Keywords: Compressible Navier-Stokes

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HYPERBOLIC HEAT CONDUCTION

YUXI HU AND REINHARD RACKE

Abstract. In this paper, we investigate the system of compressible Navier-Stokes equations with hyperbolic heat conduction, i.e., replacing the Fourier’s law by Cattaneo’s law. First, by using Kawashima’s condition on general hyperbolic parabolic systems, we show that for small relaxation timeτ, global smooth solution exists for small initial data. Moreover, asτ goes to zero, we obtain the uniform convergence of solutions of the relaxed system to that of the classical compressible Navier-Stokes equations.

Keywords: Compressible Navier-Stokes; hyperbolic heat conduction; global solution; singular limit

AMS classification code: 35B25, 76N10

1. Introduction

The compressible Navier-Stokes equations with heat conducting in Rⁿ×[0,+∞) (n ≥1) can be written in the following form







∂_tρ+ div(ρu) = 0,

∂_t(ρu) + div(ρu⊗u) +∇p= divS,

∂_t(ρ(e+ ¹₂u²)) + div(ρu(e+ ¹₂u²) +up) + divq= div(uS),

(1.1) whereρ,u= (u₁, u₂,· · · , u_n),p,S,eandqrepresent fluid density, velocity, pressure, stress tensor, specific internal energy per unit mass and heat flux, respectively. The equations (1.1)₁, (1.1)₂ and (1.1)₃ are the consequence of conservation of mass, momentum and energy, respectively.

To complete the system (1.1), we need to impose constitutive assumptions on p, S, e and q. First, we assume the fluid to be a Newtonian fluid, that is,

S=µ(∇u+ (∇u)^T) +µ⁰∇divu, (1.2) whereµand µ⁰ are the coefficient of viscosity and the second coefficient of viscosity, respectively, satisfying

µ >0, µ⁰+ 2 nµ≥0.

The heat flux q is assumed to satisfy

τ ∂_tq+q+κ∇θ = 0, (1.3)

1

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which represents Cattaneo’s law (Maxwell’s law, . . . ) and gives rise to heat waves with finite propagation speed. Here,τ >0 is the constant relaxation time andκ >0 is the constant heat conductivity. Moreover, in this paper, we consider the general equations of state and assume that the pressure p = p(ρ, θ) and e = e(ρ, θ) are smooth functions of (ρ, θ) satisfying

ρ²e_ρ(ρ, θ) =p(ρ, θ)−θp_θ(ρ, θ), (1.4) where θ denotes the absolute temperature. In particular, the case of a polytropic gas p=Rρθ, e=c_vθ is included here.

We consider the Cauchy problem for the functions

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

(ρ(x,0), u(x,0), θ(x,0), q(x,0)) = (ρ₀, u₀, θ₀, q₀). (1.5) For the limit case τ = 0, the system (1.1)-(1.3) is exactly the system of classical compressible Navier-Stokes equations, in which the relation between the heat flux and the temperature is governed by Fourier’s law,

q=−κ∇θ. (1.6)

Because of its physical importance and mathematical challenging, the well-posed theory has been widely studied for the system (1.1), (1.2) combined with Fourier’s law (1.6), see [1, 2, 3, 4, 6, 8, 11, 12, 14, 15, 16, 17, 18, 21, 23]. In particular, the local existence and uniqueness of smooth solutions was established by Serrin [21] 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 [23], Cho and Jin [1] showed that smooth solutions must blow up in finite time if the initial data has a vacuum state. The existence of global non-vacuum smooth solutions for large data is a famous open problem in fluid dynamics.

Although Fourier’s law plays an important role in experimental and applied physics, it has the drawback of an inherent infinite propagation speed of signals.

