  The currently favored scenario for the progenitor evolution of Type Ia
supernovae (SNe Ia) presumes that white dwarfs in close binary systems grow to
the Chandrasekhar limit via mass accretion from their non-degenerate
companions. However, the accreted hydrogen and/or helium usually participate
thermally unstable or even violent nuclear reactions in a geometrically
confined region, due to the compactness of the white dwarf. Since shell flashes
induced by the thermal instability might induce significant loss of mass,
efficient mass increase of white dwarfs by hydrogen and/or helium accretion has
been seriously questioned. A good understanding of the stability of
thermonuclear shell sources is therefore crucial in order to investigate the
evolution of accreting white dwarfs as SNe Ia progenitors. Here, we present a
quantitative criterion for the thermal stability of thermonuclear shell
sources, and discuss the effects of rotation on the stability of helium shell
burning in helium accreting CO white dwarfs with Mdot \approx 10^{-7} ...
10^{-6} Msun/yr. In particular, we show that, if the effects of rotation are
properly considered, helium shell sources are significantly stabilized, which
might increase the likelihood for accreting white dwarfs to grow to the
Chandrasekhar limit.
