C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade

All assertions are local in the state point and the time, so it suffices to work on a cylinder $[-h,h]\times B$ with $B$ a compact convex set with $\bar{B}\subset U$; on such a cylinder the mean value theorem [F7] applied componentwise bounds $\mathrm{\parallel }Y\mathrm{\parallel }\le M$ and makes $Y$ Lipschitz in the state variable with one constant $L$, and one chooses $h>0$ with $hM$ below the margin and $Lh<1$.

By the quantitative clause of [F1] each initial point of the cylinder carries a unique local solution on $[-h,h]$, obtained from the Picard iteration that starts at the specified constant curve and is generated by ordinary recursion, and the estimate of [F2] makes these local solutions depend uniformly continuously on the initial state.

Subtracting the Volterra equations of two solutions through nearby initial points $p,p+u$ and writing $A_u(t)={\int }_0^{1}DY(\mathrm{\Phi }(t,p)+r(\mathrm{\Phi }(t,p+u)-\mathrm{\Phi }(t,p)))dr$ gives a linear integral equation for the difference, and the uniform continuity of $DY$ on the cylinder makes $A_u\to DY(\mathrm{\Phi }(\cdot ,p))$ uniformly; the variational equation $V=I+\int DY(\mathrm{\Phi })V$ has a unique solution on each fixed compact time interval by the same local Picard argument for linear equations together with the exponential bound of [F3], and subtracting $V(t)u$ from the solution difference and applying [F3] gives an error of order $o(\mathrm{\mid }u\mathrm{\mid })$ uniformly in $t$; hence $D_p\mathrm{\Phi }=V$, the same estimate at nearby $p$ makes $V$ continuous, and ${\mathrm{\partial }}_t\mathrm{\Phi }=Y(\mathrm{\Phi })$ is continuous, so $\mathrm{\Phi }$ is jointly $C^1$.

If $Y$ is $C^2$ then $C(t,p)=DY(\mathrm{\Phi }(t,p))$ is $C^1$ in $p$ with derivative $D^{2}Y(\mathrm{\Phi })D_p\mathrm{\Phi }$; difference quotients of $V$ solve inhomogeneous linear integral equations, and the same uniform-continuity and Gronwall remainder argument converges to the solution $W$ of $W^{\mathrm{\prime }}=CW+D^{2}Y(\mathrm{\Phi })[V{e}_i,V{e}_j]$, $W(0)=0$, continuous in $(t,p)$, so $D_p^2\mathrm{\Phi }$ exists continuously; the mixed derivative is $DY(\mathrm{\Phi })D_p\mathrm{\Phi }$ and the second time derivative is $DY(\mathrm{\Phi })Y(\mathrm{\Phi })$, giving joint $C^2$ regularity, while for a $C^1$ field a single trajectory is $C^2$ in time because $x^{\mathrm{\prime }}=Y(x)$ can be differentiated once.

At a point with $Y(q)\mathrm{\ne }0$ choose a fixed linear transversal $\sigma$ to $Y(q)$; the derivative of $(t,z)\mapsto \mathrm{\Phi }(t,\sigma (z))$ at the corresponding point is invertible because its time derivative is $Y(q)\mathrm{\ne }0$ and its spatial part spans the transversal, so [F5] makes it a local $C^1$ diffeomorphism onto a $C^1$ flow box, and when $Y$ is $C^2$ the same map is $C^2$ and [F6] makes its inverse $C^2$.

Uniqueness glues the local solutions into a maximal flow $\mathrm{\Phi }$ on an open domain $D$ satisfying the flow law: covering a compact solution segment by finitely many of the common local cylinders of step 2.1 and composing them proves openness of $D$ and the stated $C^1$ regularity, respectively $C^2$ regularity when $Y$ is $C^2$, and the flow law inverts the time slices: ${\mathrm{\Phi }}_{-t}$ is the inverse of ${\mathrm{\Phi }}_t$.

Finally, if a trajectory remains in a compact set $K$ with $K\subset U$ and its maximal endpoint $T$ were finite, then the bound $\mathrm{\parallel }Y\mathrm{\parallel }\le M$ on a compact cylinder containing $K$ gives $\mathrm{\parallel }x(t)-x(s)\mathrm{\parallel }\le M\mathrm{\mid }t-s\mathrm{\mid }$, so $x(t)$ has a limit in $K$ as $t\to T$ by the completeness of [F4], and the local existence clause of step 2.1 restarts the solution past $T$, contradicting maximality; only ordinary recursion, the stated Gronwall and uniform-continuity estimates and the local inverse theorem are used, so no dependent choice or full choice principle is invoked.