CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

July 21, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

Seminar presented by Xuefeng Liu (Niigata University, Japan)

We consider the solution verification for the stationary Navier-Stokes equation over a bounded non-convex 3D domain Ω. In 1999, M.T. Nakao, et al., reported a solution existence verification example for the 2D square domain.  However, it has been a difficult problem to deal with general 2D domains and 3D domains, due to the bottleneck problem in the  a priori error estimation for the linearized NS equation. Recently, by extending the hypercircle method (Prage-Synge's theorem) to deal with the divergence-free condition in the Stokes equation, the explicit error estimation is constructed successfully based on a conforming finite element approach [arXiv:2006.02952]. Further,  we succeeded in the solution existence verification for the stationary NS equation in several nonconvex 3D domains.  In this talk, I will show the latest progress on this topic, including the rigorous estimation of the eigenvalue of Stokes operator in 3D domains.