Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum

In this paper we consider the non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. Since the strong nonlinearity and degeneracies of the equations due to the temperature equation and vanishing of density, there are a few results about global existence of classical solution to this model. In the paper, we obtain a global classical solution to the equations with large initial data and vacuum. Moreover, we get the uniqueness of the solution to this system without vacuum. The analysis is based on the assumption \begin{document}$ \kappa(\theta) = O(1+\theta^q) $\end{document} where \begin{document}$ q\geq0 $\end{document} and delicate energy estimates.

1. Introduction. In this paper, we consider non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid with vacuum, being in a thermodynamical sense perfect and polytropic. This model can describe many phenomena appeared in a large number of complex fluids such as the suspensions, animal blood, liquid crystals which can not be characterized appropriately by the Navier-Stokes system. Mathematically, the motion of 1-D compressible viscous micropolar and heat-conducting fluid, which is thermodynamically perfect and polytropic, is described by the following system of four equations in Eulerian coordinates: Here, the unknown functions ρ(x, t), u(x, t), ω(x, t), θ(x, t), P and κ denote the density, velocity, microrotation velocity, absolute temperature, pressure and coefficient of heat conduction, respectively. A is the constant microviscosity coefficient. The model of micropolar fluids is the local form of the conservation law for the mass, momentum, momentum moment and energy equations. The model of micropolar fluids, introduced by Eringen [12], has received considerable attention in the last two decades. For more background, we refer to [24] and references therein. There has been much research on the existence, uniqueness, regularity, and asymptotic respectively, (ρ, u, ω, θ)(x, 0) = (ρ 0 , u 0 , ω 0 , θ 0 )(x) in [0, 1] (2) and (u, ω, θ x )| x=0,1 = 0, t ≥ 0.
(3) We focus on the polytropic perfect and polytropic fluids and assume that where constant R > 0 are given. For convenience, we let A, R equal to 1 in this paper.
We would like to give some notations which will be used throughout the paper. (2) For p ∈ [1, ∞], L p = L p (I) denotes the L p space with the norm · L p . For k ≥ 1 and p ∈ [1, ∞], W k,p = W k,p (I) denotes the Sobolev space, whose norm is denoted as · W k,p , H k = W k,2 (I).
(3) For an integer k ≥ 0 and 0 < α < 1, let C k+α denote the Schauder space of functions on I, whose kth-order derivative is Hölder continuous with exponents α, with the norm · C k+α .
In this paper, our assumptions are the following: for q ≥ 0 and some constants C i > 0 (i = 1, 2).
Remark 1. Based on the good structure of this system, we use the Calderón-Zygmund decomposition technique to overcome the difficulty caused by possible vacuum for the case where the heat conductivity is constant. However, this technique does not apply to situation in which the conduction coefficient is not constant. For the case that q > 0, we use the ideas in [33,34] which used the weighted densitydependent test function to handle the possible singularities due to vacuum.
The theorem below is our main result.

Examples.
2.1. Preliminaries. At first, we give some necessary lemmas and corollary, they are the same as in [21,33], we omit the proof for brevity.
be a bounded domain in R, and ρ be a nonnegative function satisfies Ω ρ > 0. Then for any absolutely continuous function υ(x) ∈ Ω.
Corollary 1. Consider the same conditions as in Lemma 2.1, and in addition assume Ω = I and ρυ L 1 (I) ≤C.
Then for any l > 0, there exists a positive constant C = C(M, K, l,C) such that υ l L ∞ (I) ≤ C (υ l ) x L 2 (I) + C for any υ l ∈ H 1 (I).
The next lemma (see [21,18]), which we will be used to overcome the difficult caused by possible vacuum in the case q = 0.
Then for any α > α 0 , there exists a sequence (non-overlapping) Ω j included in Ω such that Moreover, where |Ω| denotes the Lebesgue measure of Ω.
Lemma 2.4 (see [32]). Assume X ⊂ E ⊂ Y are Banach spaces and X → → E. Then the following imbeddings are compact: 3. Proof of Theorem 1.1. In this section, we get a global solution to (1.1)-(1.3) with initial density and initial temperature having a, respectively, lower bound δ > 0 by using some a priori estimates of the solutions based on the local existence. Theorem 1.1 would be gotten after making some a priori estimates uniformly for δ and taking limit of δ → 0 + . Denote ρ δ 0 = ρ 0 + δ and θ δ 0 = θ 0 + δ for δ ∈ (0, 1). Throughout this section, we denote C to be a generic constant depending on ρ 0 , u 0 , ω 0 , θ 0 , T , and some other known constants but independent of δ for any δ ∈ (0, 1).
Proof of Theorem 3.1. We can get the local solutions of Theorem 3.1 by the successive approximations as in [6]. One can refer Appendix 2. The regularities guarantee the uniqueness (the details please refer for instance to Appendix 1). Based on it, Theorem 3.1 can be proved by some a priori estimates globally in time.
Then we have the following basic energy estimate.
The next estimate is a corollary of Lemma 3.5, which argument can be seen in [21]. For completeness, we present the proof.

