This paper aims at giving an overview of estimates in general Besov spaces for the Cauchy problem on related to the vector field . The emphasis is on the conservation or loss of regularity for the initial data. When belongs to (plus some convenient conditions depending on the functional space considered for the data), the initial regularity is preserved. On the other hand, if is slightly less regular (e.g. belongs to some limit space for which the embedding in 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 when adding a diffusion term to the transport equation.
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Raphaël Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 863–888DOI 10.4171/RMI/438