Dissipative solution to the Ericksen--Leslie system equipped with the Oseen--Frank energy
- Lasarzik, Robert
2010 Mathematics Subject Classification
- 35Q35 35K52 35R06 76A15
- Liquid crystal, Ericksen-Leslie equation, dissipative solutions, measure-valued solutions, long-time behavior, relative energy
We analyze the Ericksen-Leslie system equipped with the Oseen?Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to 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.