American Institute of Mathematical Sciences

Advanced
January  2016, 15(1): 73-90. doi: 10.3934/cpaa.2016.15.73

Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions

Jishan Fan 1, and  Fei Jiang 2,

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037
2. 

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 361000

Received  January 2015 Revised  August 2015 Published  December 2015

In this paper, we study the large-time behavior of weak solutions to the initial-boundary problem arising in a simplified Ericksen-Leslie system for nonhomogeneous incompressible flows of nematic liquid crystals with a transformation condition of trigonometric functions (called by trigonometric condition for simplicity) posed on the initial direction field in a bounded domain $\Omega\subset \mathbb{R}^2$. We show that the kinetic energy and direction field converge to zero and an equilibrium state, respectively, as time goes to infinity. Further, if the initial density is away from vacuum and bounded, then the density, and velocity and direction fields exponential decay to an equilibrium state. In addition, we also show that the weak solutions of the corresponding compressible flows converge {an equilibrium} state.
Keywords: Liquid crystals, weak solutions, exponential decay, Navier-Stokes equations., large-time behavior.
Mathematics Subject Classification: Primary: 35B41, 35Q35, 76A15; Secondary: 76D0.
Citation: Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure & Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73
References:
[1]

M. A. Abdallah, F. Jiang and Z. Tan, Decay estimates for isentropic compressible magnetohydrodynamic equations in bounded domain, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 2211-2220. doi: 10.1016/S0252-9602(12)60171-4.  Google Scholar

[2]

S. J. Ding, J. R. Huang, F. G. Xia, H. Y. Wen and R. Z. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756. doi: 10.1016/j.jfa.2013.01.011.  Google Scholar

[3]

E. Feireisl and H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96. doi: 10.1007/s002050050181.  Google Scholar

[4]

M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002. doi: 10.1137/120866476.  Google Scholar

[5]

J. L. Hineman and C. Y. Wang, Well-posedness of Nematic liquid crystal flow in $L_{u l o c}^3(R^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218. doi: 10.1007/s00205-013-0643-7.  Google Scholar

[6]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.  Google Scholar

[7]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Analysis, 252 (2013), 2678-2699. doi: 10.1137/120898814.  Google Scholar

[8]

X. P. Hu and H. Wu, Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals, Commun. Math. Sci., 11 (2013), 779-806. doi: 10.4310/CMS.2013.v11.n3.a6.  Google Scholar

[9]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Ration. Mech. Anal., 204 (2012), 285-311. doi: 10.1007/s00205-011-0476-1.  Google Scholar

[10]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differential Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036.  Google Scholar

[11]

F. Jiang, S. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397. doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[12]

F. Jiang, S. Jiang, and D. H. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451. doi: 10.1007/s00205-014-0768-3.  Google Scholar

[13]

F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132.  Google Scholar

[14]

F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 32 (2009), 2350-2367. doi: 10.1002/mma.1138.  Google Scholar

[15]

J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows,, preprint, ().   Google Scholar

[16]

J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94. doi: 10.1016/j.na.2013.12.023.  Google Scholar

[17]

X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045.  Google Scholar

[18]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.  Google Scholar

[19]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.  Google Scholar

[20]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.  Google Scholar

[21]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Cont. Dyn. S., 2 (1996), 1-22.  Google Scholar

[22]

F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three,, preprint, ().   Google Scholar

[23]

F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp.  Google Scholar

[24]

J. Y. Lin, B. S. Lai and C. Y. Wang, Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three, SIAM J. Math. Anal., 47 (2015), 2952-2983. doi: 10.1137/15M1007665.  Google Scholar

[25]

P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models, Oxford University Press, New York, 1996.  Google Scholar

[26]

Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows,, preprint, ().   Google Scholar

[27]

X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1-20.  Google Scholar

[28]

D. G. Matteis and G. E. Virga, Director libration in nematoacoustics, Physical Review E, 83 (2011), 011703. doi: 10.1103/PhysRevE.83.011703.  Google Scholar

[29]

A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.  Google Scholar

[30]

M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. doi: 10.1007/BF00752111.  Google Scholar

[31]

L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems, Ann. of Math.(2), 118 (1983), 525-571. doi: 10.2307/2006981.  Google Scholar

[32]

C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5.  Google Scholar

[33]

