The global Wasserstein gradient flow exists uniquely and is stable

Let $E_{\ast }\ge \max (F({\mu }_0),F({\nu }_0))$ be a common energy bound. The escape estimate [F2] supplies a radius $R_T$ that works for both localized flows throughout $[0,T]$. By [F1], each localized flow exists on that interval and remains in a smaller $Q_{r(T)}\subset Q_{R_T}$, so the added cutoff normal cone is inactive and each curve solves the equation constrained by $\mathrm{\Omega }$. If $S>T$, the curves constructed with a larger radius agree on $[0,T]$ by localized uniqueness [F1]; hence the finite-horizon curves patch to a well-defined global curve.

The energy identity on $[0,T]$ is inherited from [F1]. Applying the localized stability estimate on the common radius $R_T$ gives $W_2({\mu }_t,{\nu }_t)\le e^{K_{T}t}{W}_2({\mu }_0,{\nu }_0)$ for $0\le t\le T$, choosing $K_T$ to be the maximum of that localized exponent and $0$; taking the supremum gives the displayed bound. Its right side tends to zero when the initial W2 distance does, proving compact-time uniform convergence. Since $T$ was arbitrary, existence, uniqueness, energy dissipation, and stability hold globally in time.

We check uniqueness against every curve in the stated definition, without presuming its energy decreases or its support stays in a cutoff. Choose $q\in Q_1$. The neighborhood premise and derivative growth give $\mathrm{\parallel }d{\mathrm{\Phi }}_x\mathrm{\parallel }\le C(1+\mathrm{\mid }x\mathrm{\mid })$ on $\mathrm{\Omega }$, after increasing $C$ to absorb $q$. Segment integration yields $\mathrm{\parallel }\mathrm{\Phi }(x)-\mathrm{\Phi }(y)\mathrm{\parallel }\le C(1+\mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid })\mathrm{\mid }x-y\mathrm{\mid }$. Consequently every ${\mathcal{P}}_2(\mathrm{\Omega })$ law has a Bochner feature moment, and coupling Cauchy–Schwarz gives $\mathrm{\parallel }{m}_{\alpha }-m_{\beta }\mathrm{\parallel }\le C(1+(\int \mathrm{\mid }x{\mathrm{\mid }}^{2}d\alpha {)}^{1\mathrm{/}2}+(\int \mathrm{\mid }y{\mathrm{\mid }}^{2}d\beta {)}^{1\mathrm{/}2})W_2(\alpha ,\beta )$. The same growth argument for bounded potential supporting vectors shows $V$ has at most quadratic growth. For a competing $A{C}^2$ solution, second moments are uniformly bounded on $[0,T]$ by the triangle inequality with ${\delta }_q$, so its moment and forcing $\mathrm{\nabla }R(m_t)$ are continuous. The global fixed-forcing construction and superposition representation in [F4] therefore apply with $D=\mathrm{\Omega }$. Starting in $Q_{r_0}$, this competitor is concentrated throughout $[0,T]$ on a single $Q_S$, using the uniform radius estimate of [F4] for its bounded forcing. Increase $S$ slightly so the added cutoff constraint is locally inactive relative to $\mathrm{\Omega }$. Both the competitor and the constructed flow then solve the same localized model equation on a common larger cutoff, so the law uniqueness estimate in [F4] makes them equal. No energy bound on a previously unconfined competitor was assumed. This completes global uniqueness in the full stated class.