Introduction The classical compressible Navier-Stokes equations inRn,n= 2,3, are given by

MAXWELL’S LAW

YUXI HU AND REINHARD RACKE

Abstract. We investigate the compressible Navier-Stokes equations where the constitutive law for the stress tensor given by Maxwell’s law is revised to a system of relaxation equations for two parts of the tensor. The global well-posedness is proved as well as the compatibility with the classical compressible Navier-Stokes system in the sense that, for vanishing relaxation pa- rameters, the solutions to the Maxwell system are shown to converge to solutions of the classical system.

Keywords: compressible Navier-Stokes, Maxwell fluid, global existence, singular limit AMS classification code: 35B25, 76N10

1. Introduction

The classical compressible Navier-Stokes equations inRⁿ,n= 2,3, are given by







∂tρ+ div(ρu) = 0,

∂t(ρu) + div(ρu⊗u) +∇p= div(S),

∂t(ρ(e+¹₂u²)) + div(ρu(e+¹₂u²) +up)−κ4θ= div(uS),

(1.1)

with the constitutive law for a Newtonian fluid, S=µ(∇u+∇u^T −2

ndivu In) +λdivu In. (1.2) Here, ρ, u= (u1,· · ·, un), p, S, eand θ represent fluid density, velocity, pressure, stress tensor, specific internal energy per unit mass and temperature, respectively. In denotes the identity matrix inRⁿ. The equations are the consequence of conservation of mass, momentum and energy, respectively. κ, µ, λare positive constants.

Maxwell’s relaxation replaces (1.2) by the differential equation

τ ∂tS+S=µ(∇u+∇u^T− 2

ndivu In) +λdivu In, (1.3) with the relaxation parameter τ > 0. For τ → 0 we formally recover (1.2). For incompressible Navier-Stokes equations this relaxation has been discussed by Racke & Saal [20, 21] and Sch¨owe [23, 24] proving global well-posedness for small data and rigorously investigating the singular limit as τ→0.

A splitting of the tensor S was discussed by Yong [28] in the isentropic case leading to the following system with a revised Maxwell law, now for the non-isentropiccase, that we are going

1

to further investigate here:



















∂tρ+ div(ρu) = 0,

∂_t(ρu) + div(ρu⊗u) +∇p= div(S₁) +∇S₂,

∂t(ρ(e+¹₂u²)) + div(ρu(e+¹₂u²) +up)−κ4θ= div(u(S1+S2In)), τ₁∂_tS₁+S₁=µ(∇u+∇u^T −²_ndivuI_n),

τ2∂tS2+S2=λdivu,

(1.4)

whereS1is an×nsquare matrix and symmetric and traceless if it was initially, andS2is a scalar variable.

A similar revised Maxwell model was considered by Chakraborty & Sader [1] for a compressible viscoelastic fluid (isentropic case), where τ₁ counts for the shear relaxation time, and τ₂ counts for the compressional relaxation time. The importance of this model for describing high frequency limits is underlined together with the presentation of numerical experiments. The authors conclude that it provides a general formalism with which to characterize the fluid-structure interaction of nanoscale mechanical devices vibrating in simple liquids.

We consider the more complex non-isentropic case with general equations of state assuming that the pressurep=p(ρ, θ) ande=e(ρ, θ) are smooth functions of (ρ, θ) satisfying

ρ²eρ(ρ, θ) =p(ρ, θ)−θpθ(ρ, θ), (1.5) whereθdenotes the absolute temperature. In particular, the case of a polytropic gasp=Rρθ, e= cvθis included here.

We investigate the Cauchy problem for the functions

(ρ, u, θ) :Rⁿ×[0,+∞)→R⁺×Rⁿ×R⁺ with initial condition

(ρ(x,0), u(x,0), θ(x,0)) = (ρ0, u0, θ0). (1.6) In [28] a local existence result is presented exploiting a entropy dissipation structure found. Here we first present a local existence theorem in suggesting an explicit transformation to a symmetric- hyperbolic system. Moreover, we prove a global existence theorem for small data. The strategy follows our paper [7].

As second topic we consider the singular limitτ :=τ1=τ2→0, being more complex than the local in time singular limit studied in [28] for the isentropic case. For τ = 0, the relaxed system (1.4) turns into the classical Newtonian compressible Navier-Stokes system (1.1), (1.2). For the latter, because of its physical importance and mathematical challenges, the well-posedness has been widely studied, see [2, 3, 4, 5, 6, 8, 10, 11, 12, 14, 16, 17, 18, 22, 26]. In particular, the local existence and uniqueness of smooth solutions was established by Serrin [22] and Nash [18] for initial data far away from vacuum. Later, Matsumura and Nishida [16] got global smooth solutions for small initial data without vacuum. For large data, Xin [26], Cho and Jin [2] showed that smooth solutions must blow up in finite time if the initial data has a vacuum state.

