We will introduce the rudiments of a new theory of non-smooth solutions which applies to fully nonlinear PDE systems and extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. Key ingredient is the discovery of a notion of Extremum for maps which extends min-max uniquely and allows for “nonlinear passage of derivatives” to test maps. The notions supports uniqueness, existence and stability results, preserving most features of the scalar viscosity counterpart. We will also discuss applications in vector-valued Calculus of Variations in L^\infty and Hamilton-Jacobi PDE with vector solution.

In this talk, I will review briefly the general Ericksen-Leslie system modeling the hydrodynamic motion of the nematic liquid crystals proposed by Ericksen and Leslie back in 1960’s. I will focus on the mathematical analysis of a simplified version of the Ericksen-Leslie system, proposed by Lin, which is a strong coupling between the Navier Stokes equation and the transported heat flow of harmonic maps into the two sphere. I will then present some recent results on the global existence of Leray-Hopf type weak solutions in dimension two, and several well-posedness results for small initial data in various function spaces in dimension three. It is based on joint works with Fanghua Lin, Junyu Lin, Tao Huang, and Jay Hineman.

In this talk I will describe some of my recent work on the resolvent estimates in L^p for the Stokes operator in Lipschitz domains. The results, in particular, imply that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in L^p for p between 3 and 3/2. This gives an altermative answer to a conjecture of Michael Taylor.