C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$, J. Funct. Anal., 266 (2014), 5360-5376. doi: 10.1016/j.jfa.2014.02.023.  Google Scholar

[34]

D. H. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x.  Google Scholar

[35]

H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.  Google Scholar

[36]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.  Google Scholar

[37]

H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.  Google Scholar

[38]

Y. Zhou, J. S. Fan and G. Nakamura, Global strong solution to the density-dependent 2-D liquid crystal flows, Abstr. Appl. Anal., Art. ID 947291 (2013), 5pp.  Google Scholar

show all references

References:
[1]

M. A. Abdallah, F. Jiang and Z. Tan, Decay estimates for isentropic compressible magnetohydrodynamic equations in bounded domain, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 2211-2220. doi: 10.1016/S0252-9602(12)60171-4.  Google Scholar

[2]

S. J. Ding, J. R. Huang, F. G. Xia, H. Y. Wen and R. Z. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756. doi: 10.1016/j.jfa.2013.01.011.  Google Scholar

[3]

E. Feireisl and H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96. doi: 10.1007/s002050050181.  Google Scholar

[4]

M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002. doi: 10.1137/120866476.  Google Scholar

[5]

J. L. Hineman and C. Y. Wang, Well-posedness of Nematic liquid crystal flow in $L_{u l o c}^3(R^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218. doi: 10.1007/s00205-013-0643-7.  Google Scholar

[6]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.  Google Scholar

[7]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Analysis, 252 (2013), 2678-2699. doi: 10.1137/120898814.  Google Scholar

[8]

X. P. Hu and H. Wu, Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals, Commun. Math. Sci., 11 (2013), 779-806. doi: 10.4310/CMS.2013.v11.n3.a6.  Google Scholar

[9]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Ration. Mech. Anal., 204 (2012), 285-311. doi: 10.1007/s00205-011-0476-1.  Google Scholar

[10]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differential Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036.  Google Scholar

[11]

F. Jiang, S. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397. doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[12]

F. Jiang, S. Jiang, and D. H. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451. doi: 10.1007/s00205-014-0768-3.  Google Scholar

[13]

F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132.  Google Scholar

[14]

F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 32 (2009), 2350-2367. doi: 10.1002/mma.1138.  Google Scholar

[15]

J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows,, preprint, ().   Google Scholar

[16]

J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94. doi: 10.1016/j.na.2013.12.023.  Google Scholar

[17]

X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045.  Google Scholar

[18]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.  Google Scholar

[19]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.  Google Scholar

[20]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.  Google Scholar

[21]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Cont. Dyn. S., 2 (1996), 1-22.  Google Scholar

[22]

F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three,, preprint, ().   Google Scholar

[23]

F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp.  Google Scholar

[24]

J. Y. Lin, B. S. Lai and C. Y. Wang, Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three, SIAM J. Math. Anal., 47 (2015), 2952-2983. doi: 10.1137/15M1007665.  Google Scholar

[25]

P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models, Oxford University Press, New York, 1996.  Google Scholar

[26]

Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows,, preprint, ().   Google Scholar

[27]

X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1-20.  Google Scholar

[28]

D. G. Matteis and G. E. Virga, Director libration in nematoacoustics, Physical Review E, 83 (2011), 011703. doi: 10.1103/PhysRevE.83.011703.  Google Scholar

[29]

A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.  Google Scholar

[30]

M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. doi: 10.1007/BF00752111.  Google Scholar

[31]

L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems, Ann. of Math.(2), 118 (1983), 525-571. doi: 10.2307/2006981.  Google Scholar

[32]

C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5.  Google Scholar

[33]

C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$, J. Funct. Anal., 266 (2014), 5360-5376. doi: 10.1016/j.jfa.2014.02.023.  Google Scholar

[34]

D. H. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x.  Google Scholar

[35]

H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.  Google Scholar

[36]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.  Google Scholar

[37]

H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.  Google Scholar

[38]

Y. Zhou, J. S. Fan and G. Nakamura, Global strong solution to the density-dependent 2-D liquid crystal flows, Abstr. Appl. Anal., Art. ID 947291 (2013), 5pp.  Google Scholar

[1]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997
[2]

Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080
[3]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
[4]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[5]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[6]

Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[7]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003
[8]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[9]

Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246
[10]

Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013
[11]

Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115
[12]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[13]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[14]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[15]

Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361
[16]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323
[17]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615
[18]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[19]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[20]

Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy