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March  2020, 19(3): 1485-1507. doi: 10.3934/cpaa.2020075

On $ L^p $ estimates for a simplified Ericksen-Leslie system

1. 

School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China

2. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

3. 

School of Mathematics, South China University of Technology, Guangzhou 510641, China

* Corresponding author

Received  May 2019 Revised  August 2019 Published  November 2019

Fund Project: J.R. Huang is partially supported by the National Natural Science Foundation of China (Grant Nos. 11971357, 11871005, 11771155 and 11571117), and by the Natural Science Foundation of Guangdong Province (Grant No. 2019A1515011491). W.J. Wang is partially supported by the National Natural Science Foundation of China (Grant No. 11871341). H.Y. Wen is partially supported by the National Natural Science Foundation of China (Grant Nos. 11671150, 11722104), and by GDUPS (2016).

In this paper, we study Cauchy problem for a simplified Ericksen-Leslie system in three dimensions. With the initial data of small perturbation near a steady state in $ H^2 $ norm, we obtain the global well-posedness of strong solutions as well as the $ L^p(p\in[1, 6]) $ estimates. In addition, sharper decay rates for the density and the momentum are obtained.

Citation: Jinrui Huang, Wenjun Wang, Huanyao Wen. On $ L^p $ estimates for a simplified Ericksen-Leslie system. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1485-1507. doi: 10.3934/cpaa.2020075
References:
[1] R. Adams, Sobolev Spaces, Academic Press, Now York, 1975. 
[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 2003. doi: 10.1090/cln/010.

[3]

Y. S. ChenS. J. Ding and W. J. Wang, Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst., 36 (2016), 5287-5307.  doi: 10.3934/dcds.2016032.

[4]

S. J. DingJ. R. Huang and J. Y. Lin, Global existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensions, Sci. China Math., 56 (2013), 2233-2250.  doi: 10.1007/s11425-013-4620-2.

[5]

S. J. DingJ. R. HuangH. Y. Wen and R. Z. Zi, Incompressible limit of the compressible hydrodynamic flow, J. Funct. Anal., 264 (2013), 1711-1756.  doi: 10.1016/j.jfa.2013.01.011.

[6]

S. J. DingJ. R. Huang and F. G. Xia, A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension, J. Differential Equations, 255 (2013), 3848-3879.  doi: 10.1016/j.jde.2013.07.039.

[7]

S. J. DingJ. Y. LinC. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.  doi: 10.3934/dcds.2012.32.539.

[8]

S. J. DingC. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst.-Ser. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357.

[9]

R. J. DuanH. X. LiuS. J. Ukai and T. Yang, Optimal Lp-Lq convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.

[10]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.  doi: 10.1122/1.548883.

[11]

J. C. GaoQ. Tao and Z. A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033.

[12]

B. L. GuoX. Y. Xi and B. Q. Xie, Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals, J. Differential Equations, 262 (2017), 1413-1460.  doi: 10.1016/j.jde.2016.10.015.

[13]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indian Univ. Math. Journal, 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003.

[14]

D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. angew. Math. Phys., 48 (1997), 597-614.  doi: 10.1007/s000330050049.

[15]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699.  doi: 10.1137/120898814.

[16]

J. R. Huang and S. J. Ding, Compressible hydrodynamic flow of nematic liquid crystals with vacuum, J. Differential Equations, 258 (2015), 1653-1684.  doi: 10.1016/j.jde.2014.11.008.

[17]

J. R. HuangF. H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb{R}^2$, Comm. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.

[18]

T. HuangC. 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.

[19]

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

[20]

F. JiangS. 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.

[21]

S. Kawashima, Systems of a Hyperbolic-parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D Thesis, Kyoto University, 1983.

[22]

S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh, 106A (1987), 169-194.  doi: 10.1017/S0308210500018308.

[23]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbb{R}^3$, J. Differential Equations, 184 (2002), 587-619.  doi: 10.1006/jdeq.2002.4158.

[24]

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

[25]

J. LiZ. H. Xu and J. W. Zhang, Global existence of classical solutions with large oscillations and vacuum to the three-dimensional compressible nematic liquid crystal flows, J. Math. Fluid Mech., 20 (2018), 2105-2145.  doi: 10.1007/s00021-018-0400-7.

[26]

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.

[27]

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.

[28]

F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22.  doi: 10.3934/dcds.2011.31.1.

[29]

F. H. LinJ. 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.

[30]

J. Y. LinB. 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.

[31]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418.