Cattaneo’s law was among one of the physical laws describing the finite speed of heat conduction. It has been widely used in thermoelasticity which results in the second sound phenomenon, see [7, 19, 20] and the references cited therein. However, a rigorous mathematical theory for the compressible Navier-Stokes system has not been established with heat conduction described by Catteneo’s law. This is the aim of the presented paper. Note that it is not obvious that the results which hold for Fourier’s law also hold for Cattaneo’s law. Indeed and for example, Fern´andez Sare and Racke [5] showed that, for certain Timoshenko-type thermoelastic system, Fourier’s law preserve the property of exponential stability while Cattaneo’s law destroys such a property.

Moreover and naturally, it is interesting to study the relaxation limit τ → 0 for the system (1.1)-(1.5). As said above, for τ = 0 in (1.3), the system turns into

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the classical compressible Navier-Stokes equation. We will show this convergence rigorously and also obtain the convergence order with respect to τ.

Now, we introduce some notation. Denote by W^m,p(Ω), 0≤m≤ ∞, 1≤p≤ ∞, the usual Sobolev space with norm k · kW^m,p. For convenience, H^m(Ω) and L^p(Ω) stand for W^m,2(Ω) and W^0,p(Ω) with norms k · k_m and k · k_L^p, respectively. For p= 2, we denote the norm k · k_L² byk · k. Denote by G:=R⁺×Rⁿ×R⁺×Rⁿ the physical state space.

The outline of this paper is as follows. We first show a local existence theorem by transforming the system (1.1)-(1.5) into a symmetric hyperbolic-parabolic type system. Then, for small τ, we prove that the system satisfies the so-called Kawashima condition, and therefore we get a global solution for small initial data. Finally, we establish the convergence, as τ → 0, to the classical compressible Navier-Stokes system.

2. Local and global well-posedness

In this part, we consider the local and global well-posedness for the system (1.1)- (1.5). For this end, we need the following assumptions

• A.1. The initial data satisfy

{(ρ₀, u₀, θ₀, q₀)(x) :x∈Rⁿ} ⊂[ρ∗, ρ^∗]×[−C₁, C₁]ⁿ×[θ∗, θ^∗]×[−C₁, C₁]ⁿ :=G₀, where C₁ >0 as well as 0< ρ∗ <1< ρ^∗ <∞ and 0< θ∗ <1< θ^∗ <∞ are constants.

• A.2. For each given G1 satisfying G0 ⊂⊂ G1 ⊂⊂ G, ∀(ρ, u, θ, q) ∈ G1, the pressurep and the internal energye satisfy

p(ρ, θ), pθ(ρ, θ), pρ(ρ, θ), eθ(ρ, θ)> C(G1)>0, (2.1) where C(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 n ≥ 1 and 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, q0) are in H^s. Then, for each convex open subset G1 satisfying G0 ⊂⊂ G1 ⊂⊂ G, there exists T > 0 such that system (1.1)-(1.5) has unique classical solution (ρ^τ, u^τ, θ^τ, q^τ) satisfying

(ρ^τ −1, θ^τ −1, q^τ)∈C([0, T], H^s)∩C¹([0, T], H^s−1),

u^τ ∈C([0, T], H^s)∩C¹([0, T], H^s−2) (2.2) and

(ρ^τ, u^τ, θ^τ, q^τ)(x, t)∈G₁, ∀(x, t)∈Rⁿ×[0, T].

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Proof. We shall write the system (1.1) in symmetric form and use the classical theory of symmetric hyperbolic-parabolic system to prove Theorem 2.1.

By using equations (1.1)₁ and (1.2)-(1.4), the system (1.1) can be written as











∂_tρ+u∇ρ+ρdivu= 0,

ρ∂_tu+ρu∇u+p_θ∇θ+p_ρ∇ρ=µ4u+ (µ+µ⁰)∇divu,

ρe_θ∂_tθ+ρe_θu∇θ+θp_θdivu+ divq= ^µ₂|∇u+ (∇u)^T|²+µ⁰|divu|², τ ∂_tq+q+κ∇θ = 0,

(2.3)

with initial conditions

(ρ(x,0), u(x,0), θ(x,0), q(x,0)) = (ρ₀, u₀, θ₀, q₀). (2.4) Let ω:= (ρ, u, θ, q). Then we have

