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Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential
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 |
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.
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Q. Chen and C. Miao,
Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differ. Equ., 252 (2012), 2698-2724.
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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.
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Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.
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Darcy's law and diffusion for a two-fluid Euler-Maxwell system with dissipation, Math. Models Meth. Appl. Sci., 25 (2015), 2089-2151.
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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. |
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B. Huang, L. Liu and L. Zhang, Global dynamics of 3d compressible micropolar fluids with vacuum and large oscillations, preprint. |
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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.
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| [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. |
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M. E. Taylor, Partial Differential Equations, Vol. 23, Texts in Applied Mathematics, Springer-Verlag, New York, 1996.
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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. 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. |
| [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. 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. |
| [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. |
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