[32]

X. G. Liu and J. Qing, Existence of globally weak solutions to the flow of compressible liquid crystals system, Discrete Contin. Dyn. Syst., 33 (2013), 757-788.  doi: 10.3934/dcds.2013.33.757.

[33]

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

[34]

H. M. Xu, The Pointwise Estimate of Navier-Stokes Equations in Even Multi Space-dimension, Ph.D Thesis, Wuhan University, 2000.

[35]

H. M. Xu and W. K. Wang, Pointwise estimate of solutions of isentropic Navier-Stokes equations in even space-dimensions, Acta. Math. Sci., 21B (2001), 417-427.  doi: 10.1016/S0252-9602(17)30429-0.

show all references

References:
[1] R. Adams, Sobolev Spaces, Academic Press, Now York, 1975. 
[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 2003. doi: 10.1090/cln/010.

[3]

Y. S. ChenS. J. Ding and W. J. Wang, Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst., 36 (2016), 5287-5307.  doi: 10.3934/dcds.2016032.

[4]

S. J. DingJ. R. Huang and J. Y. Lin, Global existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensions, Sci. China Math., 56 (2013), 2233-2250.  doi: 10.1007/s11425-013-4620-2.

[5]

S. J. DingJ. R. HuangH. Y. Wen and R. Z. Zi, Incompressible limit of the compressible hydrodynamic flow, J. Funct. Anal., 264 (2013), 1711-1756.  doi: 10.1016/j.jfa.2013.01.011.

[6]

S. J. DingJ. R. Huang and F. G. Xia, A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension, J. Differential Equations, 255 (2013), 3848-3879.  doi: 10.1016/j.jde.2013.07.039.

[7]

S. J. DingJ. Y. LinC. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.  doi: 10.3934/dcds.2012.32.539.

[8]

S. J. DingC. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst.-Ser. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357.

[9]

R. J. DuanH. X. LiuS. J. Ukai and T. Yang, Optimal Lp-Lq convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.

[10]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.  doi: 10.1122/1.548883.

[11]

J. C. GaoQ. Tao and Z. A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033.

[12]

B. L. GuoX. Y. Xi and B. Q. Xie, Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals, J. Differential Equations, 262 (2017), 1413-1460.  doi: 10.1016/j.jde.2016.10.015.

[13]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indian Univ. Math. Journal, 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003.

[14]

D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. angew. Math. Phys., 48 (1997), 597-614.  doi: 10.1007/s000330050049.

[15]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699.  doi: 10.1137/120898814.

[16]

J. R. Huang and S. J. Ding, Compressible hydrodynamic flow of nematic liquid crystals with vacuum, J. Differential Equations, 258 (2015), 1653-1684.  doi: 10.1016/j.jde.2014.11.008.

[17]

J. R. HuangF. H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb{R}^2$, Comm. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.

[18]

T. HuangC. 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.

[19]

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

[20]

F. JiangS. 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.

[21]

S. Kawashima, Systems of a Hyperbolic-parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D Thesis, Kyoto University, 1983.

[22]

S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh, 106A (1987), 169-194.  doi: 10.1017/S0308210500018308.

[23]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbb{R}^3$, J. Differential Equations, 184 (2002), 587-619.  doi: 10.1006/jdeq.2002.4158.

[24]

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

[25]

J. LiZ. H. Xu and J. W. Zhang, Global existence of classical solutions with large oscillations and vacuum to the three-dimensional compressible nematic liquid crystal flows, J. Math. Fluid Mech., 20 (2018), 2105-2145.  doi: 10.1007/s00021-018-0400-7.

[26]

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.

[27]

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.

[28]

F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22.  doi: 10.3934/dcds.2011.31.1.

[29]

F. H. LinJ. 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.

[30]

J. Y. LinB. 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.

[31]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418.

[32]

X. G. Liu and J. Qing, Existence of globally weak solutions to the flow of compressible liquid crystals system, Discrete Contin. Dyn. Syst., 33 (2013), 757-788.  doi: 10.3934/dcds.2013.33.757.

[33]

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

[34]

H. M. Xu, The Pointwise Estimate of Navier-Stokes Equations in Even Multi Space-dimension, Ph.D Thesis, Wuhan University, 2000.

[35]

H. M. Xu and W. K. Wang, Pointwise estimate of solutions of isentropic Navier-Stokes equations in even space-dimensions, Acta. Math. Sci., 21B (2001), 417-427.  doi: 10.1016/S0252-9602(17)30429-0.

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