Corollary 2.
Under the conditions of Theorem 6, it holds that for q = 0, Proof.
where we have used the Lemmas 2.1, 3.2 and 3.5. We multiply (1) 2 by u and integrat the resulting equation to deduce Lemma 3.6. Under the conditions of Theorem 3.1, it holds that for q > 0 and for any given 0 < α < min{1, q}, where C may depend on α.
Proof. Similar to the proof in [34], multiplying (1) 4 by θ −α , integrating the resulting equation on Q T and using integration by parts, we have where we have used the Cauchy inequality, and Lemmas 3.2 and 3.3. Now we estimate the last term of (15) as follows: By (15), (16), (17) and (18), the proof of Lemma 3.6 is completed.
Lemma 3.6 implies the following corollary, which proof can be found in [33,34]. For completeness, we present the proof. Proof. By Corollary 1 and Lemma 3.2, we have Remark 2. Next, we will first complete the proof of Theorem 3.1 under the case q > 0. For the case q = 0, the following proofs are valid due to Corollary 2.
Proof. Multiplying (1) 2 by u, and integrating by parts over I, we obtain where we have used the Cauchy inequality and Lemmas 3.2, 3.3. We have q−α+1 > 1 due to q > α > 0. By Young inequality we have Integrating over (0, T ) and using Corollary 3, we complete the proof of Lemma 3.7.
Lemma 3.8. Under the conditions of Theorem 3.1, we have for any 0 ≤ t ≤ T , Proof. We multiply the second equation of (1) by u t , use integration by parts, Lemmas 2.2 and 3.3, and the Cauchy inequality, to discover Since W 1,1 → L ∞ , it follows from (1) 2 and Lemma 3.2 that

