We present  conditional existence results for  the Landau equation with
Coulomb potential. Despite lack of a comparison principle for the equation, the
proof of existence relies on barrier arguments and parabolic regularity theory. The
Landau equation arises in kinetic theory of plasma physics. It was derived by Landau
and serves as a formal approximation to the Boltzmann equation when grazing
collisions are predominant. 
We also present long-time existence results for the isotropic version of the Landau
equation with Coulomb potential.