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February  2022, 21(2): 493-515. doi: 10.3934/cpaa.2021185

Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum

Xin Zhong

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  June 2021 Published  February 2022 Early access  November 2021

Fund Project: This research was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359), the Chongqing Talent Plan for Young Topnotch Talents (No. CQYC202005074), and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2020082)

We consider an initial boundary value problem of three-dimensional (3D) nonhomogeneous magneto-micropolar fluid equations in a bounded simply connected smooth domain with homogeneous Dirichlet boundary conditions for the velocity and micro-rotational velocity and Navier-slip boundary condition for the magnetic field. We prove the global existence and exponential decay of strong solutions provided that some smallness condition holds true. Note that although the system degenerates near vacuum, there is no need to require compatibility conditions for the initial data via time weighted techniques.

Keywords: Nonhomogeneous magneto-micropolar fluid equations, exponential decay, global well-posedness, vacuum.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D03.
Citation: 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
References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.

[2]

H. Beir$\tilde{a}$o da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $L^{p}$ theory, J. Math. Fluid Mech., 12 (2010), 397-411.  doi: 10.1007/s00021-009-0295-4.

[3]

L. C. Berselli and S. Spirito, On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains, Commun. Math. Phys., 316 (2012), 171-198.  doi: 10.1007/s00220-012-1581-1.

[4]

J. L. Boldrini, M. A. Rojas-Medar and and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005.

[5] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013.  doi: 10.1007/978-1-4614-5975-0.
[6]

P. Braz e Silva, F. W. Cruz, M. Loayza and M. A. Rojas-Medar, Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach, J. Differ. Equ., 269 (2020), 1319-1348.  doi: 10.1016/j.jde.2020.01.001.

[7]

P. Braz e Silva, F. W. Cruz and M. A. Rojas-Medar, Vanishing viscosity for nonhomogeneous asymmetric fluids in $\mathbb{R}^3$: the $L^2$ case, J. Math. Anal. Appl., 420 (2014), 207-221.  doi: 10.1016/j.jmaa.2014.05.060.

[8]

P. Braz e Silva, F. W. Cruz and M. A. Rojas-Medar, Semi-strong and strong solutions for variable density asymmetric fluids in unbounded domains, Math. Methods Appl. Sci., 40 (2017), 757-774.  doi: 10.1002/mma.4006.

[9]

P. Braz e Silva, F. W. Cruz and M. A. Rojas-Medar, Global strong solutions for variable density incompressible asymmetric fluids in thin domains, Nonlinear Anal. Real World Appl., 55 (2020), 103125, 14 pp. doi: 10.1016/j.nonrwa.2020.103125.

[10]

P. Braz e Silva, F. W. Cruz, M. A. Rojas-Medar and E. G. Santos, Weak solutions with improved regularity for the nonhomogeneous asymmetric fluids equations with vacuum, J. Math. Anal. Appl., 473 (2019), 567-586.  doi: 10.1016/j.jmaa.2018.12.075.

[11]

P. Braz e Silva, E. Fernández-Cara and M. A. Rojas-Medar, Vanishing viscosity for non-homogeneous asymmetric fluids in $\mathbb{R}^3$, J. Math. Anal. Appl., 332 (2007), 833-845.  doi: 10.1016/j.jmaa.2006.10.066.

[12]

P. Braz e Silva and E. G. Santos, Global weak solutions for variable density asymmetric incompressible fluids, J. Math. Anal. Appl., 387 (2012), 953-969.  doi: 10.1016/j.jmaa.2011.10.015.

[13]

F. W. Cruz and P. Braz e Silva, Error estimates for spectral semi-Galerkin approximations of incompressible asymmetric fluids with variable density, J. Math. Fluid Mech., 21 (2019), 27 pp. doi: 10.1007/s00021-019-0405-x.

[14]

V. Girault and P. A. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5.

[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. 
[16]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.  doi: 10.1137/S0036141004442197.

[17] P. L. Lions, Mathematical Topics in Fluid Mechanics, Oxford University Press, Oxford, 1996. 
[18]

G. Łukaszewicz, On nonstationary flows of incompressible asymmetric fluids, Math. Methods Appl. Sci., 13 (1990), 219-232.  doi: 10.1002/mma.1670130304.

[19]

G. Łukaszewicz, Micropolar Fluids, Birkhäuser, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.

[20] A. Lunardi, Interpolation Theory, 3rd edition, Edizioni della Normale, Pisa, 2018.  doi: 10.1007/978-88-7642-638-4.
[21] M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2008. 
[22]

T. Tang and J. Sun, Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 6017-6026.  doi: 10.3934/dcdsb.2020377.

[23]

G. Wu and X. Zhong, Global strong solution and exponential decay of 3D nonhomogeneous asymmetric fluid equations with vacuum, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 1428-1444.  doi: 10.1007/s10473-021-0503-8.

