May  2022, 21(5): 1595-1620. doi: 10.3934/cpaa.2022033

Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry

1. 

College of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

2. 

Department of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

3. 

Laboratoire de Mathématiques et Applications, UMR CNRS 7348-SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie-Téléport 2 86962, Chasseneuil Futuroscope Cedex, France

* Corresponding Author

Received  September 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

Fund Project: This paper was partially supported by the NSFC (No. 11501199), the Young Key Teachers Project in Higher Vocational Colleges of Henan Province (No. 2020GZGG109), Young Backbone Teachers in Henan Province (No. 2018GGJS039), Incubation Fund Project of Henan Normal University (No. 2020PL17), Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003)

This paper is concerned with the global solutions of the 3D compressible micropolar fluid model in the domain to a subset of $ R^3 $ bounded with two coaxial cylinders that present the solid thermo-insulated walls, which is in a thermodynamical sense perfect and polytropic. Compared with the classical Navier-Stokes equations, the angular velocity $ w $ in this model brings benefit that is the damping term -$ uw $ can provide extra regularity of $ w $. At the same time, the term $ uw^2 $ is bad, it increases the nonlinearity of our system. Moreover, the regularity and exponential stability in $ H^4 $ also are proved.

Citation: Lan Huang, Zhiying Sun, Xin-Guang Yang, Alain Miranville. Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1595-1620. doi: 10.3934/cpaa.2022033
References:
[1]

S. Antontscv, A. Kazhikhov and V. Monakhov, Boundary Problems in Mechanics of Nonhomogeneous Fluids, Amsterdam, New York, 1990.

[2]

A. Bašić-Šiško and I. Dražić, Global solution to a one-dimensional model of viscous and heat-conducting micropolar real gas flow, J. Math. Anal. Appl., (2021), 124690. 

[3]

A. Bašić-Šiško and I. Dražić, Uniqueness of generalized solution to micropolar viscous real gas flow with homogeneous boundary conditions, Math. Meth. Appl. Sci., 44 (2021), 4330-4341.  doi: 10.1002/mma.7032.

[4]

A. BorrelliG. Giantesio and M. Patria, An exact solution for the 3D MHD stagnation-point flow of a micropolar fluid, Commun. Nonlinear Sci. Numer. Simul., 20 (2015), 121-135.  doi: 10.1016/j.cnsns.2014.04.011.

[5]

G. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal., 23 (1992), 609-634.  doi: 10.1137/0523031.

[6] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. 
[7]

G. ChenD. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Commun. Partial Differ. Equ., 25 (2000), 2233-2257.  doi: 10.1080/03605300008821583.

[8]

J. ChenC. Liang and J. Lee, Numerical simulation for unsteady compressible micropolar fluid flow, Comput. Fluids, 66 (2012), 1-9.  doi: 10.1016/j.compfluid.2012.05.015.

[9]

H. Cui and H. Yin, Stationary solutions to the one-dimensional micropolar fluid model in a half line: existence, stability and convergence rate, J. Math. Anal. Appl., 449 (2017), 464-489.  doi: 10.1016/j.jmaa.2016.11.065.

[10]

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), 21 pp. doi: 10.1186/s13661-015-0357-x.

[11]

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.

[12]

I. DražićL. Simčić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution, J. Math. Anal. Appl., 435 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.

[13]

I. DražićN. Mujaković and N. Črnjarić-Žic, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Derivation of the model and a numerical solution, Math. Comput. Simul., 140 (2017), 107-124.  doi: 10.1016/j.matcom.2017.03.006.

[14]

I. Dražić, 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry: A global existence theorem, Math. Meth. Appl. Sci., 40 (2017), 4785-4801. 

[15]

I. DražićL. Simčić and N. Mujaković, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Uniqueness of a generalized solution, Math. Meth. Appl. Sci., 40 (2017), 2686-2701. 

[16]

I. Dražić, A. Bašić-Šiško and L. Simčić, One-dimensional model and numerical solution to the viscous and heat-conducting micropolar real gas flow with homogeneous boundary conditions, 2020, Preprint.

[17]

R. Duan, Global strong solution for initial-boundary value problem of one-dimensional compressible micropolar fluids with density dependent viscosity and temperature dependent heat conductivity, Nonlinear Anal. RWA, 42 (2018), 71-92.  doi: 10.1016/j.nonrwa.2017.12.006.

[18]

A. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.

[19]

Z. Feng and C. Zhu, Global classical large solution to compressible viscous micropolar and heat-conduting fluids with vacuum, Discret. Contin. Dynam. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.

[20] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[21]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.

[22]

B. Guo and P. Zhu, Asymptotic behavior of the solution to the system for a viscous reactive gas, J. Differ. Equ., 155 (1999), 177-202.  doi: 10.1006/jdeq.1998.3578.

[23]

L. Huang and R. Lian, Exponential stability of spherically symmetric solutions for compressible viscous micropolar fluid, J. Math. Phys., 56 (2015), 071503, 12 pp. doi: 10.1063/1.4926426.

