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

Abstract

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 $\nu \ge 0$ when adding a diffusion term $-\nu \Delta u$ to the transport equation.