-
Previous Article
Local well-posedness and blow-up criteria of magneto-viscoelastic flows
- DCDS Home
- This Issue
-
Next Article
Moving planes for nonlinear fractional Laplacian equation with negative powers
Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions
1. | Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany |
2. | Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany |
We study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.
References:
[1] |
A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, New York, 1994. |
[2] |
D. Breit, E. Feireisl and M. Hofmanová,
Incompressible limit for compressible fluids with stochastic forcing, Arch. Rational Mech. Anal., 222 (2016), 895-926.
doi: 10.1007/s00205-016-1014-y. |
[3] |
C. Cavaterra, E. Rocca and H. Wu,
Global weak solution and blow-up criterion of the general Ericksen—Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57.
doi: 10.1016/j.jde.2013.03.009. |
[4] |
C. M. Dafermos,
The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., 70 (1979), 167-179.
doi: 10.1007/BF00250353. |
[5] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2016.
doi: 10.1007/978-3-662-49451-6. |
[6] |
M. Dai,
Existence of regular solutions to an Ericksen—Leslie model of the liquid crystal system, Commun. Math. Sci., 13 (2015), 1711-1740.
doi: 10.4310/CMS.2015.v13.n7.a4. |
[7] |
M. Dai, J. Qing and M. Schonbek,
Regularity of solutions to the liquid crystals systems in $\mathbb R^2$ and $\mathbb R^3$, Nonlinearity, 25 (2012), 513-532.
doi: 10.1088/0951-7715/25/2/513. |
[8] |
J. Diestel and J. J. Uhl, Jr. Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977. |
[9] |
E. Emmrich and R. Lasarzik, Existence of weak solutions to the Ericksen-Leslie model for a general class of free energies, arXiv e-prints, 1711.10277, 2017. Google Scholar |
[10] |
J. L. Ericksen,
Conservation laws for liquid crystals, J. Rheol., 5 (1961), 23-34.
doi: 10.1122/1.548883. |
[11] |
E. Feireisl,
Relative entropies in thermodynamics of complete fluid systems, Discrete Contin. Dyn. Syst., 32 (2012), 3059-3080.
doi: 10.3934/dcds.2012.32.3059. |
[12] |
E. Feireisl, Relative entropies, dissipative solutions, and singular limits of complete fluid systems, In Hyperbolic Problems: Theory, Numerics, Applications, volume 8 of AIMS on Applied Mathematics, pages 11-27. AIMS, Springfield, USA, 2014. |
[13] |
E. Feireisl, B. J. Jin and A. Novotný,
Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier—Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.
doi: 10.1007/s00021-011-0091-9. |
[14] |
E. Feireisl and A. Novotný,
Weak-strong uniqueness property for the full Navier—Stokes—Fourier system, Arch. Ration. Mech. Anal., 204 (2012), 683-706.
doi: 10.1007/s00205-011-0490-3. |
[15] |
E. Feireisl, A. Novotný and Y. Sun,
Suitable weak solutions to the Navier—Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., 60 (2011), 611-631.
doi: 10.1512/iumj.2011.60.4406. |
[16] |
J. Fischer,
A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier—Stokes equation, SIAM J. Numer. Anal., 53 (2015), 2178-2205.
doi: 10.1137/140966654. |
[17] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferential-Gleichungen, Akademie-Verlag, Berlin, 1974. |
[18] |
R. Lasarzik, Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy, arXiv: 1711.04638, Nov. 2017. Google Scholar |
[19] |
R. Lasarzik, Weak-strong uniqueness for measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank free energy, arXiv: 1711.03371, Nov. 2017. Google Scholar |
[20] |
F. M. Leslie,
Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[21] |
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. |
[22] |
F.-H. Lin and C. Liu,
Existence of solutions for the Ericksen—Leslie system, Arch. Rational Mech. Anal., 154 (2000), 135-156.
doi: 10.1007/s002050000102. |
[23] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1, The Clarendon Press, New York, 1996. |
[24] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. |
[25] |
C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966. |
[26] |
G. Prodi,
Un teorema di unicità per le equazioni di Navier—Stokes, Ann. Mat. Pura Appl.(4), 48 (1959), 173-182.
doi: 10.1007/BF02410664. |
[27] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013.
doi: 10.1007/978-3-0348-0513-1. |
[28] |
J. Serrin,
On the interior regularity of weak solutions of the Navier—Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.
doi: 10.1007/BF00253344. |
[29] |
Y.-F. Yang, C. Dou and Q. Ju,
Weak-strong uniqueness property for the compressible flow of liquid crystals, J. Differential Equations, 255 (2013), 1233-1253.
doi: 10.1016/j.jde.2013.05.011. |
[30] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
[31] |
J.-H. Zhao and Q. Liu,
Weak-strong uniqueness of hydrodynamic flow of nematic liquid crystals, Electron. J. Differential Equations, 2012 (2012), 1-16.