[24]

L. Huang and C. Kong, Global behavior for compressible viscous micropolar fluid with spherical symmetry, J. Math. Anal. Appl., 443 (2016), 1158-1178.  doi: 10.1016/j.jmaa.2016.05.056.

[25]

L. Huang and I. Dražić, Large-time behavior of solutions to the 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry, Math. Meth. Appl. Sci., 41 (2018), 7888-7905.  doi: 10.1002/mma.5250.

[26]

L. Huang and I. Drazic, Exponential stability for the compressible micropolar fluid with cylinder symmetry in $R^3$, J. Math. Phys., 60 (2019), 021507, 14 pp. doi: 10.1063/1.5017652.

[27]

L. HuangZ. Sun and X. Yang, Large time behavior of spherically symmetrical micropolar fluid in unbounded domain, Appl. Math. Optim., 84 (2021), S1607-S1638.  doi: 10.1007/s00245-021-09806-3.

[28]

A. Kazhikhov and V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.

[29]

T. Kato, Strong $L^P$-solutions of the Navier-Stokes equations in $R^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.

[30]

M. Lewicka and P. Mucha, On temporal asymptotics for the pth power viscous reactive gas, Nonlinear Anal., 57 (2004), 951-969.  doi: 10.1016/j.na.2003.12.001.

[31]

Z. Liang and F. Lin, Global mild solutions of Navier-Stokes equations, Commun. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.

[32]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A local existence theorem, Glasn. Mat., 33 (1998), 71-91. 

[33]

N. Mujaković, Nonhomogenerous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A local existence theorem, Ann. Univ. Ferrara Sez. VII Sci. Mat., 53 (2007), 361-379.  doi: 10.1007/s11565-007-0023-z.

[34]

D. MalteseM. MichálekB. Mucha PiotrA. NovotnýM. Pokorný and E. Zatorska, Existence of weak solutions for compressible Navier-Stokes equations with entropy transport, J. Differ. Equ., 261 (2016), 4448-4485.  doi: 10.1016/j.jde.2016.06.029.

[35]

I. PapautskyJ. BrazzleT. Ameel and A. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory, Sens. and Actuators A: Phys., 73 (1999), 101-108. 

[36]

Z. SunL. Huang and X. Yang, Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry, Electron. Res. Arch., 28 (2020), 861-878.  doi: 10.3934/era.2020045.

[37]

L. Wan and T. Wang, Asymptotic behavior for the one-dimensional pth power Newtonian fluid in unbounded domains, Math. Meth. Appl. Sci., 39 (2016), 1020-1025.  doi: 10.1002/mma.3539.

[38]

L. Wan and L. Zhang, Global solutions to the micropolar compressible flow with constant coefficients and vacuum, Nonlinear Anal. RWA, 51 (2020), 102990, 14 pp. doi: 10.1016/j.nonrwa.2019.102990.

[39]

T. Wang, One dimensional p-th power Newtonian fluid with temperature-dependent thermal conductivity, Commun. Pure Appl. Anal., 15 (2016), 477-494.  doi: 10.3934/cpaa.2016.15.477.

show all references

References:
[1]

S. Antontscv, A. Kazhikhov and V. Monakhov, Boundary Problems in Mechanics of Nonhomogeneous Fluids, Amsterdam, New York, 1990.

[2]

A. Bašić-Šiško and I. Dražić, Global solution to a one-dimensional model of viscous and heat-conducting micropolar real gas flow, J. Math. Anal. Appl., (2021), 124690. 

[3]

A. Bašić-Šiško and I. Dražić, Uniqueness of generalized solution to micropolar viscous real gas flow with homogeneous boundary conditions, Math. Meth. Appl. Sci., 44 (2021), 4330-4341.  doi: 10.1002/mma.7032.

[4]

A. BorrelliG. Giantesio and M. Patria, An exact solution for the 3D MHD stagnation-point flow of a micropolar fluid, Commun. Nonlinear Sci. Numer. Simul., 20 (2015), 121-135.  doi: 10.1016/j.cnsns.2014.04.011.

[5]

G. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal., 23 (1992), 609-634.  doi: 10.1137/0523031.

[6] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. 
[7]

G. ChenD. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Commun. Partial Differ. Equ., 25 (2000), 2233-2257.  doi: 10.1080/03605300008821583.

[8]

J. ChenC. Liang and J. Lee, Numerical simulation for unsteady compressible micropolar fluid flow, Comput. Fluids, 66 (2012), 1-9.  doi: 10.1016/j.compfluid.2012.05.015.

[9]

H. Cui and H. Yin, Stationary solutions to the one-dimensional micropolar fluid model in a half line: existence, stability and convergence rate, J. Math. Anal. Appl., 449 (2017), 464-489.  doi: 10.1016/j.jmaa.2016.11.065.

[10]

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), 21 pp. doi: 10.1186/s13661-015-0357-x.

