 
Summary: Stability and Bifurcation in Viscous
Incompressible Fluids
Herbert Amann
1 The Problem
We consider the motion of a viscous incompressible uid in a bounded
smooth
domain
of R 3 . It is governed by the laws of conservation of mass,
momentum, and energy given by
r v = 0 ;
@ t v + (v r)v = rp+r S + b ;
c(@ t + v r) = r q + S : D + r ;
(1:1)
respectively (e.g., [24], [27]). Here the velocity vector eld v, the pressure p,
and the absolute temperature are the unknowns. The density has been
normalized to 1, and S is the viscous part of the stress tensor, b = b(x; t; )
is the body force, c = c() > 0 is the heat capacity, q is the heat ux vector,
D := D(v) :=
1
2
