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December, 2005 Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients
Raphaël Danchin
Rev. Mat. Iberoamericana 21(3): 863-888 (December, 2005).


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.


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Raphaël Danchin. "Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients." Rev. Mat. Iberoamericana 21 (3) 863 - 888, December, 2005.


Published: December, 2005
First available in Project Euclid: 11 January 2006

zbMATH: 1098.35038
MathSciNet: MR2231013

Primary: 35B45 , 35L45 , 35Q35

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

Rights: Copyright © 2005 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.21 • No. 3 • December, 2005
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