> [!definition] > > Let $0 \in J \subset \mathbb R$ be an open interval, $E$ be a [[Banach Space|Banach space]], $U \subset E$ be [[Open Set|open]], and $f: J \times U \to E$ be a [[Time-Dependent Vector Field|time-dependent vector field]] of class $C^p$. A **local flow** for $f$ is a mapping > $ > \alpha: J_0 \times U_0 \to E > $ > such that $0 \in J_0 \subset J$, $U_0 \subset U$ are open, and for each $x \in U_0$, $\alpha_{x}: J_0 \to E$ with $t \mapsto \alpha(t, x)$ is an [[Integral Curve|integral curve]] with initial condition $x$.