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

Revista Matemática Iberoamericana (2005)

- Volume: 21, Issue: 3, page 863-888
- ISSN: 0213-2230

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topDanchin, Raphaël. "Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients.." Revista Matemática Iberoamericana 21.3 (2005): 863-888. <http://eudml.org/doc/41953>.

@article{Danchin2005,

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 ∂t + v·∇. The emphasis is on the conservation or loss of regularity for the initial data.When ∇u belongs to L1(0,T; L∞) (plus some convenient conditions depending on the functional space considered for the data), the initial regularity is preserved. On the other hand, if ∇v is slightly less regular (e.g. ∇v belogs to some limit space for which the embedding in L∞ fails), the regularity may coarsen with time. Different scenarios are possible going from linear to arbitrary 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 [11]).Besides, our techniques enable us to get estimates uniformly in v ≥ 0 when adding a diffusion term -vΔu to the transport equation.},

author = {Danchin, Raphaël},

journal = {Revista Matemática Iberoamericana},

keywords = {Ecuaciones diferenciales hiperbólicas; Problema de Cauchy; Ecuación de difusión; Fenómenos de transporte; Espacios de Besov; transport-diffusion equation; a priori estimates; regularity of solutions},

language = {eng},

number = {3},

pages = {863-888},

title = {Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients.},

url = {http://eudml.org/doc/41953},

volume = {21},

year = {2005},

}

TY - JOUR

AU - Danchin, Raphaël

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

JO - Revista Matemática Iberoamericana

PY - 2005

VL - 21

IS - 3

SP - 863

EP - 888

AB - 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 ∂t + v·∇. The emphasis is on the conservation or loss of regularity for the initial data.When ∇u belongs to L1(0,T; L∞) (plus some convenient conditions depending on the functional space considered for the data), the initial regularity is preserved. On the other hand, if ∇v is slightly less regular (e.g. ∇v belogs to some limit space for which the embedding in L∞ fails), the regularity may coarsen with time. Different scenarios are possible going from linear to arbitrary 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 [11]).Besides, our techniques enable us to get estimates uniformly in v ≥ 0 when adding a diffusion term -vΔu to the transport equation.

LA - eng

KW - Ecuaciones diferenciales hiperbólicas; Problema de Cauchy; Ecuación de difusión; Fenómenos de transporte; Espacios de Besov; transport-diffusion equation; a priori estimates; regularity of solutions

UR - http://eudml.org/doc/41953

ER -

## Citations in EuDML Documents

top- L. Miguel Rodrigues, Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid
- Raphaël Danchin, The inviscid limit for density-dependent incompressible fluids
- Qionglei Chen, Changxing Miao, Zhifei Zhang, On the uniqueness of weak solutions for the 3D Navier-Stokes equations
- Raphaël Danchin, Marius Paicu, Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux

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