Smoothing of singular Legendre transforms in renormalization group flows

We consider O(N)-symmetric potentials with a logarithmic singularity in the second field derivative. This class includes BCS and Gross Neveu potentials. Formally, the exact renormalization group equation for the Legendre transform of these potentials seems to have ill-defined initial conditions. We show that the renormalization group equation for the local potential has well-defined initial conditions and that the logarithmic singularity is smoothed rapidly in the flow. Our analysis also provides an efficient method for numerical studies.