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Dissipative solution to the Ericksen–Leslie system equipped with the Oseen–Frank energy

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Abstract

We analyze the Ericksen–Leslie system equipped with the Oseen–Frank energy in three space dimensions and introduce the concept of dissipative solutions for this system. It is shown that the expectation of a measure-valued solution, which was recently introduced by the author, is a dissipative solution. The concept of a dissipative solution itself relies on an relative energy inequality and avoids the description by parametrized measures. These solutions exist globally and fulfill the weak–strong uniqueness property. Additionally, we generalize the relative energy inequality for measure-valued as well as dissipative solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state.

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References

  1. Arsénio, D., Saint-Raymond, L.: From the Vlasov–Maxwell–Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. ArXiv e-prints (2016)

  2. Becker, R., Feng, X., Prohl, A.: Finite element approximations of the Ericksen–Leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46(4), 1704–1731 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cavaterra, C., Rocca, E., Wu, H.: Global weak solution and blow-up criterion of the general Ericksen–Leslie system for nematic liquid crystal flows. J. Differ. Equ. 255(1), 24–57 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Christof, C., Meyer, C., Walther, S., Clason, C.: Optimal control of a non-smooth semilinear elliptic equation. Math. Control Relat. Fields 8(1), 247–276 (2018)

    Article  MathSciNet  Google Scholar 

  5. Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70(2), 167–179 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2016)

    MATH  Google Scholar 

  7. De Gennes, P.G.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1974)

    MATH  Google Scholar 

  8. Diestel, J., Uhl Jr., J.J.: Vector Measures. American Mathematical Society, Providence (1977)

    Book  MATH  Google Scholar 

  9. Edwards, R.E.: Functional Analysis. Theory and Applications. Holt Rinehart and Winston, New York (1965)

    MATH  Google Scholar 

  10. Emmrich, E., Klapp, S.H., Lasarzik, R.: Nonstationary models for liquid crystals: a fresh mathematical perspective. J. Non-Newton. Fluid Mech. 259, 32–47 (2018)

    Article  MathSciNet  Google Scholar 

  11. Emmrich, E., Lasarzik, R.: Existence of weak solutions to the Ericksen–Leslie model for a general class of free energies. Math. Methods Appl. Sci. 41(16), 6492–6518 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feireisl, E.: Relative entropies in thermodynamics of complete fluid systems. Discrete Contin. Dyn. Syst. 32(9), 3059–3080 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Feireisl, E.: Relative entropies, dissipative solutions, and singular limits of complete fluid systems. In: Bressan, A., Ancona, F., Marcati, P., Marson, A. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, Volume 8 of AIMS on Applied Mathematics, pp. 11–28. AIMS, Springfield (2014)

  14. Feireisl, E., Novotný, A.: Weak–strong uniqueness property for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204(2), 683 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feireisl, E., Rocca, E., Schimperna, G.: On a non-isothermal model for nematic liquid crystals. Nonlinearity 24(1), 243–257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Feireisl, E., Rocca, E., Schimperna, G., Zarnescu, A.: On a hyperbolic system arising in liquid crystals modeling. J. Hyperbolic Differ. Equ. 15(1), 15–35 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferential-Gleichungen. Akademie-Verlag, Berlin (1974)

    MATH  Google Scholar 

  18. Grunert, K., Holden, H., Raynaud, X.: Global dissipative solutions of the two-component Camassa–Holm system for initial data with nonvanishing asymptotics. Nonlinear Anal. Real World Appl. 17, 203–244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hieber, M., Nesensohn, M., Prüss, J., Schade, K.: Dynamics of nematic liquid crystal flows: the quasilinear approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(2), 397–408 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hong, M.-C., Li, J., Xin, Z.: Blow-up criteria of strong solutions to the Ericksen–Leslie system in \(\mathbb{R}^3\). Commun. Part. Differ. Equ. 39(7), 1284–1328 (2014)

