Applied Math Seminar: Liouville-type Theorems for Steady Solutions to the Navier-Stokes System in a Slab

Friday November, 18th (4pm – 5pm)

The organizers of the Applied Math seminar would like to welcome everybody to the talk.

Presented by Dr. Jeaheang Bang (The University of Texas at San Antonio)

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Abstract : Proving Liouville-type theorems for steady solutions to the incompressible Navier-Stokes equations in the entire three-dimensional space is one of the most important open problems in this field. Instead of working on the entire space, one can consider a three-dimensional slab domain. I will present my recent work with Changfeng Gui, Yun Wang, and Chunjing Xie. In this work, we investigated 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 the proofs in detail as much as time permits.