• Previous Article
    The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries
  • CPAA Home
  • This Issue
  • Next Article
    Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential
September  2020, 19(9): 4575-4598. doi: 10.3934/cpaa.2020207

Optimal decay to the non-isentropic compressible micropolar fluids

a. 

School of Mathematics and statistics, Wuhan University, Wuhan 430072, China

b. 

Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author

Received  January 2020 Revised  April 2020 Published  June 2020

Fund Project: The second author is supported by NSF grant No.11971359, 11671309

In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropolar fluid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equilibrium state with rate $ (1+t)^{-3 / 4} $ in $ L^{2} $ norm and the micro-rotational velocity tends to the equilibrium state with the faster rate $ (1+t)^{-5 / 4} $ in $ L^{2} $ norm. The proof is based on the detailed analysis of the Green function and time-weighted energy estimates.

Citation: Lvqiao liu, Lan Zhang. Optimal decay to the non-isentropic compressible micropolar fluids. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4575-4598. doi: 10.3934/cpaa.2020207
References:
[1]

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

[2]

B. Q. DongJ. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differ. Equ., 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.

[3]

R. Duan, Global smooth flows for the compressible Euler-Maxwell system. The relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413.  doi: 10.1142/S0219891611002421.

[4]

R. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078.

[5]

R. DuanQ. Liu and C. Zhu, Darcy's law and diffusion for a two-fluid Euler-Maxwell system with dissipation, Math. Models Meth. Appl. Sci., 25 (2015), 2089-2151.  doi: 10.1142/S0218202515500530.

[6]

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

[7]

B. Huang, L. Liu and L. Zhang, Global dynamics of 3d compressible micropolar fluids with vacuum and large oscillations, preprint.

[8]

B. Huang and L. Zhang, A global existence of classical solutions to the two-dimensional vlasov-fokker-planck and magnetohydrodynamics equations with large initial data, Kinet. Relat. Models, 12 (2019), 357. doi: 10.3934/krm.2019016.

[9]

X. Huang and J. Li, Global Well-Posedness of classical solutions to the Cauchy problem of two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, preprint, arXiv: 1207.3746.

[10]

S. Kawashima, Large-time behavior of solutions for hyperbolic-parabolic systems of conservation laws, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 285-287. 

[11]

J. Li, Global well-posedness of the 1D compressible Navier-Stokes equations with constant heat conductivity and nonnegative density, arXiv e-prints, 2018.

[12]

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

[13]

Q. Liu and P. Zhang, Long-time behavior of solution to the compressible micropolar fluids with external force, Nonlinear Anal. Real World Appl., 40 (2018), 361-376.  doi: 10.1016/j.nonrwa.2017.08.007.

[14]

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

[15]

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

[16]

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

[17]

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.

[18]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.

[19]

M. E. Taylor, Partial Differential Equations, Vol. 23, Texts in Applied Mathematics, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.

[20]

M. E. Taylor, Partial Differential Equations. I, Vol. 115, Applied Mathematical Sciences, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.

[21]

V. A. Vaĭgant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[22]

Z. Wu and W. Wang, Green's function and pointwise estimate for a generalized Poisson-Nernst-Planck-Navier-Stokes model in dimension three, Z. Angew. Math. Mech., 98 (2018), 1066-1085.  doi: 10.1002/zamm.201700109.

[23]

Z. Wu and W. Wang, The pointwise estimates of diffusion wave of the compressible micropolar fluids, J. Differ. Equ., 265 (2018), 2544-2576.  doi: 10.1016/j.jde.2018.04.039.

show all references

References:
[1]

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

[2]

B. Q. DongJ. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differ. Equ., 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.

[3]

R. Duan, Global smooth flows for the compressible Euler-Maxwell system. The relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413.  doi: 10.1142/S0219891611002421.

[4]

R. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078.

[5]

R. DuanQ. Liu and C. Zhu, Darcy's law and diffusion for a two-fluid Euler-Maxwell system with dissipation, Math. Models Meth. Appl. Sci., 25 (2015), 2089-2151.  doi: 10.1142/S0218202515500530.

[6]

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

[7]

B. Huang, L. Liu and L. Zhang, Global dynamics of 3d compressible micropolar fluids with vacuum and large oscillations, preprint.

[8]

B. Huang and L. Zhang, A global existence of classical solutions to the two-dimensional vlasov-fokker-planck and magnetohydrodynamics equations with large initial data, Kinet. Relat. Models, 12 (2019), 357. doi: 10.3934/krm.2019016.

[9]

X. Huang and J. Li, Global Well-Posedness of classical solutions to the Cauchy problem of two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data, preprint, arXiv: 1207.3746.

[10]

S. Kawashima, Large-time behavior of solutions for hyperbolic-parabolic systems of conservation laws, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 285-287. 

[11]

J. Li, Global well-posedness of the 1D compressible Navier-Stokes equations with constant heat conductivity and nonnegative density, arXiv e-prints, 2018.

[12]

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

[13]

Q. Liu and P. Zhang, Long-time behavior of solution to the compressible micropolar fluids with external force, Nonlinear Anal. Real World Appl., 40 (2018), 361-376.  doi: 10.1016/j.nonrwa.2017.08.007.

[14]

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

[15]

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

[16]

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

[17]

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.

[18]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.

[19]

M. E. Taylor, Partial Differential Equations, Vol. 23, Texts in Applied Mathematics, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.

[20]

M. E. Taylor, Partial Differential Equations. I, Vol. 115, Applied Mathematical Sciences, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.

[21]

V. A. Vaĭgant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[22]

Z. Wu and W. Wang, Green's function and pointwise estimate for a generalized Poisson-Nernst-Planck-Navier-Stokes model in dimension three, Z. Angew. Math. Mech., 98 (2018), 1066-1085.  doi: 10.1002/zamm.201700109.

[23]

Z. Wu and W. Wang, The pointwise estimates of diffusion wave of the compressible micropolar fluids, J. Differ. Equ., 265 (2018), 2544-2576.  doi: 10.1016/j.jde.2018.04.039.

[1]

Xin Zhong. Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum. Communications on Pure and Applied Analysis, 2022, 21 (2) : 493-515. doi: 10.3934/cpaa.2021185

[2]

Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048

[3]

Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439

[4]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems and Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487

[5]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[6]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[7]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

[8]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[9]

Haibo Cui, Junpei Gao, Lei Yao. Asymptotic behavior of the one-dimensional compressible micropolar fluid model. Electronic Research Archive, 2021, 29 (2) : 2063-2075. doi: 10.3934/era.2020105

[10]

Zhi-Ying Sun, Lan Huang, Xin-Guang Yang. Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry. Electronic Research Archive, 2020, 28 (2) : 861-878. doi: 10.3934/era.2020045

[11]

Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062

[12]

Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations and Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019

[13]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021

[14]

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

[15]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767

[16]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure and Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657

[17]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210

[18]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189

[19]

Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193

[20]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (280)
  • HTML views (95)
  • Cited by (0)

Other articles
by authors

[Back to Top]