On Well-Posedness and Decay of Strong Solutions for 3D Incompressible Smectic-A Liquid Crystal Flows

In this paper, we study a hydrodynamic system that models smectic-A liquid crystal flow in R-3. This model consists of the Navier-Stokes equations for fluid velocity coupled with a fourth-order equation for the layer variable. The main purpose is to analyze thewell-posedness and asymptotic behavior of strong solutions. We first prove the local well-posedness through the higher-order a prior estimates of the solution and Galerkin method. Then, we establish the existence of global strong solution provided that parallel to u(0)parallel to((over dot(H)1/2) + parallel to phi(0)parallel to((over dot(H)3/2) + parallel to phi(0)parallel to((over dot(H)7/2) is sufficiently small. Finally, we showthe temporary decay estimates for the higher-order derivatives of strong solution by using the negative Sobolev norm estimates.