A⁰(ω)∂_tω+ ΣA^j(ω)∂_x_jω−ΣB^jk(ω)∂_x²_j_x_kω+L(ω)ω =g(ω, D_xω), (2.5) where Σ. . . always stands for Σⁿ_j=1, and where

A⁰(ω) =







c²

ρ 0 0 0

0 ρIn 0 0 0 0 ^ρe_θ^θ 0 0 0 0 _κθ^τ







,ΣA^jξ_j =







c²

ρu·ξ c²ξ 0 0

c²ξ^T ρ(u·ξ)I_n p_θξ^T 0 0 p_θξ ^ρe_θ^θu·ξ _θ^ξ

0 0 ^ξ_θ^T 0





 ,

ΣB^jkξ_jξ_k=







0 0 0 0

0 µI_n+ (µ+µ⁰)ξ^Tξ 0 0

0 0 0 0







, L(ω) =







0 0 0 0

0 0 0 _κθ¹ In





 ,

g(ω, D_xω) =







0 0

µ

2θ|∇u+ (∇u)^T|² +^µ_θ⁰|divu|² 0







, c² =p_ρ, ξ= (ξ₁, ξ₂,· · · , ξ_n)∈ Sⁿ⁻¹.

By assumption A.2, one can see that for all ω ∈ G₁ satisfying G₀ ⊂⊂ G₁ ⊂⊂ G, A⁰(ω) is a positive symmetric matrix and A^j(ω) is a symmetric matrix for each j, while ΣB^jk(ω)ξjξk is a semi-positive symmetric matrix. So, the local existence theorem follows from [13] (or see [10]) immediately.

Theorem 2.2. (Global existence) Let n ≥ 2 and s ≥ s₀ + 1, with s₀ ≥ [ⁿ₂] + 1, be integers. Suppose that 0< τ < _p ^2κ

θ(1,1)² and (ρ₀−1, u₀, θ₀−1, q₀)∈H^s. Then there exists a positive constants δ such that if k(ρ₀ −1, u₀, θ₀ −1, q₀)k_s ≤ δ, then there exists a global unique solution (ρ^τ, u^τ, θ^τ, q^τ) of system (1.1)-(1.5) satisfying

(ρ^τ−1, θ^τ−1, q^τ)∈C([0,∞), H^s)∩C¹([0,∞), H^s−1),

u^τ ∈C([0,∞), H^s)∩C¹([0,∞, H^s−2). (2.6)

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Proof. One can use the Kawashima condition to prove small global existence. Lin- earizing the above system around the steady state ¯ω= ( ¯ρ,u,¯ θ,¯ q) := (1,¯ 0,1,0), one has

A⁰(¯ω)∂_tω+ ΣA^j(¯ω)∂_x_jω−ΣB^jk(¯ω)∂_x²_j_x

kω+L(¯ω)ω = 0, (2.7) where

A⁰(¯ω) =







¯

c² 0 0 0 0 I_n 0 0 0 0 e¯_θ 0 0 0 0 ^τ_κ







,ΣA^j(¯ω)ξ_j =







0 ¯c²ξ 0 0

¯

c²ξ^T 0 p¯_θξ^T 0 0 p¯_θξ 0 ξ

0 0 ξ^T 0





 ,

ΣB^jk(¯ω)ξjξk =







0 0 0 0

0 µI_n+ (µ+µ⁰)ξ^Tξ 0 0

0 0 0 0







, L(¯ω) =







0 0 0 0

0 0 0 ¹_κI_n





 ,

¯

c=c(1,1), p¯_θ =p_θ(1,1), e¯_θ =e_θ(1,1), ξ = (ξ₁, ξ₂,· · · , ξ_n)∈ Sⁿ⁻¹. We choose K^j such that

ΣK^jξ_j =α







0 c¯²ξ 0 0

−ξ^T 0 0 0

0 0 0 ^κ_τξ

0 0 −^ξ_e^T

θ 0





 ,

where α >0 will be chosen later. Then, simple calculations imply

ΣK^jξ_jA⁰ =α







0 ¯c²ξ 0 0

−¯c²ξ^T 0 0 0

0 0 0 ξ

0 0 −ξ^T 0







and

ΣK^jA^kξ_jξ_k =α







¯

c⁴ 0 p¯_θ¯c² 0 0 −¯c²ξ^Tξ 0 0

0 0 ^κ_τ 0

0 −^p_¯_e^¯^θ

θξ^Tξ 0 −_e_¯¹

θξ^Tξ





 .

