Regularized continuity equations are represented by characteristic flows

Apply [F2] to $v^{\epsilon }$ and denote by $D_t$ the open set of initial points whose trajectories exist from time 0 through t, and by $E_t=X_t^{\epsilon }(D_t)$ its open image. Fix $t\in (0,T]$ and $f\in C_c^{\mathrm{\infty }}(E_t)$. For $0\le r\le t$ and $x\in D_{r,t}$, the set of states whose trajectories exist from r to t, define $\psi (r,x)=f(X^{\epsilon }(t;r,x))$; extend it by zero outside $D_{r,t}$. Since $\mathrm{supp}f$ is compactly contained in $E_t$, its backward images form a compact subset of the flow domain. Thus $\psi$ has compact spatial support uniformly in r, is continuously differentiable in x, and is absolutely continuous in r locally. The start-time derivative from [F2] and the flow identity give ${\mathrm{\partial }}_r\psi =\mathrm{\nabla }f(X(t;r,x))\cdot [-D_{x}X(t;r,x)v_r^{\epsilon }(x)]=-v_r^{\epsilon }(x)\cdot {\mathrm{\nabla }}_x\psi$ almost everywhere.

For this fixed t define the restricted finite pushforward ${\nu }_t$ on $E_t$ by ${\nu }_t(B)={\mu }_0^{\epsilon }(\{x\in D_t:X_t^{\epsilon }(x)\in B\})$ for Borel $B\subseteq E_t$. The set is Borel because $X_t^{\epsilon }$ is continuous on the open domain $D_t$; preimages preserve disjoint unions, so ${\nu }_t$ is a finite Borel measure of mass ${\mu }_0^{\epsilon }(D_t)$. The weak continuity-equation identity extends from smooth compact tests to compactly supported tests that are $C^1$ in x and absolutely continuous in r with locally integrable time derivative: multiply by a time cutoff, convolve in time and space on a fixed compact neighborhood, and pass to the limit. The terms converge because $p^{\epsilon }$ is locally bounded, $p^{\epsilon }{v}^{\epsilon }$ is locally integrable in time and space by finite action, and the test and its spatial gradient converge uniformly while its time derivative converges in $L^1$. Apply this extension to $\eta (r)\psi (r,x)$ from step 1.1 for every smooth compactly supported time cutoff $\eta$. The transport equation leaves $\int {\eta }^{\mathrm{\prime }}(r)F(r)dr=0$, so $F(r)=\int \psi (r,x)d{\mu }_r^{\epsilon }(x)$ is constant on $(0,t)$. Continuity of the flow and of ${\mu }_r^{\epsilon }$ extends the identity to r=0,t, giving $\int f(y)d{\mu }_t^{\epsilon }(y)=\int f(X_t^{\epsilon }(x))d{\mu }_0^{\epsilon }(x)$ for every $f\in C_c^{\mathrm{\infty }}(E_t)$. Uniform mollification extends equality to every $f\in C_c(E_t)$; both measures are finite, so uniform limits pass through their integrals. For a compact $K\subset E_t$, choose n sufficiently large that the closed $1\mathrm{/}n$-neighborhood of K is compactly contained in $E_t$ and set $f_n(x)=\max \{0,1-n\mathrm{dist}(x,K)\}$. Then $f_n\in C_c(E_t)$ and $f_n\downarrow {\mathbf{1}}_K$, so [F6] gives ${\nu }_t(K)={\mu }_t^{\epsilon }(K)$. By [F5], both finite Borel measures on the Polish space $E_t$ are inner regular; their agreement on compact sets therefore proves ${\nu }_t={\mu }_t^{\epsilon }{\mathrm{\mid }}_{E_t}$.

For every $r\in (0,T]$, step 2.1 gives equality of the restricted finite measures on $E_r$. Applying it to $y\mapsto \min \{N,\mathrm{\mid }{v}_r^{\epsilon }(y){\mathrm{\mid }}^2\}$ and letting $N\uparrow \mathrm{\infty }$ by [F7] yields ${\int }_{D_r}\mathrm{\mid }{v}_r^{\epsilon }(X_r^{\epsilon }(x)){\mathrm{\mid }}^{2}d{\mu }_0^{\epsilon }(x)={\int }_{E_r}\mathrm{\mid }{v}_r^{\epsilon }(y){\mathrm{\mid }}^{2}d{\mu }_r^{\epsilon }(y)\le \int \mathrm{\mid }{v}_r^{\epsilon }(y){\mathrm{\mid }}^{2}d{\mu }_r^{\epsilon }(y)$. Let $\mathcal{D}=\{(r,x):x\in D_r\}$. This domain is relatively open in $[0,T]\times {\mathbb{R}}^d$ and $(r,x)\mapsto X_r^{\epsilon }(x)$ is continuous on it: cover any compact trajectory segment through $(r,x)$ by finitely many local flow neighborhoods from [F2] and patch their continuous dependence on the initial point; time continuity follows from the integral equation. Thus the integrand is Borel; extend it by zero off $\mathcal{D}$. Tonelli [F4] then gives $\int \text{\hspace{0.17em}⁣}{\int }_0^{\min \{T,\tau (x)\}}\mathrm{\mid }{v}_r^{\epsilon }(X_r^{\epsilon }(x)){\mathrm{\mid }}^{2}drd{\mu }_0^{\epsilon }(x)\le {\int }_0^T\text{\hspace{0.17em}⁣}\int \mathrm{\mid }{v}_r^{\epsilon }(y){\mathrm{\mid }}^{2}d{\mu }_r^{\epsilon }(y)dr<\mathrm{\infty }\mathrm{.}$ Thus for ${\mu }_0^{\epsilon }$-almost every x the path has finite squared speed integral up to its maximal endpoint. Holder then makes its total path length finite. Such a path is Cauchy at any finite endpoint; it cannot escape every compact set, contradicting [F2] if an endpoint lies in $(0,T)$. At T it has a finite limit and extends continuously to T. Thus ${\mu }_0^{\epsilon }(D_T)=1$.

For each t, step 2.1 identifies ${\mu }_t^{\epsilon }{\mathrm{\mid }}_{E_t}$ with ${\nu }_t=(X_t^{\epsilon }{)}_{\mathrm{\#}}({\mu }_0^{\epsilon }{\mathrm{\mid }}_{D_t})$, where this restricted pushforward is the finite measure defined in step 2.1. Since $D_T\subseteq D_t$ and ${\mu }_0^{\epsilon }(D_T)=1$ by step 3.1, ${\nu }_t(E_t)={\mu }_0^{\epsilon }(D_t)=1$. Hence ${\mu }_t^{\epsilon }(E_t)=1$, and the equality of restricted measures extends to all Borel subsets of ${\mathbb{R}}^d$. Because $D_t$ is open and $X_t^{\epsilon }$ is continuous there, extend $X_t^{\epsilon }$ to all of ${\mathbb{R}}^d$ by setting it to zero off $D_t$; this is Borel and its pushforward is ${\nu }_t$ since ${\mu }_0^{\epsilon }(D_t)=1$. Thus ${\mu }_t^{\epsilon }=(X_t^{\epsilon }{)}_{\mathrm{\#}}{\mu }_0^{\epsilon }$. This proves the claim for every t, with the unique flow given by [F2].