148 research outputs found
Nonexistence of smooth solutions for the general compressible Ericksen -- Leslie equations in three dimensions
We prove that the smooth solutions to the Cauchy problem for the compressible
general three-dimensional Ericksen--Leslie system modeling nematic liquid
crystal flow with conserved mass, linear momentum, and dissipating total
energy, generally lose classical smoothness within a finite time.Comment: 7 pages. arXiv admin note: text overlap with arXiv:1206.2850,
arXiv:1105.2180 by other author
Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows
In this paper we investigate the three dimensional general Ericksen-Leslie
(E--L) system with Ginzburg-Landau type approximation modeling nematic liquid
crystal flows. First, by overcoming the difficulties from lack of maximum
principle for the director equation and high order nonlinearities for the
stress tensor, we prove existence of global-in-time weak solutions under
physically meaningful boundary conditions and suitable assumptions on the
Leslie coefficients, which ensures that the total energy of the E--L system is
dissipated. Moreover, for the E--L system with periodic boundary conditions, we
prove the local well-posedness of classical solutions under the so-called
Parodi's relation and establish a blow-up criterion in terms of the temporal
integral of both the maximum norm of the curl of the velocity field and the
maximum norm of the gradient of the liquid crystal director field
On the General Ericksen-Leslie System: Parodi's Relation, Well-posedness and Stability
In this paper we investigate the role of Parodi's relation in the
well-posedness and stability of the general Ericksen-Leslie system modeling
nematic liquid crystal flows. First, we give a formal physical derivation of
the Ericksen-Leslie system through an appropriate energy variational approach
under Parodi's relation, in which we can distinguish the
conservative/dissipative parts of the induced elastic stress. Next, we prove
global well-posedness and long-time behavior of the Ericksen-Leslie system
under the assumption that the viscosity is sufficiently large. Finally,
under Parodi's relation, we show the global well-posedness and Lyapunov
stability for the Ericksen-Leslie system near local energy minimizers. The
connection between Parodi's relation and linear stability of the
Ericksen-Leslie system is also discussed
Global Low-Energy Weak Solution and Large-Time Behavior for the Compressible Flow of Liquid Crystals
We consider the weak solution of the simplified Ericksen-Leslie system
modeling compressible nematic liquid crystal flows in . When the
initial data is small in and initial density is positive and essentially
bounded, we first prove the existence of a global weak solution in . The large-time behavior of a global weak solution is also established.Comment: arXiv admin note: text overlap with arXiv:1110.0053, arXiv:1004.4749
by other author
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