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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

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