Global existence of strong solutions to the micro-polar, compressible flow with density-dependent viscosities
© Chen; licensee Springer. 2011
Received: 19 March 2011
Accepted: 15 August 2011
Published: 15 August 2011
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.
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.
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:
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.
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.
The following lemma 2.2 is proved in , we omit it here, which plays crucial role for the proof of Theorem 1.1.
Now we will prove the second crucial estimates.
which completes the proof of (10), according to Gronwall's inequality.
The above inequalities together with (10) provide the proof of the second inequality.
which together with (6), (7), (10), (12), and (13) furnishes the proof of (14).
3 Proof of Theorem 1.1
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 is indebted to the referee for giving nice suggestions which have helped them improve the presentation of this article.
- Eringen AC: Theory of micropolar fluids. J Math Mech 1966, 16: 1-18.MathSciNetGoogle Scholar
- Mujakovic N: One-dimensional flow of a compressible viscous micropolar fluid: a global existence theorem. Glasnic matematički 1998, 33: 199-208.MathSciNetMATHGoogle Scholar
- Mujakovic N: Global in time estimates for one-dimensional compressible viscous micropolar fluid model. Glasnic matematički 2005, 40: 103-120.View ArticleMathSciNetMATHGoogle Scholar
- Mujakovic N: Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a global existence theorem. Math Inequalities Appl 2009, 12(3):651-662.View ArticleMathSciNetGoogle Scholar
- Liu TP, Xin ZP, Yang T: Vacuum states of compressible flow. Discrete Contin Dyn Syst 1998, 4: 1-32.MathSciNetMATHGoogle Scholar
- Chen MT: Global strong solutions for the viscous, micropolar, compressible flow. J Part Differential Equations 2011, 24: 158-164.MATHGoogle Scholar
- Amirat Y, Hamdache K: Weak solutions to the equations of motion for compressible magnetic fluids. J Math Pure Appl 2009, 91: 433-467. 10.1016/j.matpur.2009.01.015View ArticleMathSciNetMATHGoogle Scholar
- 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 2009, 349: 503-515. 10.1016/j.jmaa.2008.09.025View ArticleMathSciNetMATHGoogle Scholar
- 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 1998, 49: 774-785. 10.1007/PL00001488View ArticleMathSciNetMATHGoogle Scholar
- 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 2010, 248: 2768-2799. 10.1016/j.jde.2010.03.005View ArticleMathSciNetMATHGoogle Scholar
- Jiang S, Xin ZP, Zhang P: Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods Appl Anal 2005, 12: 239-251.MathSciNetGoogle Scholar