|
show all references
References:
[1] |
A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, New York, 1994. |
[2] |
D. Breit, E. Feireisl and M. Hofmanová,
Incompressible limit for compressible fluids with stochastic forcing, Arch. Rational Mech. Anal., 222 (2016), 895-926.
doi: 10.1007/s00205-016-1014-y. |
[3] |
C. Cavaterra, E. Rocca and H. Wu,
Global weak solution and blow-up criterion of the general Ericksen—Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57.
doi: 10.1016/j.jde.2013.03.009. |
[4] |
C. M. Dafermos,
The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., 70 (1979), 167-179.
doi: 10.1007/BF00250353. |
[5] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2016.
doi: 10.1007/978-3-662-49451-6. |
[6] |
M. Dai,
Existence of regular solutions to an Ericksen—Leslie model of the liquid crystal system, Commun. Math. Sci., 13 (2015), 1711-1740.
doi: 10.4310/CMS.2015.v13.n7.a4. |
[7] |
M. Dai, J. Qing and M. Schonbek,
Regularity of solutions to the liquid crystals systems in $\mathbb R^2$ and $\mathbb R^3$, Nonlinearity, 25 (2012), 513-532.
doi: 10.1088/0951-7715/25/2/513. |
[8] |
J. Diestel and J. J. Uhl, Jr. Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977. |
[9] |
E. Emmrich and R. Lasarzik, Existence of weak solutions to the Ericksen-Leslie model for a general class of free energies, arXiv e-prints, 1711.10277, 2017. Google Scholar |
[10] |
J. L. Ericksen,
Conservation laws for liquid crystals, J. Rheol., 5 (1961), 23-34.
doi: 10.1122/1.548883. |
[11] |
E. Feireisl,
Relative entropies in thermodynamics of complete fluid systems, Discrete Contin. Dyn. Syst., 32 (2012), 3059-3080.
doi: 10.3934/dcds.2012.32.3059. |
[12] |
E. Feireisl, Relative entropies, dissipative solutions, and singular limits of complete fluid systems, In Hyperbolic Problems: Theory, Numerics, Applications, volume 8 of AIMS on Applied Mathematics, pages 11-27. AIMS, Springfield, USA, 2014. |
[13] |
E. Feireisl, B. J. Jin and A. Novotný,
Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier—Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.
doi: 10.1007/s00021-011-0091-9. |
[14] |
E. Feireisl and A. Novotný,
Weak-strong uniqueness property for the full Navier—Stokes—Fourier system, Arch. Ration. Mech. Anal., 204 (2012), 683-706.
doi: 10.1007/s00205-011-0490-3. |
[15] |
E. Feireisl, A. Novotný and Y. Sun,
Suitable weak solutions to the Navier—Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., 60 (2011), 611-631.
doi: 10.1512/iumj.2011.60.4406. |
[16] |
J. Fischer,
A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier—Stokes equation, SIAM J. Numer. Anal., 53 (2015), 2178-2205.
doi: 10.1137/140966654. |
[17] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferential-Gleichungen, Akademie-Verlag, Berlin, 1974. |
[18] |
R. Lasarzik, Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy, arXiv: 1711.04638, Nov. 2017. Google Scholar |
[19] |
R. Lasarzik, Weak-strong uniqueness for measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank free energy, arXiv: 1711.03371, Nov. 2017. Google Scholar |
[20] |
F. M. Leslie,
Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[21] |
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. |
[22] |
F.-H. Lin and C. Liu,
Existence of solutions for the Ericksen—Leslie system, Arch. Rational Mech. Anal., 154 (2000), 135-156.
doi: 10.1007/s002050000102. |
[23] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1, The Clarendon Press, New York, 1996. |
[24] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. |
[25] |
C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966. |
[26] |
G. Prodi,
Un teorema di unicità per le equazioni di Navier—Stokes, Ann. Mat. Pura Appl.(4), 48 (1959), 173-182.
doi: 10.1007/BF02410664. |
[27] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013.
doi: 10.1007/978-3-0348-0513-1. |
[28] |
J. Serrin,
On the interior regularity of weak solutions of the Navier—Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.
doi: 10.1007/BF00253344. |
[29] |
Y.-F. Yang, C. Dou and Q. Ju,
Weak-strong uniqueness property for the compressible flow of liquid crystals, J. Differential Equations, 255 (2013), 1233-1253.
doi: 10.1016/j.jde.2013.05.011. |
[30] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
[31] |
J.-H. Zhao and Q. Liu,
Weak-strong uniqueness of hydrodynamic flow of nematic liquid crystals, Electron. J. Differential Equations, 2012 (2012), 1-16.
|
[1] |
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 |
[2] |
Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 |
[3] |
Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 |
[4] |
Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 |
[5] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[6] |
Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 |
[7] |
Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348 |
[8] |
Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 |
[9] |
Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161 |
[10] |
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 |
[11] |
Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021015 |
[12] |
Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 |
[13] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
[14] |
Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083 |
[15] |
Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 |
[16] |
Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020320 |
[17] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[18] |
Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020384 |
[19] |
Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 |
[20] |
Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]