M. S. Romanov

Existence and Uniqueness Theorems in Two-Dimensional Nematodynamics. Finite Speed of Propagation

The paper is devoted to the two-dimensional Ericksen–Leslie system describing the nematodinamics of liquid crystals. The moment of inertia of molecules is supposed to be strictly positive. The existence of the solution was proved in the case of periodic domain and in the case of bounded domain. In the last case media is supposed to adhere to the solid surface, the director vector field describing orientation of the mole-cules is constant in the neighbourhood of the boundary. The uniqueness of the strong solution was proved in both cases. Also we prove the propagation of director disturbance has finite speed. This fact shows the differ-ence between the model under consideration and models with zero moment of inertia of the molecules. The estimate of the speed of propagation depending on physical properties of the liquid crystal and the flow was obtained.

Existence and uniqueness theorems for the full three-dimensional Ericksen–Leslie system

In this paper, we study the three-dimensional Ericksen–Leslie equations for the nematodynamics of liquid crystals. We prove short time existence and uniqueness of strong solutions for the initial value problem for the periodic case and in bounded domains with both Dirichlet- and Neumann-type boundary conditions.

Fokker–Planck–Kolmogorov equations with a partially degenerate diffusion matrix

The Fokker–Planck–Kolmogorov equations with a degenerate or partially degenerate diffusion matrix are considered. The distance between probability solutions of these equations with different drift coefficients and different initial conditions is estimated. Sufficient conditions for the existence and uniqueness of probability solutions to nonlinear Fokker–Planck–Kolmogorov equations with a partially degenerate diffusion matrix are established.