕ2divu dx+C(|ϕ|2|divu|+|u|22)θddtR3ϕ2divu dx+C(|u|22+|divu|+|divu||ϕ|22), (38)

    where ϵ>0 is a sufficiently small constant. Thus (35)-(38) imply

    12ddtR3(α|u|2+(α+β)|divu|22θϕ2divu)dx+C|2u|22C((|u|22+|ϕ|22)(|divu|+1)+|divu|). (39)

    Third, applying to (19)1 and multiplying by (ϕ), we have

    (|ϕ|2)t+div(|ϕ|2u)+(γ2)|ϕ|2divu=2(ϕ)u(ϕ)(γ1)ϕϕdivu=2(ϕ)D(u)(ϕ)(γ1)ϕϕdivu. (40)

    Integrating (40) over R3, we get

    ddt|ϕ|22C(ϵ)(|D(u)|+1)|ϕ|22+ϵ|2u|22. (41)

    Adding (41) to (39), from the Gronwall's inequality and (16), we immediately obtain

    α|u(t)|222θR3ϕ2divu dx+C|ϕ(t)|22+t0|2u(s)|22dsC,

    that is

    α|u(t)|22+C|ϕ(t)|22+t0|2u(s)|22dsC+2θR3ϕ2divu dxC(1+|u|2|ϕ|2|ϕ|)C+α4|u(t)|22,

    which implies

    |u(t)|22+|ϕ(t)|22+t0|2u(s)|22dsC,0tT.

    Finally, due to ut=Luuu2θϕϕ+ψQ(u), we deduce that

    t0|ut|22dsCt0(|2u|22+|u|23|u|26+|ϕ|2|ϕ|22+|u|23|ψ|26)dsC.

    Thus we complete the proof of this lemma.

    Next, we proceed to improve the regularity of ϕ, ψ and u. First, we start with the estimates on the velocity.

    Lemma 2.4. Let (ρ,u) be the unique regular solution to the Cauchy problem (1) with (5)-(6) on [0,¯T)×R3 satisfying (16). Then

    sup0tT(|ut(t)|22+|u(t)|D2)+T0|ut|22dtC,0T<¯T, (42)

    where C only depends on C0 and ¯T.

    Proof. From the standard elliptic estimate shown in Lemma 3.3 and

    Lu=utuu2θϕϕ+ψQ(u), (43)

    one has

    |u|D2C(|ut|2+|uu|2+|ϕϕ|2+|ψQ(u)|2)C(|ut|2+|u|6|u|3+|ϕ|3|ϕ|6+|ψ|6|u|3)C(1+|ut|2+|u|6|u|122|2u|122+|u|122|2u|122)C(1+|ut|2+|2u|122)C(1+|ut|2)+12|u|D2, (44)

    where we have used Sobolev inequalities, (16), (23), (29) and Lemmas 2.1-2.3. Then we immediately obtain that

    |u|D2C(1+|ut|2). (45)

    Next, differentiating (19)3 with respect to t, it reads

    utt+Lut=(uu)t2θ(ϕϕ)t+(ψQ(u))t. (46)

    Multiplying (46) by ut and integrating over R3, one has

    12ddt|ut|22+α|ut|2212ddt|ut|22+R3(α|ut|2+(α+β)|divut|2)dx=R3((uu)tut+(ψQ(u))tut2θ(ϕϕ)tut)dx:12i=10Li. (47)

    Similarly, based on (16), (23), (29) and Lemmas 2.1-2.3, we estimate the right-hand side of (47) term by term as follows.