In order to satisfy Kawashima’s conditions from [13] (see also [22]) for proving the global existence for small data, we notice that there are no quadratic terms of the type (ρ−1, u, θ−1, q)², but essentially only of type |∇u|². Hence one only has to

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check that 1 2Σ

K^jA^kξ_jξ_k+ (K^jA^kξ_jξ_k)^T + ΣB^jkξ_jξ_k+L

=







α¯c⁴ 0 ¹₂αp¯_θ¯c² 0

0 µI_n+ ((µ+µ⁰)−α¯c²)ξ^Tξ 0 −α_2¯^p^¯_e^θ

θξ^Tξ

1

2αp¯_θ¯c² 0 α^κ_τ 0

0 −α_2¯^p^¯_e^θ

θξ^Tξ 0 _κ¹I_n−α_¯_e¹

θξ^Tξ







=:M

is a positive definite matrix for any ξ = (ξ₁, ξ₂,· · · , ξ_n) ∈ Sⁿ⁻¹. In fact, let η = (η₁, η₂, η₃, η₄)∈R⁸ where η₁, η₃ ∈R¹ and η₂, η₄ ∈R³. Then we have

ηM η^T = (η₁α¯c⁴+1

2η₃αp¯_θ¯c²)η₁+ (η₂N +η₄(−α p¯_θ

2¯e_θξ^Tξ))η₂+ (1

2η₁α¯p_θ¯c²+η₃ακ

τ)η₃+ (−η₂α p¯_θ

2¯e_θξ^Tξ+η₄Q)η₄, where N = µI_n+ ((µ+µ⁰)−α¯c²)ξ^Tξ and Q = _κ¹I_n−α_¯_e¹

θξ^Tξ. If 0 < τ < ^2κ_p_¯2 θ

, we know that

4:= ¯p²_θ−4κ τ <0.

So, for any α >0, we have

α¯c⁴η₁²+α¯c²p¯_θη₁η₃+ακ

τη₃² >0.

On the other hand, for small α, we know that the matrix N and Q are positive definite. So, we can choose α small enough such that

η₂N η₂−αη₄ p¯_θ 2¯eθ

ξ^Tξη₂+η₄Qη₄ >0.

Therefore, for sufficiently small α, we derive that ηM η^T > 0 for any η ∈R⁸ which implies that M is positive definite. This completes the proof.

3. Relaxation limit: convergence for τ →0

In this part, we show the uniform convergence of the relaxed system (τ > 0) to the classical compressible Navier-Stokes equations (τ = 0). To this end, we need the natural compatibility condition on the initial data, that is,

q₀ =−κ∇θ₀,

which be assumed in this section. Let G₁ be given satisfying G₀ ⊂⊂ G₁ ⊂⊂ G.

DefineT_τ = sup{T >0; (ρ^τ−1, v^τ, θ^τ−1, q^τ)∈C([0, T], H^s), (ρ^τ, v^τ, θ^τ, q^τ)(x, t)∈ G₁, ∀(x, t)∈Rⁿ×[0, T]}. Then we have the following theorem.

Theorem 3.1. Let(ρ, u, θ)be a smooth solution to the classical compressible Navier- Stokes equations 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)

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with T∗ >0 finite. Then there are positive constants τ₀ and C such that for τ ≤τ₀, k(ρ^τ, u^τ, θ^τ)(t,·)−(ρ, u, θ)(t,·)k_s≤Cτ (3.1) and

k(q^τ +κ∇θ)(t,·)k_s ≤Cτ¹² (3.2) for t ∈[0,min{T∗, T_τ}), where C does not depend on τ.

