WIAS Preprint No. 2535, (2018)

Approximation and optimal control of dissipative solutions to the Ericksen--Leslie system



Authors

  • Lasarzik, Robert

2010 Mathematics Subject Classification

  • 35A35 35Q35 49J20 76A15

Keywords

  • Liquid crystal, Ericksen--Leslie equation, weak-strong-uniqueness, dissipative solutions, optimal control

DOI

10.20347/WIAS.PREPRINT.2535

Abstract

We analyze the Ericksen--Leslie system equipped with the Oseen--Frank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measure-valued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relative simple scheme, which fulfills the norm-restriction on the director in every step. We introduce a semi-discrete scheme and derive an approximated version of the relative-energy inequality for solutions of this scheme. Passing to the limit in the semi-discretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, show the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semi-continuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity.

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