June  2019, 39(6): 3069-3097. doi: 10.3934/dcds.2019127

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

School of Mathematics, South China University of Technology, Guangzhou 510641, China

* Corresponding author: Changjiang Zhu

Received  February 2018 Revised  December 2018 Published  February 2019

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 $ \kappa(\theta) = O(1+\theta^q) $ where $ q\geq0 $ and delicate energy estimates.

Citation: Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127
References:
[1]

M. T. Chen, Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.   Google Scholar

[2]

M. T. Chen, Blow up criterion for viscous, compressible micropolar fluids with vacuum, Nonlinear Anal.RWA., 13 (2012), 850-859.  doi: 10.1016/j.nonrwa.2011.08.021.  Google Scholar

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M. T. ChenB. Huang and J. W. Zhang, Blow up criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum, Nonlinear Anal., 79 (2013), 1-11.  doi: 10.1016/j.na.2012.10.013.  Google Scholar

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Q. L. Chen and C. X. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar

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Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

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S. J. DingH. Y. Wen and C. J. Zhu, Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum, J. Differential Equations, 251 (2011), 1696-1725.  doi: 10.1016/j.jde.2011.05.025.  Google Scholar

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I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A global existence theorem, Bound. Value Probl., 2015 (2015), 21pp. doi: 10.1186/s13661-015-0357-x.  Google Scholar

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I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution, J. Math. Anal. Appl., 431 (2015), 545-568.  doi: 10.1016/j.jmaa.2015.06.002.  Google Scholar

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I. DražićL. Simčić and N. Mujakovi'c, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: regularity of the solution, J. Math. Anal. Appl., 438 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.  Google Scholar

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Z. L. Liang and S. Q. Wu, Classical solution to 1D viscous polytropic perfect fluids with constant diffusion coefficients and vacuum, Z. Angew. Math. Phys., 68 (2017), Art. 22, 20 pp. doi: 10.1007/s00033-017-0767-y.  Google Scholar

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Q. Q. Liu and P. X. Zhang, Optimal time decay of the compressible micropolar fluids, J. Differential Equations, 260 (2016), 7634-7661.  doi: 10.1016/j.jde.2016.01.037.  Google Scholar

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G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston. MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[25]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: a global existence theorem, Glas. Mat., 33 (1998), 199-208.   Google Scholar

[26]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution, Rad. Mat., 10 (2001), 181-193.   Google Scholar

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N. Mujaković, Global in time estimates for one-dimensional compressible viscous micropolar fluid model, Glas. Mat. Ser. III, 40 (2005), 103-120.  doi: 10.3336/gm.40.1.10.  Google Scholar

[28]

N. Mujaković, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: Regularity of the solution, Bound. Value Probl., 2008 (2008), Art. ID 189748, 15 pp. doi: 10.1155/2008/189748.  Google Scholar

[29]

N. Mujaković, Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a global existence theorem, Math. Inequal. Appl., 12 (2009), 651-662.  doi: 10.7153/mia-12-49.  Google Scholar

[30]

B. Nowakowski, Large time existence of strong solutions to micropolar equations in cylindrical domains, Nonlinear Anal. RWA., 14 (2013), 635-660.  doi: 10.1016/j.nonrwa.2012.07.023.  Google Scholar

[31]

Y. M. QinT. G. Wang and G. L. Hu, The Cauchy problem for a 1D compressible viscous micropolar fluid model: analysis of the stabilization and the regularity, Nonlinear Anal. RWA., 13 (2012), 1010-1029.  doi: 10.1016/j.nonrwa.2010.10.023.  Google Scholar

[32]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar

[33]

H. Y. Wen and C. J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.  doi: 10.1137/120877829.  Google Scholar

[34]

H. Y. Wen and C. J. Zhu, Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, SIAM J. Math. Anal., 102 (2014), 498-545.  doi: 10.1016/j.matpur.2013.12.003.  Google Scholar

show all references

References:
[1]

M. T. Chen, Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.   Google Scholar

[2]

M. T. Chen, Blow up criterion for viscous, compressible micropolar fluids with vacuum, Nonlinear Anal.RWA., 13 (2012), 850-859.  doi: 10.1016/j.nonrwa.2011.08.021.  Google Scholar