Theorem 3.1 in particular implies that T_τ is independent of τ.

Theorem 3.2. Under the condition of Theorem 3.1, for any G₁ satisfying

G₀∪G˜ ⊂⊂G₁ ⊂⊂G, (3.3)

where G˜ ={∪(ρ, u, θ,−κ∇θ)(x, t), (x, t)∈Rⁿ×[0, T∗]}, we have that Tτ > T∗ holds for τ > 0 sufficiently small.

Proof. The proof of Theorem 3.2 follows the arguments in [24]. For the reader’s convenience we sketch it here.

Suppose, there is aG₁satisfying (3.3) and a sequence{τ_k}k≥1such that lim

k→∞τ_k = 0 and T_τ_k =T_τ_k(G₁)≤T∗. Then there exists ˜G satisfying

∪_x,t{(ρ, u, θ,−κ∇θ),(x, t)∈Rⁿ×[0, T_∗]} ⊂⊂G˜ ⊂⊂G₁. (3.4) Moreover, from Sobolev’s imbedding theorem and Theorem 3.1, we deduce that

|(ρ^τ, u^τ, θ^τ, q^τ)−(ρ, u, θ,−κ∇θ)|

≤Ck(ρ^τ, u^τ, θ^τ, q^τ)−(ρ, u, θ,−κ∇θ)ks ≤Cτ¹².

Thus, there exists a k such that (ρ^τ, u^τ, θ^τ, q^τ)∈G˜ for all (x, t)∈Rⁿ×[0, T_τ_k). On the other hand, it follows from

k(ρ^τ −1, u^τ, θ^τ −1, q^τ)k_s

≤ k(ρ^τ, u^τ, θ^τ, q^τ)−(ρ, u, θ,−κ∇θ)k_s+k(ρ−1, u, θ−1,−κ∇θ)k_s

≤Cτ

1 2

k +k(ρ−1, u, θ−1,−κ∇θ)k_s

thatk(ρ^τ−1, u^τ, θ^τ−1, q^τ)k_s is uniformly bounded with respect tot∈[0, T_τ_k). Now, we could apply Theorem 2.1 at a timetless thanT_τ_k (k is fixed here) to continue the solution beyond Tτ_k(G1) which is a contradiction and the proof is completed.

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]. There- fore, we can establish a convergence results for any fixed T >0 for small data.

Remark 3.2. A global convergence theorem on [0,∞) can not be obtained from the above results. One would have to spend considerably more efforts to get this result (if at all).

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Proof of Theorem 3.1:

We introduce the variable q:=−κ∇θ and define ρ^d:= ρ^τ−ρ

τ , u^d= u^τ−u

τ , θ^d= θ^τ −θ

τ , q^d = q^τ−q

τ . (3.5)

We want to show that, for small τ and for t <min{T∗, T_τ}, k(ρ^d, u^d, θ^d)(t,·)k_s ≤C, k√

τ q^d(t,·)k_s≤C, (3.6) where C > 0 denotes constants not depending on τ or t. The equations for the difference (ρ^d, u^d, θ^d, q^d) can be written as

∂tρ^d+u^τ∇ρ^d+u^d∇ρ+ρ^τdivu^d+ρ^ddivu= 0, (3.7) ρ^τ∂_tu^d+ρ^d∂_tu+ρ^τu^τ∇u^d+p^τ_θ∇θ^d+p^τ_ρ∇ρ^d (3.8)

+ 1 τ

(ρ^τu^τ−ρu)∇u+ (p^τ_ρ−p_ρ)∇ρ+ (p^τ_θ −p_θ)∇θ =µ4u^d+ (µ+µ⁰)∇divu^d, ρ^τe^τ_θ∂_tθ^d+ρ^τe^τ_θu^τ∇θ^d+θ^τp^τ_θdivu^d+ divq^d (3.9)

