Abstract

What boundedness assumptions on the solution of an equation cause it to vanish everywhere? This is the question that a Liouville type theorem seeks to answer. For reasons that will become apparent, these results are intimately related to problems of regularity and uniqueness. In this talk, we will consider Liouville type theorems for equations arising in the study of incompressible fluid dynamics. In particular, we will give a criterion for smooth solutions of the stationary equations of magneto-hydrodynamics (MHD) to be identically zero. The result in question is a refinement of previous ones, all of which systematically required stronger integrability and the additional assumption of finite Dirichlet integral, and builds on the earlier works of Seregin, Chae and Weng, and Chae and Wolf.