## IMA Postdoc Seminar (March 23, 2010)

**Speaker:**Xu, Xiang (Department of Mathematics, Penn. State University)

**Title:** Global existence and long time behavior of the general Ericksen-Leslie
system.

**Abstract:** The Ericksen-Leslie system modeling the flow of nematic liquid
crystals is a coupled system
consisting of Navier-Stokes equations and kinematic transport equations for
molecular orientation.
First, through different energetic variational approaches, we get a physical
derivation of the system,
and distinguish conservative and dissipative parts of the induced stress terms.
Then the existence
of global classical solutions is proved, under the assumption of one large
viscosity coefficient. Furthermore,
by a suitable type Lojaciewicz-Simon inequality, we find the convergence of the
classical solutions to steady
states as time tends to infinity and get the estimate on the convergence rate.
Finally, we study the
well-posedness of the system when the initial data is near a functional
minimizer, and partly reveal the
relation between Parodi's condition and certain stability property of the
liquid crystal system.