The localized Wasserstein gradient flow exists, satisfies EVI, and is unique

Form the label Hilbert space $K=L^2({\mu }_0;{\mathbb{R}}^d)$ and the initial vector $X_0(s)=s$. This Euclidean Borel label space is countably generated, so its $d$-vector $L^2$ space is separable under the generalized label-space interface. It has finite $E_r$ because $F({\mu }_0)<\mathrm{\infty }$. [F1] yields the unique Hilbert gradient flow $X_t$ for all $t\ge 0$. Define ${\mu }_t=(X_t{)}_{\mathrm{\#}}{\mu }_0$. By [F2] this curve solves the localized continuity equation with the prescribed minimum-norm velocity and is in $A{C}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$.

The Hilbert energy identity in [F1] is $E_r(X_t)-E_r(X_s)=-{\int }_s^t\mathrm{\parallel }{\dot{X}}_q{\mathrm{\parallel }}_K^{2}dq$. By [F2], ${\dot{X}}_q(s)=v_q(X_q(s))$ almost everywhere and $E_r(X_q)=F({\mu }_q)$. The change-of-variables identity for the pushforward therefore gives the displayed Wasserstein energy identity.

Fix $\eta \in {\mathcal{P}}_2(Q_r)$ and a time where [F4] applies. Choose an optimal coupling ${\gamma }_t$ of ${\mu }_t,\eta$. The derivative estimate [F4] gives ${\displaystyle \frac{1}{2}\frac{d}{dt}{W}_2^2({\mu }_t,\eta )\le \int (x-y)\cdot v_t(x)d{\gamma }_t=-\int g_{{\mu }_t}(x)\cdot (x-y)d{\gamma }_t\mathrm{.}}$ By [F3], $F(\eta )\ge F({\mu }_t)+\int g_{{\mu }_t}(x)\cdot (y-x)d{\gamma }_t+\frac{{\lambda }_r}{2}{W}_2^2({\mu }_t,\eta )$. Combining the inequalities proves the EVI.

If $\mu$ and $\nu$ are two such solutions, [F5] gives their stability estimate; equal initial data imply $W_2({\mu }_t,{\nu }_t)=0$ for every $t$, hence equality of the measures and uniqueness. The construction in step 1.1 proves existence.