Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. Existing methods are unable to verify global stability for the many flows that seem to be stable despite these energies growing transiently. I will present a broadly applicable method to verify global stability of such flows. This method uses polynomial optimization computations to construct non-quadratic Lyapunov functions that decrease monotonically. I will present an application of the method to 2D Couette flow, where it verifies global stability at Reynolds numbers above the energy stability threshold found by Orr in 1907. This is joint work with Federico Fuentes and Sergei Chernyshenko.
David Goluskin is an Assistant Professor of Mathematics at the University of Victoria in British Columbia, Canada. From 2014 to 2017 he was a James Van Loo Postdoctoral Fellow in the Mathematics Department at the University of Michigan. He holds a PhD in Applied Mathematics from Columbia University. Dr. Goluskin’s research centers on fluid dynamics and related nonlinear dynamical PDEs and ODEs. Much of his work on fluids has focused on thermal convection, incorporating direct numerical simulation as well as theoretical analysis of topics including nonlinear stability and a priori estimates of average quantities. His recent focus has been the development of broadly applicable mathematical methods based on computational convex optimization, especially polynomial optimization.