Instability of Fluid Flows

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Shvydkoy0604050 Instability of fluid motion is the main subject of thisproposal. Although it was a classical and rather experimentalbranch of the hydrodynamics in the early days, it now stands asan actively developing area of modern analytical fluid dynamics,and it provides many challenging problems to mathematicians. Themotion of an ideal incompressible fluid is governed by the Eulerequation. Because of the particular structure of the nonlinearitypresent in the equation it appears to be notoriously difficult toanswer even the most basic questions in a rigorous mathematicalway. One of these questions is to justify the Lyapunovlinearization method for stationary fluid flows. In particular,if the Euler equation linearized about a given equilibrium hasunstable spectrum, does this imply that the equilibrium isunstable in the nonlinear sense, say, in the basic energy norm?The investigator studies this and other questions related toinstability and spectra. These include: the question of findingunstable shortwave perturbations for general fluid flows;analysis of the spectrum for the Euler equations and therelationship with the spectrum of weakly viscous flows in thelimit of vanishing viscosity. The investigator examines theseproblems using modern WKB-type asymptotic analysis for the Eulerequations, pseudo-differential calculus, as well as newlydeveloped connections between the cocycle theory and fluiddynamics. The questions addressed in this particular project areclosely related to the fundamental problems of environmentalscience such as weather prediction, climate change, and oceanicmotion. Instability of large fluid masses is inherent in thenature of those processes, while large scale instabilities oftengrow out of small scales. The investigator presents mechanismsfor such small scale instabilities and gives them a precisemathematical description. Understanding instabilities in fluidflows is important for a broad range of questions in atmosphericscience and geophysics.