−µ(∇u^τ + (∇u^τ)^T +∇u+ (∇u)^T)∇u^d−µ⁰(divu^τ + divu)divu^d

= 1

τ {(ρ^τe^τ_θ −ρe_θ)∂_tθ+ (ρ^τe^τ_θu^τ −ρe_θu)∇θ+ (θ^τp^τ_θ−θp_θ)divu}

τ ∂_tq^d+q^d+κ∇θ^d =−∂_tq. (3.10)

In order to use energy methods, we rewrite the above system as

∂_tρ^d+u^τ∇ρ^d+ρ^τdivu^d=−u^d∇ρ−ρ^ddivu=:f₁, (3.11)

∂_tu^d+u^τ∇u^d+ p^τ_θ

ρ^τ∇θ^d+ p^τ_ρ

ρ^τ∇ρ^d− 1

ρ^τ(µ4u^d+ (µ+µ⁰)∇divu^d)

=− 1

ρ^τρ^du_t− 1 τ ρ^τ

(ρ^τu^τ −ρu)∇u+ (p^τ_ρ−p_ρ)∇ρ+ (p^τ_θ −p_θ)∇θ =:f₂, (3.12)

∂_tθ^d+u^τ∇θ^d+θ^τp^τ_θ

ρ^τe^τ_θdivu^d+ 1

ρ^τe^τ_θdivq^d

= 1

τ ρ^τe^τ_θ {(ρ^τe^τ_θ −ρe_θ)∂_tθ+ (ρ^τe^τ_θu^τ −ρe_θu)∇θ+ (θ^τp^τ_θ −θp_θ)divu}

+ µ

ρ^τe^τ_θ(∇u^τ + (∇u^τ)^T +∇u+ (∇u)^T)∇u^d+ µ⁰

ρ^τe^τ_θ(divu^τ + divu)divu^d =:f₃, (3.13)

τ ∂_tq^d+q^d+κ∇θ^d =−∂_tq=:f₄. (3.14)

Define

E := sup

0≤t≤T

k(ρ, u, θ)k_H^s+3+ sup

0≤t≤T

kρ_tk_H^s+2+ sup

0≤t≤T

k(u_t, θ_t)k_H^s+1 (3.15)

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and

E^d:= sup

0≤t≤T

k(ρ^d, u^d, θ^d,√

τ q^d)k_H^s. (3.16) Note that

E ≤C (3.17)

and

k(ρ^τ, u^τ, θ^τ)k_s ≤C+τ E^d, kq^τk_s≤C+√

τ E^d. (3.18)

We need the following two lemmas to prove our theorem.

Lemma 3.3. For 0≤ |α| ≤s, we have the following estimates

k∇^αf₁k ≤CE^d, (3.19) k∇^αf₂k ≤C(E^d+τ(E^d)²), (3.20) k∇^αf3k ≤C(E^d+τ(E^d)²+τ²(E^d)³+k∇^α+1u^dk+τ E^dk∇^α+1u^dk). (3.21) Proof. By Sobolev’s imbedding theorem and Moser inequalities, using (3.17), we have

k∇^αf₁k=k∇^α(−u^d∇ρ−ρ^ddivuk

≤ k∇ρk_L^∞k∇^αu^dk+ku^dk_L^∞k∇^α+1ρk+kdivuk_L^∞k∇^αρ^dk+kρ^dk_L^∞k∇^α+1uk

≤CE^d.

Therefore, (3.19) holds.

Remember that both (ρ, u, θ) and (ρ^τ, u^τ, θ^τ, q^τ) take values in a convex compact subset of the state space, we have

∇^α(− 1 ρ^τρ^dut)

≤ ku_tk_L^∞

∇^α(ρ^d ρ^τ)

+

ρ^d ρ^τ L^∞

k∇^αu_tk

≤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−p_ρ)∇ρ+ (p^τ_θ −p_θ)∇θ

≤C

∇^α 1

ρ^τ(ρ^d+θ^d)(∇ρ+∇^θ)

≤C(E^d+τ(E^d)²).

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Thus, (3.20) holds.

