It is often implicitly presumed that linear stability is equivalent to nonlinear stability in practical settings for physically relevant systems. However, even the very early linear studies of Rayleigh and Kelvin seemed in contradiction with experimental observations, particularly the famous experiments of Reynolds in 1883. Indeed, in his experiment, Reynolds pumped fluid through a pipe under various conditions and demonstrated that at sufficiently high Reynolds number, the laminar equilibrium becomes spontaneously unstable.
Despite the observed nonlinear instability, to this day, there is no evidence of any linear instabilities in 3D pipe flow for any finite Reynolds number. Perhaps even more troubling is that even for laminar flows for which the linearization has unstable eigenvalues at high Reynolds numbers, experiments and computer simulations normally display instabilities which are different than those predicted by the linear theory (and at lower Reynolds numbers). These phenomena are known in fluid mechanics as subcritical transition or by-pass transition, and are completely ubiquitous in 3D hydrodynamics. These are very important questions were theory, numerics and experiments should be combined.
The foundations of hydrodynamic stability will be used and applied in engineering problems such as crowd motion, traffic flow, power flow, distribution, environmental flows. To account for possible uncertainties in system models, noise, imperfectness and incomplete information will be considered. Robustness and distributionnally robustness approaches will be used to examine various stability criteria. The emphasis will be also on the development of stability and computationally efficient framework for partial-integro-differential equations of mean-field type and learning algorithms in science, engineering, and social science problems.
Price of anarchy, stability and simplicity of distributionally robust interactive dynamical of McKean-Vlasov type will be investigated. This will enable risk-aware engineering technologies by including variance, quantile and high order moments of the system performance, speed and stability.