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Nematic Liquid Crystal Systems and Magneto-Hydrodynamics System:The Properties of Their Solutions


We study a simplified system for the flow of Nematic Liquid Crystals (LCD) in the cases of non-constant density and constant density with certain initial and boundary conditions. This is a coupled non-parabolic dissipative dynamic system.

For the system with non-constant density in the whole space $\mathbb{R}^3$, we establish the global existence of weak solutions.

For the system with non-constant density on a bounded domain in two dimensional and three dimensional cases, we establish the regularity and uniqueness of solutions. We obtain that, in 2D, the global regularity with general data; in 3D, the global regularity with small initial data and a local (short time) regularity with large data. In addition, with more smoothness assumption on initial data, we obtain the uniqueness both for dimension 2 and 3 cases. The main tools are higher order energy estimates (so called Ladyzhenskaya estimate), Gagliardo-Nirenberg interpolation inequalities and the frozen coefficient method.

For the system with constant density in the whole space $\mathbb{R}^3$ with small initial data,

we study the large time behavior of solutions and obtain algebraic decay rate for the solutions in

energy space. The main ingredient to derive decay is Fourier splitting method which was originally introduced by M. Schonbek to study the large time behavior of solutions to Navier-Stokes equations.

We also study the magneto-hydrodynamics system (MHD). We

demonstrate that the solutions to the Cauchy problem for the three

dimensional incompressible MHD system can

develop different types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$.

Particularly the magnetic field can develop norm inflation in short

time even when the velocity remains small and vice verse.

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