Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hit-conf.imec.msu.ru/2012/lectures/Chernyshenko.doc Дата изменения: Sun Jun 14 09:32:09 2015 Дата индексирования: Sat Apr 9 23:43:30 2016 Кодировка:

Sum of Squares of polynomials technique for stability and bounds in fluid
dynamics


S. Chernyshenko (Imperial College London) and P. Goulart (ETH Zurich)


This talk will introduce a new method for proving global stability of
fluid flows using recently developed optimization methods based on sum-of-
squares (SOS) decomposition to construct a polynomial Lyapunov function.
The method is first explained for the case of finite-dimensional
approximations of fluid systems obtained by truncated Galerkin expansions,
and an example is given. We then show how this approach can be extended to
full (infinite dimensional) Navier-Stokes systems using robust optimization
techniques. Crucially, this extension requires solving only linear infinite-
dimensional eigenvalue problems and finite-dimensional sum-of-squares
optimization problems. We then show that subject to minor technical
constraints, a general polynomial Lyapunov function is always guaranteed to
provide better results than the classical energy methods in determining a
lower-bound on the maximum Reynolds number for which a flow is globally
stable, if the flow does remain globally stable for Reynolds numbers at
least slightly beyond the energy stability limit. Such polynomial functions
can be searched for efficiently using the SOS technique. We then
demonstrate how a similar technique can be used to obtain bounds for the
energy dissipation rate and other functionals in turbulent flows.