We will show the convergence of solutions to the relaxed system (1.4) to the the classical system (1.1), (1.2) rigorously and also obtain the convergence order with respect toτ. The energy method is used extending [7, 28].

To summarize the main new contributions, we mention

• a first discussion of the non-isentropic compressible Navier-Stokes equations with revised Maxwell’s law,

• the proof of global well-posedness via finding appropriate symmetric structures,

• the description of the singular limit to the classical Newtonian case in terms of order of convergence in the relaxation parameterτ.

The paper is organized as follows. In Section 2 we prove the local well-posedness as well as a global existence result for small data for the Cauchy problem (1.4), (1.6). The singular limit as τ →0 is subject of Section 3, where a convergence result is proved. In the Appendix in Section 4, we provide Moser-type inequalities.

Finally, we introduce some notation. W^m,p =W^m,p(Rⁿ),0 ≤m ≤ ∞,1 ≤p ≤ ∞, denotes the usual Sobolev space with normk · km,p. For convenience,H^m andL^p stand forW^m,2(Ω) and W^0,p(Ω) with normsk · kmandk · kL^p, respectively. Forp= 2, we denote the normk · kL² byk · k.

2. Local and Global Well-Posedness

In this part, we prove the local and the global well-posedness for the Cauchy problem (1.4), (1.6). For this we need the following assumptions A.1 and A.2. As in [7] we try to transform the system with symmetrizers to finally be able to apply the results from Kawashima, see [13] or [25].

• A.1. The initial data satisfy

{(ρ₀, u₀, θ₀, S₁₀, S₂₀)(x) :x∈Rⁿ} ⊂[ρ_∗, ρ^∗]×[−C₁, C₁]ⁿ×[θ_∗, θ^∗]×[−C₁, C₁]^n×n×[−C₁, C₁]

=:G0,

whereC1>0 as well as 0< ρ_∗<1< ρ^∗<∞and 0< θ_∗<1< θ^∗<∞are constants.

• A.2. For each givenG1satisfyingG0⊂⊂G1⊂⊂G,∀(ρ, u, θ, S1, S2)∈G1, the pressure p and the internal energyesatisfy

p(ρ, θ), pθ(ρ, θ), pρ(ρ, θ), eθ(ρ, θ)> C(G1)>0, whereC(G1) is a positive constants depending on G1.

For the standard assumption A.2 see for example [9, 17].

Theorem 2.1. (Local existence) Let s ≥s0+ 1 with s0 ≥[ⁿ₂] + 1 be integers. Suppose that the Assumptions A.1 and A.2 hold and that the initial data (ρ0−1, u0, θ0−1, S10, S20) are in H^s. Then, for each convex open subset G1 satisfying G0⊂⊂G1 ⊂⊂G, there exists Tex>0 such that the system (1.4)has an unique classical solution (ρ, u, θ, S1, S2)satisfying

((ρ−1, u, S₁, S₂)∈C([0, T_ex], H^s)∩C¹([0, T_ex], H^s−1),

θ−1∈C([0, Tex], H^s)∩C¹([0, Tex], H^s−2) (2.1) and

(ρ, u, θ, S₁, S₂)(x, t)∈G₁, ∀(x, t)∈Rⁿ×[0, T_ex].

Proof. First we consider the three-dimensional case n = 3. Using (1.5), we rewrite the system (1.4) as



















∂tρ+u∇ρ+ρdivu= 0,

ρ∂_tu+ρu∇u+p_θ∇θ+p_ρ∇ρ= divS₁+∇S₂, ρeθ∂tθ+ρeθu∇θ+θpθdivu=κ4θ+ (S1+S2I3)∇u, τ₁∂_tS₁+S₁=µ(∇u+ (∇u)^T −²₃divuI₃),

τ2∂tS2+S2=λdivu.

(2.2)

Without loss of generality, we assumeS₁to take the following form:

S1=





a11 a12 a13

a12 a22 a23

a₁₃ a₂₃ −a11−a₂₂



. (2.3)

Letω= (ρ, u, a11, a12, a13, a22, a23, S2). Then, we have







A₀(ω)ω_t+

3

P

j=1

A_j(ω)∂_xjω+L(ω)ω=f₁(ω, θ,∇θ), ρeθ∂tθ−κ4θ=f2(ω, θ,∇ω,∇θ).

