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

  • Robert Lasarzik
Article
  • 27 Downloads

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.

Keywords

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

Mathematics Subject Classification

35Q35 35K55 35R06 35R45 76A15 

Notes

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Weierstrass InstituteBerlinGermany

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