[3]

M. T. ChenB. Huang and J. W. Zhang, Blow up criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum, Nonlinear Anal., 79 (2013), 1-11.  doi: 10.1016/j.na.2012.10.013.  Google Scholar

[4]

M. T. ChenX. Y. Xu and J. W. Zhang, Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum, Commun. Math. Sci., 13 (2015), 225-247.  doi: 10.4310/CMS.2015.v13.n1.a11.  Google Scholar

[5]

Q. L. Chen and C. X. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar

[6]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[7]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[8]

S. J. DingH. Y. Wen and C. J. Zhu, Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum, J. Differential Equations, 251 (2011), 1696-1725.  doi: 10.1016/j.jde.2011.05.025.  Google Scholar

[9]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A global existence theorem, Bound. Value Probl., 2015 (2015), 21pp. doi: 10.1186/s13661-015-0357-x.  Google Scholar

[10]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution, J. Math. Anal. Appl., 431 (2015), 545-568.  doi: 10.1016/j.jmaa.2015.06.002.  Google Scholar

[11]

I. DražićL. Simčić and N. Mujakovi'c, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: regularity of the solution, J. Math. Anal. Appl., 438 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.  Google Scholar

[12]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.   Google Scholar

[13]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[14]

E. Feireisl, On the motion of a viscous, compressible and heat-conducting fluid, Indiana Univ. Math. J., 53 (2004), 1705-1738.  doi: 10.1512/iumj.2004.53.2510.  Google Scholar

[15]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[16] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.   Google Scholar
[17]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[18]

Q. Han and F. H. Lin, Elliptic Partial Differential Equations, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[19]

D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Am. Math. Soc., 303 (1987), 169-181.  doi: 10.2307/2000785.  Google Scholar

[20]

X. D. HuangJ. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Commun. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar

[21]

Z. L. Liang and S. Q. Wu, Classical solution to 1D viscous polytropic perfect fluids with constant diffusion coefficients and vacuum, Z. Angew. Math. Phys., 68 (2017), Art. 22, 20 pp. doi: 10.1007/s00033-017-0767-y.  Google Scholar

[22]

P. L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, Oxford University Press on Demand, 1998.  Google Scholar

[23]

Q. Q. Liu and P. X. Zhang, Optimal time decay of the compressible micropolar fluids, J. Differential Equations, 260 (2016), 7634-7661.  doi: 10.1016/j.jde.2016.01.037.  Google Scholar

[24]

G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston. MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[25]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: a global existence theorem, Glas. Mat., 33 (1998), 199-208.   Google Scholar

[26]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution, Rad. Mat., 10 (2001), 181-193.   Google Scholar

[27]

N. Mujaković, Global in time estimates for one-dimensional compressible viscous micropolar fluid model, Glas. Mat. Ser. III, 40 (2005), 103-120.  doi: 10.3336/gm.40.1.10.  Google Scholar

[28]

N. Mujaković, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: Regularity of the solution, Bound. Value Probl., 2008 (2008), Art. ID 189748, 15 pp. doi: 10.1155/2008/189748.  Google Scholar

[29]

N. Mujaković, Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a global existence theorem, Math. Inequal. Appl., 12 (2009), 651-662.  doi: 10.7153/mia-12-49.  Google Scholar

[30]

B. Nowakowski, Large time existence of strong solutions to micropolar equations in cylindrical domains, Nonlinear Anal. RWA., 14 (2013), 635-660.  doi: 10.1016/j.nonrwa.2012.07.023.  Google Scholar

[31]

Y. M. QinT. G. Wang and G. L. Hu, The Cauchy problem for a 1D compressible viscous micropolar fluid model: analysis of the stabilization and the regularity, Nonlinear Anal. RWA., 13 (2012), 1010-1029.  doi: 10.1016/j.nonrwa.2010.10.023.  Google Scholar

[32]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar

[33]

H. Y. Wen and C. J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.  doi: 10.1137/120877829.  Google Scholar

[34]

H. Y. Wen and C. J. Zhu, Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, SIAM J. Math. Anal., 102 (2014), 498-545.  doi: 10.1016/j.matpur.2013.12.003.  Google Scholar

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