Special Colloquium
Roman Shvydkoy
UIC
Shortwave instabilities of ideal fluid and the cocycle theory
Abstract: Shortwave instabilities of an ideal fluid flow are instabilities generated by
highly oscillating localized wavepackets, which propagate according to a
finite-dimensional system of ODEs. Analysis of the cocycle (fundamental matrix
solution) generated by this system of ODEs establishes an equivalence relation
between the essential spectrum of the linearized Euler equation and all possible
shortwave instabilities that can occur in a given steady flow. Thus, studying
the essential spectrum of the Euler equation we can understand more about how an
ideal fluid can get unstable. We will survey various results that have been
obtained via the cocycle approach. Those include inherent instability of 3D
flows with periodic streamlines, pseudo-differential structure of the Euler and
Navier-Stokes semigroups, spectrum under vanishing viscosity limit. We will show
that typically the essential spectrum of the Euler equation is a vertical solid
band or a ladder.
Monday January 23, 2006 at 4:00 PM in SEO 636