  The purpose of this short note is to demonstrate uniform logarithmic Sobolev
inequalities for the mean field gradient particle systems associated to an
energy functional that is convex in the flat sense. A defective log-Sobolev
inequality was already established implicitly in a previous joint work with F.
Chen and Z. Ren [arXiv:2212.03050 [math.PR]]. It remains only to tighten it by
a uniform Poincar\'e inequality, which we prove by the method in a recent work
of Guillin, W. Liu, L. Wu and C. Zhang [Ann. Appl. Probab., 32(3):1590-1614,
2022]. As an application, we show that the particle system exhibits the
concentration of measure phenomenon in the long time.
