Instability of fluid motion is the main subject of this proposal. Although it was a classical and rather experimental branch of the hydrodynamics in the early days, it now stands as an actively developing area of modern analytical fluid dynamics, and it provides many challenging problems to mathematicians. The motion of an ideal incompressible fluid is governed by the Euler equation. Because of the particular structure of the nonlinearity present in the equation it appears to be notoriously difficult to answer even the most basic questions in a rigorous mathematical way. One of these questions is to justify the Lyapunov linearization method for stationary fluid flows. In particular, if the Euler equation linearized about a given equilibrium has unstable spectrum, does this imply that the equilibrium is unstable in the nonlinear sense, say, in the basic energy norm? The investigator studies this and other questions related to instability and spectra. These include: the question of finding unstable shortwave perturbations for general fluid flows; analysis of the spectrum for the Euler equations and the relationship with the spectrum of weakly viscous flows in the limit of vanishing viscosity. The investigator examines these problems using modern WKB-type asymptotic analysis for the Euler equations, pseudo-differential calculus, as well as newly developed connections between the cocycle theory and fluid dynamics.
The questions addressed in this particular project are closely related to the fundamental problems of environmental science such as weather prediction, climate change, and oceanic motion. Instability of large fluid masses is inherent in the nature of those processes, while large scale instabilities often grow out of small scales. The investigator presents mechanisms for such small scale instabilities and gives them a precise mathematical description. Understanding instabilities in fluid flows is important for a broad range of questions in atmospheric science and geophysics.
