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- ItemWeak-strong uniqueness for the general Ericksen-Leslie system in three dimensions(Springfield, Mo. : American Institute of Mathematical Sciences, 2018) Emmrich, Etienne; Lasarzik, RobertWe study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.
- ItemExistence of weak solutions to a dynamic model for smectic-A liquid crystals under undulations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Emmrich, Etienne; Lasarzik, RobertA nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smectic-A liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the three-dimensional case.
- ItemAn existence result and evolutionary [Gamma]-convergence for perturbed gradient systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bacho, Aras; Emmrich, Etienne; Mielke, AlexanderWe consider the initial-value problem for the perturbed gradient flows, where a differential inclusion is formulated in terms of a subdifferential of an energy functional, a subdifferential of a dissipation potential and a more general perturbation, which is assumed to be continuous and to satisfy a suitable growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique.