January  2019, 18(1): 1-13. doi: 10.3934/cpaa.2019001

Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$

School of Science, Jiangnan University, Wuxi, 214122, China

* Corresponding author

Received  May 2017 Revised  November 2017 Published  August 2018

Fund Project: This work is partially supported by NSFC (Grant No. 11401258), NSF of Jiangsu Province (grant No. BK20170172) and China Postdoctoral Science Foundation (grant No. 2015M581689).

In this paper, for a nematic liquid crystal system, we address the space-time decay properties of strong solutions in the whole space $\mathbb{R}^3$. Based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates, we obtain the higher order derivative estimates for such system.

Citation: Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001
References:
[1]

C. AmroucheV. GiraultM. Schonbek and T. Schonbek, Pointwidse decay of solutions and of higher derivatives to Navier-Stokes equations, SIAM J. Math. Anal., 31 (2000), 740-753.  doi: 10.1137/S0036141098346177.  Google Scholar

[2]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math., 53 (1984), 259-275.   Google Scholar

[3]

M. C. CaldererD. GolovatyF. Lin and C. Liu, Time evolution of nematic liquid crystals with variable degree of orientation, SIAM J. Math. Anal., 33 (2002), 1033-1047.  doi: 10.1137/S0036141099362086.  Google Scholar

[4]

M. Dai and M. Schonbek, Asymptotic behavior of solutions to Liquid crystal systems in $\mathbb{R}^3$, Comm. Partial Differential Equations, 37 (2012), 2138-2164.  doi: 10.1080/03605302.2012.729172.  Google Scholar

[5]

M. Dai and M. Schonbek, Asymptotic behavior of solutions to the Liquid crystal system in $H^M(\mathbb{R}^3)$, SIAM J. Math. Anal., 46 (2014), 3131-3150.  doi: 10.1137/120895342.  Google Scholar

[6]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.   Google Scholar

[7]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392.   Google Scholar

[8]

J. Fan and J. Li, Regularity criteria for the strong solutions to the Ericksen-Leslie system in $\mathbb{R}^3$, J. Math. Anal. Appl., 425 (2015), 695-703.  doi: 10.1016/j.jmaa.2014.12.063.  Google Scholar

[9]

J. Fan and Y. Zhou, A regularity criterion for a 3D density-dependent incompressible liquid crystals model, Appl. Math. Lett., 58 (2016), 119-124.  doi: 10.1016/j.aml.2016.02.002.  Google Scholar

[10]

J. FanF. S. AlzahraniT. HayatG. Nakamura and Y. Zhou, Global regularity for the 2D liquid crystal model with mixed partial viscosity, Anal. Appl. (Singap.), 13 (2015), 185-200.  doi: 10.1142/S0219530514500481.  Google Scholar

[11]

A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969.  Google Scholar

[12]

W. GuJ. Fan and Y. Zhou, Regularity criteria for some simplified non- isothermal models for nematic liquid crystals, Comput. Math. Appl., 72 (2016), 2839-2853.  doi: 10.1016/j.camwa.2016.10.006.  Google Scholar

[13]

M. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[14]

Z. Jiang and M. Zhu, The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping, Math. Methods Appl. Sci., 35 (2012), 97-102.  doi: 10.1002/mma.1540.  Google Scholar

[15]

Z. Jiang and J. Fan, Time decay rate for two 3D magnetohydrodynamics-α models, Math. Methods Appl. Sci., 37 (2014), 838-845.  doi: 10.1002/mma.2840.  Google Scholar

[16]

I. Kukavica, Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222.  doi: 10.1512/iumj.2001.50.2084.  Google Scholar

[17]

I. Kukavica, On the weighted decay for solutions of the Navier-Stokes system, Nonlinear Anal., 70 (2009), 2466-2470.  doi: 10.1016/j.na.2008.03.031.  Google Scholar

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I. Kukavica and J. J. Torres, Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303.  doi: 10.1088/0951-7715/19/2/003.  Google Scholar

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I. Kukavica and J. J. Torres, Weighted $L^p$ decay for solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 819-831.  doi: 10.1080/03605300600781659.  Google Scholar

[20]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.  Google Scholar

[21]