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

∇^α 1

ρ^τe^τ_θ(ρ^τe^τ_θ −ρe_θ)θ_t

≤

∇^α 1

τ e^τ_θ(e^τ_θ −e_θ)

+

∇^α ρ^d

ρ^τe^τ_θe_θθ_t

≤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−ρe_θu)∇θ

≤C(E^d+τ(E^d)²).

Moreover, we have

∇^α µ

ρ^τe^τ_θ(∇u^τ + (∇u^τ)^T +∇u+∇u^T)∇u^d

≤C

∇^α µ

ρ^τe^τ_θ((∇u+τ∇u^d)∇u^d)

≤ k∇uk_L^∞k∇^α( µ

ρ^τe^τ_θ∇u^d)k+k µ

ρ^τe^τ_θ∇u^dk_L^∞k∇uk +τk µ

ρ^τe^τ_θkL^∞k∇^α(∇u^d:∇u^d)k+τk∇u^d :∇u^dkL^∞k∇^α( µ ρ^τe^τ_θ)k

≤C(E^d+τ(E^d)²+τ²(E^d)³ +k∇^α+1u^dk+τ E^dk∇^α+1u^dk).

Similarly, we get

∇^α µ⁰

ρ^τe^τ_θ(divu^τ + divu)divu^d

≤C(E^d+τ(E^d)²+τ²(E^d)³+k∇^α+1u^dk+τ E^dk∇^α+1u^dk).

So, (3.21) holds and this completes the proof.

Lemma 3.4. we have d

dt(E^d)² ≤C(1 + (E^d)²+τ(E^d)³+τ²(E^d)⁴). (3.22)

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Proof. Applying ∇^α to the equations (3.11)-(3.14) and multiplying the result by

p^τ_ρ

ρ^τ∇^αρ^d,ρ^τ∇^αu^d, ^ρ^τ_θ^eτ^τ^θ∇^αθ^d, _κθ¹τ∇^αq^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

κθ^τ(∇^αq^d)²

dx +

Z

µ(∇^α+1u^d)²+ (µ+µ⁰)(∇^αdivu^d)²+ 1

κθ^τ(∇^αq^d)²

dx

≤

4

X

i=1

(F_i+T_i) +

9

X

i=1

G_i +D, (3.23)

where F₁ =

Z

∇^αf₁p^τ_ρ

ρ^τ∇^αρ^ddx, F₂ = Z

∇^αf₂ρ^τ∇^αu^ddx, F₃ =

Z

∇^αf₃ρ^τe^τ_θ

θ^τ ∇^αθ^ddx, F₄ = Z

∇^αf₄∇^αq^d κθ^τ dx, 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, T₄ = Z

1 κθ^τ

t

(∇^αq^d)²dx, D=

Z

∇^α 1

ρ^τ(µ4u^d+ (µ+µ⁰)∇divu^d)

− 1

ρ^τ∇^α(µ4u^d+ (µ+µ⁰)∇divu^d)

dx, 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(u^τ∇θ^d)ρ^τe^τ_θ

θ^τ ∇^αθ^ddx, G₇ =

Z

∇^α θ^τp^τ_θ

ρ^τe^τ_θdivu^d ρ^τe^τ_θ

θ^τ ∇^αθ^ddx, G₈ =

Z

∇^α 1

ρ^τe^τ_θdivq^d ρ^τe^τ_θ

θ^τ ∇^αθ^ddx, G₉ = Z

∇^α(κ∇θ^d)∇^αq^d κθ^τ dx.

From Lemma 3.3, we know that

F_i ≤C((E^d)²+τ(E^d)³+τ²(E^d)⁴) +εk∇^α+1u^dk²,

C depending on ε, for each i = 1,2,3 and F₄ ≤ C+εk∇^αq^dk² (here we use that q_t=−κ∇θ_t and kθ_tk_s+1 ≤C). Now, we estimateG_i for each i.