(2.4)

Here, f1(ω, θ,∇θ) = (0, pθ∇θ,0,0,0,0,0,0),f2(ω, θ,∇ω,∇θ) =S∇u−ρeθu∇θ−θpθdivuand A0(ω) = diag

pρ

ρ, ρ, ρ, ρ,3τ1

4µ,τ1

µ,τ1

µ,3τ1

4µ,τ1

µ,τ2

λ

, L(ω) = diag

0,0,0,0, 3 4µ,1

µ, 1 µ, 3

4µ, 1 µ,1

λ

,

3

X

j=1

Aj(ω)ξj =









p_ρ

ρuξ pρξ 01×5 0

p_ρξ^T ρuξI₃ C_3×5(ξ) −ξ^T 0_5×1 D_5×3(ξ) 0_5×5 0

0 −ξ 0_1×5 0







 ,

where

C_3×5(ξ) =





−ξ1 −ξ2 −ξ3 0 0

0 −ξ₁ 0 −ξ₂ −ξ₃

ξ₃ 0 ξ₃−ξ₁ 0 −ξ₂



, D_5×3(ξ) =













−ξ1 ξ2

2 ξ3

2

−ξ2 −ξ1 0

−ξ₃ 0 −ξ₁

ξ₁

2 −ξ2 ξ₃ 2

0 −ξ3 −ξ2













for eachξ∈S³.

Note that the matrix

3

P

j=1

Ajξj is not symmetric. Therefore, the theory of symmetric hy- perbolic parabolic system does not apply directly. Fortunately, we can perform a transforma- tion to overcome this problem. Let b11 := ^a11+a₂ ²², b22 := ^a11−a₂ ²². This particularly implies a11=b11+b22, a22=b11−b22. Let ˜ω:= (ρ, u, b11, a12, a13, b22, a23, S2). Then system (2.4) can be rewritten as







A˜₀(˜ω)˜ω_t+

3

P

j=1

A˜_j(˜ω)∂_xjω˜+ ˜L(˜ω)˜ω=f₁(˜ω, θ,∇θ), ρeθ∂tθ−κ4θ=f2(˜ω,∇˜ω, θ,∇θ).

(2.5)

Here, f1(˜ω, θ,∇θ) = (0, pθ∇θ,0,0,0,0,0,0),f2(,∇ω, θ,˜ ∇θ) =S∇u−ρeθu∇θ−θpθdivuand A˜₀(˜ω) = diag

p_ρ

ρ, ρ, ρ, ρ,3τ₁ µ ,τ₁

µ,τ₁ µ,τ₁

µ,τ₁ µ,τ₂

λ

,L(˜˜ ω) = diag

0,0,0,0, 3 µ,1

µ,1 µ,1

µ, 1 µ,1

λ

and

3

X

j=1

A˜_j(˜ω)ξ_j =









pρ

ρuξ pρξ 0 0

pρξ^T ρuξI3 C˜_3×5(ξ) −ξ^T

0 D˜_5×3(ξ) 0 0

0 −ξ 0 0







 ,

where

C˜_3×5(ξ) =





−ξ1 −ξ2 −ξ3 −ξ1 0

−ξ2 −ξ1 0 ξ2 −ξ3

2ξ3 0 −ξ1 0 −ξ2



,D˜_5×3(ξ) =













−ξ1 −ξ2 2ξ3

−ξ2 −ξ1 0

−ξ3 0 −ξ1

−ξ1 ξ2 0 0 −ξ3 −ξ2











 .

Note that ˜C_3×5(ξ) = ˜D^T_5×3(ξ) for each ξ ∈ S³. Therefore, the system (2.5) is a symmetric hyperbolic parabolic system and the local existence theorem follows, see [15, 13, 19].

In the two-dimensional case n= 2, we only remark that one can easily check that the system can be written in a symmetric form immediately. This is different from the 3-d case, for which we needed further transformations to get a system in a symmetric form.

Remark 2.1. In the isentropic case, Yong [28] proved a local existence theorem by checking that the system satisfies an entropy dissipation condition. A global existence theorem is not proved. In contrast to[28], our method is to write out the corresponding system explicitly for each component, see (2.4)and to try to find a symmetrizer explicitly, see (2.5). This methods allow us to deal with the non-isentropic case and more importantly, to get the global solutions by checking the so called Kawashima condition, see Theorem 2.2 below.

Theorem 2.2. (Global existence) Let s ≥ s₀+ 1 with s₀ ≥ [ⁿ₂] + 1 be integers. Suppose that the initial data satisfy (ρ0−1, u0, θ0−1, S10, S20)∈H^s. Then there exists a positive constant δ such that ifk(ρ0−1, u0, θ0−1, S10, S20)ks≤δ, there exists a global unique solution(ρ, u, θ, S1, S2) satisfying

((ρ−1, u, S1, S2)∈C([0,∞), H^s)∩C¹([0,∞), H^s−1),

(θ−1)∈C([0,∞), H^s)∩C¹([0,∞, H^s−2). (2.6) Proof. Again the interesting case is the tree-dimensional casen= 3.

