  In this paper, we introduce a class of new logarithmic curvature flow. The
flows are designed to embrace the monotonicity of the related functional, and
the convergence of this flow would tackle the solvability of the weighted
Christoffel-Minkowski problem, but a full proof scheme is missing, the key
factor of forming this phenomenon lies in the establishment of the upper bound
of the principal curvature, which essentially depends on finding a clean
condition on smooth positive function defined on the unit sphere $\sn$. Except
for obtaining this tricky estimate, we get all the other a priori estimates and
hope that this note can attract wide attention to this interesting issue.
