How are eigenvalues related to linear stability analysis?
- M. Renardy, Nonlinear stability of flows of Jeffreys fluids at low
Weissenberg numbers, Arch. Rat. Mech. Anal. 132 (1995), 37-48.
- M. Renardy, Spectrally determined growth is generic, Proc. Amer.
Math. Soc. 124 (1996), 2451-2453
In stability studies in fluid mechanics, the following two premises are
usually taken for granted:
- (1)The stability to small disturbances follows from linear stability.
- (2) Linear stability can be decided by looking at the spectrum.
While both principles are well justified for Newtonian flows, the
situation
is much more uncertain for viscoelastic flows, while involve hyperbolic
partial
differential equations. Indeed, abstract counterexamples where linear
stability
is not determined by the spectrum have been known since the 1950s, and
our recent
work has shown that this happens for quite natural problems
such as
a lower
order perturbation of the wave equation. In Renardy's paper, a positive
result
is obtained for flows of viscoelastic flows with constitutive laws of
Jeffreys
type, under the assumption that the Weissenberg number is sufficiently
small.
In this
case, it is shown that both 1 and 2 hold. In Renardy's paper, the abstract
issue
of spectrally determined growth is investigated from a new angle. While
in
general the growth rate of solutions is not determined from the spectrum,
one may ask whether spectrally determined growth is the rule or the
exception.
One may thus consider a family of evolution problems du/dt=(A+B)u,
where
B varies over a class of operators, say all bounded operators. The result
is that, if A is the infinitesimal generator of a
C_0-semigroup
in a Hilbert space, then the set of all operators B for which spectrally
determined growth fails is of first category.
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