Applications of optimization and optimal control to some fundamental problems in mathematical hydrodynamics
Location: MSRI: Online/Virtual
Optimization and optimal dynamical control can be used to investigate the precision of rigorous analytical estimates (inequalities) for solutions of partial differential equations. Even when constituent estimates are demonstrably sharp, the result of a sequence of applications of such inequalities need not be, leaving uncertainty in the accuracy of the ultimate result of the analysis. We examine classical bounds on enstrophy and palinstrophy amplification in the 2D and 3D Navier-Stokes and Burgers’ equations and discover that the best known instantaneous growth rates estimates for these quantities are indeed sharp. Integrating instantaneous estimates on growth rates in time may – as in the 2D Navier-Stokes case – but does not always – as in the Burgers case – produce sharp estimates in which case optimal control techniques must be brought to bear to determine the actual extreme behavior of the nonlinear dynamics. The relevant question for 3D Navier-Stokes equations, i.e., the $1M Millennium Prize Problem, remains unresolved although work is in progress to apply these tools to address it. Some of the research reported here is joint with Lu Lu, Indiana University Mathematics Journal 57, 2693-2727 (2008) and some with Diego Ayala & Thilo Simon, Journal of Fluid Mechanics 837, 839-857 (2018).