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September  2018, 38(9): 4617-4635. doi: 10.3934/dcds.2018202

Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions

Etienne Emmrich 1,, and  Robert Lasarzik 2,

1. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany
2. 

Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany

* Corresponding author

Received  November 2017 Revised  April 2018 Published  June 2018

Fund Project: This work was funded by CRC 901 Control of self-organizing nonlinear systems: Theoretical methods and concepts of application (Project A8).

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.

Keywords: Liquid crystal, Ericksen-Leslie equation, existence, weak solution, weak-strong uniqueness.
Mathematics Subject Classification: Primary: 35Q35, 35D30; Secondary: 35K52, 76A15.
Citation: Etienne Emmrich, Robert Lasarzik. Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4617-4635. doi: 10.3934/dcds.2018202
References:
[1]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Diestel and J. J. Uhl, Jr. Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferential-Gleichungen, Akademie-Verlag, Berlin, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1, The Clarendon Press, New York, 1996.  Google Scholar

[24]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  Google Scholar

[25]

C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.  Google Scholar

[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.  Google Scholar

[27]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Diestel and J. J. Uhl, Jr. Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferential-Gleichungen, Akademie-Verlag, Berlin, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1, The Clarendon Press, New York, 1996.  Google Scholar

[24]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  Google Scholar

[25]

C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.  Google Scholar

[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.  Google Scholar

[27]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[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.   Google Scholar

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