F. M. Leslie, Theory of flow phenomena in liquid crystals, in Advances in Liquid Crystals (Vol 4, G. Brown ed.), Academic press, New york, 1979, 1-81. Google Scholar

[22]

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

[23]

F. Lin and C. Liu, Nonparabolic dissipative ssytems modelling the flow of liquid crystals, Comm. Pure. Appl. Math., 489 (1995), 501-537.  doi: 10.1002/cpa.3160480503.  Google Scholar

[24]

S. Liu and X. Xu, Global existence and temporal decay for the nematic liquid crystal flows, J. Math. Anal. Appl., 426 (2015), 228-246.  doi: 10.1016/j.jmaa.2015.01.001.  Google Scholar

[25]

Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal folws, Math. Nachr., 289 (2016), 678-692.  doi: 10.1002/mana.201400313.  Google Scholar

[26]

T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac., 43 (2000), 541-557.   Google Scholar

[27]

C. Qian, Remarks on the regularity criterion for the nematic liquid crystal folws in $\mathbb{R}^3$, Appl. Math. Computation, 274 (2016), 679-689.  doi: 10.1016/j.amc.2015.11.007.  Google Scholar

[28]

M. E. 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

[29]

M. E. Schonbek, Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Diff. Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.  Google Scholar

[30]

M. Schonbek and T. Schonbek, On the boundedness and decay of moments of solutions to the Navier-Stokes equations, Adv. Differential Equations, 5 (2000), 861-898.   Google Scholar

[31]

S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789.  doi: 10.1016/S0362-546X(98)00070-4.  Google Scholar

[32]

C. 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]

S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270 (2016), 2168-2187.  doi: 10.1016/j.jfa.2016.01.021.  Google Scholar

[34]

S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Methods Appl. Sci., 39 (2016), 4398-4418.  doi: 10.1002/mma.3868.  Google Scholar

[35]

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

[36]

Y. Zhou, A remark on the decay of solutions to the 3-D Navier-Stokes equations, Math. Methods Appl. Sci., 30 (2007), 1223-1229.  doi: 10.1002/mma.841.  Google Scholar

[37]

Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071.  doi: 10.1088/0951-7715/21/9/008.  Google Scholar

[38]

Y. Zhou and J. Fan, A regularity criterion for the nematic liquid crystal flows, J. Inequal. Appl., 2010, Art. ID 589697, 9 pp. doi: 10.1155/2010/589697.  Google Scholar

show all references

References:
[1]

C. AmroucheV. GiraultM. Schonbek and T. Schonbek, Pointwidse decay of solutions and of higher derivatives to Navier-Stokes equations, SIAM J. Math. Anal., 31 (2000), 740-753.  doi: 10.1137/S0036141098346177.  Google Scholar

[2]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math., 53 (1984), 259-275.   Google Scholar

[3]

M. C. CaldererD. GolovatyF. Lin and C. Liu, Time evolution of nematic liquid crystals with variable degree of orientation, SIAM J. Math. Anal., 33 (2002), 1033-1047.  doi: 10.1137/S0036141099362086.  Google Scholar

[4]

M. Dai and M. Schonbek, Asymptotic behavior of solutions to Liquid crystal systems in $\mathbb{R}^3$, Comm. Partial Differential Equations, 37 (2012), 2138-2164.  doi: 10.1080/03605302.2012.729172.  Google Scholar

[5]

M. Dai and M. Schonbek, Asymptotic behavior of solutions to the Liquid crystal system in $H^M(\mathbb{R}^3)$, SIAM J. Math. Anal., 46 (2014), 3131-3150.  doi: 10.1137/120895342.  Google Scholar

[6]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.   Google Scholar

[7]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392.   Google Scholar

[8]

J. Fan and J. Li, Regularity criteria for the strong solutions to the Ericksen-Leslie system in $\mathbb{R}^3$, J. Math. Anal. Appl., 425 (2015), 695-703.  doi: 10.1016/j.jmaa.2014.12.063.  Google Scholar

[9]

J. Fan and Y. Zhou, A regularity criterion for a 3D density-dependent incompressible liquid crystals model, Appl. Math. Lett., 58 (2016), 119-124.  doi: 10.1016/j.aml.2016.02.002.  Google Scholar