    L10=R3(uu)tut dx=R3((utu)ut+(uut)ut)dx=R3(utD(u)ut12(ut)2divu)dxC|D(u)||ut|22,L11=R3(ψQ(u))tut dx=R3ψQ(u)tut dx+R3ψtQ(u)ut dx=R3ψQ(u)tut dxR3divuQ(u)ut dx+R3uψdiv(Q(u)ut)dxC(|ψ|6|ut|2|ut|3+|2u|2|Q(u)||ut|2+|ψ|6|u|6|2u|2|ut|6+|ψ|6|u|6|Q(u)|6|ut|2)C(|ut|2|ut|122|ut|122+|2u|2|Q(u)||ut|2+|u|6|ut|2+|2u|2|ut|2)α8|ut|22+C(1+|D(u)|)(|ut|22+|u|2D2),L12=R32θ(ϕϕ)tut dx=θR3(ϕ2)tdivutdx=2θR3ϕϕtdivutdx=2θR3ϕ(uϕ+γ12ϕdivu)divutdx=θR3(uϕ2+(γ1)ϕ2divu)divutdx (48)
    \begin{equation} \begin{split} = &-\frac{\theta(\gamma-1)}{2}\int_{\mathbb{R}^3} \phi^2 (\text{div} u)^2_t dx-\theta\int_{\mathbb{R}^3} u\cdot\nabla \phi^2\text{div}u_t dx\\ = &-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx +\theta(\gamma-1)\int_{\mathbb{R}^3} u\phi \phi_t (\text{div} u)^2 dx\\ &-\theta\int_{\mathbb{R}^3} u\cdot \nabla \phi^2 \text{div}u_t dx\\ = &-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx -\theta(\gamma-1)\int_{\mathbb{R}^3} u\phi (u\cdot\nabla\phi) (\text{div} u)^2 dx\\ &-\frac{\theta(\gamma-1)^2}{2}\int_{\mathbb{R}^3} u\phi^2 (\text{div} u)^3 dx-\theta\int_{\mathbb{R}^3} u\cdot \nabla \phi^2 \text{div}u_t dx\\ = &-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx +\frac{\theta(\gamma-1)}{2}\int_{\mathbb{R}^3} u\phi^2 \nabla(\text{div}u)^2 dx\\ &+\frac{\theta(\gamma-1)(3-\gamma)}{2}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^3 dx -\theta\int_{\mathbb{R}^3} u\cdot \nabla \phi^2 \text{div}u_t dx\\ \le&-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx + C\big(|u|_\infty| \phi|^2_\infty|\nabla u|_2|\nabla^2 u|_2\\ &+|\phi|^2_\infty|D( u)|_\infty|\nabla u|^2_2+|\phi|_\infty|\nabla \phi|_2|u|_\infty|\nabla u_t|_2\big) \\ \le&-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx\\ &+ C\big(|u|_\infty| |\nabla^2 u|_2+|D( u)|_\infty+|u|_\infty|\nabla u_t|_2\big) \\\\ \end{split} \end{equation} (49)
    \begin{equation} \begin{split} \le&-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx + \frac{\alpha}{4}|\nabla u_t|^2_2\\ &+ C(1+|D( u)|_\infty+|u|^2_{D^2}),\\ \end{split} \end{equation} (50)

    where we also used Hölder's inequality, Young's inequality and

    \begin{equation} |u|_{\infty}\le C|u|_{W^{1,3}}\le C(|u|_2^{\frac12}|\nabla u|_2^{\frac12}+|\nabla u|^{\frac12}|\nabla^2 u|^{\frac12}). \end{equation} (51)

    It is clear from (47)-(50) and (45) that

    \begin{equation} \begin{split} & \frac{d}{dt}(|u_t|^2_2+|\phi \text{div}u|^2_2)+|\nabla u_t|^2_2\le C(1+|\nabla^2 u|_2+ |D( u)|_\infty)|u_t|^2_2. \end{split} \end{equation} (52)

    Integrating (52) over (\tau,t) (\tau \in( 0,t)) , we have

    \begin{equation} \begin{split} &|u_t(t)|^2_2+|\phi \text{div}u(t)|^2_2+\int_\tau^t|\nabla u_t(s)|^2_2 ds\\ \le & |u_t(\tau)|^2_2+|\phi \text{div}u(\tau)|^2_2+C\int_{\tau}^t \Big(\big(1+|\nabla^2 u|_2 +|D( u)|_\infty\big)|u_t|^2_2\Big)(s) ds. \end{split} \end{equation} (53)

