In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to two of my previous posts, universal-decay-rate-of-the-fisher-information-along-the-heat-flow and improved-regularization-for-lambda-convex-gradient-flows.

**Fact 0:**the quadratic Wasserstein distance $W_2$ induces a (formal) Riemannian structure on the space of probability measures, which gives a meaning to Wasserstein gradients $\operatorname{grad}_{W_2}F(\rho)$ of a functional $F:\mathcal P(\mathbb T^d)\to\mathbb R$ at a point $\rho$**Fact 1:**the heat flow $\partial_t\rho_t=\Delta\rho_t$ is the Wasserstein gradient flow $$ \dot\rho_t=-\operatorname{grad}_{W_2}H(\rho_t) $$ of the Boltzmann entropy $$ H(\rho)=\int_{\mathbb T^d}\rho\log\rho $$**Fact 2:**the Boltzmann entropy is $\lambda$-(displacement) convex for some $\lambda$. Its dissipation functional is the Fisher information, $$ F(\rho):=\|\operatorname{grad}_{W_2} H(\rho)\|^2_{\rho}=\int _{\mathbb T^d}|\nabla\log\rho|^2 \rho $$**Fact 3:**for abstract metric gradient flows (in the sense of [AGS]) and $\lambda$-convex functionals $\Phi:X\to\mathbb R\cup\{\infty\}$ one expects a smoothing effect for gradient flows $\dot x_t=-\operatorname{grad}\Phi(x_t)$ in the form \begin{equation} |\nabla\Phi(x_t)|^2\leq \frac{C_\lambda}{t} \Big[\Phi(x_0)-\inf_X\Phi\Big] \tag{R} \end{equation} at least for small times, where $C_\lambda$ depends only on $\lambda$ but not on $x_0$ see e.g. [AG, Proposition 3.22 (iii)].**Fact 3':**with the same notation as in Fact 3, an alternative regularization can be stated as \begin{equation} |\nabla\Phi(x_t)|^2 \leq \frac{1}{2e^{\lambda t}-1}|\nabla\Phi(y)|^2 +\frac{1}{(\int_0^te^{\lambda s}ds)^2} dist^2(x_0,y), \,\, \forall y\in X \tag{R'} \end{equation}**Fact 4:**in the Torus the Fisher information decays at a universal rate, i-e there is $C=C_d$ depending on the dimension only such that, for all $\rho_0\in \mathcal P(\mathbb T^d)$ and $t>0$, the solution $\rho_t$ of the heat flow emanating from $\rho_0$ satisfies \begin{equation} F(\rho_t)\leq \frac{C}{t} \tag{*} \end{equation} This follows from the Li-Yau inequality [LY], see this post of mine and F. Baudoin's answer.

Question: is there more to ($*$) than just the convexity of the Boltzmann functional? If the driving functional were upper-bounded $\Phi(x_0)\leq C$ (for all $x_0\in X$) in the regularization estimate (R) then we would immediately get the universal decay $|\nabla \Phi(x_t)|^2\leq \frac{C}{t}$. However, in the specific context of Facts 0-2 it is clearly not true that the Boltzmann entropy is upper-bounded. In fact there are many probability measures with infinite entropy, take e.g. any Dirac mass. Since (R) is optimal, I guess that one cannot simply deduce (*) from general $\lambda$-convexity arguments, and there is more than meets the eyes. But is there any connection? Note that both the Li-Yau inequality and the displacement convexity of the Boltzmann entropy strongly rely on the nonnegative Ricci curvature of the underlying torus.

I tried desperately to use any modified regularization estimate (e.g. R' and variants thereof instead of R), but to no avail so far. I am starting to believe that there is no direct implication, and that the work of Li-Yau is really profoundly ad-hoc (don't get me wrong, I just mean that their results cannot be generalized for abstract gradient-flows, and that their result/proof really leverages the specific structure and setting of the heat flow in Riemannian manifolds, not just any gradient flow). I would immensely appreciate any input or insight!

[AG] Ambrosio, L., & Gigli, N. (2013). A user’s guide to optimal transport. In Modelling and optimisation of flows on networks (pp. 1-155). Springer, Berlin, Heidelberg.

[AGS] Ambrosio, L., Gigli, N., & Savaré, G. (2008). Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media.

[LY] Li, P., & Yau, S. T. (1986). On the parabolic kernel of the Schrödinger operator. Acta Mathematica, 156, 153-201.

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