Well-posedness of the full Ericksen–Leslie model of nematic liquid crystals

Abstract

The Ericksen–Leslie model of nematic liquid crystals is a coupled system between the Navier–Stokes and the Ginzburg–Landau equations. We show here the local well-posedness for this problem for any initial data regular enough, by a fixed point approach relying on some weak continuity properties in a suitable functional setting. By showing the existence of an appropriate local Lyapunov functional, we also give sufficient conditions for the global existence of the solution, and some stability conditions.