Traveling waves of director fields

The heat flow of harmonic maps from an infinitely long vertical cylinder of radius R, $\Omega= \{(x_1,x_2,x_3): x_12+x_22<R2\}$, to the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ is the vector equation: \[ u_t \,=\, \Delta u + | \nabla u |^2 u \quad\text{ in }\Omega\times \mathbb{R}\,. \] Here $u: \Omega \times \mathbb{R} \longrightarrow \mathbb{S}^2$ is a vector field in $\mathbb{R}^3$ with $|u| = 1$, $\Delta u = (\Delta u_1, \Delta u_2, \Delta u_3)$ and $|\nabla u|^2 = \sum\limits_{i,\,j=1}^{3} \left| \frac{\partial u_i}{\partial x_j} \right|^2$.<br />
This equation is the simplest possible one within a class of evolution equations for director fields which naturally arise in the study of the orientation of nematic liquid crystals and microscopic magnetic dipoles in ferromagnetic materials.

We show and discuss several results concerning the construction and the properties of axially symmetric traveling wave solutions for the previous equation with one point singularity on the vertical axis of the cylinder $\Omega$. Such a kind of solutions can be used to prove new interesting non-uniqueness results for the weak solutions of the flow.