Flow stability theory is a huge and well-established part of fluid dynamics, with Helmholtz, Kelvin, Rayleigh and Reynolds among its pioneers. The following is limited to what is relevant here.
In the majority of applications steady flows are better than unsteady flows. Steady flows are usually associated with smaller fuel consumption, less fatigue, and less noise. Theoretically, a steady flow always exists as a solution of the governing equations. However, in engineering applications fluid flows are usually unsteady, or even turbulent, because the corresponding steady flow is unstable, that is, if disturbed it will never return to the steady state. Flow control aims at stabilising fluid flows. Technology progress is making more complicated control approaches, like, for example, feedback control, more and more feasible. The bulk of the work on hydrodynamic stability is concerned with infinitesimal perturbations. This allows to represent the solution of the governing equations as a sum of the steady solution and a small perturbation, and neglect the nonlinear terms. The resulting linear problem is much easier to solve.
Linear stability of canonical flows is now largely a closed area of research, with the centre of gravity shifted to developing efficient numerical methods applicable to complex flows encountered in practice. Importantly, linear stability analysis can reveal instability but cannot prove stability. This is because steady flows are often stable with respect to infinitesimally small perturbations but unstable with respect to finite perturbations. Moreover, the finite amplitude required to destabilise the flow is often small. Hence, it is the stability with respect to finite disturbances, which is also called global stability, and the control of finite disturbances that represent major practical interest. Typically, flows are stable if the Reynolds number R is small enough, but there is a value Rl above which the flow is unstable with respect to arbitrary small perturbations, i.e. if R > Rl . On the other hand, there is a critical Reynolds number Rc such that if R < Rc the flow is stable with respect to disturbances of any amplitude. Finding Rc is difficult since this is a nonlinear problem.
Progress in understanding instability with respect to finite disturbances was made in the late eighties in the works on non-modal stability, but no methods for determining Rc have yet emerged from these works. Serrin (1959) demonstrated that for each incompressible flow in a closed domain there is Re such that the energy of an arbitrary perturbation decreases monotonously if R < Re. This, of course, means that the flow is globally stable. Moreover, Serrin demonstrated that the problem of determining Re can be reduced to a linear eigenvalue problem similar to the eigenvalue problems arising in the linear stability theory. Naturally, Re ≤ Rc ≤ Rl . In many cases the difference between Re and Rl is large: for example, for a pipe flow Rl is infinite. Therefore, while finding Re and Rl is relatively easy, more powerful methods are required for estimating Rc.