Well-posedness of the Ericksen-Leslie system

In this paper, we prove the local well-posedness of the Ericksen-Leslie system, and the global well-posednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg-Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.Comment: 16 page

Long-range coupling of prefrontal cortex and visual (MT) or polysensory (STP) cortical areas in motion perception

To investigate how, where and when moving auditory cues interact with the perception of object-motion during self-motion, we conducted psychophysical, MEG, and fMRI experiments in which the subjects viewed nine textured objects during simulated forward self-motion. On each trial, one object was randomly assigned its own looming motion within the scene. Subjects reported which of four labeled objects had independent motion within the scene in two conditions: (1) visual information only and (2) with additional moving- auditory cue. In MEG, comparison of the two conditions showed: (i) MT activity is similar across conditions, (ii) late after the stimulus presentation there is additional activity in the auditory cue condition ventral to MT, (iii) with the auditory cue, the right auditory cortex (AC) shows early activity together with STS, (iv) these two activities have different time courses and the STS signals occur later in the epoch together with frontal activity in the right hemisphere, (v) for the visual-only condition activity in PPC (posterior parietal cortex) is stronger than in the auditory-cue condition. fMRI conducted for visual-only condition reveals activations in a network of parietal and frontal areas and in MT. In addition, Dynamic Granger Causality analysis showed for auditory cues a strong connection of the AC with STP but not with MT suggesting binding of visual and auditory information at STP. Also, while in the visual-only condition PFC is connected with MT, in the auditory-cue condition PFC is connected to STP (superior temporal polysensory) area. These results indicate that PFC allocates attention to the “object” as a whole, in STP to a moving visual-auditory object, and in MT to a moving visual object.Accepted manuscrip

Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals

The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution

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 $\mu_4$ 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