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

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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Correspondence to 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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Keywords

  • Liquid crystal
  • Ericksen–Leslie equation
  • Dissipative solutions
  • Measure-valued solutions
  • Long-time behavior
  • Relative energy

Mathematics Subject Classification

  • 35Q35
  • 35K55
  • 35R06
  • 35R45
  • 76A15