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G₁ = Z

(∇^α(u^τ∇ρ^d)−u^τ∇^α+1ρ^d)p^τ_ρ

ρ^τ∇^αρ^d+u^τp^τ_ρ

ρ^τ ∇^α+1ρ^d∇^αρ^d

dx

≤C(k∇ρ^dk_L^∞k∇^αu^τk+k∇u^τk_L^∞k∇^αρ^dk)k∇^αρ^dk_s+k∇

u^τp^τ_ρ ρ^τ

k_L^∞k∇^αρ^dk²

≤C((E^d)²+τ(E^d)³).

G6 can be estimated the same way.

G₂+G₄

= Z

(∇^α(ρ^τdivu^d)−ρ^τdiv∇^αu^d)p^τ_ρ

ρ^τ∇^αρ^ddx +

Z

∇^α(p^τ_ρ

ρ^τ∇ρ^d)ρ^τ∇^αu^d−p^τ_ρ∇^α+1ρ^d∇^αu^d

dx

≤Ck(divu^d,∇ρ^d)k_L^∞(k∇^αρ^τk+k∇(p^τ_ρ

ρ^τ)k)(k∇^αρ^dk+k∇^αu^dk) +k(∇ρ^d,∇p^τ_ρ,p^τ_ρ

ρ^τ)k_L^∞k∇^αu^dkk∇^αρ^dk ≤C((E^d)²+τ(E^d)³).

G₅+G₇ and can also be estimated similarly, while G₈+G₉

= Z

∇^α 1

ρ^τe^τ_θdivq^d ρ^τe^τ_θ

θ^τ ∇^αθ^d− 1

θ^τ∇^αdivq^d∇^αθ^d− ∇(1

θ^τ)∇^αθ^d∇^αq^d

dx

≤C(kdivq^dk_L^∞k∇^α( 1

ρ^τe^τ_θ)kk∇^αθ^dk+k(∇( 1

ρ^τe^τ_θ),∇( 1

θ^τ))k_L^∞k∇^αq^dkk∇^αθ^dk)

≤εk∇^αq^dk²+C((E^d)²+τ(E^d)³ +τ²(E^d)⁴), G₃ ≤Ck∇^α(u^τ∇u^d)kk∇^αu^dk

≤C(ku^τk_L^∞k∇^α+1u^dk+k∇u^dk_L^∞k∇^αu^τk)k∇^αu^dk

≤εk∇^α+1u^dk²+C(ε)((E^d)²+τ(E^d)³) and

|T_i| ≤ k(ρ^τ_t, θ_t^τ)k_L^∞(E^d)² ≤C(1 +τk(ρ^d_t, θ_t^d)k_L^∞)(E^d)²

≤C(1 +τ(E^d+τ(E^d)²+kq^dk_s))(E^d)²

≤C((E^d)²+τ(E^d)³+τ²(E^d)⁴) +εkq^dk²_s.

By choosing ε sufficiently small and using assumption A.2, we conclude that (3.22)

holds and this completes the proof.

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Using Lemma 3.4, we can show that E^d is uniformly bounded. In fact, from Lemma 3.4, we derive that

d

dtg ≤C(1 +b g+τ²g²), (3.24) for some C >b 0, where g := (E^d)². We assume a priori that

g ≤e²^CT^b −1. (3.25)

We will show that g ≤ ¹₂(e²^CT^b −1) holds if we choose τ small enough such that τ² ≤ ¹

e²^CT^b −1. This justifies the a priori estimate (3.25) and thus proves our result.

In fact, if τ² ≤ ¹

e²^CT^b −1, we get g ≤ _τ¹2 and thus τ²g² ≤ g. Hence, the inequality (3.24) turns into

d

dtg ≤C(1 + 2g).b (3.26)

Solving the above inequality, we immediately get g ≤ ¹₂(e²^CT^b −1). This finishes the proof of the main Theorem 3.1.

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Yuxi Hu, LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.R.

China, yxhu86@163.com

Reinhard Racke, Department of Mathematics and Statistics, University of Konstanz, 78457 Kon- stanz, Germany, reinhard.racke@uni-konstanz.de