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 Parodiʼs 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.

Global weak solution and blow-up criterion of the general Ericksen–Leslie system for nematic liquid crystal flows / C. Cavaterra, E. Rocca, H. Wu. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 255:1(2013 Jul), pp. 24-57.

Global weak solution and blow-up criterion of the general Ericksen–Leslie system for nematic liquid crystal flows

C. Cavaterra
Primo
;
E. Rocca
Secondo
;
2013

Abstract

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 Parodiʼs 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.
Nematic liquid crystal flow ; Ericksen–Leslie system ; Existence of weak solutions ; Blow up criterion
Settore MAT/05 - Analisi Matematica
lug-2013
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/222754
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