# Revista Matemática Iberoamericana

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**Volume 21, Issue 3, 2005, pp. 863–888**

**DOI: 10.4171/RMI/438**

Published online: 2005-12-31

Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients

Raphaël Danchin^{[1]}(1) Université Paris 12 – Val de Marne, Créteil, France

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.

*Keywords: *Transport equation, transport-diffusion equation, estimates in Besov spaces, almost Lipschitz vectorfield, loss of regularity

Danchin Raphaël: Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. *Rev. Mat. Iberoam.* 21 (2005), 863-888. doi: 10.4171/RMI/438