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. 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.

arXiv commentarXiv admin note: text overlap with arXiv:1711.03371, arXiv:2001.01509

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• Analysis of a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model
• Analysis and numerical approximation of energy-variational solutions to the Ericksen--Leslie equations
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