doi: 10.3934/dcdsb.2021257
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Global existence of weak solutions to inhomogeneous Doi-Onsager equations

1. 

School of Mathematical Science, Peking University, Beijing 100871, China

2. 

School of Mathematics, Hunan University, Changsha 410082, China

* Corresponding author: Jianfeng Zhou

Received  March 2021 Revised  August 2021 Early access November 2021

In this paper, we study the inhomogeneous Doi-Onsager equations with a special viscous stress. We prove the global existence of weak solutions in the case of periodic regions without considering the effect of the constraint force arising from the rigidity of the rods. The key ingredient is to show the convergence of the nonlinear terms, which can be reduced to proving the strong compactness of the moment of the family of number density functions. The proof is based on the propagation of strong compactness by studying a transport equation for some defect measure, L2-estimates for a family of number density functions, and energy dissipation estimates.

Citation: Wenji Chen, Jianfeng Zhou. Global existence of weak solutions to inhomogeneous Doi-Onsager equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021257
References:
[1]

N. Arcozzi, Riesz transforms on compact Lie groups, spheres and Gauss space, Ark. Mat., 36 (1998), 201-231.  doi: 10.1007/BF02384766.  Google Scholar

[2]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[3]

J. M. Ball, E. Feireisl and F. Otto, Mathematical Thermodynamics of Complex Fluids, Springer, Cham, Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2017.  Google Scholar

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations II convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[5]

J. W. Barrett and E. Süeli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci., 21 (2011), 1211-1289.  doi: 10.1142/S0218202511005313.  Google Scholar

[6]

X. Chen and J.-G. Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions, J. Differential Equations, 254 (2013), 2764-2802.  doi: 10.1016/j.jde.2013.01.005.  Google Scholar

[7]

A. Cotsiolis and N. Labropoulos, Sharp Nash inequalities on the unit sphere: The influence of symmetries, Nonlinear Anal., 75 (2012), 612-624.  doi: 10.1016/j.na.2011.08.063.  Google Scholar

[8]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[9]

R. J. DiPerna and A. Majda, Reduced Hausdorff dimension and concentration-cancellation for two dimensional incompressible flow, J. Amer. Math. Soc., 1 (1988), 59-95.  doi: 10.2307/1990967.  Google Scholar

[10]

M. Doi, Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases, Journal of Polymer, 19 (1981), 229-243.  doi: 10.1002/pol.1981.180190205.  Google Scholar

[11]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1988. Google Scholar

[12]

W. E. and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods Appl. Anal., 13 (2006), 181-198.  doi: 10.4310/MAA.2006.v13.n2.a5.  Google Scholar

[13]

L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 74. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. doi: 10.1090/cbms/074.  Google Scholar

[14]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. doi: 10.1093/acprof:oso/9780198528388.003.0002.  Google Scholar

[15]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18] P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[19]

P.-L. Lions and N. Masmoudi, Global existence of weak solutions to some micro-macro models, C. R. Math. Acad. Sci. Paris, 345 (2007), 15-20.  doi: 10.1016/j.crma.2007.05.011.  Google Scholar

[20]

P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146.  doi: 10.1142/S0252959900000170.  Google Scholar

[21]

Y. Liu and W. Wang, The small Deborah number limit of the Doi-Onsager equation without hydrodynamics, J. Funct. Anal., 275 (2018), 2740-2793.  doi: 10.1016/j.jfa.2018.07.013.  Google Scholar

[22]

W. Maier and A. Saupe, Eine einfache molekulare theorie des nematischen kristallinflüssigen Zustandes, Zeitschrift für Naturforschung A, 13 (1958), 564–566. doi: 10.1515/zna-1958-0716.  Google Scholar

[23]

G. Marrucci and F. Greco, The elastic constants of Maier-Saupe rodlike molecule nematics, Molecular Crystals and Liquid Crystals, 206 (1991), 17-30.  doi: 10.1080/00268949108037714.  Google Scholar

[24]

N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500.  doi: 10.1007/s00222-012-0399-y.  Google Scholar

[25]

L. Onsager, The effects of shape on the interaction of colloidal particles, Annals of the New York Academy of Sciences, 51 (1949), 627-659.  doi: 10.1111/j.1749-6632.1949.tb27296.x.  Google Scholar

[26]

A. D. Rey and T. Tsuji, Recent advances in theoretical liquid crystal rheology, Macromolecular Theory and Simulations, 7 (1998), 623-639.   Google Scholar

[27]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations: Classical Theory, Cambridge University Press, 2016. doi: 10.1017/CBO9781139095143.  Google Scholar

[28]

O. Sieber, Existence of global weak solutions to an inhomogeneous Doi model for active liquid crystals, preprint, arXiv: 2006.16832. Google Scholar

[29]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[31]

T. Tsuji and A. D. Rey, Orientation mode selection mechanisms for sheared nematic liquid crystalline materials, Physical Review E, 57 (1998), 5609.  doi: 10.1103/PhysRevE.57.5609.  Google Scholar

[32]

Q. WangW. E. C. Liu and P. Zhang, Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential, Physical Review E, 65 (2002), 051504.  doi: 10.1103/PhysRevE.65.051504.  Google Scholar

[33]

W. WangL. Zhang and P. Zhang, Modelling and computation of liquid crystals, Acta Numer., 30 (2021), 765-851.  doi: 10.1017/S0962492921000088.  Google Scholar

[34]

W. WangP. Zhang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation, Comm. Pure Appl. Math., 68 (2015), 1326-1398.  doi: 10.1002/cpa.21549.  Google Scholar

[35]

