# American Institute of Mathematical Sciences

August  2018, 38(8): 4133-4162. doi: 10.3934/dcds.2018180

## Global regularity for the 2D micropolar equations with fractional dissipation

 1 College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China 2 School of Mathematical Science, Anhui University, Hefei 230601, China 3 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA 4 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China 5 Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, China

* Corresponding author: Zhuan Ye

Received  December 2017 Revised  March 2018 Published  May 2018

Fund Project: Dong was partially supported by the National Natural Science Foundation of China (NO. 11571240; No.11271019) and Research Fund of Shenzhen University (No. 2017056). Wu was partially supported by NSF grants DMS 1209153 and DMS 1624146, by the AT & T Foundation at Oklahoma State University (OSU) and by NSFC (No.11471103, a grant awarded to Professor Baoquan Yuan). Xu was partially supported by the National Natural Science Foundation of China (No. 11371059; No. 11471220; No. 11771045). Ye was supported by the Foundation of Jiangsu Normal University (No. 16XLR029), the Natural Science Foundation of Jiangsu Province (No. BK20170224) and the National Natural Science Foundation of China (No. 11701232)

Micropolar equations, modeling micropolar fluid flows, consist of coupled equations obeyed by the evolution of the velocity $u$ and that of the microrotation $w$. This paper focuses on the two-dimensional micropolar equations with the fractional dissipation $(-Δ)^{α} u$ and $(-Δ)^{β}w$, where $0<α, β<1$. The goal here is the global regularity of the fractional micropolar equations with minimal fractional dissipation. Recent efforts have resolved the two borderline cases $α = 1$, $β = 0$ and $α = 0$, $β = 1$. However, the situation for the general critical case $α+β = 1$ with $0<α<1$ is far more complex and the global regularity appears to be out of reach. When the dissipation is split among the equations, the dissipation is no longer as efficient as in the borderline cases and different ranges of $α$ and $β$ require different estimates and tools. We aim at the subcritical case $\alpha+\beta>1$ and divide $\alpha\in (0,1)$ into five sub-intervals to seek the best estimates so that we can impose the minimal requirements on $\alpha$ and $\beta$. The proof of the global regularity relies on the introduction of combined quantities, sharp lower bounds for the fractional dissipation and delicate upper bounds for the nonlinearity and associated commutators.

Citation: Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180
##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-16830-7. [2] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976. [3] D. Chamorro and P. G. Lemarié-Rieusset, Quasi-geostrophic equation, nonlinear Bernstein inequalities and α-stable processes, Rev. Mat. Iberoam., 28 (2012), 1109-1122. doi: 10.4171/RMI/705. [4] Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724. doi: 10.1016/j.jde.2011.09.035. [5] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. [6] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. [7] S. C. Cowin, Polar fluids, Phys. Fluids, 11 (1968), 1919-1927. doi: 10.1063/1.1692219. [8] C. Doering and J. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511608803. [9] B. Dong and Z. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete Cont. Dyn. Sys., 23 (2009), 765-784. doi: 10.3934/dcds.2009.23.765. [10] B. Dong, J. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523. doi: 10.1016/j.jde.2016.11.029. [11] B. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213. doi: 10.1016/j.jde.2010.03.016. [12] M. E. Erdogan, Polar effects in the apparent viscosity of suspension, Rheol. Acta, 9 (1970), 434-438. doi: 10.1007/BF01975413. [13] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. [14] A. C. Eringen, Micropolar fluids with stretch, Int. J. Engng. Eci., 7 (1969), 115-127. doi: 10.1016/0020-7225(69)90026-3. [15] G. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of micropolar fluid equations, Int. J. Engrg. Sci., 15 (1977), 105-108. doi: 10.1016/0020-7225(77)90025-8. [16] L. Grafakos, Loukas Modern Fourier Analysis, Third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8. [17] Q. Jiu, C. Miao, J. Wu and Z. Zhang, The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454. doi: 10.1137/140958256. [18] T. Kato and G. Ponce, Commutator estimates and the Euler and the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704. [19] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. American Mathematical Society, 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0. [20] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [21] P. G. Lemarie-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics Series, CRC Press, 2002. doi: 10.1201/9781420035674. [22] G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5. [23] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [24] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics, Science Press, Beijing, China, 2012 (in Chinese). [25] S. Popel, A. Regirer and P. Usick, A continuum model of blood flow, Biorheology, 11 (1974), 427-437. [26] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. doi: 10.1515/9783110812411. [27] V. K. Stokes, Theories of Fluids with Microstructure, Springer, New York, 1984. doi: 10.1007/978-3-642-82351-0. [28] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343. [29] H. Triebel, Theory of Function Spaces II, Birkhauser Verlag, 1992. doi: 10.1007/978-3-0346-0419-2. [30] L. Xue, Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777. doi: 10.1002/mma.1491. [31] K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207. doi: 10.3934/dcds.2015.35.2193. [32] Z. Ye and X. Xu, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J. Differential Equations, 260 (2016), 6716-6744. doi: 10.1016/j.jde.2016.01.014. [33] B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci., 30B (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7.

