For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2\_{ac}(\mathbb{R}^d)$), let
$$
\mathcal{H}(\rho)=\int\_{\mathbb{R}^d}\rho\log \rho \,dx
$$
be the Boltzmann entropy.



> 
> **Question:** Is $\mathcal{H}$ continuous with respect to the quadratic Wasserstein distance $\mathcal{W}\_2$? i-e is it true that
>  $$
>  \mathcal{W}\_2(\rho\_n,\rho)\to 0
>  \quad\Rightarrow\quad
>  \mathcal{H}(\rho\_n)\to\mathcal{H}(\rho)\qquad ???
>  $$
> 
> 
> 


I guess this should be known and written down somewhere. I looked at the classical references (Villani's books [topics...] and [old and new...], also in [Ambrosio-Gigli-Savaré]) but surprisingly enough I couldn't find the precise statement. 


It is well known that $\mathcal{H}$ is $0$-displacement convex (in the sense of McCann). By analogy with the Euclidean setting, where any convex function $\phi:\mathbb{R}^d\to \mathbb{R}$ is continuous, I suspect that any displacement convex functional $\mathcal{F}:\mathcal{P}\_2(\mathbb{R}^d)\to\mathbb{R}$ should be continuous with respect to the wasserstein distance $\mathcal{W}\_2$. But maybe not...


This question arised in the context of my research: I am trying to construct weak solutions to some system of PDEs by means of the Jordan-Kinderlehrer-Otto/DeGiorgi's minimizing scheme, and I need the above statement in some technical step. Unfortunately the mere lower semi-continuity would not be enough for my purpose, I really need full continuity.



continuity of the Boltzmann entropy in the Wasserstein metric
Let $A\_n=\bigcup\_{i=0}^{n-1}[2i/(2n),(2i+1)/2n)$, $\rho\_n=2\chi\_{A\_n}$, $A=[0,1]$ and $\rho=\chi\_{[0,1]}$. 
Then it seems pretty clear that $\mathcal W\_2(\rho\_n,\rho)\to 0$. On the other hand $H(\rho\_n)\not\to H(\rho)$, right?


