  We consider by a combination of analytical and numerical techniques some
basic questions regarding the relations between inviscid and viscous stability
and existence of a convex entropy. Specifically, for a system possessing a
convex entropy, in particular for the equations of gas dynamics with a convex
equation of state, we ask: (i) can inviscid instability occur? (ii) can there
occur viscous instability not detected by inviscid theory? (iii) can there
occur the ---necessarily viscous--- effect of Hopf bifurcation, or "galloping
instability"? and, perhaps most important from a practical point of view, (iv)
as shock amplitude is increased from the (stable) weak-amplitude limit, can
there occur a first transition from viscous stability to instability that is
not detected by inviscid theory? We show that (i) does occur for strictly
hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of
state, while (ii) and (iii) do occur for an artifically constructed system with
convex viscosity-compatible entropy. We do not know of an example for which
(iv) occurs, leaving this as a key open question in viscous shock theory,
related to the principal eigenvalue property of Sturm Liouville and related
operators. In analogy with, and partly proceeding close to, the analysis of
Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria
for shock (in)stability in the form of necessary and sufficient conditions on
the equation of state.
