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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, 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

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