On the General Ericksen-Leslie System: Parodi's Relation, Well-posedness and Stability

Details On the General Ericksen-Leslie System: Parodi's Relation, Well-posedness and Stability On the General Ericksen-Leslie System: Parodi's Relation, Well-posedness and Stability

2011-05-11

arxiv.org/abs/1105.2180v5

Mathematics

Analysis of PDEs

Scientific paper

In this paper we investigate the interplay between Parodi's relation and the well-posedness as well as stability of the general Ericksen-Leslie system modeling nematic liquid crystal flows. First, we give a formal physical derivation of the E-L system through an appropriate energetic variational approach under Parodi's relation, in which we can distinguish conservative/dissipative parts of the induced elastic stress. Next, we prove the well-posedness as well as long-time behavior of the E-L system assuming that the viscosity $\mu_4$ is sufficiently large. Finally, under Parodi's relation, we show the well-posedness and Lyapunov stability for the E-L system near local energy minimizers. We also discuss the connection between Poradi's relation and linear stability of the E-L system.

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