H. Yu and P. Zhang, A kinetic hydrodynamic simulation of microstructure of liquid crystal polymers in plane shear flow, J. Non-Newton. Fluid Mech., 141 (2007), 116-127.  doi: 10.1016/j.jnnfm.2006.09.005.  Google Scholar

[36]

H. Zhang and P. Zhang, On the new multiscale rodlike model of polymeric fluids, SIAM J. Math. Anal., 40 (2008), 1246-1271.  doi: 10.1137/050640795.  Google Scholar

show all references

References:
[1]

N. Arcozzi, Riesz transforms on compact Lie groups, spheres and Gauss space, Ark. Mat., 36 (1998), 201-231.  doi: 10.1007/BF02384766.  Google Scholar

[2]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[3]

J. M. Ball, E. Feireisl and F. Otto, Mathematical Thermodynamics of Complex Fluids, Springer, Cham, Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2017.  Google Scholar

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations II convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[5]

J. W. Barrett and E. Süeli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci., 21 (2011), 1211-1289.  doi: 10.1142/S0218202511005313.  Google Scholar

[6]

X. Chen and J.-G. Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions, J. Differential Equations, 254 (2013), 2764-2802.  doi: 10.1016/j.jde.2013.01.005.  Google Scholar

[7]

A. Cotsiolis and N. Labropoulos, Sharp Nash inequalities on the unit sphere: The influence of symmetries, Nonlinear Anal., 75 (2012), 612-624.  doi: 10.1016/j.na.2011.08.063.  Google Scholar

[8]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[9]

R. J. DiPerna and A. Majda, Reduced Hausdorff dimension and concentration-cancellation for two dimensional incompressible flow, J. Amer. Math. Soc., 1 (1988), 59-95.  doi: 10.2307/1990967.  Google Scholar

[10]

M. Doi, Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases, Journal of Polymer, 19 (1981), 229-243.  doi: 10.1002/pol.1981.180190205.  Google Scholar

[11]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1988. Google Scholar

[12]

W. E. and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods Appl. Anal., 13 (2006), 181-198.  doi: 10.4310/MAA.2006.v13.n2.a5.  Google Scholar

[13]

L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 74. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. doi: 10.1090/cbms/074.  Google Scholar

[14]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. doi: 10.1093/acprof:oso/9780198528388.003.0002.  Google Scholar

[15]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18] P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[19]

P.-L. Lions and N. Masmoudi, Global existence of weak solutions to some micro-macro models, C. R. Math. Acad. Sci. Paris, 345 (2007), 15-20.  doi: 10.1016/j.crma.2007.05.011.  Google Scholar

[20]

P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146.  doi: 10.1142/S0252959900000170.  Google Scholar

[21]

Y. Liu and W. Wang, The small Deborah number limit of the Doi-Onsager equation without hydrodynamics, J. Funct. Anal., 275 (2018), 2740-2793.  doi: 10.1016/j.jfa.2018.07.013.  Google Scholar

[22]

W. Maier and A. Saupe, Eine einfache molekulare theorie des nematischen kristallinflüssigen Zustandes, Zeitschrift für Naturforschung A, 13 (1958), 564–566. doi: 10.1515/zna-1958-0716.  Google Scholar

[23]

G. Marrucci and F. Greco, The elastic constants of Maier-Saupe rodlike molecule nematics, Molecular Crystals and Liquid Crystals, 206 (1991), 17-30.  doi: 10.1080/00268949108037714.  Google Scholar

[24]

N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500.  doi: 10.1007/s00222-012-0399-y.  Google Scholar

[25]

L. Onsager, The effects of shape on the interaction of colloidal particles, Annals of the New York Academy of Sciences, 51 (1949), 627-659.  doi: 10.1111/j.1749-6632.1949.tb27296.x.  Google Scholar

[26]

A. D. Rey and T. Tsuji, Recent advances in theoretical liquid crystal rheology, Macromolecular Theory and Simulations, 7 (1998), 623-639.   Google Scholar

[27]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations: Classical Theory, Cambridge University Press, 2016. doi: 10.1017/CBO9781139095143.  Google Scholar

[28]

O. Sieber, Existence of global weak solutions to an inhomogeneous Doi model for active liquid crystals, preprint, arXiv: 2006.16832. Google Scholar

[29]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[31]

T. Tsuji and A. D. Rey, Orientation mode selection mechanisms for sheared nematic liquid crystalline materials, Physical Review E, 57 (1998), 5609.  doi: 10.1103/PhysRevE.57.5609.  Google Scholar

[32]

Q. WangW. E. C. Liu and P. Zhang, Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential, Physical Review E, 65 (2002), 051504.  doi: 10.1103/PhysRevE.65.051504.  Google Scholar

[33]

W. WangL. Zhang and P. Zhang, Modelling and computation of liquid crystals, Acta Numer., 30 (2021), 765-851.  doi: 10.1017/S0962492921000088.  Google Scholar

[34]

W. WangP. Zhang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation, Comm. Pure Appl. Math., 68 (2015), 1326-1398.  doi: 10.1002/cpa.21549.  Google Scholar

[35]

H. Yu and P. Zhang, A kinetic hydrodynamic simulation of microstructure of liquid crystal polymers in plane shear flow, J. Non-Newton. Fluid Mech., 141 (2007), 116-127.  doi: 10.1016/j.jnnfm.2006.09.005.  Google Scholar

[36]

H. Zhang and P. Zhang, On the new multiscale rodlike model of polymeric fluids, SIAM J. Math. Anal., 40 (2008), 1246-1271.  doi: 10.1137/050640795.  Google Scholar

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