Abstract
Weakly nonlinear theory of fluid motion is discussed in relation with pattern formation phenomena in fluids. In nonlinear stability theory of parallel flows, Landau-type equation is obtained by growth-rate expansion method around the critical point, while in subcritical case it is obtained by amplitude expansion method in which the solvability condition is replaced by an exact definition of the 'amplitude'. In the case of falling film flow, amplitude equations derived so far do not agree well with experiments, and a new expansion method of Pade type proposed by Ooshida is successful although the method is not yet fully understood in mathematical sense. In the Benard convection problem, an order-parameter equation describes the weakly nonlinear stage, and some reduced amplitude equations are derived from this equation. Even in strongly nonlinear stage, pattern formation phenomena are often observed as in the two-dimensional turbulence on a rotating sphere where a circumpolar jet appears from a turbulent initial flow field.
