國立清華大學 數學系

Abstract: In 2010, Chi Hin Chan and Magdalena Czubak observed for the first time the non-uniqueness phenomena of finite energy Navier-Stokes flows on a 2D-hyperbolic space. This kind of non-uniqueness phenomena occurring in finite energy Navier-Stokes flows on a negatively curved simply connected manifold is in sharp contrast to the well-known, classical uniqueness theorem for finite energy Navier-Stokes flows in the 2D Euclidean space. Recently, Magdalena Czubak and I have worked out a self-contained Leray-Hopf theory for finite energy weak solutions to the Navier-Stokes equation in the setting of a 2D hyperbolic space. This Leray-Hopf theory not only resolves the dilemma arising from the non-uniqueness phenomena observed in our previous work in 2010, but also includes all weakly divergence free finite energy vector fields into the class of admissible finite energy initial datum for the Cauchy problem of the Navier-Stokes equation. In this talk, we will discuss these recent works by Magdalena Czubak and myself, as well as some recent progress on this topic, such as the Louville-type theorem for stationary Navier-Stokes equation in the setting of 3D hyperbolic space as obtained by Magdalena Czubak and myself in 2015. Tea Time: 3:30PM, R707