SPECTRUM OF SMALL OSCILLATIONS OF AN IDEAL FLUID AND LYAPUNOV EXPONENTS

Essential spectral radius of small oscillations in an ideal incompress ible fluid can be determined explicitly via generalized Lyapunov expon ents associated with the equilibrium flow. We prove that the maximal g rowth rate of a solution to a certain system of ODE's (bicharacteristi c amplitude equation) equals logarithm of the essential spectral radiu s of the evolution operator corresponding to linearized Euler equation . Thus some dynamical systems invariants of the flow, which can be com puted effectively from the Lagrangian trajectories, are responsible fo r hydrodynamic instability due to the essential spectrum.