Supernovae: an example of complexity in the physics of compressible
  fluids
  The supernovae are typical complex phenomena in fluid mechanics with very
different time scales. We describe them in the light of catastrophe theory,
assuming that successive equilibria between pressure and gravity present a
saddle-node bifurcation. In the early stage we show that the loss of
equilibrium may be described by a generic equation of the Painlev\'e I form. In
the final stage of the collapse, just before the divergence of the central
density, we show that the existence of a self-similar collapsing solution
compatible with the numerical observations imposes that the gravity forces are
stronger than the pressure ones. This situation differs drastically in its
principle from the one generally admitted where pressure and gravity forces are
assumed to be of the same order. Our new self-similar solution (based on the
hypothesis of dominant gravity forces) which matches the smooth solution of the
outer core solution, agrees globally well with our numerical results. Whereas
some differences with the earlier self-similar solutions are minor, others are
very important. For example, we find that the velocity field becomes singular
at the collapse time, diverging at the center, and decreasing slowly outside
the core, whereas previous works described a finite velocity field in the core
which tends to a supersonic constant value at large distances. This discrepancy
should be important for explaining the emission of remnants in the
post-collapse regime. Finally we describe the post-collapse dynamics, when mass
begins to accumulate in the center, also within the hypothesis that gravity
forces are dominant.
