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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

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

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

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

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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