ZEFU FENG AND CHANGJIANG ZHU
Owing to (20), we use (1) 4 and integration by parts to conclude Substituting (21) into (19), we have Multiplying the third equation of (1) by the ω t , and integrating the resulting equation, it gives Adding (24) to (22), we obtain that Integrating the above result on (0, T ) and using the Cauchy inequality, we receive that (26) After the third term of the right side is absorbed by the left, we have Here we have used Lemma 3.2 and Young's inequality on the second term of the right side. Obviously, we only need to handel the terms about θ in (27). To do this, we have to use the third equation of (1). For convenience, we denote E = (u 2 x + ω 2 x + ω 2 ) in the following context. Multiplying (1) 3 by θ 0 κ(ξ)dξ and using integration by parts over I, we have Using Corollary 3 and (A 3 ), we have Substituting (29) into (28) and using the Hölder inequality, the Cauchy inequality, and Lemma 3.3, we have Integrating over (0, T ), and using (A 3 ), Lemma 3.2, and Corollary 3, we get Using (27), (30), Corollary 3, Lemma 3.7, and the Gronwall inequality, we complete the proof.
Corollary 4 (see [34]). Under the conditions of Theorem 3.1, we have Proof. By Corollary 1 and Cauchy inequality, we obtain which together with Lemma 3.8, we complete the proof of Corollary 4.
Lemma 3.9 (see [33]). Under the conditions of Theorem 3.1, we have for any 0 ≤ t ≤ T , Lemma 3. 10. Under the conditions of Theorem 3.1, it holds that for any 0 ≤ t ≤ T , Proof. Squaring both right and left of the third equation of (1) and integrating over Q T , we have where we have used Lemma 3.8, the Sobolev inequality, the Cauchy inequality.
Lemma 3.11. Under the conditions of Theorem3.1, it holds that for any 0 ≤ t ≤ T , Proof. Differentiating the second equation of (1) with respect to t, we obtain Multiplying (31) by u t and integrating the resulting equation over I, similar to the proof in the [33] we have Multiplying the fourth equation of (1) by ( θ 0 κ(ξ)dξ) t , integrating the resulting equation over I, and using integration by parts, Lemmas 2.2, 3.3, 3.8 and the Cauchy inequality, we have for any ε > 0 The first term of the right side can be absorbed by the left. After that, combining Lemmas 2.2, 3.3, 3.8 implies Integrating over (0, T ) and using (A3), Lemmas 2.2, 3.8, and 3.9, Corollary 4 and the Cauchy inequality, similar to proof we obtain Differentiating the third equation of (1) with respect to t, we have Multiplying (35) by ω t and integrating the resulting equation over I, we have The first term of right side is absored by the left. Integrating over (0, T ) and using Lemma 3.8, we have Multiplying (34) by 2C, adding the resulting inequality to (32) and (37), taking ε = 1 4C 2 , we have Using the Gronwall inequality and Lemmas 3.9, 3.10, we complete the proof of Lemma 3.11.
Combing Corollary 1 and Lemmas 3.2, 3.11, we have the following corollary immediately.
Lemma 3.12. Under the conditions of Theorem 3.1, it holds that for any 0 ≤ t ≤ T , Proof. Since the estimate of u and ρ is similar to the proof in [33], for brevity, we omit the detail. We only need to get the estimate of ω. Differentiating the third equation of (1) with respect to x, we obtain By Lemmas 2.2, 3.3 and 3.10, Corollaries 5 and 6 and the Sobolev inequality and Cauchy inequality, we have Integrating (46) over [0, T ] and using Lemma 3.11, we obtain This proves Lemma 3.12.
The next two corollaries are needed in our analysis, whose proof is available in [33].
Proof. From (44) and Lemma 3.13, we have which, combining Corollary 5, Lemma 3.11, and the Sobolev inequality, we obtain Differentiating (1) 4 with respect to x, we have

ZEFU FENG AND CHANGJIANG ZHU
By (49), (50), (51), Lemmas 3.12, 3.13, and Corollary 6, we have Integrating it over (0, T ) and using Corollaries 7 and 8, we have The next lemma, which plays a key role in getting H 3 estimates of θ in the following.
Lemma 3.14 (see [33]). Under the conditions of Theorem 3.1, it holds that To get H 3 estimates of θ, we also need to use the next lemma.
Proof. Multiply (47) by ρ 2 (κθ t ) t , and using integration by parts, we have This implies This, along with Lemmas 3.11, 3.13, Corollaries 7, 8 and compatibility conditions (5), we have This completes the proof.
Corollary 10. Under the conditions of Theorem 3.1, we have for any 0 ≤ t ≤ T , Proof. The proof of this lemma see [33], we omit the detail here.
The next two lemmas will be used to get H 3 estimates of u and ω.
Lemma 3.16 (see [33]). Under the conditions of Theorem 3.1, it holds that for any Lemma 3.17. Under the conditions of Theorem 3.1, it holds that for any 0 ≤ t ≤ T , Proof. Similarly to Lemma 3.16, multiplying (35) by ρ 2 ω tt , and integrating over I, we have .
After the first term of the right side is absorbed by the left, we obtain Integrating this inequality on both sides over (0, T ), and using Lemma 3.11, 3.12, 3.14 and Corollaries 6, 9, we have completed the proof.
Corollary 11. Under the conditions of Theorem 3.1, it holds that for any 0 ≤ t ≤ T , Proof. Differentiating the second equation of (1) with respect to x, we obtain By Lemmas 2.2, 3.3, 3.9-3.12 and 3.14, and Corollaries 5, 6 and 9, and the Sobolev inequality, we have

