JournalsrmiVol. 21, No. 3pp. 863–888

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

  • Raphaël Danchin

    Université Paris 12 – Val de Marne, Créteil, France
Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients cover
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Abstract

This paper aims at giving an overview of estimates in general Besov spaces for the Cauchy problem on t=0t=0 related to the vector field t+v\partial_t+v\cdot\nabla. The emphasis is on the conservation or loss of regularity for the initial data. When v\nabla v belongs to L1(0,T;L)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 v\nabla v is slightly less regular (e.g. v\nabla v belongs to some limit space for which the embedding in LL^\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 ν0\nu\geq0 when adding a diffusion term νΔu-\nu\Delta u to the transport equation.

Cite this article

Raphaël Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 863–888

DOI 10.4171/RMI/438