[24]

Z. Ye, Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6725-6743.  doi: 10.3934/dcdsb.2019164.

[25]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity, Comput. Math. Appl., 76 (2018), 2304-2314.  doi: 10.1016/j.camwa.2018.08.041.

[26]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.

show all references

References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.

[2]

H. Beir$\tilde{a}$o da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $L^{p}$ theory, J. Math. Fluid Mech., 12 (2010), 397-411.  doi: 10.1007/s00021-009-0295-4.

[3]

L. C. Berselli and S. Spirito, On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains, Commun. Math. Phys., 316 (2012), 171-198.  doi: 10.1007/s00220-012-1581-1.

[4]

J. L. Boldrini, M. A. Rojas-Medar and and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005.

[5] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013.  doi: 10.1007/978-1-4614-5975-0.
[6]

P. Braz e Silva, F. W. Cruz, M. Loayza and M. A. Rojas-Medar, Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach, J. Differ. Equ., 269 (2020), 1319-1348.  doi: 10.1016/j.jde.2020.01.001.

[7]

P. Braz e Silva, F. W. Cruz and M. A. Rojas-Medar, Vanishing viscosity for nonhomogeneous asymmetric fluids in $\mathbb{R}^3$: the $L^2$ case, J. Math. Anal. Appl., 420 (2014), 207-221.  doi: 10.1016/j.jmaa.2014.05.060.

[8]

P. Braz e Silva, F. W. Cruz and M. A. Rojas-Medar, Semi-strong and strong solutions for variable density asymmetric fluids in unbounded domains, Math. Methods Appl. Sci., 40 (2017), 757-774.  doi: 10.1002/mma.4006.

[9]

P. Braz e Silva, F. W. Cruz and M. A. Rojas-Medar, Global strong solutions for variable density incompressible asymmetric fluids in thin domains, Nonlinear Anal. Real World Appl., 55 (2020), 103125, 14 pp. doi: 10.1016/j.nonrwa.2020.103125.

[10]

P. Braz e Silva, F. W. Cruz, M. A. Rojas-Medar and E. G. Santos, Weak solutions with improved regularity for the nonhomogeneous asymmetric fluids equations with vacuum, J. Math. Anal. Appl., 473 (2019), 567-586.  doi: 10.1016/j.jmaa.2018.12.075.

[11]

P. Braz e Silva, E. Fernández-Cara and M. A. Rojas-Medar, Vanishing viscosity for non-homogeneous asymmetric fluids in $\mathbb{R}^3$, J. Math. Anal. Appl., 332 (2007), 833-845.  doi: 10.1016/j.jmaa.2006.10.066.

[12]

P. Braz e Silva and E. G. Santos, Global weak solutions for variable density asymmetric incompressible fluids, J. Math. Anal. Appl., 387 (2012), 953-969.  doi: 10.1016/j.jmaa.2011.10.015.

[13]

F. W. Cruz and P. Braz e Silva, Error estimates for spectral semi-Galerkin approximations of incompressible asymmetric fluids with variable density, J. Math. Fluid Mech., 21 (2019), 27 pp. doi: 10.1007/s00021-019-0405-x.

[14]

V. Girault and P. A. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5.

[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. 
[16]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.  doi: 10.1137/S0036141004442197.

[17] P. L. Lions, Mathematical Topics in Fluid Mechanics, Oxford University Press, Oxford, 1996. 
[18]

G. Łukaszewicz, On nonstationary flows of incompressible asymmetric fluids, Math. Methods Appl. Sci., 13 (1990), 219-232.  doi: 10.1002/mma.1670130304.

[19]

G. Łukaszewicz, Micropolar Fluids, Birkhäuser, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.

[20] A. Lunardi, Interpolation Theory, 3rd edition, Edizioni della Normale, Pisa, 2018.  doi: 10.1007/978-88-7642-638-4.
[21] M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2008. 
[22]

T. Tang and J. Sun, Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 6017-6026.  doi: 10.3934/dcdsb.2020377.

[23]

G. Wu and X. Zhong, Global strong solution and exponential decay of 3D nonhomogeneous asymmetric fluid equations with vacuum, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 1428-1444.  doi: 10.1007/s10473-021-0503-8.

[24]

Z. Ye, Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6725-6743.  doi: 10.3934/dcdsb.2019164.

[25]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity, Comput. Math. Appl., 76 (2018), 2304-2314.  doi: 10.1016/j.camwa.2018.08.041.

[26]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.

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