We have studied the two-dimensional axisymmetric Ericksen–Leslie equations with the non-zero inertial constant. The existence and uniqueness of the global weak solution is proved under certain assumptions. In the neighborhood of the solid surface we derived the boundary layer equations and show that the effective viscosity of the liquid in the boundary layer decreases as the inertial constant grows. Under the assumption that the Reynolds number is large enough we prove the solution to the boundary layer equations to be close to the solution of the original problem in the Sobolev space.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

We introduce the three-dimensional Eringen system of equations for the nematodynamics of liquid crystals, announce the short time existence and uniqueness of strong solutions for the one-dimensional problem in the periodic case, and show the continuous dependence of the solution on the initial data.

Using a metric which interpolates between the Kantorovich metric and the total variation norm we estimate the distance between solutions to Fokker–Planck–Kolmogorov equations with degenerate diffusion matrices. Some relations between the degeneracy of the diffusion matrix and the regularity of the drift coefficient are analysed. Applications to nonlinear Fokker–Planck–Kolmogorov equations are given.

Рассматриваются уравнения Фоккера-Планка-Колмогорова с вырожденной или частично вырожденной матрицей диффузии. Получены оценки расстояния между вероятностными решениями уравнений Фоккера-Планка-Колмогорова с различны­ми коэффициентами сноса и различными начальными условиями. Установлены до­статочные условия существования и единственности вероятностных решений нели­нейных уравнений Фоккера-Планка-Колмогорова с частично вырожденной матри­цей диффузии.

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.

We study the homogenization problem for the system of equations of dynamics of a mixture of liquid crystals with random structure. We consider a simplified form of the Ericksen–Leslie equations for an incompressible medium with inhomogeneous density with random structure. Under the assumption that randomness is statistically homogeneous and ergodic, we construct the limit problem and prove almost sure convergence of solutions of the original problem to the solution of the limit (homogenized) problem.

In this paper, we study the full three-dimensional Ericksen-Leslie system of equations for the nematodynamics of liquid crystals. We announce the short-time existence and uniqueness of strong solutions for the initial value problem in the periodic case and in a bounded domain with Dirichlet- and Neumann-type boundary conditions. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

In this paper we study the two dimensional Ericksen–Leslie equations for the nematodynamics of liquid crystals if the moment of inertia of the molecules does not vanish. We prove short time existence and uniqueness of strong solutions for the initial value problem in two situations: the space-periodic problem and the case of a bounded domain with spatial Dirichlet boundary conditions on the Eulerian velocity and the cross product of the director field with its time derivative. We also show that the speed of propagation of the director field is finite and give an upper bound for it.

We study uniqueness of families of probability measures solving the Cauchy problem for nonlinear Fokker–Planck–Kolmogorov equation with unbounded coefficients. Sufficient conditions for uniqueness are indicated and examples of non-uniqueness are constructed.

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.

This paper is concerned with the homogenization of the equations describing a magnetohydrodynamic boundary layer flow past a flat plate, the flow being subjected to velocities caused by injection and suction. The fluid is assumed incompressible, viscous and electrically conducting with a magnetic field applied transversally to the direction of the flow. The velocities of injection and suction and the applied magnetic field are represented by rapidly oscillating functions according to several scales. We derive the homogenized equations, prove convergence results and establish error estimates in a weighted Sobolev norm and in C 0 -norm. We also examine the asymptotic behavior of the solutions of the equations governing a boundary layer flow past a rough plate with a locally periodic oscillating structure.

In this paper, we study the homogenization problem for equations of magnetohydrodynamic boundary layer of pseudo-plastic fluid. It is assumed that the external flow velocity and the external magnetic field are described by oscillating functions and the frequency depends on a small parameter. In von Mises variables and in Cartesian variables, we construct the homogenized problem, establish strong convergence of solutions in a special norm, and estimate the rate of this convergence. We show that in von Mises variables the convergence rates in different norms are of different orders of smallness.