LetU = (ρ, u, θ, b11, a12, a13, b22, a23, S2). Linearizing the system (2.5) around the steady state U¯ = ( ¯ρ,u,¯ θ,¯¯b11,¯a12,a¯13,¯b22,¯a23,S¯2) = (1,0,1,0,0,0,0,0,0), one gets

B₀( ¯U)∂_tU+

3

X

j=1

B_j( ¯U)∂_xjU+

3

X

j=1 3

X

k=1

D_jk( ¯U)∂_xjxkU+L( ¯U)U = 0. (2.7)

Here,B₀( ¯U) = diagn

¯

p_ρ,1,1,1,e¯_θ,^3τ_µ¹,^τ_µ¹,^τ_µ¹,^τ_µ¹,^τ_µ¹,^τ_λ²o

,L( ¯U) = diagn

0,0,0,0,0,_µ³,¹_µ,_µ¹,¹_µ,_µ¹,¹_λo ,

3

P

j=1 3

P

k=1

Djk( ¯U)ξjξk= diag{0,0,0,0, κ,0,0,0,0,0,0} and

3

X

j=1

Bj( ¯U)ξj=













0 p¯ρξ 0 0_1×5 0

¯

pρξ^T 0_3×3 p¯θξ^T A_3×5(ξ) ξ^T

0 p¯θξ 0 0_1×5 0

0_5×1 A^T_3×5(ξ) 0 0_5×5 0

0 ξ 0 0_1×5 0











 ,

where ¯p_ρ:=p_ρ(1,1),e¯_θ:=e_θ(1,1) and A_3×5(ξ) =





−ξ1 −ξ2 −ξ3 −ξ1 0

−ξ2 −ξ1 0 ξ2 −ξ3

2ξ3 0 −ξ1 0 −ξ2



. Define

3

X

j=1

Kjξj =α













0 ¯c²ξ 0 0 0

−ξ^T 0 0 P3×5 0

0 0 0 0 0

0 −(P M)^T_5×3 0 0 0

0 0 0 0 0













(2.8)

where ¯c :=√

¯

pρ and M = diagn

3τ

µ,^τ_µ,^τ_µ,^τ_µ,^τ_µo

. The positive parameterαand the matrixP_3×5 will be determined later. A simple calculation gives

3

X

j=1

KjξjA0=α













0 ¯c²ξ 0 0 0

−¯c²ξ^T 0 0 P M 0

0 0 0 0 0

0 −(P M)^T 0 0 0

0 0 0 0 0













, (2.9)

which is an anti-symmetric square matrix. On the other hand, we have

Q:= 1 2

3

X

j=1 3

X

k=1

K_jξ_jA_kξ_k+ (K_jξ_jA_kξ_k)^T +

3

X

j=1 3

X

k=1

D_jkξ_jξ_k+L

=α











¯

c⁴ 0 ^¯c₂²p_θ ^c¯₂²ξ(A−P M) ^¯c₂²

0 ¹₂(P A^T+AP^T)−¯c²ξ^Tξ 0 0 0

¯ c2

2pθ 0 ^κ_α −^pθ₂ξP M 0

¯ c2

2(A^T−(P M)^T)ξ^T 0 −^pθ₂(P M)^Tξ^T _α¹J−¹₂((P M)^TA+A^TP M)−¹₂(P M)^Tξ^T

¯ c2

2 0 0 −¹₂ξ(P M) _αλ¹









 ,

whereJ = diagn

3

µ,_µ¹,_µ¹,_µ¹,_µ¹o

. We need to show that the matrixQis a symmetric positive definite matrix in order to explore the theory of symmetric hyperbolic parabolic system, see [13, 25]. Let η= (η₁, η₂, η₃, η₄, η₅) whereη₁, η₃, η₅∈R¹ andη₃∈R³, η₄∈R⁵. Then we have

ηQη^T =

¯ c⁴η₁+1

2¯c²p_θη₃+1

2¯c²η₄(A^Tξ^T−(P M)^Tξ^T) +1 2¯c²η₅

η₁ +

η₂(1

2(P A^T +AP^T)−c¯²ξ^Tξ)

η^T₂ +

1

2¯c²p_θη₁+ 1

ακη₃−pθ

2η₄(P M)^Tξ^T

η₃ +

1

2¯c²η1(ξA−ξP M)−pθ

2η3ξP M +η4

1 αJ−1

2((P M)^TA+A^TP M)

−1 2η5ξP M

η₄^T +

1

2¯c²η1−1

2η4(P M)^Tξ^T + 1 αλη5

η5

= ¯c⁴η²₁+ κ αη²₃+η4

1 αJ−1

2((P M)^TA+A^TP M)

η^T₄ + 1 αλη₅²

+ ¯c²p_θη₁η₃+ ¯c²η₄(A^Tξ^T −(P M)^Tξ^T)η₁+ ¯c²η₁η₅−p_θη₄(P M)^Tξ^Tη₃−η₅ξP M η₄^T +η2

1

2(P A^T +AP^T)−¯c²ξ^Tξ

η^T₂.

