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doi: 10.3934/dcdsb.2021296
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Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum

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

Received  January 2021 Revised  November 2021 Early access December 2021

Fund Project: This research was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359), Exceptional Young Talents Project of Chongqing Talent (No. cstc2021ycjh-bgzxm0153), and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2020082)

We study the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity in the whole plane $ \mathbb{R}^2 $. We derive the global existence and uniqueness of strong solutions if the initial density decays not too slowly at infinity. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Our method relies upon the delicate weighted energy estimates and the structural characteristics of the system under consideration.

Citation: Xin Zhong. Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021296
References:
[1]

J. L. BoldriniM. A. Rojas-Medar 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.

[2]

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

[3]

P. Braz e SilvaF. 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.

[4]

P. Braz e SilvaF. 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.

[5]

P. Braz e SilvaF. 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.  doi: 10.1016/j.nonrwa.2020.103125.

[6]

P. Braz e SilvaF. W. CruzM. 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.

[7]

P. Braz e SilvaE. 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.

[8]

P. Braz e SilvaL. Friz and M. A. Rojas-Medar, Exponential stability for magneto-micropolar fluids, Nonlinear Anal., 143 (2016), 211-223.  doi: 10.1016/j.na.2016.05.015.

[9]

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.

[10]

R. CoifmanP. L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. 

[11]

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.

[12]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.  doi: 10.1007/s002050050025.

[13]

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

[14]

A. C. Eringen, Microcontinuum Field Theories. Ⅰ. Foundations and Solids, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0555-5.

[15]

J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), 7, 37pp.  doi: 10.1007/s40818-019-0064-5.

[16] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996. 
[17]

L. Liu and X. Zhong, Global existence and exponential decay of strong solutions for 2D nonhomogeneous micropolar fluids with density-dependent viscosity, J. Math. Phys., 62 (2021), 061508.  doi: 10.1063/5.0055689.

[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. Theory and Applications, Birkhäuser, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.

[20]

B. LüX. Shi and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.

[21]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. 

[22] E. M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. 
[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 micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.

[26]

X. Zhong, Strong solutions to the Cauchy problem of two-dimensional nonhomogeneous micropolar fluid equations with nonnegative density, Dyn. Partial Differ. Equ., 18 (2021), 49-69.  doi: 10.4310/DPDE.2021.v18.n1.a4.

show all references

References:
[1]

J. L. BoldriniM. A. Rojas-Medar 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.

[2]

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

[3]

P. Braz e SilvaF. 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.

[4]

P. Braz e SilvaF. 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.

[5]

P. Braz e SilvaF. 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.  doi: 10.1016/j.nonrwa.2020.103125.

[6]

P. Braz e SilvaF. W. CruzM. 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.

[7]

P. Braz e SilvaE. 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.

[8]

P. Braz e SilvaL. Friz and M. A. Rojas-Medar, Exponential stability for magneto-micropolar fluids, Nonlinear Anal., 143 (2016), 211-223.  doi: 10.1016/j.na.2016.05.015.

[9]

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.

[10]

R. CoifmanP. L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. 

[11]

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.

[12]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.  doi: 10.1007/s002050050025.

[13]

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

[14]

A. C. Eringen, Microcontinuum Field Theories. Ⅰ. Foundations and Solids, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0555-5.

[15]

J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), 7, 37pp.  doi: 10.1007/s40818-019-0064-5.

[16] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996. 
[17]

L. Liu and X. Zhong, Global existence and exponential decay of strong solutions for 2D nonhomogeneous micropolar fluids with density-dependent viscosity, J. Math. Phys., 62 (2021), 061508.  doi: 10.1063/5.0055689.

[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. Theory and Applications, Birkhäuser, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.

[20]

B. LüX. Shi and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.

[21]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. 

[22] E. M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. 
[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 micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.

[26]

X. Zhong, Strong solutions to the Cauchy problem of two-dimensional nonhomogeneous micropolar fluid equations with nonnegative density, Dyn. Partial Differ. Equ., 18 (2021), 49-69.  doi: 10.4310/DPDE.2021.v18.n1.a4.

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