CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

December 3, 2020 from 10:15 to 10:30 (Montreal/EST time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted existence proofs for Navier-Stokes equations on an unbounded strip with obstacle

Seminar presented by Jonathan Wunderlich (Karlsruhe Institute of Technology, Germany)

The incompressible stationary 2D Navier-Stokes equations are considered on an unbounded strip domain with a compact obstacle. In order to get existence and error bounds (in a Sobolev space) for a solution, an approximate solution (using divergence-free finite elements), a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution are computed. For the latter, bounds for the essential spectrum and for eigenvalues play a crucial role, especially for the eigenvalues “close to” zero. Note that, on an unbounded domain, the only possible method for computing the desired norm bound appears to be via eigenvalue bounds. In this way, the first rigorous existence proof for the Navier-Stokes problem under consideration is obtained.