From the above formula, by choosingαsufficiently small, we see that the positive definiteness of Q is equivalent to the positive definiteness of ¹₂(P A^T +AP^T)−¯c²ξ^Tξ. Therefore, our aim is to choose P such that ¹₂(P A^T+AP^T)−¯c²ξ^Tξ is a positive definite matrix for eachξ∈S³. Let

P= ¯c²





−ξ1 −ξ2 −ξ3 −ξ1 0

−ξ2 −ξ1 0 ξ2 −ξ3

ξ₃ 0 −ξ1 0 −ξ2



.

One easily calculates 1

2(P A^T +AP^T)−c¯²ξ^Tξ= ¯c²





1 0 −³₂ξ1ξ3

0 1 −³₂ξ2ξ3

−³₂ξ1ξ3 −³₂ξ2ξ3 1



.

Note that the first and second leading principal minors of the above matrix is 1. The third leading principal minors is

1−9

4ξ²₂ξ₃²−9

4ξ²₁ξ²₃= 1−9

4ξ₃²(1−ξ²₃).

Define a functionf(x) = 1−⁹₄x(1−x),0≤x≤1. It is not difficult to see that min0≤x≤1f(x) = f ¹₂

= ₁₆⁷ >0. Therefore, the matrix ¹₂(P A^T +AP^T)−¯c²ξ^Tξis is a positive definite matrix for eachξ∈S³. So, Kawashima’s condition follows and the proof is completed.

Remark 2.2. We note that a smallness condition on the L^p-norm of the initial data is not nec- essary since there are no quadratic terms of the type |(ρ−1, u, θ−1, S1, S2)|² in our system, see [13]. In fact, one can see that in our system (2.2)the nonlinear terms appear in the formU· ∇U.

We also have that the conditionsn≥3 ands≥s₀+ 2 there are changed inton≥2ands≥s₀+ 1 here since there are no quadratic terms of this type.

Remark 2.3. Kawashima’s results also imply decay properties of the solutions, that is, k(ρ−1, u, θ−1, S1, S2)k_s−(s0+1)→0, as t→ ∞.

Moreover, for n= 3, if we further assumes≥s0+ 2 and k(ρ−1, u, θ−1, S1, S2)kL^p ≤δ where p∈[1,³₂], then the solutions have the following decay

k(ρ−1, u, θ−1, S1, S2)ks−1≤C(1 +t)^{−32(p1−12)}k(ρ0−1, u0, θ0−1, S10, S20)ks−1,p, where the constant C is neither depending ont nor on the data.

3. Convergence Results

In this part, we show the compatibility of the revised Maxwell law with the Newtonian law.

This has been done for a similar singular limit in the isentropic case in [28], and for a singular limit for compressible Navier-Stokes equations with hyperbolic heat conduction in [7]. There and here, the energy method combined with sophisticated estimates of the nonlinear terms is used.

For simplicity, we assumeτ₁ =τ₂ ≡τ. We shall show the uniform convergence of the system (1.4) to the classical compressible Navier-Stokes system asτ go to zero. To this end, we need the following natural compatibility condition on the initial data, that is we assume

S10=µ(∇u0+ (∇u0)^T −2

ndivu0In), S20=λdivu0. (3.1) Denote by (ρ^τ, u^τ, θ^τ, S₁^τ, S₂^τ) the solutions given by Theorem 2.1 withG₁satisfyingG₀⊂⊂G₁⊂⊂

G. Denote

Tτ= sup{T >0,(ρ^τ−1, u^τ, θ^τ−1, S₁^τ, S₂^τ)∈C([0, T], H^s),(ρ^τ, u^τ, θ^τ, S₁^τ, S₂^τ)∈G1}. Then we have the following theorem.

Theorem 3.1. Let(ρ, u, θ)be a smooth solution to the classical compressible Navier-Stokes equa- tions with (ρ(x,0), u(x,0), θ(x,0)) = (ρ0, u0, θ0)satisfying

ρ∈C([0, T∗], H^s+3)∩C¹([0, T∗], H^s+2),(u, θ)∈C([0, T∗], H^s+3)∩C¹([0, T∗], H^s+1) with T_∗>0 (finite). Then there are positive constantsτ₀ andC such that forτ ≤τ₀,

k(ρ^τ, u^τ, θ^τ)(t,·)−(ρ, u, θ)(t,·)ks≤Cτ (3.2)

and

kS^τ₁(t,·)−µ

∇u+ (∇u)^T − 2 ndivuI_n

(t,·)k_s≤Cτ¹², kS₂^τ(t,·)−λdivu(t,·)k_s≤Cτ¹² (3.3) fort∈[0,min{T∗, Tτ}), whereC does not depend on τ.

