This article is concerned with global strong solutions of the micro-polar, compressible flow with density-dependent viscosity coefficients in one-dimensional bounded intervals. The important point in this article is that the initial density may vanish in an open subset.
Theory of micro-polar, compressible flow was first introduced by Eringen , describing the compressible fluids with randomly oriented particles suspended in the medium when the deformation of fluid particles is ignored. The governing equations in Eulerian coordinate take the form as
where ρ = ρ (t, x) denotes the density of the fluid, u = u(t, x) is the velocity, w = w(t, x) is the micro-rotational velocity, θ = θ (t, x) is the temperature, e = e(t, x) is the internal energy, p = p(ρ, θ) is the pressure. μ = μ (ρ, θ), ν = ν (ρ,θ), and λ = λ (ρ, θ) are the viscosities of the fluid, and κ is the heat conductivity.
There are several articles that have considered the above micro-polar, compressible flow, with the viscosity being constant satisfying some physical meaning. Here, we only refer the reader to [2-4], wherein the global existence was established for (1), with the condition that the initial density needs to be bounded a way from zero.
In view of their being physically important, the viscosities are not constants. In this article, we consider a simpler model (2) below. For the physical consideration, in the case of isothermal flow,  introduce the viscosities depending on the density ρ for isentropic flow. For the micro-polar, compressible flow, the model meets the following conditions:
with the initial and boundary conditions:
The pressure p is determined by p(ρ) = aργ, where a is some positive constant and γ > 1, and we normalize a = 1 in the rest of this article. The viscosities tend to depend on the density ρ, i.e. μ (ρ), ν (ρ), and λ (ρ), satisfying
where μ1 and λ1 are positive constants.
Our main concern here is to show the existence of global strong solution for the initial boundary value problem (2)-(3). It is worth emphasizing that the initial density may vanish in an open subset, and the viscosity coefficients μ, ν, and λ depend on density ρ.
Some of the relevant studies in this direction can be summarized as follows. When the viscosity μ, ν, and λ are constants, the global strong solution is established by Chen in  where the vacuum is also allowed. We also refer the reader for a detailed description of three-dimensional micro-polar, compressible flow under the effect of magnetic field, in respect of which global weak solution was established by Amirat and Hamdache in .
Without the randomly oriented particles suspended in the fluid, i.e., when w = 0, the compressible Navier-Stokes equation with density-dependent viscosity, Wen and Yao  proved the global strong solution in one dimension, which generalized Hoff's study  (dealing with the case of constant viscosity coefficient); for the free boundary, the existence of global weak solutions, we refer the readers to Guo and Zhu , and Jiang, Xin and Zhang  and references therein.
The aim of this article is to consider the micro-polar, compressible flow with density-dependent viscosities, in the spirit of .
Now, we state our main result:
Theorem 1.1. Assume that the viscosity μ (ρ), ν (ρ), and λ (ρ) satisfy (4), with the initial data ρ0 ∈ H1 (0, 1), . Then, there exists a global strong solutions (ρ, u, w) to the initial boundary value problem (2)-(3) such that for all T ∈ (0, + ∞),
This article is organized as follows. In Section 2, we derive some uniform estimates for the proof of the main Theorem 1.1, which do not depend on the lower bound of the density. We shall complete the proof of Theorem 1.1 in Section 3.
Notations Throughout this article, we denote C, a generic positive constant, depending only on ρ0, u0, w0, and the time T, but independent of lower bounds of the initial density; we will also use the following simplified notations for the standard Sobolev spaces:
2 Uniform estimates
The following lemma provides standard (energy) estimates which can be obtained by multiplying (2)2 by u and (2)3 by w, and then integrating over (0, T) × (0, 1), with the help of (2)1.
Lemma 2.1. Under the conditions of Theorem 1.1, we have
The following lemma 2.2 is proved in , we omit it here, which plays crucial role for the proof of Theorem 1.1.
Lemma 2.2. Under the conditions of Theorem 1.1, we have
Now we will prove the second crucial estimates.
Lemma 2.3. Under the conditions of Theorem 1.1, we have
Proof. Equation (9) can be obtained via , and so we focus on the proof of (10). From (2)3, we have
Multiplying the above equality by wt, integrating the resultant equality with respect to x over [0, 1], with the help of Young's inequality and (2)1, one gets
Using Sobolev inequalities, (8) and (9), we have
Substituting the above estimates into (11), choosing δ = 1/6, and then integrating with respect to t over (0, t), we get
which completes the proof of (10), according to Gronwall's inequality.
Lemma 2.4. Under the conditions of Theorem 1.1, we have
Proof. The first inequality has been proved in ; now, we consider the second inequality. By virtue of (8), then
The above inequalities together with (10) provide the proof of the second inequality.
Lemma 2.5. Under the conditions of Theorem 1.1, we have
Proof. For the proof of (13), see . From (2), (4), and (8), we have
which together with (6), (7), (10), (12), and (13) furnishes the proof of (14).
3 Proof of Theorem 1.1
In this section, we prove the global existence of strong solutions to the problems (2)-(3) by applying the a priori estimates established in the previous section. One of the main issues is the non-vanishing characteristic of the density in the approximate solutions. To this end, we modify the initial data, and choose the smooth approximate function such that
Now, we consider the initial-boundary value problems (2)-(3) with the initial data (ρ0, u0, w0) replaced by . By virtue of Lemmas 2.1-2.5, we could conclude that
We emphasize that C does not depend on the parameter ε, i.e., the lower bound of the initial density. Then by the standard argument of compactness, we conclude from (2)-(3) that there exists a global strong solution, details of which are omitted here.
The author declares that they have no competing interests.
The author is indebted to the referee for giving nice suggestions which have helped them improve the presentation of this article.
Amirat, Y, Hamdache, K: Weak solutions to the equations of motion for compressible magnetic fluids. J Math Pure Appl. 91, 433–467 (2009). Publisher Full Text
Wen, HY, Yao, L: Global existence of strong solutions of the Navier-Stokes equations for isentropic compressible fluids with density-dependent viscosity. J Math Anal Appl. 349, 503–515 (2009). Publisher Full Text
Hoff, D: Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states. Z Angew Math Phys. 49, 774–785 (1998). Publisher Full Text
Guo, ZH, Zhu, CJ: Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum. J Differential Equations. 248, 2768–2799 (2010). Publisher Full Text