    MATH  Google Scholar 

  22. Hong, M.-C., Xin, Z.: Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in \(\mathbb{R}^2\). Adv. Math. 231(3–4), 1364–1400 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Kinderlehrer, D., Pedregal, P.: Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115(4), 329–365 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kinderlehrer, D., Pedregal, P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4(1), 59–90 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kristensen, J., Rindler, F.: Characterization of generalized gradient Young measures generated by sequences in \({\varvec {W}}^{1,1}\) and BV. Arch. Ration. Mech. Anal. 197(2), 539–598 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Lasarzik, R.: Approximation and optimal control of dissipative solutions to the Ericksen–Leslie system. WIAS, Preprint 2535 (2018). https://doi.org/10.20347/WIAS.PREPRINT.2535

  27. Lasarzik, R.: Measure-valued solutions to the Ericksen–Leslie model equipped with the Oseen–Frank energy. Nonlinear Anal. 179, 146–183 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lasarzik, R.: Weak–strong uniqueness for measure-valued solutions to the Ericksen–Leslie model equipped with the Oseen–Frank free energy. J. Math. Anal. Appl. 470(1), 36–90 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28(4), 265–283 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lin, F.-H., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48(5), 501–537 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lin, F.-H., Liu, C.: Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2(1), 1–22 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lin, F.-H., Liu, C.: Existence of solutions for the Ericksen–Leslie system. Arch. Ration. Mech. Anal. 154(2), 135–156 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1. Dunod, Paris (1968)

    MATH  Google Scholar 

  34. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications. I, II. J. Math. Kyoto Univ. 34(2), 391–427, 429–461 (1994)

  35. Lions, P.-L.: Mathematical Topics in Fluid Mechanics, vol. 1. The Clarendon Press, New York (1996)

    MATH  Google Scholar 

  36. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  37. Mielke, A.: On evolutionary \(\Gamma \)-convergence for gradient systems. In: Muntean, A., Rademacher, J., Zagaris, A. (eds.) Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Volume 3 of the Lecture Notes in Applied Mathematics and Mechanics, pp. 187–249. Springer, Berlin (2016)

    Chapter  Google Scholar 

  38. Olmsted, P.D., Goldbart, P.: Theory of the nonequilibrium phase transition for nematic liquid crystals under shear flow. Phys. Rev. A 41, 4578–4581 (1990)

    Article  Google Scholar 

  39. Roubíček, T.: Relaxation in Optimization Theory and Variational Calculus. Volume 4 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin (1997)

    Book  MATH  Google Scholar 

  40. Roubíček, T.: Optimality conditions for nonconvex variational problems relaxed in terms of Young measures. Kybernetika 34(3), 335–347 (1998)

    MathSciNet  MATH  Google Scholar 

  41. Roubíček, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005)

    MATH  Google Scholar 

  42. Roubíček, T., Hoffmann, K.-H.: About the concept of measure-valued solutions to distributed parameter systems. Math. Methods Appl. Sci. 18(9), 671–685 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  43. Villani, C.: Optimal Transport. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  44. Vorotnikov, D.A.: Dissipative solutions for equations of viscoelastic diffusion in polymers. J. Math. Anal. Appl. 339(2), 876–888 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, W., Zhang, P., Zhang, Z.: Well-posedness of the Ericksen–Leslie system. Arch. Ration. Mech. Anal. 210(3), 837–855 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zeidler, E.: Nonlinear Functional Analysis and Its Applications: III: Variational Methods and Optimization. Springer, New York (2013)

    Google Scholar 

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  1. Weierstrass Institute, Mohrenstr. 39, 10117, Berlin, Germany

    Robert Lasarzik

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Lasarzik, R. Dissipative solution to the Ericksen–Leslie system equipped with the Oseen–Frank energy. Z. Angew. Math. Phys. 70, 8 (2019). https://doi.org/10.1007/s00033-018-1053-3

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