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

Danchin, 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 -