[11]

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.

[12]

I. DražićL. Simčić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution, J. Math. Anal. Appl., 435 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.

[13]

I. DražićN. Mujaković and N. Črnjarić-Žic, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Derivation of the model and a numerical solution, Math. Comput. Simul., 140 (2017), 107-124.  doi: 10.1016/j.matcom.2017.03.006.

[14]

I. Dražić, 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry: A global existence theorem, Math. Meth. Appl. Sci., 40 (2017), 4785-4801. 

[15]

I. DražićL. Simčić and N. Mujaković, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Uniqueness of a generalized solution, Math. Meth. Appl. Sci., 40 (2017), 2686-2701. 

[16]

I. Dražić, A. Bašić-Šiško and L. Simčić, One-dimensional model and numerical solution to the viscous and heat-conducting micropolar real gas flow with homogeneous boundary conditions, 2020, Preprint.

[17]

R. Duan, Global strong solution for initial-boundary value problem of one-dimensional compressible micropolar fluids with density dependent viscosity and temperature dependent heat conductivity, Nonlinear Anal. RWA, 42 (2018), 71-92.  doi: 10.1016/j.nonrwa.2017.12.006.

[18]

A. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.

[19]

Z. Feng and C. Zhu, Global classical large solution to compressible viscous micropolar and heat-conduting fluids with vacuum, Discret. Contin. Dynam. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.

[20] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[21]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.

[22]

B. Guo and P. Zhu, Asymptotic behavior of the solution to the system for a viscous reactive gas, J. Differ. Equ., 155 (1999), 177-202.  doi: 10.1006/jdeq.1998.3578.

[23]

L. Huang and R. Lian, Exponential stability of spherically symmetric solutions for compressible viscous micropolar fluid, J. Math. Phys., 56 (2015), 071503, 12 pp. doi: 10.1063/1.4926426.

[24]

L. Huang and C. Kong, Global behavior for compressible viscous micropolar fluid with spherical symmetry, J. Math. Anal. Appl., 443 (2016), 1158-1178.  doi: 10.1016/j.jmaa.2016.05.056.

[25]

L. Huang and I. Dražić, Large-time behavior of solutions to the 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry, Math. Meth. Appl. Sci., 41 (2018), 7888-7905.  doi: 10.1002/mma.5250.

[26]

L. Huang and I. Drazic, Exponential stability for the compressible micropolar fluid with cylinder symmetry in $R^3$, J. Math. Phys., 60 (2019), 021507, 14 pp. doi: 10.1063/1.5017652.

[27]

L. HuangZ. Sun and X. Yang, Large time behavior of spherically symmetrical micropolar fluid in unbounded domain, Appl. Math. Optim., 84 (2021), S1607-S1638.  doi: 10.1007/s00245-021-09806-3.

[28]

A. Kazhikhov and V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.

[29]

T. Kato, Strong $L^P$-solutions of the Navier-Stokes equations in $R^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.

[30]

M. Lewicka and P. Mucha, On temporal asymptotics for the pth power viscous reactive gas, Nonlinear Anal., 57 (2004), 951-969.  doi: 10.1016/j.na.2003.12.001.

[31]

Z. Liang and F. Lin, Global mild solutions of Navier-Stokes equations, Commun. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.

[32]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A local existence theorem, Glasn. Mat., 33 (1998), 71-91. 

[33]

N. Mujaković, Nonhomogenerous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A local existence theorem, Ann. Univ. Ferrara Sez. VII Sci. Mat., 53 (2007), 361-379.  doi: 10.1007/s11565-007-0023-z.

[34]

D. MalteseM. MichálekB. Mucha PiotrA. NovotnýM. Pokorný and E. Zatorska, Existence of weak solutions for compressible Navier-Stokes equations with entropy transport, J. Differ. Equ., 261 (2016), 4448-4485.  doi: 10.1016/j.jde.2016.06.029.

[35]

I. PapautskyJ. BrazzleT. Ameel and A. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory, Sens. and Actuators A: Phys., 73 (1999), 101-108. 

[36]

Z. SunL. Huang and X. Yang, Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry, Electron. Res. Arch., 28 (2020), 861-878.  doi: 10.3934/era.2020045.

[37]

L. Wan and T. Wang, Asymptotic behavior for the one-dimensional pth power Newtonian fluid in unbounded domains, Math. Meth. Appl. Sci., 39 (2016), 1020-1025.  doi: 10.1002/mma.3539.

[38]

L. Wan and L. Zhang, Global solutions to the micropolar compressible flow with constant coefficients and vacuum, Nonlinear Anal. RWA, 51 (2020), 102990, 14 pp. doi: 10.1016/j.nonrwa.2019.102990.

[39]

T. Wang, One dimensional p-th power Newtonian fluid with temperature-dependent thermal conductivity, Commun. Pure Appl. Anal., 15 (2016), 477-494.  doi: 10.3934/cpaa.2016.15.477.

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