Robust transitivity and topological mixing for $C^1$-flows

We prove that nontrivial homoclinic classes of $C^r$-generic flows are topologically mixing. This implies that given $\Lambda$, a nontrivial $C^1$-robustly transitive set of a vector field $X$, there is a $C^1$-perturbation $Y$ of $X$ such that the continuation $\Lambda_Y$ of $\Lambda$ is a topologically mixing set for $Y$. In particular, robustly transitive flows become topologically mixing after $C^1$-perturbations. These results generalize a theorem by Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows whose nontrivial homoclinic classes are topologically mixing is not open and dense, in general.