  Many conservative partial differential equations correspond to geodesic
equations on groups of diffeomorphisms. Stability of their solutions can be
studied by examining sectional curvature of these groups: negative curvature in
all sections implies exponential growth of perturbations and hence suggests
instability, while positive curvature suggests stability. In the first part of
the paper we survey what we currently know about the curvature-stability
relation in this context and provide detailed calculations for several
equations of continuum mechanics associated to Sobolev $H^0$ and $H^1$
energies. In the second part we prove that in most cases (with some notable
exceptions) the sectional curvature assumes both signs.