Theorem 3.1 in particular implies thatT_τ is independent ofτ, see [7, 27, 28].

Theorem 3.2. Under the condition of Theorem 3.1, for any G₁ satisfying G0∪G˜0⊂⊂G1⊂⊂G,

whereG˜₀=

∪(ρ, u, θ, µ(∇u+ (∇u)^T−_n²divuI_n), λdivu)(x, t), (x, t)∈Rⁿ×[0, T_∗] , we have that T_τ > T_∗ holds forτ >0sufficiently small.

Remark 3.1. We note that if the initial data are sufficiently small, there exists a global solution for classical compressible Navier-Stokes equations, see [16]. Therefore, we can establish a convergence results for any fixed interval [0, Tex]for small data.

Proof. (of Theorem 3.1) We introduce the variablesS₁⁰:=µ(∇u+∇u^T −_n²divuIn), S⁰₂ :=λdivu and define

ρ^d:= ρ^τ−ρ

τ , u^d:= u^τ−u

τ , θ^d:= θ^τ−θ

τ , S₁^d:= S₁^τ−S₁⁰

τ , S^d₂ :=S₂^τ−S₂⁰

τ . (3.4)

Our aim is to show that, for smallτ and fort <min{T_∗, T_τ}, k(ρ^d, u^d, θ^d)(t,·)ks≤C, k√

τ(S₁^d, S₂^d)(t,·)ks≤C, (3.5) whereC >0 denotes constants not depending onτort. The equations for the difference variables (ρ^d, u^d, θ^d, S₁^d, S₂^d) can be written as



























∂tρ^d+u^τ∇ρ^d+ρ^τdivu^d=−u^d∇ρ−ρ^ddivu=:f1,

∂_tu^d+u^τ∇u^d+^p_ρ^τθ_τ∇θ^d+^p

τ ρ

ρ^τ∇ρ^d−_ρ¹_τ(div(S₁^d) +∇S₂^d)

=−_ρ¹τρ^du_t−_{τ ρ}¹τ

(ρ^τu^τ−ρu)∇u+ (p^τ_ρ−p_ρ)∇ρ+ (p^τ_θ−p_θ)∇θ =:f₂,

∂tθ^d+u^τ∇θ^d+^θ_ρ^ττ^pe^τ_θ^τθdivu^d−_ρτ¹e^τ_θκ4θ^d−_ρτ¹e^τ_θ(S₁^τ∇u^d+S₁^d∇u+S₂^τdivu^d+S₂^ddivu)

= _{τ ρ}¹τe^τ_θ {(ρ^τe^τ_θ−ρeθ)∂tθ+ (ρ^τe^τ_θu^τ−ρeθu)∇θ+ (θ^τp^τ_θ−θpθ)divu}=:f3, τ ∂tS₁^d+S₁^d−µ(∇u^d+ (∇u^d)^T −_n²divu^dIn) =−∂tS⁰₁=:f4,

τ ∂tS₂^d+S₂^d−λdivu^d=−∂tS⁰₂=:f5.

(3.6)

Here, we note that the expression for f3 is different from that in our previous paper [7],f3 there does not include the term“_ρτ¹e^τ_θ(S₁^τ∇u^d+· · ·)”. This is due to the fact that the velocity u^d in our case does not have enough dissipation compared to that in [7] and the term k∇^s+1u^dk can not be controlled. Instead, we shall use the dissipation ofθ to control such terms in the following estimates. Now we define

E:= sup

0≤t≤T

k(ρ, u, θ)ks+3+ sup

0≤t≤T

kρtks+2+ sup

0≤t≤T

k(ut, θt)ks+1

and

E^d:= sup

0≤t≤T

k(ρ^d, u^d, θ^d,√ τ S^d₁,√

τ S₂^d)ks. Note that

E≤C (3.7)

and

k(ρ^τ, u^τ, θ^τ)ks≤C+τ E^d,k(S₁^τ, S₂^τ)ks≤C+√

τ E^d. (3.8)

Here and in the sequel we often use the Moser-type inequalities from Lemma 4.1 in the Appendix.

We also need the following two lemmas in order to continue the proof of Theorem 3.1. We will use the letterC to denote various positive constants.

