Variational wave equations in liquid crystals

The goal of the project is to use mathematical models of liquid crystals to calculate how waves propagate in such media. These models are non-linear systems of partial differential equations, and cannot be solved in ?closed form?. Therefore one must use a numerical method to gain some understanding of how the crystals behave. This is closely tied to the notion of ?well-posedness? of the model, i.e., whether it is possible to rigorously define a solution, and whether small variation in the initial conditions or parameters will lead to large variations in the solutions. In the past year we have worked chiefly along two main directions. The first concerns the numerical simulation of models for waves in liquid crystals, where the crystals are confined in a finite domain, and where there are restrictions on the behaviour of the crystals on the boundary of the domain. The is a an important problem, since modern LDC screen consist of many small container with crystals whose orientation is controlled via the orientation of the crystals at the boundary. For this problem we have developed numerical methods which reproduce experimental results. The other main direction concerns the convergence of numerical methods for non-linear wave equations which are ?similar? to liquid crystal models in the sense that they contain some, but not all, features of the full models. We have shown convergence of numerical schemes for several such equations, and the hope is that this will aid the understanding of the full liquid crystal models.

Modeling using elastic continuum theory and the Oseen-Franck free energy density, the equations describing waves in stationary liquid crystals are second order nonlinear wave equations. We aim to study these equations under various assumptions; planar on e-dimensional waves, planar two-dimensional waves, general one-dimensional waves, etc. We aim at establishing a mathematical theory for well-posedness of the initial value problem for these equations, both in the case of artificial and of physical wave sp eed functions. Furthermore, we shall study the convergence of numerical methods for these models. We also aim at developing accurate higher order numerical schemes (e.g. based on finite volume approximations) for variational waves.