Existence theorems for periodic solutions to partial differential equations with applications in hydrodynamics

The thesis looks at a number of existence theorems that prove the existence of small-amplitude periodic solutions to systems of partial differential equations. The existence theorems we consider are the Hopf bifurcation theorem, the Lyapunov centre theorem, the Weinstein-Moser theorem, and extensions of these theorems; the Hopf-Iooss bifurcation theorem, the Lyapunov-Iooss centre theorem and the Weinstein-Moser-Iooss theorem, respectively. The theorems have been derived so that they are applicable to functional analytical problems, and have been represented in a coherent and uniform manner in order to bridge the fundamental structure common to them all. Applications of these theorems, in this standardised form, are then applied in a systematic way to two particular hydrodynamical problems; the water wave problem and the Navier-Stokes equations. The classic water wave problem concerns the irrotational flow of a perfect fluid of unit density, subject to the forces of gravity and surface tension. We apply the Lyapunov-Iooss centre theorem to prove the existence of doubly-periodic waves; a doubly-periodic wave is a travelling wave that possess spatially periodic profiles in two different horizontal directions. Fundamental to our approach is the spatial dynamics formulation. The spatial dynamics formulation involves formulating a system of partial differential equations, defined on some spatial domain, as a dynamical system where one of the unbounded spatial variables plays the role of time. We catalogue a variety of parameter values for which it is possible to obtain doubly periodic waves, and we conclude with an existence result for doubly periodic waves under specific parameter restrictions. The Navier-Stokes equations in an exterior domain models the flow of an incompressible, viscous fluid past an obstacle. We apply the Hopf-Iooss bifurcation theorem to the defining equations to determine the existence of time-periodic waves. Our approach involves a careful examination of the Oseen problem to which we apply a 'cut-off' technique. This technique is used to constructs a solution to the Oseen problem using the respective solutions to the Oseen problem on a bounded domain and free space (the existence of which are well established). Time-periodic solutions are established using the Hopf-Iooss bifurcation theorem provided certain spectral conditions are met. The verification of the conditions may only be possible numerically, and so beyond the scope of our investigation.