Abstract: In this paper, we give a flowbox theorem for the Lipschitz vector fields on a Banach spcace. We prove:if X is a Lipschitz vector field with a Lipschitz constant L, then there is a constant r₀ associated to L only such that for any regular point x of X, there is a flowbox with size r₀||X(x)||, and the Lipschitz constants of the respected lipeomorphism in the flowbox theorem has a uniform bound.

Key words: Lipschitz vector fields, Flowbox theorem, Gronwall inequality