HYDRODYNAMIC STABILITY WITHOUT EIGENVALUES

Fluid flows that are smooth at low speeds become unstable and then tur bulent at higher speeds. This phenomenon has traditionally been invest igated by linearizing the equations of flow and testing for unstable e igenvalues of the linearized problem, but the results of such investig ations agree poorly in many cases with experiments. Nevertheless, line ar effects play a central role in hydrodynamic instability. A reconcil iation of these findings with the traditional analysis is presented ba sed on the ''pseudospectra'' of the linearized problem, which imply th at small perturbations to the smooth flow may be amplified by factors on the order of 10(5) by a linear mechanism even though all the eigenm odes decay monotonically. The methods suggested here apply also to oth er problems in the mathematical sciences that involve nonorthogonal ei genfunctions.