Lemma 3.3. For0≤ |α| ≤s, we have the following estimates

k∇^αf1k ≤CE^d,k∇^αf2k ≤C(E^d+τ(E^d)²),k∇^αf3k ≤C(E^d+τ(E^d)²). (3.9) Proof. The proof of Lemma 3.3 can be found in our previous paper [7], we recall it here for completeness. First, by Sobolev’s imbedding theorem and the Moser-type inequalities, using (3.7), we have

k∇^αf₁k=k∇^α(−u^d∇ρ−ρ^ddivuk

≤ k∇ρkL^∞k∇^αu^dk+ku^dkL^∞k∇^α+1ρk+kdivukL^∞k∇^αρ^dk+kρ^dkL^∞k∇^α+1uk ≤CE^d. Remember that both (ρ, u, θ) and (ρ^τ, u^τ, θ^τ, S₁^τ, S₂^τ) take values in a convex compact subset of the state space, we have

∇^α(−1 ρ^τρ^dut)

≤ kutkL^∞

∇^α(ρ^d ρ^τ)

+

ρ^d ρ^τ _L∞

k∇^αutk

≤CE^d+Ckρ^dkL^∞k∇^αρ^τk ≤C(E^d+τ(E^d)²).

Similarly, we have

∇^α 1

τ ρ^τ(ρ^τu^τ−ρu)∇u

≤C(E^d+τ(E^d)²).

Recalling that p(ρ, θ) is a smooth function of (ρ, θ) and using the mean value theorem, we obtain

∇^α 1

τ ρ^τ (p^τ_ρ−pρ)∇ρ+ (p^τ_θ−pθ)∇θ

≤C

∇^α 1

ρ^τ(ρ^d+θ^d)(∇ρ+∇θ)

≤C(E^d+τ(E^d)²).

By assumption A.2 and using the mean value theorem, we have

∇^α 1

τ ρ^τe^τ_θ(ρ^τe^τ_θ−ρe_θ)θ_t

≤

∇^α 1

τ e^τ_θ(e^τ_θ−eθ)

+

∇^α ρ^d

ρ^τe^τ_θeθθt

≤C(E^d+τ(E^d)²), where we used the fact that

k∇^α(ρ^τe^τ_θ)k ≤ kρ^τkL^∞k∇^αe^τ_θk+ke^τ_θkL^∞k∇ρ^τk ≤C+τ E^d. Similarly, we get

∇^α 1

τ ρ^τe^τ_θ(θ^τp^τ_θ−θpθ)divu

≤C(E^d+τ(E^d)²) and

∇^α 1

τ ρ^τe^τ_θ(ρ^τe^τ_θu^τ−ρeθu)∇θ

≤C(E^d+τ(E^d)²).

This completes the proof of Lemma 3.3.

Lemma 3.4. We have for τ≤1 that d

dt(E^d)²≤C(1 + (E^d)²+τ(E^d)⁴). (3.10) Proof. Applying∇^αto the equations (3.6) and multiplying the result by ^p

τ ρ

ρ^τ∇^αρ^d,ρ^τ∇^αu^d,^ρτ_θ^eτ^τθ∇^αθ^d,

1

2µ∇^αS₁^d, _λ¹∇^αS₂^d, respectively, we get 1

2 d dt

Z p^τ_ρ

ρ^τ(∇^αρ^d)²+ρ^τ(∇^αu^d)²+ρ^τe^τ_θ

θ^τ (∇^αθ^d)²+ τ

2µ(∇^αS₁^d)²+τ

λ(∇^αS₂^d)²

dx +

Z κ

θ^τ(∇^α+1θ^d)²+ 1

2λ(∇^αS₁^d)²+ 1

µ(∇^αS^d₂)²

dx

≤

5

X

i=1

Fi+

3

X

i=1

Ti+

10

X

i=1

Gi+D+N, (3.11)

where F₁=

Z

∇^αf₁p^τ_ρ

ρ^τ∇^αρ^ddx, F₂= Z

∇^αf₂ρ^τ∇^αu^ddx, F3=

Z

∇^αf3

ρ^τe^τ_θ

θ^τ ∇^αθ^ddx, F4= Z

∇^αf4∇^αS₁^ddx, F5= Z

∇^αf5∇^αS₂^ddx T₁=

Z p^τ_ρ ρ^τ

t

(∇^αρ^d)²dx, T₂= Z

ρ^τ_t(∇^αu^d)²dx, T₃=

Z ρ^τe^τ_θ θ^τ

t

(∇^αθ^d)²dx, D=

Z

∇^α( κ

ρ^τe^τ_θ4θ^d)− κ

ρ^τe^τ_θ∇^α(4θ^d) ρ^τe^τ_θ

θ^τ ∇^αθ^d+∇(κ

θ^τ)∇^α+1θ^d∇^αθ^d

dx, N=

Z

∇^α 1

ρ^τe^τ_θ(S₁^τ∇u^d+S₁^d∇u+S₂^τdivu^d+S₁^ddivu)

