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June  2018, 38(6): 2879-2910. doi: 10.3934/dcds.2018124

Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces

Qunyi Bie 1,, , Haibo Cui 2, , Qiru Wang 3, and  Zheng-An Yao 3,

1. 

College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China
2. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
3. 

School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

* Corresponding author: qybie@126.com

Received  July 2017 Published  April 2018

Fund Project: Research Supported by the NNSF of China (Grant Nos.11271379, 11271381, 11671406, 11601164 and 11701325), the National Basic Research Program of China (973 Program) (Grant No. 2010CB808002), the Natural Science Foundation of Fujian Province of China (Grant Nos. 2016J05010 and 2017J05007) and the Scientific Research Funds of Huaqiao University (Grant No.15BS201)

The present paper is devoted to the compressible nematic liquid crystal flow in the whole space $ \mathbb{R}^N\,(N≥ 2)$. Here we concentrate on the incompressible limit in the $ L^p$ type critical Besov spaces setting. We first establish the existence of global solutions in the framework of $ L^p$ type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the ill prepared data case. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.

Keywords: Liquid crystal flow, $ L^p$ type critical Besov spaces, global existence, incompressible limit.
Mathematics Subject Classification: Primary: 35Q35, 76N10; Secondary: 35B40.
Citation: Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011.

[2]

Q. Bie, H. Cui, Q. Wang and Z. Yao, Global existence and incompressible limit in critical spaces for compressible flow of liquid crystals, Z. Angew. Math. Phys., 68 (2017), 113.

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246. doi: 10.24033/asens.1404.

[4]

J. Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.

[5]

J. Y. Chemin, I. Gallagher, D. Iftimie, J. Ball and D. Welsh, Perfect Incompressible Fluids, Clarendon Press Oxford, 1998.

[6]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[7]

R. Danchin, Zero mach number limit for compressible flows with periodic boundary conditions, Am. J. Math., 124 (2002), 1153-1219. doi: 10.1353/ajm.2002.0036.

[8]

R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27-75. doi: 10.1016/S0012-9593(01)01085-0.

[9]

R. Danchin, Fourier Analysis Methods for Compressible Flows, Topics on compressible Navier-Stokes equations, états de la recherche SMF, Chambéry 2012.

[10]

R. Danchin and L. He, The incompressible limit in $ {L}^{p}$ type critical spaces, Math. Ann., 366 (2016), 1365-1402. doi: 10.1007/s00208-016-1361-x.

[11]

R. Danchin and L. He, The Oberbeck-Boussinesq approximation in critical spaces, Asymptotic Anal., 84 (2013), 61-102.

[12]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, P. Roy. Soc. Edinb. A, 455 (1986), 2271-2279. doi: 10.1098/rspa.1999.0403.

[13]

B. Desjardins, E. Grenier, P. L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equationswith dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471. doi: 10.1016/S0021-7824(99)00032-X.

[14]

S. Ding, J. Huang, H. Wen and R. 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.

[15]

S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.

[16]

S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371.

[17]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883.

[18]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.

[19]

T. Hagstrom and J. Lorenz, All-time existence of classical solutions for slightly compressible flows, SIAM J. Math. Anal., 29 (1998), 652-672. doi: 10.1137/S0036141097315312.

[20]

Y. Hao and X. Liu, Incompressible limit of a compressible liquid crystals system, Acta Math. Sci. Ser. B, 33 (2013), 781-796. doi: 10.1016/S0252-9602(13)60038-7.

[21]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2.

[22]

B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differential Equations, 251 (2011), 2262-2295. doi: 10.1016/j.jde.2011.06.013.

[23]

D. Hoff, The zero-mach limit of compressible flows, Comm. Math. Phys., 192 (1998), 543-554. doi: 10.1007/s002200050308.

[24]

M. 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.

[25]

T. Huang, C. Wang and H. 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.

[26]

T. Huang, C. Wang and H. 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.

[27]

F. Jiang, S. Jiang and D. 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.

[28]

F. Jiang, S. Jiang and D. 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.

[29]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.

[30]

H. O. Kreiss, J. Lorenz and M. J. Naughton, Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations, Adv. Appl. Math., 12 (1991), 187-214. doi: 10.1016/0196-8858(91)90012-8.

[31]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.

[32]

F. 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.

[33]

F. Lin, J. 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.

[34]

F. 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.

[35]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571. doi: 10.1002/cpa.21583.

