  The aim of this paper is to study in detail the relations between strong
chain recurrence for flows and Lyapunov functions. For a continuous flow on a
compact metric space, uniformly Lipschitz continuous on the compact subsets of
the time, we first make explicit a Lipschitz continuous Lyapunov function
strict -that is strictly decreasing- outside the strong chain recurrent set of
the flow. This construction extends to flows some recent advances of Fathi and
Pageault in the case of homeomorphisms; moreover, it improves Conley's result
about the existence of a continuous Lyapunov function strictly decreasing
outside the chain recurrent set of a continuous flow. We then present two
consequences of this theorem. From one hand, we characterize the strong chain
recurrent set in terms of Lipschitz continuous Lyapunov functions. From the
other hand, in the case of a flow induced by a vector field, we establish a
sufficient condition for the existence of a $\mathcal{C}^{1,1}$ strict Lyapunov
function and we also discuss various examples. Moreover, for general continuous
flows, we show that the strong chain recurrent set has only one strong chain
transitive component if and only if the only Lipschitz continuous Lyapunov
functions are the constants. Finally, we provide a necessary and sufficient
condition to guarantee that the strong chain recurrent set and the chain
recurrent one coincide.