    From the momentum equations (19)_3 , we obtain

    \begin{equation} \begin{split} |u_t(\tau)|_2 \le & C\big(|u\cdot \nabla u|_2+|\phi\nabla\phi|_2+|L u|_2+|\psi\cdot Q(u)|_2\big)(\tau)\\ \le & C\big( |u|_\infty |\nabla u|_2+|\phi|_\infty|\nabla \phi|_2+|u|_{D^2}+|\psi|_6|\nabla u|_3\big)(\tau), \end{split} \end{equation} (54)

    which, together with the definition of regular solution, gives

    \begin{equation} \begin{split} &\lim \sup\limits_{\tau\rightarrow 0}|u_t(\tau)|_2\\ \le& C\big( |u_0|_\infty |\nabla u_0|_2+|\phi_0|_\infty|\nabla \phi_0|_2+|u_0|_{D^2}+|\psi_0|_6|\nabla u_0|_3\big)\le C_0. \end{split} \end{equation} (55)

    Letting \tau \rightarrow 0 in (53), applying the Gronwall's inequality, (16) and Lemma 2.3, we arrive at

    \begin{equation} |u_t(t)|^2_2+|u(t)|_{D^2}^2+\int_0^t|\nabla u_t(s)|^2_2 ds\le C, \quad 0\le t\le T. \end{equation} (56)

    This completes the proof of this lemma.

    The following lemma gives bounds of \nabla \phi and \nabla^2 u .

    Lemma 2.5. Let ( \rho, u) be the unique regular solution to the Cauchy problem (1) with (5)-(6) on [0,\overline{T})\times \mathbb{R}^3 satisfying (16). Then

    \begin{equation} \begin{split} &\sup\limits_{0\leq t\leq T}\big(\|\phi(t)\|_{W^{1,6}}+|\phi_t(t)|_6\big)+\int_0^T|u(t)|^2_{D^{2,6}} dt\leq C, \quad 0\leq T < \overline{T}, \end{split} \end{equation} (57)

    where C only depends on C_0 and \overline{T} .

    Proof. First, taking q = 6 in Lemma 2.1, combing with (29) we have

    \begin{equation*} \sup\limits_{0\leq t\leq T} \|\phi(t)\|_{W^{1,6}}\le C,\quad 0\leq T < \overline{T}. \end{equation*}

    Second, one has

    \begin{equation} \begin{split} |\phi_t|_6 = & \big|u\cdot\nabla\phi+\frac{\gamma-1}{2}\phi\text{div}u\big|_6\\ \le& C(|\nabla\phi|_6|u|_{\infty}+|\phi|_{\infty}|\text{div}u|_6)\le C, \end{split} \end{equation} (58)

    where we have used Lemmas 2.3-2.4 and (29) .

    Third, according to

    \begin{equation} Lu = -u_t-u\cdot\nabla u -2\theta\phi\nabla \phi+\psi\cdot Q(u), \end{equation} (59)

    and the standard elliptic estimate shown in Lemma 3.3, one has

    \begin{equation} \begin{split} |\nabla ^2 u|_6 \le& C(|u_t|_6+|u\cdot \nabla u|_6+|\phi\nabla\phi|_6+|\psi \cdot Q( u)|_6)\\ \le & C(|\nabla u_t|_2+|u|_{\infty} |\nabla u|_6+|\phi|_{\infty}|\nabla\phi|_6+|\psi|_{6} |Q( u)|_{\infty})\\ \le & C(1+|\nabla u_t|_2+ |D( u)|^{\frac14}_2 |\nabla D(u)|^{\frac34}_6)\\ \le & C(1+|\nabla u_t|_2+|\nabla^2 u|^{\frac34}_6)\\ \le & C(1+|\nabla u_t|_2)+\frac12|\nabla^2 u|_6, \end{split} \end{equation} (60)

    where we have used (16), (23), (29), Lemmas 2.1-2.4 and

    \begin{equation} \begin{split} & |\text{div}u|_{\infty}\le C|D( u)|_\infty,\\ & |D( u)|_\infty\leq C|D( u)|^{\frac14}_2 |\nabla D(u)|^{\frac34}_6. \end{split} \end{equation} (61)

    Thus, (60) implies that

    \begin{equation} |\nabla^2 u|_6 \le C(1+|\nabla u_t|_2). \end{equation} (62)

    Combing (62) with Lemma 2.4, one has

    \begin{equation} \begin{split} \int_0^t|u(s)|^2_{D^{2,6}} ds \le& C\int_0^t(1+ |\nabla u_t(s)|^2_2) ds\le C,\quad 0\le t\leq T. \end{split} \end{equation} (63)

    The proof of this lemma is completed.