[36]

J. Lin, B. Lai and C. 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.

[37]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, volume 3, Walter de Gruyter, 1996.

[38]

D. Wang and C. Yu, Incompressible limit for the compressible flow of liquid crystals, J. Math. Fluid Mech., 16 (2014), 771-786. doi: 10.1007/s00021-014-0185-2.

[39]

F. Xu, S. Hao and J. Yuan, Well-posedness for the density-dependent incompressible flow of liquid crystals, Math. Meth. Appl. Sci., 38 (2015), 2680-2702. doi: 10.1002/mma.3248.

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011.

[2]

Q. Bie, H. Cui, Q. Wang and Z. Yao, Global existence and incompressible limit in critical spaces for compressible flow of liquid crystals, Z. Angew. Math. Phys., 68 (2017), 113.

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246. doi: 10.24033/asens.1404.

[4]

J. Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.

[5]

J. Y. Chemin, I. Gallagher, D. Iftimie, J. Ball and D. Welsh, Perfect Incompressible Fluids, Clarendon Press Oxford, 1998.

[6]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[7]

R. Danchin, Zero mach number limit for compressible flows with periodic boundary conditions, Am. J. Math., 124 (2002), 1153-1219. doi: 10.1353/ajm.2002.0036.

[8]

R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27-75. doi: 10.1016/S0012-9593(01)01085-0.

[9]

R. Danchin, Fourier Analysis Methods for Compressible Flows, Topics on compressible Navier-Stokes equations, états de la recherche SMF, Chambéry 2012.

[10]

R. Danchin and L. He, The incompressible limit in $ {L}^{p}$ type critical spaces, Math. Ann., 366 (2016), 1365-1402. doi: 10.1007/s00208-016-1361-x.

[11]

R. Danchin and L. He, The Oberbeck-Boussinesq approximation in critical spaces, Asymptotic Anal., 84 (2013), 61-102.

[12]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, P. Roy. Soc. Edinb. A, 455 (1986), 2271-2279. doi: 10.1098/rspa.1999.0403.

[13]

B. Desjardins, E. Grenier, P. L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equationswith dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471. doi: 10.1016/S0021-7824(99)00032-X.

[14]

S. Ding, J. Huang, H. Wen and R. 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.

[15]

S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.

[16]

S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371.

[17]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883.

[18]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.

[19]

T. Hagstrom and J. Lorenz, All-time existence of classical solutions for slightly compressible flows, SIAM J. Math. Anal., 29 (1998), 652-672. doi: 10.1137/S0036141097315312.

[20]

Y. Hao and X. Liu, Incompressible limit of a compressible liquid crystals system, Acta Math. Sci. Ser. B, 33 (2013), 781-796. doi: 10.1016/S0252-9602(13)60038-7.

[21]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2.

[22]

B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differential Equations, 251 (2011), 2262-2295. doi: 10.1016/j.jde.2011.06.013.

[23]

D. Hoff, The zero-mach limit of compressible flows, Comm. Math. Phys., 192 (1998), 543-554. doi: 10.1007/s002200050308.

[24]

M. 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.

[25]

T. Huang, C. Wang and H. 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.

[26]

T. Huang, C. Wang and H. 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.

[27]

F. Jiang, S. Jiang and D. 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.

[28]

F. Jiang, S. Jiang and D. 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.

[29]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.

[30]

H. O. Kreiss, J. Lorenz and M. J. Naughton, Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations, Adv. Appl. Math., 12 (1991), 187-214. doi: 10.1016/0196-8858(91)90012-8.

[31]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.

[32]

F. 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.

[33]

F. Lin, J. 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.

[34]

F. 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.

[35]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571. doi: 10.1002/cpa.21583.

[36]

J. Lin, B. Lai and C. 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.

[37]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, volume 3, Walter de Gruyter, 1996.

[38]

D. Wang and C. Yu, Incompressible limit for the compressible flow of liquid crystals, J. Math. Fluid Mech., 16 (2014), 771-786. doi: 10.1007/s00021-014-0185-2.

[39]

F. Xu, S. Hao and J. Yuan, Well-posedness for the density-dependent incompressible flow of liquid crystals, Math. Meth. Appl. Sci., 38 (2015), 2680-2702. doi: 10.1002/mma.3248.

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