Weak and classical solutions of equations of motion for third grade fluids

Bernard, Jean Marie. "Weak and classical solutions of equations of motion for third grade fluids." ESAIM: Mathematical Modelling and Numerical Analysis 33.6 (2010): 1091-1120. <http://eudml.org/doc/197589>.

@article{Bernard2010,
abstract = { This paper shows that the decomposition method with special basis, introduced by Cioranescu and Ouazar, allows one to prove global existence in time of the weak solution for the third grade fluids, in three dimensions, with small data. Contrary to the special case where $\vert\alpha_1+\alpha_2\vert\le(24\nu\beta)^\{1/2\}$, studied by Amrouche and Cioranescu, the H1 norm of the velocity is not bounded for all data. This fact, which led others to think, in contradiction to this paper, that the method of decomposition could not apply to the general case of third grade, complicates substantially the proof of the existence of the solution. We also prove further regularity results by a method similar to that of Cioranescu and Girault for second grade fluids. This extension to the third grade fluids is not straightforward, because of a transport equation which is much more complex. },
author = {Bernard, Jean Marie},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Galerkin method; special basis; energy estimates.; decomposition method; weak solution; classical solution; third grade fluid; global existence of solution; small initial data; regularity; energy estimate},
language = {eng},
month = {3},
number = {6},
pages = {1091-1120},
publisher = {EDP Sciences},
title = {Weak and classical solutions of equations of motion for third grade fluids},
url = {http://eudml.org/doc/197589},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Bernard, Jean Marie
TI - Weak and classical solutions of equations of motion for third grade fluids
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 6
SP - 1091
EP - 1120
AB - This paper shows that the decomposition method with special basis, introduced by Cioranescu and Ouazar, allows one to prove global existence in time of the weak solution for the third grade fluids, in three dimensions, with small data. Contrary to the special case where $\vert\alpha_1+\alpha_2\vert\le(24\nu\beta)^{1/2}$, studied by Amrouche and Cioranescu, the H1 norm of the velocity is not bounded for all data. This fact, which led others to think, in contradiction to this paper, that the method of decomposition could not apply to the general case of third grade, complicates substantially the proof of the existence of the solution. We also prove further regularity results by a method similar to that of Cioranescu and Girault for second grade fluids. This extension to the third grade fluids is not straightforward, because of a transport equation which is much more complex.
LA - eng
KW - Galerkin method; special basis; energy estimates.; decomposition method; weak solution; classical solution; third grade fluid; global existence of solution; small initial data; regularity; energy estimate
UR - http://eudml.org/doc/197589
ER -