16 research outputs found
Thermodynamical Consistent Modeling and Analysis of Nematic Liquid Crystal Flows
The general Ericksen-Leslie system for the flow of nematic liquid crystals is
reconsidered in the non-isothermal case aiming for thermodynamically consistent
models. The non-isothermal model is then investigated analytically. A fairly
complete dynamic theory is developed by analyzing these systems as quasilinear
parabolic evolution equations in an -setting. First, the existence of
a unique, local strong solution is proved. It is then shown that this solution
extends to a global strong solution provided the initial data are close to an
equilibrium or the solution is eventually bounded in the natural norm of the
underlying state space. In these cases, the solution converges exponentially to
an equilibrium in the natural state manifold
Uniqueness of weak solutions for the general Ericksen-Leslie system with Ginzburg-Landau penalization in T^2
The Ericksen-Leslie system is a fundamental hydrodynamic model that describes
the evolution of incompressible liquid crystal flows of nematic type. In this
paper, we prove the uniqueness of global weak solutions to the general
Ericksen-Leslie system with a Ginzburg-Landau type approximation in a two
dimensional periodic domain. The proof is based on some delicate energy
estimates for the difference of two weak solutions within a suitable functional
framework that is less regular than the usual one at the natural energy level,
combined with the Osgood lemma involving a specific double-logarithmic type
modulus of continuity. We overcome the essential mathematical difficulties
arising from those highly nonlinear terms in the Leslie stress tensor and in
particular, the lack of maximum principle for the director equation due to the
stretching effect of the fluid on the director field. Our argument makes full
use of the coupling structure as well as the dissipative nature of the system,
and relies on some techniques from harmonic analysis and paradifferential
calculus in the periodic setting
On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D
We introduce a system of equations
that models a non-isothermal magnetoviscoelastic fluid. We show that the
model is thermodynamically consistent,
and that the critical points of the entropy functional with prescribed energy
correspond exactly with the equilibria of the system.
The system is investigated in the framework of quasilinear parabolic systems
and shown to be
locally well-posed in an -setting. Furthermore, we prove that
constant equilibria are normally stable. In particular, we show that
solutions
that start close to a constant equilibrium exist globally and converge
exponentially fast to a (possibly different)
constant equilibrium. Finally, we establish that the negative entropy serves
as a strict Lyapunov functional
and we then show that every solution that is eventually bounded in the
topology of the natural state
space exists globally and converges to the set of equilibria.Comment: 36 page