This paper aims at giving an overview of estimates in general Besov
spaces for the Cauchy problem on $t=0$ related to the vector field
$\partial_t+v\cdot\nabla$. The emphasis is on the conservation or loss of
regularity for the initial data.
When $\nabla v$ belongs to $L^1(0,T;L^\infty)$ (plus some
convenient conditions depending on the functional space considered
for the data), the initial regularity is preserved. On the other
hand, if $\nabla v$ is slightly less regular (e.g. $\nabla v$
belongs to some limit space for which the embedding in $L^\infty$
fails), the regularity may coarsen with time. Different scenarios
are possible going from linear to arbitrarily small loss of
regularity. This latter result will be used in a forthcoming paper
to prove global well-posedness for two-dimensional incompressible
density-dependent viscous fluids
(see [Danchin, R.: Local theory in critical spaces for compressible viscous and
heat-conductive gases. Comm. Partial Differential Equations 26 (2001),
1183-1233, and Erratum, 27 (2002), 2531-2532.]).
Besides, our techniques enable us to get estimates uniformly in $\nu\geq0$ when
adding a diffusion term $-\nu\Delta u$ to the transport equation.