It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows in metric spaces one has
Theorem Let $(X,d)$ be a "nice" metric space, and $\Phi:X\to\mathbb R\cup\{\infty\}$ a convex function. If $t\mapsto u_t$ is a solution to the gradient-flow $$ "\dot u_t=-\nabla\Phi(u_t)" $$ then for any $w\in X$ there holds $$ |\partial\Phi|^2(u_t) \leq \frac{1}{t^2} d^2(u_0,w)+|\partial\Phi|^2(w). $$ In particular if $\Phi$ attains a minimum at a point $w$ (with $|\partial\Phi|(w)=0$) then this quantifies an instantaneous regularization: The slope decreases at $t=0$ at least as \begin{equation} \label{eq:regularization} |\partial\Phi|^2(u_t) \leq \frac{C}{t^2} \tag{R1} \end{equation} wih $C=d^2(u_0,w)$ depending only on the initial datum. Moreover if $\Phi$ has modulus of convexity $\lambda>0$ then \begin{equation} \label{eq:regularization2} |\partial\Phi|^2(u_t) \leq e^{-2\lambda t} |\partial\Phi|^2(u_0) \tag{R2} \end{equation}
Here $|\partial\Phi|(v)$ denotes the metric slope at a point $v\in X$, and the ODE $"\dot u_t=-\nabla\Phi(u_t)"$ should be understood in the abstract metric sense, for example in the sense of Energy-Dissipation-Inequality - see [AGS]. Please allow me to remain sloppy here and dispense from the technical details/assumptions in order to avoid burdening the exposition (see e.g. [AGS, Theorem 11.2.1] for a rigorous statement in a precise context). This is easy to prove for example in the framework of Hilbert spaces.
Question:
The instantaneous regularization \eqref{eq:regularization} holds if the driving functional is merely convex, i-e has a modulus of convexity $0$. Can we strengthen this estimate under the stronger hypothesis that $\Phi$ has a modulus of convexity $\lambda>0$? Typically one would expect a better rate $$ |\partial\Phi|^2(u_t)\leq \frac{C}{t}\qquad ??? $$ for some $C>0$ possibly depending on $u_0,\lambda$. Of course \eqref{eq:regularization2} is already much stronger (and optimal), but it requires the initial regularity $|\partial \Phi|(u_0)<\infty$ to make sense. What if $u_0\not\in D(\partial\Phi)$?
Actually any improvement of the form $|\partial\Phi|^2(u_t)\leq \frac{C}{t^{2-\epsilon}}$ with $\epsilon>0$ would suffice for my purpose.
For a non-trivial example of this improved regularity, take the (quadratic) Wasserstein space $(\mathcal P(\mathbb T^d),\mathcal W_2)$ on the flat torus, and take as a driving functional the entropy $\Phi(u)=H(u)=\int_{\mathbb T^d} u\log u\,dx$ (with the convention that $H(u)+\infty$ whenever $u$ is not absolutely continuous w.r.t the Lebesgue measure $dx$). The the gradient flow is the linear heat equation $\partial_t u=\Delta u$, and the (squared) slope is nothing but the Fisher information $|\partial H|^2(u)= F(u)=\int_{\mathbb T^d} \frac{|\nabla u|^2}{u}=\int_{\mathbb T^d}|\nabla\log u|^2 u$. From \eqref{eq:regularization} we already know that $F(u_t)\leq \frac{C}{t^2}$. However from the Li-Yau inequality [2] (see my previous post) it is know that the Fisher information decays at the BETTER universal rate $F(u_t)\leq \frac{C}{t}$ and in this case \eqref{eq:regularization} is clearly suboptimal.
I would be happy with an aswer even in Hilbert spaces (the statement is probably vacuous in finite dimension, though)
[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.: On the parabolic kernel of the Schrödinger operator. Acta Mathematica 156(1), 153–201 (1986)
