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Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces
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 |
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.
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. Chen, S. 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. Ding, J. 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. Ding, J. R. Huang, H. 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. Ding, J. 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. Ding, J. Y. Lin, C. 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. Ding, C. 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. Duan, H. X. Liu, S. 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. Gao, Q. 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. Guo, X. 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. Huang, F. 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. 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. |
[19] |
T. Huang, C. 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. 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. |
[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. Li, Z. 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. 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. |
[30] |
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. |
[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. Chen, S. 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. Ding, J. 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. Ding, J. R. Huang, H. 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. Ding, J. 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. Ding, J. Y. Lin, C. 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. Ding, C. 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. Duan, H. X. Liu, S. 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. Gao, Q. 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. Guo, X. 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. Huang, F. 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. 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. |
[19] |
T. Huang, C. 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. 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. |
[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. Li, Z. 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. 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. |
[30] |
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. |
[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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