Abstract: Linear stability theory is widely used in fluid mechanics to predict when a particular base flow becomes unstable. The critical values of the parameters of the problem (for example, the critical Reynolds numbers) are usually calculated by solving the corresponding linearized eigenvalue problem. The structure of the critical motion which takes place just above the threshold can also be analyzed by means of the linear stability theory. However, the evolution of the most unstable disturbance above the threshold cannot be described by the linear theory.
Weakly nonlinear theories are used in such cases to derive an amplitude evolution equation (or a system of evolution equations). The coefficients of such equations (one example is the complex Ginzburg-Landau equation) can be calculated by means of the method of multiple scales.
In this plenary talk we discuss both linear and weakly nonlinear models of fluid motion in detail. Two examples are considered: stability of shallow water flows and stability of transient rapidly decelerated flow in a circular pipe. The results of joint work with Prof. M.S. Ghidaoui are discussed.

Brief Biography of the Speaker:
Andrei Kolyshkin received his undergraduate degree in Applied Mathematics in 1976 at the Riga Technical University. In 1981 he received a Ph.D in differential equations and mathematical physics at the University of St. Petersburg. Andrei Kolyshkin is currently a full professor at the Department of Engineering Mathematics at the Riga Technical University. His current research interests include investigation of stability problems in fluid mechanics with applications to open-channel flows and transient flows in hydraulic systems and mathematical models for eddy current testing. He is the co-author of three monographs published by Academic Press and CRM. Andrei Kolyshkin has participated in more than 30 international conferences and has published more than 50 papers in refereed journals since 1990. As a visiting professor and visiting researcher he spent a few years at the University of Ottawa and Hong Kong University of Science and Technology.