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

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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 - A, 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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