    Lemma 2.5 implies that

    \begin{equation} \int_0^t|\nabla u(\cdot, s)|_\infty ds\le C, \end{equation} (64)

    for any t\in [0, \overline{T}) with C>0 a finite number. Noting that (19) is essentially a parabolic-hyperbolic system, it is then standard to derive other higher order estimates for the regularity of the regular solutions. We will show this fact in the following lemma.

    Lemma 2.6. Let ( \rho, u) be the unique regular solution to the Cauchy problem (1) with (5)-(6) on [0,\overline{T})\times \mathbb{R}^3 satisfying (16). Then

    \begin{equation*} \begin{split} \sup\limits_{0\leq t\leq T}\big(|\phi(t)|^2_{D^2}&+|\psi(t)|^2_{D^1}+\|\phi_t(t)\|^2_1+|\psi_t(t)|^2_{2}\big)\\ &+\int_{0}^{T}\Big(|u(t)|^2_{D^{3}}+|\phi_{tt}(t)|^2_{2}\Big)\mathit{\text{d}}t\leq C, \quad 0\leq T < \overline{T}, \end{split} \end{equation*}

    where C only depends on C_0 and \overline{T} .

    Proof. From (19)_3 and Lemma 3.3, we have

    \begin{equation} \begin{split} |u|_{D^3} \le& C\big(|u_t|_{D^1}+|u\cdot \nabla u|_{D^1}+|\phi\nabla\phi|_{D^1}+|\psi \cdot Q(u)|_{D^1}\big)\\ \le & C\big(|u_t|_{D^1}+|u|_{\infty} |\nabla^2 u|_{2}+|\nabla u|_6|\nabla u|_3+|\psi|_6 |\nabla^2 u|_3\\ &+|\nabla\phi|_6|\nabla\phi|_3+|\phi|_{\infty}|\nabla^2\phi|_2+|\nabla\psi|_2|D(u)|_{\infty}\big)\\ \le & C\big(1+|u_t|_{D^1}+| \phi|_{D^2} +| u|^{\frac12}_{D^3}+|\psi|_{D^1} |D( u)|_\infty\big)\\ \le & C\big(1+|u_t|_{D^1}+| \phi|_{D^2}+|\psi|_{D^1} |D( u)|_\infty\big)+\frac12 | u|_{D^3}, \end{split} \end{equation} (65)

    where we have used Young's inequality, Lemma 2.5, (16), (23), (29) and (61). Thus (65) offers that

    \begin{equation} |u|_{D^3} \le C\big(1+|u_t|_{D^1}+| \phi|_{D^2}+|D( u)|_\infty|\psi|_{D^1}\big). \end{equation} (66)

    Next, applying \partial_i ( i = 1,2,3 ) to (19)_2 with respect to x , we obtain

    \begin{equation} \begin{split} &(\partial_i \psi)_t+\sum\limits_{l = 1}^3 A_l \partial_l\partial_i \psi+B\partial_i \psi+\partial_i \nabla \text{div} u \\ = &\big(-\partial_i(B\psi)+B\partial_i \psi\big)+\sum\limits_{l = 1}^3 \big(-\partial_i(A_l) \partial_l \psi\big). \end{split} \end{equation} (67)

    Multiplying (67) by 2(\partial_i\psi)^{\top} , integrating over \mathbb{R}^3 , and then summing over i , noting that A_l ( l = 1,2,3 ) are symmetric, it is not difficult to show that

    \begin{equation} \begin{split} \frac{d}{dt}|\nabla \psi|^2_2 \le & C\int_{\mathbb{R}^3}\Big( |\text{div} A| |\nabla\psi|^2 +|\nabla^3 u| |\nabla\psi| +|\nabla\psi|^2 |\nabla u|\\ &+|\partial_i(B\psi)-B\partial_i \psi | |\nabla\psi|\Big) dx\\ \le& C\big(\big|\text{div}A\big|_\infty|\nabla \psi|^2_2+|\nabla^3 u|_2|\nabla \psi|_2+|\nabla\psi|^2_2 |\nabla u|_{\infty}\\ &+|\partial_i(B\psi)-B\partial_i \psi |_2|\nabla\psi|_2\big), \end{split} \end{equation} (68)

