Nonlinear Semigroups and Their Perturbations in Hydrodynamics. Three Examples

In this paper we present some mutual relations between semigroup theory in the context of the theory of infinite dimensional dynamical systems and the mathematical theory of hydrodynamics. These mutual relations prove to be very fruitful, enrich both fields and help to understand behaviour of solutions of both infinite dimensional dynamical systems and hydrodynamical equations. We confine ourselves to present these connections on some recent developments in the important problem of heat transport in incompressible fluids which features all main aspects of chaotic dynamics. To be specific, we consider the Rayleigh–Bénard problem for the two and three-dimensional Boussinesq systems for the Navier–Stokes and micropolar fluids and two-dimensional thermomicropolar fluid. Each of the three examples is remarkably distinct from the other two in the context of the semigroup theory.