Let $h \in C^2_{\mathrm{ub}}(\mathbb{R}^{2n})$, where $C_{\mathrm{ub}}^k$ consists of $C^k$-functions that are bounded and uniformly continuous along with their derivatives up to $k$th-order.

It is clear that the Hamiltonian vector field $X$ is $C^1$ and globally Lipschitz, hence the Hamiltonian flow $\Phi_t$ exists for all times.

Since $X$ is $C^1$, so is the flow. Can one say something similar regarding the uniform continuity? Is the flow of a function $h$ as described above uniformly continuous?

Any help would be much appreciated.