I am considering the minimizing movement scheme related to the gradient of entropy functional in 2-Wasserstein space. The problem is to minimize the following functional for each fixed $\eta$ which is a probability density w.r.t. $(\mathbb{R}^d,Leb)$ with finite second moments: $$\int\rho\log\rho dx+W_2^2(\rho,\eta),$$ among all probability densities $\rho$(so $\rho dx\ll Leb$) with finite second moments. Now I need to show the existence of a minimizer to this problem.

So first we choose a minimizing sequence $\rho_n$, which gives that $W_2^2(\rho_n,\eta)$ are uniformly bounded. Since the second moments can be bounded by the 2-Wasserstein distance, we know the second moments of $\rho_n$ are uniformly bounded, so they are tight(and also uniformly integrable). This gives a subsequence $\rho_{n_k}$ converging weakly to some probability measure $\mu$. Now we need to show $\mu\ll Leb$ and has finite second moment.

For the second part I used Skorokhod's theorem to find $X_n\sim\rho_n$ and $X\sim\mu$ with $X_n\overset{a.s.}{\rightarrow}X$. Then Fatou's lemma gives $\mathbb{E}X^2\leq\liminf_{n\rightarrow\infty}\mathbb{E}X_n^2<\infty$.

But I have no idea how to show $\mu\ll Leb$: we can find counterexamples if we only have $X_n$ converges a.s. and in $L^1$. We might need other observations; or it is possible that the limit of the minimizing sequence of this problem is not absolutely continuous w.r.t. Lebesgue measure?