  The nuclear mean-field potential arising from the $^{12}$C+$^{12}$C
interaction at the low energies relevant for the astrophysical carbon burning
process has been constructed within the double-folding model, using the
realistic nuclear ground-state density of the $^{12}$C nucleus and the
effective M3Y nucleon-nucleon (NN) interaction constructed from the G-matrix of
the Paris (free) NN potential. To explore the nuclear medium effect, both the
original density independent M3Y-Paris interaction and its density dependent
CDM3Y6 version have been used in the folding model calculation of the
$^{12}$C+$^{12}$C potential. The folded potentials at the different energies
were used in the optical model description of the elastic $^{12}$C+$^{12}$C
scattering at the energies around and below the Coulomb barrier, as well as in
the barrier penetration model to estimate the fusion cross section and
astrophysical $S$ factor of the $^{12}$C+$^{12}$C reactions at the low
energies. The obtained results are in good agreement with experimental data
over a wide range of energies.