ZEFU FENG AND CHANGJIANG ZHU
Similarly, differentiating the third equation of (1) in x, we have Therefore, using the previous estimates, we immediately have the following lemma.
Lemma 3.18. There exists a uniform constant C such that the following estimate holds for the solution on [0, T ) × R, for any T > 0, Corollary 12. Under the conditions of Theorem 3.1, there is a positive constant C δ depending on δ such that for any (x, t) ∈ Q T , it holds that From (59), (60), (31), (35) and (47) we have The proof of Theorem 3.1 is completed.

Remark 3.
We can't get the uniqueness of the solution to this system with vacuum due to our boundary condition (3) and the term (κ(θ x )θ x ) x . In particular, for the temperature equation, we can't make all terms θ L 2 become I ρ 1θ 2 dx on the right hand of inequality, which leads us to not use Gronwall's inequality. However, we obtain the uniqueness of this system without vacuum. The detailed proof of uniqueness is given in Appendix.
Acknowledgments. The authors would like to thank the anonymous referees for their helpful suggestions and kindly comments. The research was supported by the National Natural Science Foundation of China #11771150 and #11331005.
Appendix. Appendix 1. In this appendix, we will give the detailed proofs of uniqueness of the solution to the compressible viscous micropolar fluid without vacuum.

Appendix 2.
In this appendix, we will give a sketch proof of existence of the local solution to the compressible viscous micropolar fluid.
Firstly, we consider the following linearized problem: where v is a known vector field on (0, T ) × I,θ is a known scalar. Lemma 3.19. Assume that ρ 0 , u 0 , ω 0 , θ 0 v andθ satisfy the properties Then there exists a classical solution (ρ, u, θ, ω) to the initial boundary value problem (72) such that Proof. The existence and regularity of a solution ρ of the linear hyperbolic problem see [7,Lemma 1]. Moreover, it follows from the representation formula (2.2) in [7, The existence and regularity of solutions ω, θ and then u to the corresponding linear parabolic problems have been well known and we omit the details. For instance, we may apply a semi-discrete Galerkin method.
Then we derive local (in time) a priori estimates for classical solutions to the linearized problem (72), which are independent of a lower bound δ of the initial density ρ 0 . Let (ρ 0 , u 0 , θ 0 , ω 0 ) be a given initial data satisfying the hypotheses of Lemma 3.19, and let us choose any fixed c 0 so that Moreover let v andθ satisfying the regularity stated in Lemma 3.19, and assume that v(0) = u 0 ,θ(0) = θ 0 and sup 0≤t≤T * Now let we derive some a priori estimates for the solution (ρ, u, θ, ω) which are independent of δ.
The estimate the density ρ please refer [7, lemma 5] sup 0≤t≤T * Then we get the estimate of ω by using the parabolic and elliptic theory.
To derive estimate for θ, we differentiate Eq 72 4 with respect to t and obtain (77) Then multiplying this by θ t , integrating over I and using transport equation, noticing that κ(θ) > 1 we have (78) Making use of (74), (75) and (76), we can estimate each term I j = I j (t) for 0 ≤ t ≤ min (T * , T 1 ) as follows: Substituting these estimates into (78), we have L 2 ), then using the method in [6] and noting the Corollary 1, we derive estimates for θ as follows Then we obtain the estimate of u by using the similar method in [6], we omit the detail for brevity.
Furthermore, it follows from (81) that the limit (ρ, u, ω, theta) satisfies the following regularity estimate: (98) It is now easy to show that (ρ, u, ω, θ) is a classical solution to the original nonlinear problem (1). This proves the local existence of a classical solution of Theorem 3.1.