[10]

J. FanF. S. AlzahraniT. HayatG. Nakamura and Y. Zhou, Global regularity for the 2D liquid crystal model with mixed partial viscosity, Anal. Appl. (Singap.), 13 (2015), 185-200.  doi: 10.1142/S0219530514500481.  Google Scholar

[11]

A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969.  Google Scholar

[12]

W. GuJ. Fan and Y. Zhou, Regularity criteria for some simplified non- isothermal models for nematic liquid crystals, Comput. Math. Appl., 72 (2016), 2839-2853.  doi: 10.1016/j.camwa.2016.10.006.  Google Scholar

[13]

M. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[14]

Z. Jiang and M. Zhu, The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping, Math. Methods Appl. Sci., 35 (2012), 97-102.  doi: 10.1002/mma.1540.  Google Scholar

[15]

Z. Jiang and J. Fan, Time decay rate for two 3D magnetohydrodynamics-α models, Math. Methods Appl. Sci., 37 (2014), 838-845.  doi: 10.1002/mma.2840.  Google Scholar

[16]

I. Kukavica, Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222.  doi: 10.1512/iumj.2001.50.2084.  Google Scholar

[17]

I. Kukavica, On the weighted decay for solutions of the Navier-Stokes system, Nonlinear Anal., 70 (2009), 2466-2470.  doi: 10.1016/j.na.2008.03.031.  Google Scholar

[18]

I. Kukavica and J. J. Torres, Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303.  doi: 10.1088/0951-7715/19/2/003.  Google Scholar

[19]

I. Kukavica and J. J. Torres, Weighted $L^p$ decay for solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 819-831.  doi: 10.1080/03605300600781659.  Google Scholar

[20]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.  Google Scholar

[21]

F. M. Leslie, Theory of flow phenomena in liquid crystals, in Advances in Liquid Crystals (Vol 4, G. Brown ed.), Academic press, New york, 1979, 1-81. Google Scholar

[22]

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

[23]

F. Lin and C. Liu, Nonparabolic dissipative ssytems modelling the flow of liquid crystals, Comm. Pure. Appl. Math., 489 (1995), 501-537.  doi: 10.1002/cpa.3160480503.  Google Scholar

[24]

S. Liu and X. Xu, Global existence and temporal decay for the nematic liquid crystal flows, J. Math. Anal. Appl., 426 (2015), 228-246.  doi: 10.1016/j.jmaa.2015.01.001.  Google Scholar

[25]

Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal folws, Math. Nachr., 289 (2016), 678-692.  doi: 10.1002/mana.201400313.  Google Scholar

[26]

T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac., 43 (2000), 541-557.   Google Scholar

[27]

C. Qian, Remarks on the regularity criterion for the nematic liquid crystal folws in $\mathbb{R}^3$, Appl. Math. Computation, 274 (2016), 679-689.  doi: 10.1016/j.amc.2015.11.007.  Google Scholar

[28]

M. E. 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

[29]

M. E. Schonbek, Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Diff. Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.  Google Scholar

[30]

M. Schonbek and T. Schonbek, On the boundedness and decay of moments of solutions to the Navier-Stokes equations, Adv. Differential Equations, 5 (2000), 861-898.   Google Scholar

[31]

S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789.  doi: 10.1016/S0362-546X(98)00070-4.  Google Scholar

[32]

C. 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]

S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270 (2016), 2168-2187.  doi: 10.1016/j.jfa.2016.01.021.  Google Scholar

[34]

S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Methods Appl. Sci., 39 (2016), 4398-4418.  doi: 10.1002/mma.3868.  Google Scholar

[35]

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

[36]

Y. Zhou, A remark on the decay of solutions to the 3-D Navier-Stokes equations, Math. Methods Appl. Sci., 30 (2007), 1223-1229.  doi: 10.1002/mma.841.  Google Scholar

[37]

Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071.  doi: 10.1088/0951-7715/21/9/008.  Google Scholar

[38]

Y. Zhou and J. Fan, A regularity criterion for the nematic liquid crystal flows, J. Inequal. Appl., 2010, Art. ID 589697, 9 pp. doi: 10.1155/2010/589697.  Google Scholar

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