    where \text{div}A = \sum\limits_{l = 1}^3\partial_{l}A_l . When |\zeta| = 1 , choosing r = 2,\ a = 3 , b = 6 in (83), we have

    \begin{equation} \begin{split} |\partial_i(B\psi)-B\partial_i \psi |_2& = |D^\zeta(B\psi)-BD^\zeta \psi|_2\leq C|\nabla^2 u|_3|\psi|_6\\ &\le C|\nabla^2 u|_2^{\frac12}|\nabla^3 u|_2^{\frac12}\le C|\nabla^3 u|_2^{\frac12}. \end{split} \end{equation} (69)

    Thus

    \begin{equation} \begin{split} \frac{d}{dt}|\nabla \psi|^2_2\leq& C\big(|\nabla u|_\infty|\nabla \psi|^2_2 +|\nabla^3 u|_2|\nabla \psi|_2+|\nabla^3 u |^{\frac12}_2|\nabla\psi|_2\big). \end{split} \end{equation} (70)

    Combining (70) with (66) and Lemma 3.1, we have

    \begin{equation} \begin{split} \frac{d}{dt}|\psi|^2_{D^1}\le & C(1+|\nabla u|_\infty)|\psi|^2_{D^1}+C(1+|\nabla^3 u|_2)|\psi|_{D^1}\\ \leq& C(1+|\nabla u|_\infty)|\psi|^2_{D^1}+C(1+|\phi|^2_{D^2}+|\nabla u_t|^2_2). \end{split} \end{equation} (71)

    On the other hand, let \nabla \phi = G = (G^{(1)},G^{(2)}, G^{(3)})^\top . Applying \nabla^2 to (19)_1 , we have

    \begin{equation} \begin{split} 0 = &(\nabla G)_t+\nabla(\nabla u\cdot G)+\nabla (\nabla G \cdot u)+\frac{\gamma-1}{2}\nabla(G\text{div}u +\phi \nabla \text{div}u\big), \end{split} \end{equation} (72)

    similarly to the previous step, we multiply (72) by 2\nabla G and integrate it over {\mathbb {R}}^3 to derive

    \begin{equation} \begin{split} \frac{d}{dt}|G|^2_{D^1} \le & C\int_{\mathbb{R}^3} \big(|\nabla^2 u| |G|+ |\nabla u| |\nabla G| +|\nabla \phi| |\nabla^2 u|+|\phi| |\nabla^3 u| \big)|\nabla G| dx\\ \le & C\big(|G|_6|\nabla^2 u|_3+|\nabla u|_\infty |\nabla G|_2+|\phi|_\infty|\nabla^3 u|_2\big)|\nabla G|_2\\ \le & C\big(|\nabla^2 u|_2^{\frac12}|\nabla^3 u|_2^{\frac12}+|\nabla u|_\infty |G|_{D^1}+| u|_{D^3}\big)| G|_{D^1}\\ \le & C\big(|u|_{D^3}^{\frac12}+|u|_{D^3}\big)| G|_{D^1}+C|\nabla u|_\infty | G|_{D^1}^2\\ \le & C(1+| u|_{D^3})| G|_{D^1}+C|\nabla u|_\infty | G|_{D^1}^2\\ \le & C(1+|u_t|_{D^1}+| \phi|_{D^2}+|D( u)|_\infty|\nabla\psi|_2)| G|_{D^1}+C|\nabla u|_\infty | G|_{D^1}^2\\ \le & C(1+|u_t|_{D^1}+|\nabla u|_\infty|\psi|_{D^1})| G|_{D^1}+C(1+|\nabla u|_\infty)| G|_{D^1}^2\\ \le & C(1+|\nabla u|_\infty)(|G|^2_{D^1}+|\psi|^2_{D^1})+C(1+|\nabla u_t|^2_2), \end{split} \end{equation} (73)

    where we have used the Young's inequality, (29) and (66). This estimate, together with (71), gives that

    \begin{equation} \begin{split} \frac{d}{dt}(|G|^2_{D^1}+|\psi|^2_{D^1}) \leq& C(1+|\nabla u|_\infty)(|G|^2_{D^1}+|\psi|^2_{D^1})+C(1+|\nabla u_t|^2_2). \end{split} \end{equation} (74)

    Then the Gronwall's inequality, (42), (64) and (74) imply

    \begin{equation} \begin{split} |\phi(t)|^2_{D^2}+|\psi(t)|^2_{D^1}\le C, \quad 0\leq t\leq T. \end{split} \end{equation} (75)

    Combing (75) with (66) and Lemma 2.4, one has

    \begin{equation} \int_{0}^{t}|u(s)|^2_{D^{3}}ds\le C\int_{0}^{t}(1+|\nabla u_t(s)|^2_2) ds\leq C, \quad 0\leq t\leq T. \end{equation} (76)

    Finally, using the following relations

    \begin{equation} \begin{split} \psi_t = &-\nabla (u \cdot \psi)-\nabla \text{div} u,\ \ \ \phi_t = -u\cdot \nabla \phi-\frac{\gamma-1}{2}\phi\text{div} u,\\ \phi_{tt} = &-u_t\cdot \nabla \phi-u\cdot \nabla \phi_t-\frac{\gamma-1}{2}\phi_t\text{div} u-\frac{\gamma-1}{2}\phi\text{div} u_t, \end{split} \end{equation} (77)

    according to Hölder's inequality, (16), (29), Lemmas 2.1-2.5, one has

    \begin{equation} \begin{split} |\psi_t|_2\le& C(|\nabla u \cdot \psi|_2+|u\cdot \nabla\psi|_2+|\nabla \text{div} u|_2)\\ \le& C(|\nabla u|_3|\psi|_6+|u|_{\infty}|\nabla\psi|_2+|\nabla^2 u|_2)\le C,\\ |\phi_t|_2\le& C(|u\cdot \nabla \phi|_2+|\phi\text{div} u|_2)\\ \le& C(|u|_{\infty}|\nabla\phi|_2+|\phi|_{\infty}|\nabla u|_2)\le C,\\ |\nabla\phi_t|_2 \le & C(|\nabla(u\cdot \nabla\phi)|_2+|\nabla(\phi\text{div} u)|_2)\\ \le& C(|\nabla u\cdot \nabla \phi|_2+| \nabla^2 \phi\cdot u|_2+|\nabla\phi\text{div} u|_2+|\phi\nabla\text{div} u|_2)\\ \le & C(|\nabla u|_3 |\nabla \phi|_6+|u|_{\infty} |\nabla^2\phi|_2+|\nabla\phi|_6|\nabla u|_3+|\phi|_{\infty}|\nabla^2 u|_2)\\ \le& C,\\ |\phi_{tt}|_2\le& C(|u_t\cdot \nabla \phi|_2+|u\cdot \nabla \phi_t|_2+|\phi_t\text{div} u|_2+|\phi\text{div} u_t|_2)\\ \le & C(|u_t|_{6}|\nabla\phi|_3+|u|_{\infty}|\nabla\phi_t|_2+|\phi_t|_6|\nabla u|_3+|\phi|_{\infty}|\nabla u_t|_2)\\ \le & C(1+|\nabla u_t|_2). \end{split} \end{equation} (78)

    Thus

    \begin{equation*} \sup\limits_{0\leq t\leq T}\big(\|\phi_t(t)\|^2_1+|\psi_t(t)|^2_{2}\big)\leq C, \end{equation*}

    and according to (42), one has

    \begin{equation*} \int_{0}^{T}|\phi_{tt}(t)|^2_{2} dt \le \int_{0}^{T}\big(1+|\nabla u_{t}(t)|^2_{2}\big) dt \leq C. \end{equation*}

    The proof of this lemma is completed.

    Now we know from Lemmas 2.1-2.6 that, if the regular solution (\rho, u)(x,t) exists up to the time {\overline{T}}>0 , with the maximal time {\overline{T}}<+\infty such that the assumption (16) holds, then

    \begin{equation*} (\rho^{\frac{\gamma-1}{2}},\nabla \log\rho,u)|_{t = \overline{T}} = \lim\limits_{t\rightarrow \overline{T}}(\rho^{\frac{\gamma-1}{2}},\nabla \log\rho,u) \end{equation*}

    satisfies the conditions imposed on the initial data (13) . If we solve the system (1) with the initial time {\overline{T}} , then Theorem 1.1 ensures that (\rho, u)(x,t) extends beyond {\overline{T}} as the unique regular solution. This contradicts to the fact that {\overline{T}} is the maximal existence time. We thus complete the proof of Theorem 1.3.

    In this subsection, we present some important lemmas which are frequently used in our previous proof. The first one is the well-known Gagliardo-Nirenberg inequality, which can be found in [9].

    Lemma 3.1. [9] Let r\in (1,+\infty) and \ h\in W^{1,p}(\mathbb{R}^3) \cap L^r(\mathbb{R}^3) . Then the following inequality holds for some constant C(c,p,r)

    \begin{equation} |h|_q\leq C|\nabla h|^c_p |h|^{1-c}_r, \end{equation} (79)

    where

    \begin{equation} c = \big(\frac{1}{r}-\frac{1}{q}\big)\big(\frac{1}{r}-\frac{1}{p}+\frac{1}{3}\big)^{-1}, \quad 0\le c\le 1. \end{equation} (80)

    If p< 3 , then q\in [r,\frac{3p}{3-p}] when r<\frac{3p}{3-p} ; and q\in [\frac{3p}{3-p},r] when r\geq\frac{3p}{3-p} . If p = 3 , then q\in [r,+\infty) . If p> 3 , then q\in [r,+\infty] .

    Some common versions of this inequality can be written as

    \begin{equation} \begin{split} |f|_3\leq C|f|^{\frac12}_{2}|\nabla f|^{\frac12}_{2},\quad |f|_6\leq C|\nabla f|_{2},\quad |f|_\infty\leq C|f|^{\frac14}_{2}|\nabla f|^{\frac34}_{6}, \end{split} \end{equation} (81)

    which have be used frequently in our previous proof.

    The second one can be found in Majda [17], and we omit its proof.

    Lemma 3.2. [17] Let positive constants r , a and b satisfy the relation

    \frac{1}{r} = \frac{1}{a}+\frac{1}{b}

    and 1\leq a,\ b, \ r\leq +\infty . \forall s\geq 1 , if f, g \in W^{s,a} (\mathbb{R}^3)\cap W^{s,b}(\mathbb{R}^3) , then we have

    \begin{equation} |D^s(fg)-f D^s g|_r\leq C_s\big(|\nabla f|_a |D^{s-1}g|_b+|D^s f|_b|g|_a\big), \end{equation} (82)
    \begin{equation} |D^s(fg)-f D^s g|_r\leq C_s\big(|\nabla f|_a |D^{s-1}g|_b+|D^s f|_a|g|_b\big), \end{equation} (83)

    where C_s> 0 is a constant depending only on s , and \nabla^s f means that the set of all elements of \partial^{\zeta} f with |\zeta| = s .

    The third one is on the regularity estimates for Lam \acute{ \text{e}} operator. For the elliptic problem

    \begin{equation} \begin{cases} -\alpha\Delta u-(\alpha+\beta)\nabla \text{div} u = f, \\[10pt] u\rightarrow 0 \quad \text{as} \ |x|\rightarrow +\infty, \end{cases} \end{equation} (84)

    one has

    Lemma 3.3. [21] If u\in D^{k,q} with 1< q< +\infty is a weak solution to the problem (84), then

    \begin{equation} |u|_{D^{k+2,q}} \leq C |f|_{D^{k,q}}, \end{equation} (85)

    where k is an integer and the constant C>0 depend on \alpha, \beta and q . Moreover, if u\in D^{k,q} is a weak solution to the following problem

    \begin{equation} -\Delta u = f,\quad u\to 0\quad \mathit{\text{as}} \ |x|\rightarrow +\infty, \end{equation} (86)

    then (85) holds and if f = \mathit{\text{div}} g , we also have

    \begin{equation} |u|_{D^{1,q}} \leq C |g|_{L^q}. \end{equation} (87)

    The proof can be obtained via the classical estimates from harmonic analysis, which can be found in [21] or [22]. We omit it here.

