Characteristic existence and total-variation preservation for regularized signed vorticity

Let $M=\mathrm{\mid }{\omega }_0\mathrm{\mid }({\mathbb{R}}^2)$. If $M=0$, [F7] implies ${\omega }_0=0$, the represented curve is zero by [F2], and $u_t=0$ has the identity flow; all claims follow. Hence suppose $M>0$. By [A1], write ${\omega }_0={\omega }_0^{+}-{\omega }_0^{-}$ with mutually singular positive parts. The definition [F2] supplies the label probability ${\lambda }_0$ on $E={\mathbb{R}}^2\times \{+1,-1\}$, with ${\lambda }_0(A\times \{s\})={\omega }_0^s(A)\mathrm{/}M$; [F8] and the assumed total-variation moment give its finite first moment.

On a subinterval $[a,a+T^{\mathrm{\prime }}]$, use the weighted path space over $(E,{\lambda }_0)$ and fix the attained initial label map $Z_a$ (with $Z_0(x,s)=x$ at the first interval). The constant-in-time path $Z_a$ shows the space is nonempty. Define $\mathrm{\Phi }(Z{)}_t(x,s)=Z_a(x,s)+M{\int }_a^t{\int }_{E}r{K}_{\epsilon }(Z_u(x,s)-Z_u(y,r))d{\lambda }_0(y,r)du\mathrm{.}$ Linear growth and the integrable path supremum make this a well-defined continuous-path map; [F9] supplies measurability of the paired kernel integrand. The Lipschitz bound in [F1] gives $\mathrm{\parallel }\mathrm{\Phi }Z-\mathrm{\Phi }W\mathrm{\parallel }\le 2M{L}_{\epsilon }{T}^{\mathrm{\prime }}\mathrm{\parallel }Z-W\mathrm{\parallel }$ on each interval. Choose $T^{\mathrm{\prime }}<1\mathrm{/}(2M{L}_{\epsilon })$ (any finite $T^{\mathrm{\prime }}$ if $L_{\epsilon }=0$); [F3] gives a unique fixed point. Its endpoint has finite first moment by the weighted path norm, so the same construction restarts there; finitely many intervals cover $[0,T]$.

The fixed point defines the curve in [F2]. The positive and negative label masses remain fixed, and its signed integral defines $u_t(x)=\int K_{\epsilon }(x-y)d{\omega }_t(y)$. The integrable path supremum and [F6] give dominated continuity in $t$; [F1] gives spatial Lipschitz constant at most $M{L}_{\epsilon }$ and at most linear growth.

By [F4], the velocity field has a unique maximal flow from each point. The linear-growth estimate and [F5] bound every trajectory on $[0,T]$ by a finite constant depending on its initial point, $T$, $M$, and $\epsilon$. A bounded trajectory has a limit at any finite endpoint because its derivative is bounded on the resulting compact time-state set; local existence at that endpoint extends it. Hence the maximal flow ${\varphi }_t$ exists for all $[0,T]$. Each label path from step 2.1 solves this same ODE from its initial position, so ODE uniqueness gives $Z_t(x,s)={\varphi }_t(x)$ for ${\lambda }_0$-almost every label. Solving the time-reversed equation gives an inverse; Gronwall [F5] applied to two initial states in both time orientations makes the forward and inverse maps Lipschitz on each finite interval. Thus each time map is a homeomorphism and both Jordan parts follow that common position flow.

The initial Jordan parts are mutually singular by [A1]. The common homeomorphism in step 4.1 transports a witnessing disjoint Borel partition to a disjoint Borel partition, so their pushforwards remain mutually singular. They are therefore the Jordan parts of ${\omega }_t$; their masses are unchanged by pushforward under the bijection. Thus $\mathrm{\mid }{\omega }_t\mathrm{\mid }({\mathbb{R}}^2)={\omega }_0^{+}({\mathbb{R}}^2)+{\omega }_0^{-}({\mathbb{R}}^2)=M$ by [F7]. Every solution in [F2] is a fixed point of the same contraction on each subinterval, so uniqueness in [F3] gives uniqueness of the represented curve.