  We obtain a $C^1$-generic subset of the incompressible flows in a closed
three-dimensional manifold where Pesin's entropy formula holds thus
establishing the continuous-time version of \cite{T}. Moreover, in any compact
manifold of dimension larger or equal to three we obtain that the metric
entropy function and the integrated upper Lyapunov exponent function are not
continuous with respect to the $C^1$ Whitney topology. Finally, we establish
the $C^2$-genericity of Pesin's entropy formula in the context of Hamiltonian
four-dimensional flows.
