  The uncertainty relation is a fundamental limit in quantum mechanics and is
of great importance to quantum information processing as it relates to quantum
precision measurement. Due to interactions with the surrounding environment, a
quantum system will unavoidably suffer from decoherence. Here, we investigate
the dynamic behaviors of the entropic uncertainty relation of an atom-cavity
interacting system under a bosonic reservoir during the crossover between
Markovian and non-Markovian regimes. Specifically, we explore the dynamic
behavior of the entropic uncertainty relation for a pair of incompatible
observables under the reservoir-induced atomic decay effect both with and
without quantum memory. We find that the uncertainty dramatically depends on
both the atom-cavity and the cavity-reservoir interactions, as well as the
correlation time, $\tau$, of the structured reservoir. Furthermore, we verify
that the uncertainty is anti-correlated with the purity of the state of the
observed qubit-system. We also propose a remarkably simple and efficient way to
reduce the uncertainty by utilizing quantum weak measurement reversal.
Therefore our work offers a new insight into the uncertainty dynamics for
multi-component measurements within an open system, and is thus important for
quantum precision measurements.
