Speaker: David Goluskin, University of Victoria
Date: June 3, 2021
Title: Variational analysis of fluid equations with help from computational optimization
Abstract: Various types of statements about dynamical ODEs and PDEs can be proved by finding 'auxiliary functions', meaning scalar-valued functions whose time evolution obeys certain inequalities that imply the desired result. One type of auxiliary function is a Lyapunov function, which satisfies inequalities guaranteeing that the function is positive and decreasing in time, and which implies nonlinear stability of a particular state. Other types of auxiliary functions, subject to different inequalities, imply other statements such as bounds on time-averaged or instantaneous values of a chosen quantity of interest. In many cases involving the Navier-Stokes equations and other fluid models, all auxiliary functions used in the past have been quadratic integrals, and the relevant inequality conditions can be checked by solving a quadratic variational problem. When auxiliary functions are quadratic integrals, showing stability amounts to the 'energy method', and bounding quantities of interest amounts to the 'background method'. I will explain why it is technically difficult to generalize to auxiliary functions that are not quadratic integrals, in part because one confronts intractable variational problems that must be relaxed. I will present one way of meeting these difficulties that involves the computational solution of relaxed variational problems using a type of convex optimization involving sum-of-squares polynomials. I will show two examples in which this approach has been applied: global stability of Couette flow beyond the energy stability threshold, and bounds on mean energy in the 1D Kuramoto-Sivashinsky equation.