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.  Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  Google Scholar

[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.  Google Scholar

[7]

S. C. Cowin, Polar fluids, Phys. Fluids, 11 (1968), 1919-1927.  doi: 10.1063/1.1692219.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. DongJ. 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.  Google Scholar

[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.  Google Scholar

[12]

M. E. Erdogan, Polar effects in the apparent viscosity of suspension, Rheol. Acta, 9 (1970), 434-438.  doi: 10.1007/BF01975413.  Google Scholar

[13]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.   Google Scholar

[14]

A. C. Eringen, Micropolar fluids with stretch, Int. J. Engng. Eci., 7 (1969), 115-127.  doi: 10.1016/0020-7225(69)90026-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Q. JiuC. MiaoJ. 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.  Google Scholar

[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.  Google Scholar

[19]

C. E. KenigG. 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.  Google Scholar

[20]

C. E. KenigG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.  Google Scholar

[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). Google Scholar

[25]

S. Popel, A. Regirer and P. Usick, A continuum model of blood flow, Biorheology, 11 (1974), 427-437.   Google Scholar

[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.  Google Scholar

[27]

V. K. Stokes, Theories of Fluids with Microstructure, Springer, New York, 1984. doi: 10.1007/978-3-642-82351-0.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[29]

H. Triebel, Theory of Function Spaces II, Birkhauser Verlag, 1992. doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  Google Scholar

[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.  Google Scholar

[7]

S. C. Cowin, Polar fluids, Phys. Fluids, 11 (1968), 1919-1927.  doi: 10.1063/1.1692219.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. DongJ. 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.  Google Scholar

[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.  Google Scholar

[12]

M. E. Erdogan, Polar effects in the apparent viscosity of suspension, Rheol. Acta, 9 (1970), 434-438.  doi: 10.1007/BF01975413.  Google Scholar

[13]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.   Google Scholar

[14]

A. C. Eringen, Micropolar fluids with stretch, Int. J. Engng. Eci., 7 (1969), 115-127.  doi: 10.1016/0020-7225(69)90026-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Q. JiuC. MiaoJ. 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.  Google Scholar

[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.  Google Scholar

[19]

C. E. KenigG. 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.  Google Scholar

[20]

C. E. KenigG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.  Google Scholar

[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). Google Scholar

[25]

S. Popel, A. Regirer and P. Usick, A continuum model of blood flow, Biorheology, 11 (1974), 427-437.   Google Scholar

[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.  Google Scholar

[27]

V. K. Stokes, Theories of Fluids with Microstructure, Springer, New York, 1984. doi: 10.1007/978-3-642-82351-0.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[29]

H. Triebel, Theory of Function Spaces II, Birkhauser Verlag, 1992. doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Regularity region
[1]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083

[2]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034

[3]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[4]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[5]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[6]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[7]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[8]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[9]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289

[10]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[11]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[12]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[13]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[14]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287

[15]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[16]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[17]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[18]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[19]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312

[20]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (194)
  • HTML views (212)
  • Cited by (13)

Other articles
by authors

[Back to Top]