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Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one
Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system
1. | School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, China |
2. | Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China |
3. | LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing, 100871 |
References:
[1] |
H. Abels, G. Dolzmann and Y. Liu, Well-posedness of a fully-coupled Navier-Stokes/$Q$-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077.
doi: 10.1137/130945405. |
[2] |
H. Abels, G. Dolzmann and Y. Liu, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions,, preprint, ().
|
[3] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Fundamental Principles of Mathematical Sciences, 343, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[4] |
J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525 (2010), 1-11.
doi: 10.1080/15421401003795555. |
[5] |
A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford Engrg. Sci. Ser., 36, Oxford University Press, Oxford, New York, 1994. |
[6] |
P. G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. |
[7] |
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. |
[8] |
J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34.
doi: 10.1122/1.548883. |
[9] |
I. Fatkullin and V. Slastikov, Critical points of the Onsager functional on a sphere, Nonlinearity, 18 (2005), 2565-2580.
doi: 10.1088/0951-7715/18/6/008. |
[10] |
J. Feng, C. V. Chaubal and L. G. Leal, Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers?, J. Rheol., 42 (1998), 1095-1119.
doi: 10.1122/1.550920. |
[11] |
J. Feng, G. Sgalari and L. G. Leal, A theory for flowing nematic polymers with orientational distortion, J. Rheol., 44 (2000), 1085-1101.
doi: 10.1122/1.1289278. |
[12] |
J. Han, Y. Luo, W. Wang, P. Zhang and Z. Zhang, From microscopic theory to macroscopic theory: A systematic study on modeling for liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 741-809.
doi: 10.1007/s00205-014-0792-3. |
[13] |
J. Huang and S. Ding, Global well-posedness for a coupled incompressible Navier-Stokes and Q-tensor system,, preprint, ().
|
[14] |
J. Huang, F. H Lin and C. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb R^2$, Commun. Math. Phys., 331 (2014), 805-850.
doi: 10.1007/s00220-014-2079-9. |
[15] |
N. Kuzuu and M. Doi, Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation, J. Phys. Soc. Japan, 52 (1983), 3486-3494.
doi: 10.1143/JPSJ.52.3486. |
[16] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[17] |
F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156.
doi: 10.1007/s002050000102. |
[18] |
H. Liu, H. Zhang and P. Zhang, Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential, Comm. Math. Sci., 3 (2005), 201-218.
doi: 10.4310/CMS.2005.v3.n2.a7. |
[19] |
A. Majumdar and A. Zarnescu, Landau-De Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280.
doi: 10.1007/s00205-009-0249-2. |
[20] |
N. J. Mottram and C. J. P. Newton, Introduction to $Q$-tensor theory,, preprint, ().
|
[21] |
M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and $Q$-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67.
doi: 10.1007/s00205-011-0443-x. |
[22] |
M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and $Q$-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049.
doi: 10.1137/10079224X. |
[23] |
O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. France, 31 (1970), 581-584.
doi: 10.1051/jphys:01970003107058100. |
[24] |
T. Qian and P. Sheng, Generalized hydrodynamic equations for nematic liquid crystals, Phys. Rev. E, 58 (1998), 7475-7485.
doi: 10.1103/PhysRevE.58.7475. |
[25] |
H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Birkhäuser Verlag, Basel, Boston, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[26] |
M. Wang and W. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differ. Equ., 51 (2014), 915-962.
doi: 10.1007/s00526-013-0700-y. |
[27] |
W. Wang, P. 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. |
[28] |
W. Wang, P. Zhang and Z. Zhang, Well-posedness of the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 210 (2013), 837-855.
doi: 10.1007/s00205-013-0659-z. |
[29] |
W. Wang, P. Zhang and Z. Zhang, Rigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theory, SIAM J. Math. Anal., 47 (2015), 127-158.
doi: 10.1137/13093529X. |
show all references
References:
[1] |
H. Abels, G. Dolzmann and Y. Liu, Well-posedness of a fully-coupled Navier-Stokes/$Q$-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077.
doi: 10.1137/130945405. |
[2] |
H. Abels, G. Dolzmann and Y. Liu, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions,, preprint, ().
|
[3] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Fundamental Principles of Mathematical Sciences, 343, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[4] |
J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525 (2010), 1-11.
doi: 10.1080/15421401003795555. |
[5] |
A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford Engrg. Sci. Ser., 36, Oxford University Press, Oxford, New York, 1994. |
[6] |
P. G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. |
[7] |
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. |
[8] |
J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34.
doi: 10.1122/1.548883. |
[9] |
I. Fatkullin and V. Slastikov, Critical points of the Onsager functional on a sphere, Nonlinearity, 18 (2005), 2565-2580.
doi: 10.1088/0951-7715/18/6/008. |
[10] |
J. Feng, C. V. Chaubal and L. G. Leal, Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers?, J. Rheol., 42 (1998), 1095-1119.
doi: 10.1122/1.550920. |
[11] |
J. Feng, G. Sgalari and L. G. Leal, A theory for flowing nematic polymers with orientational distortion, J. Rheol., 44 (2000), 1085-1101.
doi: 10.1122/1.1289278. |
[12] |
J. Han, Y. Luo, W. Wang, P. Zhang and Z. Zhang, From microscopic theory to macroscopic theory: A systematic study on modeling for liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 741-809.
doi: 10.1007/s00205-014-0792-3. |
[13] |
J. Huang and S. Ding, Global well-posedness for a coupled incompressible Navier-Stokes and Q-tensor system,, preprint, ().
|
[14] |
J. Huang, F. H Lin and C. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb R^2$, Commun. Math. Phys., 331 (2014), 805-850.
doi: 10.1007/s00220-014-2079-9. |
[15] |
N. Kuzuu and M. Doi, Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation, J. Phys. Soc. Japan, 52 (1983), 3486-3494.
doi: 10.1143/JPSJ.52.3486. |
[16] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[17] |
F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156.
doi: 10.1007/s002050000102. |
[18] |
H. Liu, H. Zhang and P. Zhang, Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential, Comm. Math. Sci., 3 (2005), 201-218.
doi: 10.4310/CMS.2005.v3.n2.a7. |
[19] |
A. Majumdar and A. Zarnescu, Landau-De Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280.
doi: 10.1007/s00205-009-0249-2. |
[20] |
N. J. Mottram and C. J. P. Newton, Introduction to $Q$-tensor theory,, preprint, ().
|
[21] |
M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and $Q$-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67.
doi: 10.1007/s00205-011-0443-x. |
[22] |
M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and $Q$-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049.
doi: 10.1137/10079224X. |
[23] |
O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. France, 31 (1970), 581-584.
doi: 10.1051/jphys:01970003107058100. |
[24] |
T. Qian and P. Sheng, Generalized hydrodynamic equations for nematic liquid crystals, Phys. Rev. E, 58 (1998), 7475-7485.
doi: 10.1103/PhysRevE.58.7475. |
[25] |
H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Birkhäuser Verlag, Basel, Boston, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[26] |
M. Wang and W. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differ. Equ., 51 (2014), 915-962.
doi: 10.1007/s00526-013-0700-y. |
[27] |
W. Wang, P. 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. |
[28] |
W. Wang, P. Zhang and Z. Zhang, Well-posedness of the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 210 (2013), 837-855.
doi: 10.1007/s00205-013-0659-z. |
[29] |
W. Wang, P. Zhang and Z. Zhang, Rigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theory, SIAM J. Math. Anal., 47 (2015), 127-158.
doi: 10.1137/13093529X. |
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