Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system

In dimension three, the existence of global weak solutions to the axisymmetric simplified Ericksen-Leslie system without swirl is established. This is achieved by analyzing weak convergence of solutions of the axisymmetric Ginzburg-Landau approximated solutions as the penalization parameter tends to zero. The proof relies on the one hand on the use of a blow-up argument to rule out energy concentration off the z-axis, which exploits the topological restrictions of the axisymmetry. On the other hand, possible limiting energy concentrations on the z-axis can be dealt by a cancellation argument at the origin. Once more, the axisymmetry plays a substantial role. We will also show that the set of axisymmetric solutions without swirl (u, d) to the simplified Ericksen-Leslie system is compact under weak convergence in L^ₜL²ₓ L²ₜH¹ₓ.