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September  2020, 19(9): 4575-4598. doi: 10.3934/cpaa.2020207

Optimal decay to the non-isentropic compressible micropolar fluids

Lvqiao liu a, and  Lan Zhang b,,

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.

Keywords: Green function, optimal decay, micropolar fluid.
Mathematics Subject Classification: Primary: 35Q35, 76D03; Secondary: 86A10.
Citation: Lvqiao liu, Lan Zhang. Optimal decay to the non-isentropic compressible micropolar fluids. Communications on Pure & 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.  Google Scholar

[2]

B. Q. Dong, J. 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.  Google Scholar

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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.  Google Scholar

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R. Duan, Q. 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.  Google Scholar

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

[7]

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

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

B. Q. Dong, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. Duan, Q. 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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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