Seminars & Colloquia Calendar

Abstract: I will present on my recent work with Changfeng Gui, Yun Wang, and Chunjing Xie. In this work, Liouville-type theorems for the steady incompressible Navier-Stokes system are investigated for solutions in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. For the no-slip boundary conditions, we proved that any bounded solution is trivial if it is axisymmetric or ru^r is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big in L^infty space. For the periodic boundary conditions, we proved that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or ru^r decays to zero as r tends to infinity. The proofs are based on energy estimates, and the key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions. During the talk, I will present on the proofs in detail as much as time permits.