    Now we show that, via introducing new variables

    \begin{equation} \phi = \rho^{\frac{\gamma-1}{2}}, \; \psi = \nabla \log \rho = \frac{2}{\gamma-1}\nabla \phi/\phi, \end{equation} (88)

    the system (1) can be rewritten as

    \begin{equation} \begin{cases} \phi_t+\frac{\gamma-1}{2}\phi\text{div} u+u\cdot\nabla\phi = 0,\\[10pt] \psi_t+\nabla(u\cdot\psi)+\nabla\text{div} u = 0,\\[10pt] u_t+u\cdot\nabla u+2\theta\phi\nabla\phi+Lu = \psi\cdot Q(u). \end{cases} \end{equation} (89)

    Proof. First, from the momentum equation, one has

    \begin{equation*} \begin{split} \rho u_t+\rho u\cdot\nabla u+\nabla P-\rho\text{div}\big(\alpha(\nabla u+&\nabla u^{\top}) +\beta\text{div} u \mathbb{I}_3\big)\\ = &\nabla\rho\cdot [\alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u\mathbb{I}_3], \end{split} \end{equation*}

    where P = A\rho^{\gamma} , divide both side by \rho , one has

    \begin{equation*} \begin{split} u_t+u\cdot\nabla u+A\gamma\rho^{\gamma-2}\nabla\rho-\text{div}\big(\alpha(\nabla u&+\nabla u^{\top}) +\beta\text{div} u \mathbb{I}_3\big)\\ = &\frac{\nabla\rho}{\rho}\cdot [\alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u\mathbb{I}_3]. \end{split} \end{equation*}

    Denote

    \begin{align*} Lu& = -\text{div}\big(\alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u \mathbb{I}_3\big),\\ Q(u)& = \alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u\mathbb{I}_3, \quad \theta = \frac{A\gamma}{\gamma-1}, \end{align*}

    we have

    \begin{equation} u_t+u\cdot\nabla u+2\theta\phi\nabla\phi+Lu = \psi \cdot Q(u). \end{equation} (90)

    Second, for \psi = \nabla\log\rho , one has

    \begin{equation} \begin{split} \psi_t & = (\nabla\log\rho)_t = \nabla(\log\rho)_t = \nabla\Big(\frac{\rho_t}{\rho}\Big) = \nabla\Big(\frac{-\text{div}(\rho u)}{\rho}\Big)\\ & = \nabla\big(\frac{-\nabla\rho\cdot u-\rho\text{div} u}{\rho}\big) = -\nabla(\nabla\log\rho\cdot u+\text{div} u)\\ & = -\nabla\text{div} u-u\cdot\nabla(\nabla\log\rho)-\nabla\log\rho\cdot\nabla u^{\top}\\ & = -\nabla\text{div} u-u\cdot\nabla\psi-\psi\cdot\nabla u^{\top}\\ & = -\nabla\text{div} u-\nabla(u\cdot\psi). \end{split} \end{equation} (91)

    Third, for \phi = \rho^{\frac{\gamma-1}{2}} , one has

    \begin{equation} \begin{split} \phi_t & = (\rho^{\frac{\gamma-1}{2}})_t = \frac{\gamma-1}{2}\rho^{\frac{\gamma-3}{2}}\rho_t\\ & = \frac{\gamma-1}{2}\rho^{\frac{\gamma-1}{2}}\frac{\rho_t}{\rho} = \frac{\gamma-1}{2}\rho^{\frac{\gamma-1}{2}}\frac{-\text{div}(\rho u)}{\rho}\\ & = \frac{\gamma-1}{2}\phi\frac{-\rho\text{div} u-\nabla\rho\cdot u}{\rho}\\ & = -\frac{\gamma-1}{2}\phi\text{div} u-u\cdot\nabla \phi. \end{split} \end{equation} (92)

    Combing (90)-(92) together, we complete the proof of the transformation.

    The author sincerely appreciates Dr. Shengguo Zhu for his very helpful suggestions and discussions on the problem solved in this paper. The research of Y. Cao was supported in part by China Scholarship Council 201806230126 and National Natural Science Foundation of China under Grants 11571232.

    Conflict of Interest: The authors declare that they have no conflict of interest.



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