show all references

##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-16830-7. [2] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976. [3] D. Chamorro and P. G. Lemarié-Rieusset, Quasi-geostrophic equation, nonlinear Bernstein inequalities and α-stable processes, Rev. Mat. Iberoam., 28 (2012), 1109-1122. doi: 10.4171/RMI/705. [4] Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724. doi: 10.1016/j.jde.2011.09.035. [5] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. [6] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. [7] S. C. Cowin, Polar fluids, Phys. Fluids, 11 (1968), 1919-1927. doi: 10.1063/1.1692219. [8] C. Doering and J. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511608803. [9] B. Dong and Z. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete Cont. Dyn. Sys., 23 (2009), 765-784. doi: 10.3934/dcds.2009.23.765. [10] B. Dong, J. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523. doi: 10.1016/j.jde.2016.11.029. [11] B. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213. doi: 10.1016/j.jde.2010.03.016. [12] M. E. Erdogan, Polar effects in the apparent viscosity of suspension, Rheol. Acta, 9 (1970), 434-438. doi: 10.1007/BF01975413. [13] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. [14] A. C. Eringen, Micropolar fluids with stretch, Int. J. Engng. Eci., 7 (1969), 115-127. doi: 10.1016/0020-7225(69)90026-3. [15] G. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of micropolar fluid equations, Int. J. Engrg. Sci., 15 (1977), 105-108. doi: 10.1016/0020-7225(77)90025-8. [16] L. Grafakos, Loukas Modern Fourier Analysis, Third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8. [17] Q. Jiu, C. Miao, J. Wu and Z. Zhang, The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454. doi: 10.1137/140958256. [18] T. Kato and G. Ponce, Commutator estimates and the Euler and the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704. [19] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. American Mathematical Society, 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0. [20] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [21] P. G. Lemarie-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics Series, CRC Press, 2002. doi: 10.1201/9781420035674. [22] G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5. [23] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [24] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics, Science Press, Beijing, China, 2012 (in Chinese). [25] S. Popel, A. Regirer and P. Usick, A continuum model of blood flow, Biorheology, 11 (1974), 427-437. [26] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. doi: 10.1515/9783110812411. [27] V. K. Stokes, Theories of Fluids with Microstructure, Springer, New York, 1984. doi: 10.1007/978-3-642-82351-0. [28] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343. [29] H. Triebel, Theory of Function Spaces II, Birkhauser Verlag, 1992. doi: 10.1007/978-3-0346-0419-2. [30] L. Xue, Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777. doi: 10.1002/mma.1491. [31] K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207. doi: 10.3934/dcds.2015.35.2193. [32] Z. Ye and X. Xu, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J. Differential Equations, 260 (2016), 6716-6744. doi: 10.1016/j.jde.2016.01.014. [33] B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci., 30B (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7.
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