∇^αθdx, G₁=

Z

∇^α(u^τ∇ρ^d)p^τ_ρ

ρ^τ∇^αρ^ddx, G₂= Z

∇^α(ρ^τdivu^d)p^τ_ρ

ρ^τ∇^αρ^ddx, G3=

Z

∇^α(u^τ∇u^d)ρ^τ∇^αu^ddx, G4= Z

∇^α(p^τ_ρ

ρ^τ∇ρ^d)ρ^τ∇^αu^ddx, G5=

Z

∇^α(p^τ_θ

ρ^τ∇θ^d)ρ^τ∇^αu^ddx, G6= Z

∇^α 1

ρ^τ(divS₁^d+∇S^d₂)

ρ^τ∇^αu^ddx, G7=

Z

∇^α(u^τ∇θ^d)ρ^τe^τ_θ

θ^τ ∇^αθ^ddx, G8= Z

∇^α θ^τp^τ_θ

ρ^τe^τ_θdivu^d ρ^τe^τ_θ

θ^τ ∇^αθ^ddx, G9=

Z 1 2∇^α

∇u^d+ (∇u^d)^T −2

ndivu^dIn

∇^αS₁^ddx, G10= Z

∇^α(divu^d)∇^αS₂^ddx.

In the sequel, we keep in mind that some inequalities such as the Cauchy-Schwarz inequality, the H¨older inequality or the Moser-type inequalities will be frequently used without being mentioned explicitly (for exemplarily detailed estimates see (3.12) - (3.16) below). From Lemma 3.3 we know that

F_i≤C((E^d)²+τ(E^d)³),

for each i= 1,2,3 and F4+F5 ≤C(ε) +ε(k∇^αS₁^dk²+k∇^αS₂^dk²) (for ε >0, with C(ε) at most depending on ε; here we use the fact that ∂t(S₁⁰) =µ(∇ut+ (∇ut)^T −²₃divutI), ∂tS₂⁰ =λdivut

andkutks+1≤C). Moreover, we have

|D| ≤

∇^α κ

ρ^τe^τ_θ4θ^d

− κ

ρ^τe^τ_θ∇^α(4θ^d)

ρ^τe^τ_θ θ^τ ∇^αθ^d

+

∇κ

θ^τ

L^∞

k∇^α+1θ^dkk∇^αθ^dk

≤C

∇ κ

ρ^τe^τ_θ

_L∞

k∇^α−1∆θ^dk+k∆θ^dkL^∞

∇^α κ

ρ^τe^τ_θ

ρ^τe^τ_θ θ^τ ∇^αθ^d

(Moser inequalities Lemma 4.1 (ii)) +

εk∇^α+1θ^dk²+C(ε)

∇κ

θ^τ

2 L^∞

k∇^αθ^dk²

≡A+ [B] (3.12)

with

B≤εk∇^α+1θ^dk²+C(ε) (E^d)²+τ(E^d)³+τ²(E^d)⁴

(3.13) and

A ≤

∇ κ

ρ^τe^τ_θ

_L∞

k∇^α+1θ^dk

ρ^τe^τ_θ θ^τ ∇^αθ^d

+k∆θ^dkL^∞

∇^α κ

ρ^τe^τ_θ

ρ^τe^τ_θ θ^τ ∇^αθ^d

≡ A1+A2. (3.14)

We have

A1≤εk∇^α+1θ^dk²+C(ε) (E^d)²+τ(E^d)³+τ²(E^d)⁴

(3.15) and

A₂≤C(E^d)²

∇^α κ

ρ^τe^τ_θ

≤C(E^d)²(1 +τ E^d), (3.16) where we used

∇^α 1

ρ^τ

≤C 1 kρ^τk²_L∞

k∇^αρ^τk ≤Ckρ^τks

which follows from the Moser-type inequalities Lemma 4.1 (i). Summarizing (3.12) - (3.16) we have

|D| ≤εk∇^α+1θ^dk²+C(ε) (E^d)²+τ(E^d)³+τ²(E^d)⁴

. (3.17)

The term N can be divided into two terms:

N = Z

∇^α 1

ρ^τe^τ_θ(S^τ₁∇u^d+S₁^d∇u+S^τ₂divu^d+S₁^ddivu)

∇^αθdx

= Z

∇^α 1

ρ^τe^τ_θ(S^τ₁∇u^d+S₂^τdivu^d)

∇^αθdx+ Z

∇^α 1

ρ^τe^τ_θ(S^d₁∇u+S₂^ddivu)

∇^αθdx

=:N1+N2.

We estimate the termN₁ as follows: forα= 0, we have

|N1|=

Z 1

ρ^τe^τ_θ S₁^τ∇u^d+S₂^τdivu^d θ^ddx

≤ k(ρ^τ, θ^τ, S₁^τ, S₂^τ)kL^∞k∇u^dkkθ^dk ≤C((E^d)²+√

τ(E^d)³);